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In theoretical physics, the Madelung equations, or the equations of quantum hydrodynamics, are Erwin Madelung's alternative formulation of the Schrödinger equation for a spinless non relativistic particle, written in terms of hydrodynamical variables, similar to the Navier–Stokes equations of fluid dynamics. The derivation of the Madelung equations is similar to the de Broglie–Bohm formulation, which represents the Schrödinger equation as a quantum Hamilton–Jacobi equation. In both cases the hydrodynamic interpretations are not equivalent to Schrodinger's equation without the addition of a quantization condition.
Recently, the extension to the relativistic case with spin was done by having the Dirac equation written with hydrodynamic variables. In the relativistic case, the Hamilton–Jacobi equation is also the guidance equation, which therefore does not have to be postulated.
== History ==
In the fall of 1926, Erwin Madelung reformulated Schrödinger's quantum equation in a more classical and visualizable form resembling hydrodynamics. His paper was one of numerous early attempts at different approaches to quantum mechanics, including those of Louis de Broglie and Earle Hesse Kennard. The most influential of these theories was ultimately de Broglie's through the 1952 work of David Bohm now called Bohmian mechanics.
In 1994 Timothy C. Wallstrom showed that an additional ad hoc quantization condition must be added to the Madelung equations to reproduce Schrodinger's work. His analysis paralleled earlier work by Takehiko Takabayashi on the hydrodynamic interpretation of Bohmian mechanics. The mathematical foundations of the Madelung equations continue to be a topic of research.
== Equations ==
The Madelung equations are quantum Euler equations:
∂
t
ρ
m
+
∇
⋅
(
ρ
m
v
)
=
0
,
d
v
d
t
=
∂
t
v
+
v
⋅
∇
v
=
−
1
m
∇
(
Q
+
V
)
,
{\displaystyle {\begin{aligned}&\partial _{t}\rho _{m}+\nabla \cdot (\rho _{m}\mathbf {v} )=0,\\[4pt]&{\frac {d\mathbf {v} }{dt}}=\partial _{t}\mathbf {v} +\mathbf {v} \cdot \nabla \mathbf {v} =-{\frac {1}{m}}\mathbf {\nabla } (Q+V),\end{aligned}}}
where
v
{\displaystyle \mathbf {v} }
is the flow velocity,
ρ
m
=
m
ρ
=
m
|
ψ
|
2
{\displaystyle \rho _{m}=m\rho =m|\psi |^{2}}
is the mass density,
Q
=
−
ℏ
2
2
m
∇
2
ρ
ρ
=
−
ℏ
2
2
m
∇
2
ρ
m
ρ
m
{\displaystyle Q=-{\frac {\hbar ^{2}}{2m}}{\frac {\nabla ^{2}{\sqrt {\rho }}}{\sqrt {\rho }}}=-{\frac {\hbar ^{2}}{2m}}{\frac {\nabla ^{2}{\sqrt {\rho _{m}}}}{\sqrt {\rho _{m}}}}}
is the Bohm quantum potential,
V is the potential from the Schrödinger equation.
The Madelung equations answer the question whether
v
(
x
,
t
)
{\displaystyle \mathbf {v} (\mathbf {x} ,t)}
obeys the continuity equations of hydrodynamics and, subsequently, what plays the role of the stress tensor.
The circulation of the flow velocity field along any closed path obeys the auxiliary quantization condition
Γ
≐
∮
m
v
⋅
d
l
=
2
π
n
ℏ
{\textstyle \Gamma \doteq \oint {m\mathbf {v} \cdot d\mathbf {l} }=2\pi n\hbar }
for all integers n.
== Derivation ==
The Madelung equations are derived by first writing the wavefunction in polar form
ψ
(
x
,
t
)
=
R
(
x
,
t
)
e
i
S
(
x
,
t
)
/
ℏ
,
{\displaystyle \psi (\mathbf {x} ,t)=R(\mathbf {x} ,t)e^{iS(\mathbf {x} ,t)/\hbar },}
with
R
≥
0
{\displaystyle R\geq 0}
and
S
{\displaystyle S}
both real and
ρ
(
x
,
t
)
=
ψ
(
x
,
t
)
∗
ψ
(
x
,
t
)
=
R
2
(
x
,
t
)
,
{\displaystyle \rho (\mathbf {x} ,t)=\psi (\mathbf {x} ,t)^{*}\psi (\mathbf {x} ,t)=R^{2}(\mathbf {x} ,t),}
the associated probability density. Substituting this form into the probability current gives:
J
=
ℏ
2
m
i
(
ψ
∗
∇
ψ
−
ψ
∇
ψ
∗
)
=
1
m
ρ
(
x
,
t
)
∇
S
(
x
,
t
)
=
ρ
(
x
,
t
)
v
(
x
,
t
)
,
{\displaystyle \mathbf {J} ={\frac {\hbar }{2mi}}(\psi ^{*}\nabla \psi -\psi \nabla \psi ^{*})={\frac {1}{m}}\rho (\mathbf {x} ,t)\nabla S(\mathbf {x} ,t)=\rho (\mathbf {x} ,t)\mathbf {v} (\mathbf {x} ,t),}
where the flow velocity is expressed as
v
(
x
,
t
)
=
1
m
∇
S
(
x
,
t
)
.
{\displaystyle \mathbf {v} (\mathbf {x} ,t)={\frac {1}{m}}\nabla S(\mathbf {x} ,t).}
However, the interpretation of
v
{\displaystyle \mathbf {v} }
as a "velocity" should not be taken too literal, because a simultaneous exact measurement of position and velocity would necessarily violate
the uncertainty principle.
Next, substituting the polar form into the Schrödinger equation
i
ℏ
∂
∂
t
ψ
(
x
,
t
)
=
[
−
ℏ
2
2
m
∇
2
+
V
(
x
)
]
ψ
(
x
,
t
)
,
{\displaystyle i\hbar {\frac {\partial }{\partial t}}\psi (\mathbf {x} ,t)=\left[{\frac {-\hbar ^{2}}{2m}}\nabla ^{2}+V(\mathbf {x} )\right]\psi (\mathbf {x} ,t),}
and performing the appropriate differentiations, dividing the equation by
e
i
S
(
x
,
t
)
/
ℏ
{\displaystyle e^{iS(\mathbf {x} ,t)/\hbar }}
and separating the real and imaginary parts, one obtains a system of two coupled partial differential equations:
∂
t
R
(
x
,
t
)
+
1
m
∇
R
(
x
,
t
)
⋅
∇
S
(
x
,
t
)
+
1
2
m
R
(
x
,
t
)
Δ
S
(
x
,
t
)
=
0
,
∂
t
S
(
x
,
t
)
+
1
2
m
[
∇
S
(
x
,
t
)
]
2
+
V
(
x
)
=
ℏ
2
2
m
Δ
R
(
x
,
t
)
R
(
x
,
t
)
.
{\displaystyle {\begin{aligned}&\partial _{t}R(\mathbf {x} ,t)+{\frac {1}{m}}\nabla R(\mathbf {x} ,t)\cdot \nabla S(\mathbf {x} ,t)+{\frac {1}{2m}}R(\mathbf {x} ,t)\Delta S(\mathbf {x} ,t)=0,\\&\partial _{t}S(\mathbf {x} ,t)+{\frac {1}{2m}}\left[\nabla S(\mathbf {x} ,t)\right]^{2}+V(\mathbf {x} )={\frac {\hbar ^{2}}{2m}}{\frac {\Delta R(\mathbf {x} ,t)}{R(\mathbf {x} ,t)}}.\end{aligned}}}
The first equation corresponds to the imaginary part of Schrödinger equation and can be interpreted as the continuity equation. The second equation corresponds to the real part and is also referred to as the quantum Hamilton-Jacobi equation.
Multiplying the first equation by
2
R
{\displaystyle 2R}
and calculating the gradient of the second equation results in the Madelung equations:
∂
t
ρ
(
x
,
t
)
+
∇
⋅
[
ρ
(
x
,
t
)
v
(
x
,
t
)
]
=
0
,
d
d
t
v
(
x
,
t
)
=
∂
t
v
(
x
,
t
)
+
[
v
(
x
,
t
)
⋅
∇
]
v
(
x
,
t
)
=
−
1
m
∇
[
V
(
x
)
−
ℏ
2
2
m
Δ
ρ
(
x
,
t
)
ρ
(
x
,
t
)
]
=
−
1
m
∇
[
V
(
x
)
+
Q
(
x
,
t
)
]
.
{\displaystyle {\begin{aligned}&\partial _{t}\rho (\mathbf {x} ,t)+\nabla \cdot \left[\rho (\mathbf {x} ,t)v(\mathbf {x} ,t)\right]=0,\\&{\frac {d}{dt}}\mathbf {v} (\mathbf {x} ,t)=\partial _{t}v(\mathbf {x} ,t)+\left[v(\mathbf {x} ,t)\cdot \nabla \right]v(\mathbf {x} ,t)=-{\frac {1}{m}}\nabla \left[V(\mathbf {x} )-{\frac {\hbar ^{2}}{2m}}{\frac {\Delta {\sqrt {\rho (\mathbf {x} ,t)}}}{\sqrt {\rho (\mathbf {x} ,t)}}}\right]=-{\frac {1}{m}}\nabla \left[V(\mathbf {x} )+Q(\mathbf {x} ,t)\right].\end{aligned}}}
with quantum potential
Q
(
x
,
t
)
=
−
ℏ
2
2
m
Δ
ρ
(
x
,
t
)
ρ
(
x
,
t
)
.
{\displaystyle Q(\mathbf {x} ,t)=-{\frac {\hbar ^{2}}{2m}}{\frac {\Delta {\sqrt {\rho (\mathbf {x} ,t)}}}{\sqrt {\rho (\mathbf {x} ,t)}}}.}
Alternatively, the quantum Hamilton-Jacobi equation can be written in a form similar to the Cauchy momentum equation:
d
d
t
v
=
f
−
1
ρ
m
∇
⋅
p
Q
,
{\displaystyle {\frac {d}{dt}}\mathbf {v} =\mathbf {f} -{\frac {1}{\rho _{m}}}\nabla \cdot \mathbf {p} _{Q},}
with an external force defined as
f
(
x
)
=
−
1
m
∇
V
(
x
)
,
{\displaystyle \mathbf {f} (\mathbf {x} )=-{\frac {1}{m}}\nabla V(\mathbf {x} ),}
and a quantum pressure tensor
p
Q
=
−
(
ℏ
/
2
m
)
2
ρ
m
∇
⊗
∇
ln
ρ
m
.
{\displaystyle \mathbf {p} _{Q}=-(\hbar /2m)^{2}\rho _{m}\nabla \otimes \nabla \ln \rho _{m}.}
The integral energy stored in the quantum pressure tensor is proportional to the Fisher information, which accounts for the quality of measurements. Thus, according to the Cramér–Rao bound, the Heisenberg uncertainty principle is equivalent to a standard inequality for the efficiency of measurements.
=== Quantum energies ===
The thermodynamic definition of the quantum chemical potential
μ
=
Q
+
V
=
1
ρ
m
H
^
ρ
m
{\displaystyle \mu =Q+V={\frac {1}{\sqrt {\rho _{m}}}}{\widehat {H}}{\sqrt {\rho _{m}}}}
follows from the hydrostatic force balance above:
∇
μ
=
m
ρ
m
∇
⋅
p
Q
+
∇
V
.
{\displaystyle \nabla \mu ={\frac {m}{\rho _{m}}}\nabla \cdot \mathbf {p} _{Q}+\nabla V.}
According to thermodynamics, at equilibrium the chemical potential is constant everywhere, which corresponds straightforwardly to the stationary Schrödinger equation. Therefore, the eigenvalues of the Schrödinger equation are free energies, which differ from the internal energies of the system. The particle internal energy is calculated as
ε
=
μ
−
tr
(
p
Q
)
m
ρ
m
=
−
ℏ
2
8
m
(
∇
ln
ρ
m
)
2
+
U
{\displaystyle \varepsilon =\mu -\operatorname {tr} (\mathbf {p} _{Q}){\frac {m}{\rho _{m}}}=-{\frac {\hbar ^{2}}{8m}}(\nabla \ln \rho _{m})^{2}+U}
and is related to the von Weizsäcker correction of density functional theory.
== See also ==
Classical limit
De Broglie–Bohm theory
Magnetohydrodynamics
Pilot wave theory
Quantum potential
Quantum hydrodynamics
WKB approximation
== Notes ==
== References ==
Białynicki-Birula, Iwo; Cieplak, Marek; Kaminski, Jerzy (1992). Theory of Quanta. New York: Oxford University Press, USA. ISBN 0-19-507157-3.
Heifetz, Eyal; Cohen, Eliahu (2015). "Toward a Thermo-hydrodynamic Like Description of Schrödinger Equation via the Madelung Formulation and Fisher Information". Foundations of Physics. 45 (11): 1514–1525. arXiv:1501.00944. doi:10.1007/s10701-015-9926-1. ISSN 0015-9018.
Kragh, Helge; Carazza, Bruno (2000). "Classical Behavior of Macroscopic Bodies from Quantum Principles: Early Discussions". Archive for History of Exact Sciences. 55 (1): 43–56. doi:10.1007/s004070000018. ISSN 0003-9519.
Madelung, E. (1926). "Eine anschauliche Deutung der Gleichung von Schrödinger". Die Naturwissenschaften (in German). 14 (45): 1004–1004. doi:10.1007/BF01504657. ISSN 0028-1042.
Madelung, E. (1927). "Quantentheorie in hydrodynamischer Form". Zeitschrift für Physik (in German). 40 (3–4): 322–326. doi:10.1007/BF01400372. ISSN 1434-6001.
Reginatto, Marcel (1998-09-01). "Derivation of the equations of nonrelativistic quantum mechanics using the principle of minimum Fisher information". Physical Review A. 58 (3): 1775–1778. doi:10.1103/PhysRevA.58.1775. ISSN 1050-2947.
Sakurai, J. J.; Napolitano, Jim (2020). Modern Quantum Mechanics. Cambridge: Cambridge University Press. ISBN 978-1-108-47322-4.
Schönberg, M. (1954). "On the hydrodynamical model of the quantum mechanics". Il Nuovo Cimento. 12 (1): 103–133. doi:10.1007/BF02820368. ISSN 0029-6341.
Tsekov, Roumen (2011). "Quantum diffusion". Physica Scripta. 83 (3). arXiv:1001.1071. doi:10.1088/0031-8949/83/03/035004. ISSN 0031-8949.
Tsekov, Roumen (2009). "Dissipative Time Dependent Density Functional Theory". International Journal of Theoretical Physics. 48 (9): 2660–2664. arXiv:0903.3644. doi:10.1007/s10773-009-0054-6. ISSN 0020-7748.
von Weizsäcker, Carl F. (1935). "Zur Theorie der Kernmassen". Zeitschrift für Physik (in German). 96 (7–8): 431–58. doi:10.1007/bf01337700. ISSN 0044-3328.
Wyatt, Robert E. (2005). Quantum Dynamics with Trajectories. New York: Springer Science & Business Media. ISBN 0-387-22964-7. | Wikipedia/Madelung_equations |
In physics, a hidden-variable theory is a deterministic model which seeks to explain the probabilistic nature of quantum mechanics by introducing additional, possibly inaccessible, variables.
The mathematical formulation of quantum mechanics assumes that the state of a system prior to measurement is indeterminate; quantitative bounds on this indeterminacy are expressed by the Heisenberg uncertainty principle. Most hidden-variable theories are attempts to avoid this indeterminacy, but possibly at the expense of requiring that nonlocal interactions be allowed. One notable hidden-variable theory is the de Broglie–Bohm theory.
In their 1935 EPR paper, Albert Einstein, Boris Podolsky, and Nathan Rosen argued that quantum entanglement might imply that quantum mechanics is an incomplete description of reality. John Stewart Bell in 1964, in his eponymous theorem proved that correlations between particles under any local hidden variable theory must obey certain constraints. Subsequently, Bell test experiments have demonstrated broad violation of these constraints, ruling out such theories. Bell's theorem, however, does not rule out the possibility of nonlocal theories or superdeterminism; these therefore cannot be falsified by Bell tests.
== Motivation ==
Macroscopic physics requires classical mechanics which allows accurate predictions of mechanical motion with reproducible, high precision. Quantum phenomena require quantum mechanics, which allows accurate predictions of statistical averages only. If quantum states had hidden-variables awaiting ingenious new measurement technologies, then the latter (statistical results) might be convertible to a form of the former (classical-mechanical motion).
This classical mechanics description would eliminate unsettling characteristics of quantum theory like the uncertainty principle. More fundamentally however, a successful model of quantum phenomena with hidden variables implies quantum entities with intrinsic values independent of measurements. Existing quantum mechanics asserts that state properties can only be known after a measurement. As N. David Mermin puts it:It is a fundamental quantum doctrine that a measurement does not, in general, reveal a pre-existing value of the measured property. On the contrary, the outcome of a measurement is brought into being by the act of measurement itself...
In other words, whereas a hidden-variable theory would imply intrinsic particle properties, in quantum mechanics an electron has no definite position and velocity to even be revealed.
== History ==
=== "God does not play dice" ===
In June 1926, Max Born published a paper, in which he was the first to clearly enunciate the probabilistic interpretation of the quantum wave function, which had been introduced by Erwin Schrödinger earlier in the year. Born concluded the paper as follows:Here the whole problem of determinism comes up. From the standpoint of our quantum mechanics there is no quantity which in any individual case causally fixes the consequence of the collision; but also experimentally we have so far no reason to believe that there are some inner properties of the atom which conditions a definite outcome for the collision. Ought we to hope later to discover such properties ... and determine them in individual cases? Or ought we to believe that the agreement of theory and experiment—as to the impossibility of prescribing conditions for a causal evolution—is a pre-established harmony founded on the nonexistence of such conditions? I myself am inclined to give up determinism in the world of atoms. But that is a philosophical question for which physical arguments alone are not decisive.Born's interpretation of the wave function was criticized by Schrödinger, who had previously attempted to interpret it in real physical terms, but Albert Einstein's response became one of the earliest and most famous assertions that quantum mechanics is incomplete:Quantum mechanics is very worthy of respect. But an inner voice tells me this is not the genuine article after all. The theory delivers much but it hardly brings us closer to the Old One's secret. In any event, I am convinced that He is not playing dice.Niels Bohr reportedly replied to Einstein's later expression of this sentiment by advising him to "stop telling God what to do."
=== Early attempts at hidden-variable theories ===
Shortly after making his famous "God does not play dice" comment, Einstein attempted to formulate a deterministic counter proposal to quantum mechanics, presenting a paper at a meeting of the Academy of Sciences in Berlin, on 5 May 1927, titled "Bestimmt Schrödinger's Wellenmechanik die Bewegung eines Systems vollständig oder nur im Sinne der Statistik?" ("Does Schrödinger's wave mechanics determine the motion of a system completely or only in the statistical sense?"). However, as the paper was being prepared for publication in the academy's journal, Einstein decided to withdraw it, possibly because he discovered that, contrary to his intention, his use of Schrödinger's field to guide localized particles allowed just the kind of non-local influences he intended to avoid.
At the Fifth Solvay Congress, held in Belgium in October 1927 and attended by all the major theoretical physicists of the era, Louis de Broglie presented his own version of a deterministic hidden-variable theory, apparently unaware of Einstein's aborted attempt earlier in the year. In his theory, every particle had an associated, hidden "pilot wave" which served to guide its trajectory through space. The theory was subject to criticism at the Congress, particularly by Wolfgang Pauli, which de Broglie did not adequately answer; de Broglie abandoned the theory shortly thereafter.
=== Declaration of completeness of quantum mechanics, and the Bohr–Einstein debates ===
Also at the Fifth Solvay Congress, Max Born and Werner Heisenberg made a presentation summarizing the recent tremendous theoretical development of quantum mechanics. At the conclusion of the presentation, they declared:[W]hile we consider ... a quantum mechanical treatment of the electromagnetic field ... as not yet finished, we consider quantum mechanics to be a closed theory, whose fundamental physical and mathematical assumptions are no longer susceptible of any modification...
On the question of the 'validity of the law of causality' we have this opinion: as long as one takes into account only experiments that lie in the domain of our currently acquired physical and quantum mechanical experience, the assumption of indeterminism in principle, here taken as fundamental, agrees with experience.Although there is no record of Einstein responding to Born and Heisenberg during the technical sessions of the Fifth Solvay Congress, he did challenge the completeness of quantum mechanics at various times. In his tribute article for Born's retirement he discussed the quantum representation of a macroscopic ball bouncing elastically between rigid barriers. He argues that such a quantum representation does not represent a specific ball, but "time ensemble of systems". As such the representation is correct, but incomplete because it does not represent the real individual macroscopic case. Einstein considered quantum mechanics incomplete "because the state function, in general, does not even describe the individual event/system".
=== Von Neumann's proof ===
John von Neumann in his 1932 book Mathematical Foundations of Quantum Mechanics had presented a proof that there could be no "hidden parameters" in quantum mechanics. The validity of von Neumann's proof was questioned by Grete Hermann in 1935, who found a flaw in the proof. The critical issue concerned averages over ensembles. Von Neumann assumed that a relation between the expected values of different observable quantities holds for each possible value of the "hidden parameters", rather than only for a statistical average over them. However Hermann's work went mostly unnoticed until its rediscovery by John Stewart Bell more than 30 years later.
The validity and definitiveness of von Neumann's proof were also questioned by Hans Reichenbach, and possibly in conversation though not in print by Albert Einstein. Reportedly, in a conversation circa 1938 with his assistants Peter Bergmann and Valentine Bargmann, Einstein pulled von Neumann's book off his shelf, pointed to the same assumption critiqued by Hermann and Bell, and asked why one should believe in it. Simon Kochen and Ernst Specker rejected von Neumann's key assumption as early as 1961, but did not publish a criticism of it until 1967.
=== EPR paradox ===
Einstein argued that quantum mechanics could not be a complete theory of physical reality. He wrote,
Consider a mechanical system consisting of two partial systems A and B which interact with each other only during a limited time. Let the ψ function [i.e., wavefunction] before their interaction be given. Then the Schrödinger equation will furnish the ψ function after the interaction has taken place. Let us now determine the physical state of the partial system A as completely as possible by measurements. Then quantum mechanics allows us to determine the ψ function of the partial system B from the measurements made, and from the ψ function of the total system. This determination, however, gives a result which depends upon which of the physical quantities (observables) of A have been measured (for instance, coordinates or momenta). Since there can be only one physical state of B after the interaction which cannot reasonably be considered to depend on the particular measurement we perform on the system A separated from B it may be concluded that the ψ function is not unambiguously coordinated to the physical state. This coordination of several ψ functions to the same physical state of system B shows again that the ψ function cannot be interpreted as a (complete) description of a physical state of a single system.
Together with Boris Podolsky and Nathan Rosen, Einstein published a paper that gave a related but distinct argument against the completeness of quantum mechanics. They proposed a thought experiment involving a pair of particles prepared in what would later become known as an entangled state. Einstein, Podolsky, and Rosen pointed out that, in this state, if the position of the first particle were measured, the result of measuring the position of the second particle could be predicted. If instead the momentum of the first particle were measured, then the result of measuring the momentum of the second particle could be predicted. They argued that no action taken on the first particle could instantaneously affect the other, since this would involve information being transmitted faster than light, which is impossible according to the theory of relativity. They invoked a principle, later known as the "EPR criterion of reality", positing that: "If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of reality corresponding to that quantity." From this, they inferred that the second particle must have a definite value of both position and of momentum prior to either quantity being measured. But quantum mechanics considers these two observables incompatible and thus does not associate simultaneous values for both to any system. Einstein, Podolsky, and Rosen therefore concluded that quantum theory does not provide a complete description of reality.
Bohr answered the Einstein–Podolsky–Rosen challenge as follows:
[The argument of] Einstein, Podolsky and Rosen contains an ambiguity as regards the meaning of the expression "without in any way disturbing a system." ... [E]ven at this stage [i.e., the measurement of, for example, a particle that is part of an entangled pair], there is essentially the question of an influence on the very conditions which define the possible types of predictions regarding the future behavior of the system. Since these conditions constitute an inherent element of the description of any phenomenon to which the term "physical reality" can be properly attached, we see that the argumentation of the mentioned authors does not justify their conclusion that quantum-mechanical description is essentially incomplete."
Bohr is here choosing to define a "physical reality" as limited to a phenomenon that is immediately observable by an arbitrarily chosen and explicitly specified technique, using his own special definition of the term 'phenomenon'. He wrote in 1948:
As a more appropriate way of expression, one may strongly advocate limitation of the use of the word phenomenon to refer exclusively to observations obtained under specified circumstances, including an account of the whole experiment.
This was, of course, in conflict with the EPR criterion of reality.
=== Bell's theorem ===
In 1964, John Stewart Bell showed through his famous theorem that if local hidden variables exist, certain experiments could be performed involving quantum entanglement where the result would satisfy a Bell inequality. If, on the other hand, statistical correlations resulting from quantum entanglement could not be explained by local hidden variables, the Bell inequality would be violated. Another no-go theorem concerning hidden-variable theories is the Kochen–Specker theorem.
Physicists such as Alain Aspect and Paul Kwiat have performed experiments that have found violations of these inequalities up to 242 standard deviations. This rules out local hidden-variable theories, but does not rule out non-local ones. Theoretically, there could be experimental problems that affect the validity of the experimental findings.
Gerard 't Hooft has disputed the validity of Bell's theorem on the basis of the superdeterminism loophole and proposed some ideas to construct local deterministic models.
== Bohm's hidden-variable theory ==
In 1952, David Bohm proposed a hidden variable theory. Bohm unknowingly rediscovered (and extended) the idea that Louis de Broglie's pilot wave theory had proposed in 1927 (and abandoned) – hence this theory is commonly called "de Broglie-Bohm theory". Assuming the validity of Bell's theorem, any deterministic hidden-variable theory that is consistent with quantum mechanics would have to be non-local, maintaining the existence of instantaneous or faster-than-light relations (correlations) between physically separated entities.
Bohm posited both the quantum particle, e.g. an electron, and a hidden 'guiding wave' that governs its motion. Thus, in this theory electrons are quite clearly particles. When a double-slit experiment is performed, the electron goes through either one of the slits. Also, the slit passed through is not random but is governed by the (hidden) pilot wave, resulting in the wave pattern that is observed.
In Bohm's interpretation, the (non-local) quantum potential constitutes an implicate (hidden) order which organizes a particle, and which may itself be the result of yet a further implicate order: a superimplicate order which organizes a field. Nowadays Bohm's theory is considered to be one of many interpretations of quantum mechanics. Some consider it the simplest theory to explain quantum phenomena. Nevertheless, it is a hidden-variable theory, and necessarily so. The major reference for Bohm's theory today is his book with Basil Hiley, published posthumously.
A possible weakness of Bohm's theory is that some (including Einstein, Pauli, and Heisenberg) feel that it looks contrived. (Indeed, Bohm thought this of his original formulation of the theory.) Bohm said he considered his theory to be unacceptable as a physical theory due to the guiding wave's existence in an abstract multi-dimensional configuration space, rather than three-dimensional space.
== Recent developments ==
In August 2011, Roger Colbeck and Renato Renner published a proof that any extension of quantum mechanical theory, whether using hidden variables or otherwise, cannot provide a more accurate prediction of outcomes, assuming that observers can freely choose the measurement settings. Colbeck and Renner write: "In the present work, we have ... excluded the possibility that any extension of quantum theory (not necessarily in the form of local hidden variables) can help predict the outcomes of any measurement on any quantum state. In this sense, we show the following: under the assumption that measurement settings can be chosen freely, quantum theory really is complete".
In January 2013, Giancarlo Ghirardi and Raffaele Romano described a model which, "under a different free choice assumption [...] violates [the statement by Colbeck and Renner] for almost all states of a bipartite two-level system, in a possibly experimentally testable way".
== See also ==
== References ==
== Bibliography ==
Peres, Asher; Zurek, Wojciech (1982). "Is quantum theory universally valid?". American Journal of Physics. 50 (9): 807–810. Bibcode:1982AmJPh..50..807P. doi:10.1119/1.13086.
Jammer, Max (1985). "The EPR Problem in Its Historical Development". In Lahti, P.; Mittelstaedt, P. (eds.). Symposium on the Foundations of Modern Physics: 50 years of the Einstein–Podolsky–Rosen Gedankenexperiment. Singapore: World Scientific. pp. 129–149.
Fine, Arthur (1986). The Shaky Game: Einstein Realism and the Quantum Theory. Chicago: University of Chicago Press. | Wikipedia/Hidden-variables_theory |
In abstract algebra, in particular in the theory of nondegenerate quadratic forms on vector spaces, the finite-dimensional real and complex Clifford algebras for a nondegenerate quadratic form have been completely classified as rings. In each case, the Clifford algebra is algebra isomorphic to a full matrix ring over R, C, or H (the quaternions), or to a direct sum of two copies of such an algebra, though not in a canonical way. Below it is shown that distinct Clifford algebras may be algebra-isomorphic, as is the case of Cl1,1(R) and Cl2,0(R), which are both isomorphic as rings to the ring of two-by-two matrices over the real numbers.
== Notation and conventions ==
The Clifford product is the manifest ring product for the Clifford algebra, and all algebra homomorphisms in this article are with respect to this ring product. Other products defined within Clifford algebras, such as the exterior product, and other structure, such as the distinguished subspace of generators V, are not used here. This article uses the (+) sign convention for Clifford multiplication so that
v
2
=
Q
(
v
)
1
{\displaystyle v^{2}=Q(v)1}
for all vectors v in the vector space of generators V, where Q is the quadratic form on the vector space V. We will denote the algebra of n × n matrices with entries in the division algebra K by Mn(K) or End(Kn). The direct sum of two such identical algebras will be denoted by Mn(K) ⊕ Mn(K), which is isomorphic to Mn(K ⊕ K).
== Bott periodicity ==
Clifford algebras exhibit a 2-fold periodicity over the complex numbers and an 8-fold periodicity over the real numbers, which is related to the same periodicities for homotopy groups of the stable unitary group and stable orthogonal group, and is called Bott periodicity. The connection is explained by the geometric model of loop spaces approach to Bott periodicity: their 2-fold/8-fold periodic embeddings of the classical groups in each other (corresponding to isomorphism groups of Clifford algebras), and their successive quotients are symmetric spaces which are homotopy equivalent to the loop spaces of the unitary/orthogonal group.
== Complex case ==
The complex case is particularly simple: every nondegenerate quadratic form on a complex vector space is equivalent to the standard diagonal form
Q
(
u
)
=
u
1
2
+
u
2
2
+
⋯
+
u
n
2
,
{\displaystyle Q(u)=u_{1}^{2}+u_{2}^{2}+\cdots +u_{n}^{2},}
where n = dim(V), so there is essentially only one Clifford algebra for each dimension. This is because the complex numbers include i by which −uk2 = +(iuk)2 and so positive or negative terms are equivalent. We will denote the Clifford algebra on Cn with the standard quadratic form by Cln(C).
There are two separate cases to consider, according to whether n is even or odd. When n is even, the algebra Cln(C) is central simple and so by the Artin–Wedderburn theorem is isomorphic to a matrix algebra over C.
When n is odd, the center includes not only the scalars but the pseudoscalars (degree n elements) as well. We can always find a normalized pseudoscalar ω such that ω2 = 1. Define the operators
P
±
=
1
2
(
1
±
ω
)
.
{\displaystyle P_{\pm }={\frac {1}{2}}(1\pm \omega ).}
These two operators form a complete set of orthogonal idempotents, and since they are central they give a decomposition of Cln(C) into a direct sum of two algebras
C
l
n
(
C
)
=
C
l
n
+
(
C
)
⊕
C
l
n
−
(
C
)
,
{\displaystyle \mathrm {Cl} _{n}(\mathbf {C} )=\mathrm {Cl} _{n}^{+}(\mathbf {C} )\oplus \mathrm {Cl} _{n}^{-}(\mathbf {C} ),}
where
C
l
n
±
(
C
)
=
P
±
C
l
n
(
C
)
.
{\displaystyle \mathrm {Cl} _{n}^{\pm }(\mathbf {C} )=P_{\pm }\mathrm {Cl} _{n}(\mathbf {C} ).}
The algebras Cln±(C) are just the positive and negative eigenspaces of ω and the P± are just the projection operators. Since ω is odd, these algebras are mixed by α (the linear map on V defined by v ↦ −v):
α
(
C
l
n
±
(
C
)
)
=
C
l
n
∓
(
C
)
,
{\displaystyle \alpha \left(\mathrm {Cl} _{n}^{\pm }(\mathbf {C} )\right)=\mathrm {Cl} _{n}^{\mp }(\mathbf {C} ),}
and therefore isomorphic (since α is an automorphism). These two isomorphic algebras are each central simple and so, again, isomorphic to a matrix algebra over C. The sizes of the matrices can be determined from the fact that the dimension of Cln(C) is 2n. What we have then is the following table:
The even subalgebra Cl[0]n(C) of Cln(C) is (non-canonically) isomorphic to Cln−1(C). When n is even, the even subalgebra can be identified with the block diagonal matrices (when partitioned into 2 × 2 block matrices). When n is odd, the even subalgebra consists of those elements of End(CN) ⊕ End(CN) for which the two pieces are identical. Picking either piece then gives an isomorphism with Cln[0](C) ≅ End(CN).
=== Complex spinors in even dimension ===
The classification allows Dirac spinors and Weyl spinors to be defined in even dimension.
In even dimension n, the Clifford algebra Cln(C) is isomorphic to End(CN), which has its fundamental representation on Δn := CN. A complex Dirac spinor is an element of Δn. The term complex signifies that it is the element of a representation space of a complex Clifford algebra, rather than that is an element of a complex vector space.
The even subalgebra Cln0(C) is isomorphic to End(CN/2) ⊕ End(CN/2) and therefore decomposes to the direct sum of two irreducible representation spaces Δ+n ⊕ Δ−n, each isomorphic to CN/2. A left-handed (respectively right-handed) complex Weyl spinor is an element of Δ+n (respectively, Δ−n).
=== Proof of the structure theorem for complex Clifford algebras ===
The structure theorem is simple to prove inductively. For base cases, Cl0(C) is simply C ≅ End(C), while Cl1(C) is given by the algebra C ⊕ C ≅ End(C) ⊕ End(C) by defining the only gamma matrix as γ1 = (1, −1).
We will also need Cl2(C) ≅ End(C2). The Pauli matrices can be used to generate the Clifford algebra by setting γ1 = σ1, γ2 = σ2. The span of the generated algebra is End(C2).
The proof is completed by constructing an isomorphism Cln+2(C) ≅ Cln(C) ⊗ Cl2(C). Let γa generate Cln(C), and
γ
~
a
{\displaystyle {\tilde {\gamma }}_{a}}
generate Cl2(C). Let ω = i
γ
~
1
γ
~
2
{\displaystyle {\tilde {\gamma }}_{1}{\tilde {\gamma }}_{2}}
be the chirality element satisfying ω2 = 1 and
γ
~
a
{\displaystyle {\tilde {\gamma }}_{a}}
ω + ω
γ
~
a
{\displaystyle {\tilde {\gamma }}_{a}}
= 0. These can be used to construct gamma matrices for Cln+2(C) by setting Γa = γa ⊗ ω for 1 ≤ a ≤ n and Γa = 1 ⊗
γ
~
a
−
n
{\displaystyle {\tilde {\gamma }}_{a-n}}
for a = n + 1, n + 2. These can be shown to satisfy the required Clifford algebra and by the universal property of Clifford algebras, there is an isomorphism Cln(C) ⊗ Cl2(C) → Cln+2(C).
Finally, in the even case this means by the induction hypothesis Cln+2(C) ≅ End(CN) ⊗ End(C2) ≅ End(CN+1). The odd case follows similarly as the tensor product distributes over direct sums.
== Real case ==
The real case is significantly more complicated, exhibiting a periodicity of 8 rather than 2, and there is a 2-parameter family of Clifford algebras.
=== Classification of quadratic forms ===
Firstly, there are non-isomorphic quadratic forms of a given degree, classified by signature.
Every nondegenerate quadratic form on a real vector space is equivalent to an isotropic quadratic form:
Q
(
u
)
=
u
1
2
+
⋯
+
u
p
2
−
u
p
+
1
2
−
⋯
−
u
p
+
q
2
{\displaystyle Q(u)=u_{1}^{2}+\cdots +u_{p}^{2}-u_{p+1}^{2}-\cdots -u_{p+q}^{2}}
where n = p + q is the dimension of the vector space. The pair of integers (p, q) is called the signature of the quadratic form. The real vector space with this quadratic form is often denoted Rp,q. The Clifford algebra on Rp,q is denoted Clp,q(R).
A standard orthonormal basis {ei} for Rp,q consists of n = p + q mutually orthogonal vectors, p of which have norm +1 and q of which have norm −1.
=== Unit pseudoscalar ===
Given a standard basis {ei} as defined in the previous subsection, the unit pseudoscalar in Clp,q(R) is defined as
ω
=
e
1
e
2
⋯
e
n
.
{\displaystyle \omega =e_{1}e_{2}\cdots e_{n}.}
This is both a Coxeter element of sorts (product of reflections) and a longest element of a Coxeter group in the Bruhat order; this is an analogy. It corresponds to and generalizes a volume form (in the exterior algebra; for the trivial quadratic form, the unit pseudoscalar is a volume form), and lifts reflection through the origin (meaning that the image of the unit pseudoscalar is reflection through the origin, in the orthogonal group).
To compute the square ω2 = (e1e2⋅⋅⋅en)(e1e2⋅⋅⋅en), one can either reverse the order of the second group, yielding sgn(σ)e1e2⋅⋅⋅enen⋅⋅⋅e2e1, or apply a perfect shuffle, yielding sgn(σ)e1e1e2e2⋅⋅⋅enen. These both have sign (−1)⌊n/2⌋ = (−1)n(n−1)/2, which is 4-periodic (proof), and combined with eiei = ±1, this shows that the square of ω is given by
ω
2
=
(
−
1
)
n
(
n
−
1
)
2
(
−
1
)
q
=
(
−
1
)
(
p
−
q
)
(
p
−
q
−
1
)
2
=
{
+
1
p
−
q
≡
0
,
1
mod
4
−
1
p
−
q
≡
2
,
3
mod
4.
{\displaystyle \omega ^{2}=(-1)^{\frac {n(n-1)}{2}}(-1)^{q}=(-1)^{\frac {(p-q)(p-q-1)}{2}}={\begin{cases}+1&p-q\equiv 0,1\mod {4}\\-1&p-q\equiv 2,3\mod {4}.\end{cases}}}
Note that, unlike the complex case, it is not in general possible to find a pseudoscalar that squares to +1.
=== Center ===
If n (equivalently, p − q) is even, the algebra Clp,q(R) is central simple and so isomorphic to a matrix algebra over R or H by the Artin–Wedderburn theorem.
If n (equivalently, p − q) is odd then the algebra is no longer central simple but rather has a center which includes the pseudoscalars as well as the scalars. If n is odd and ω2 = +1 (equivalently, if p − q ≡ 1 (mod 4)) then, just as in the complex case, the algebra Clp,q(R) decomposes into a direct sum of isomorphic algebras
Cl
p
,
q
(
R
)
=
Cl
p
,
q
+
(
R
)
⊕
Cl
p
,
q
−
(
R
)
,
{\displaystyle \operatorname {Cl} _{p,q}(\mathbf {R} )=\operatorname {Cl} _{p,q}^{+}(\mathbf {R} )\oplus \operatorname {Cl} _{p,q}^{-}(\mathbf {R} ),}
each of which is central simple and so isomorphic to a matrix algebra over R or H.
If n is odd and ω2 = −1 (equivalently, if p − q ≡ −1 (mod 4)) then the center of Clp,q(R) is isomorphic to C and can be considered as a complex algebra. As a complex algebra, it is central simple and so isomorphic to a matrix algebra over C.
=== Classification ===
All told there are three properties which determine the class of the algebra Clp,q(R):
signature mod 2: n is even/odd: central simple or not
signature mod 4: ω2 = ±1: if not central simple, center is R ⊕ R or C
signature mod 8: the Brauer class of the algebra (n even) or even subalgebra (n odd) is R or H
Each of these properties depends only on the signature p − q modulo 8. The complete classification table is given below. The size of the matrices is determined by the requirement that Clp,q(R) have dimension 2p+q.
It may be seen that of all matrix ring types mentioned, there is only one type shared by complex and real algebras: the type M2m(C). For example, Cl2(C) and Cl3,0(R) are both determined to be M2(C). It is important to note that there is a difference in the classifying isomorphisms used. Since the Cl2(C) is algebra isomorphic via a C-linear map (which is necessarily R-linear), and Cl3,0(R) is algebra isomorphic via an R-linear map, Cl2(C) and Cl3,0(R) are R-algebra isomorphic.
A table of this classification for p + q ≤ 8 follows. Here p + q runs vertically and p − q runs horizontally (e.g. the algebra Cl1,3(R) ≅ M2(H) is found in row 4, column −2).
=== Symmetries ===
There is a tangled web of symmetries and relationships in the above table.
Cl
p
+
1
,
q
+
1
(
R
)
=
M
2
(
Cl
p
,
q
(
R
)
)
Cl
p
+
4
,
q
(
R
)
=
Cl
p
,
q
+
4
(
R
)
{\displaystyle {\begin{aligned}\operatorname {Cl} _{p+1,q+1}(\mathbf {R} )&=\mathrm {M} _{2}(\operatorname {Cl} _{p,q}(\mathbf {R} ))\\\operatorname {Cl} _{p+4,q}(\mathbf {R} )&=\operatorname {Cl} _{p,q+4}(\mathbf {R} )\end{aligned}}}
Going over 4 spots in any row yields an identical algebra.
From these Bott periodicity follows:
Cl
p
+
8
,
q
(
R
)
=
Cl
p
+
4
,
q
+
4
(
R
)
=
M
2
4
(
Cl
p
,
q
(
R
)
)
.
{\displaystyle \operatorname {Cl} _{p+8,q}(\mathbf {R} )=\operatorname {Cl} _{p+4,q+4}(\mathbf {R} )=M_{2^{4}}(\operatorname {Cl} _{p,q}(\mathbf {R} )).}
If the signature satisfies p − q ≡ 1 (mod 4) then
Cl
p
+
k
,
q
(
R
)
=
Cl
p
,
q
+
k
(
R
)
.
{\displaystyle \operatorname {Cl} _{p+k,q}(\mathbf {R} )=\operatorname {Cl} _{p,q+k}(\mathbf {R} ).}
(The table is symmetric about columns with signature ..., −7, −3, 1, 5, ...)
Thus if the signature satisfies p − q ≡ 1 (mod 4),
Cl
p
+
k
,
q
(
R
)
=
Cl
p
,
q
+
k
(
R
)
=
Cl
p
−
k
+
k
,
q
+
k
(
R
)
=
M
2
k
(
Cl
p
−
k
,
q
(
R
)
)
=
M
2
k
(
Cl
p
,
q
−
k
(
R
)
)
.
{\displaystyle \operatorname {Cl} _{p+k,q}(\mathbf {R} )=\operatorname {Cl} _{p,q+k}(\mathbf {R} )=\operatorname {Cl} _{p-k+k,q+k}(\mathbf {R} )=\mathrm {M} _{2^{k}}(\operatorname {Cl} _{p-k,q}(\mathbf {R} ))=\mathrm {M} _{2^{k}}(\operatorname {Cl} _{p,q-k}(\mathbf {R} )).}
== See also ==
Dirac algebra Cl1,3(C)
Pauli algebra Cl3,0(R)
Spacetime algebra Cl1,3(R)
Clifford module
Spin representation
== References ==
== Sources ==
Budinich, Paolo; Trautman, Andrzej (1988). The Spinorial Chessboard. Springer Verlag. ISBN 978-3-540-19078-3.
Lawson, H. Blaine; Michelsohn, Marie-Louise (2016). Spin Geometry. Princeton Mathematical Series. Vol. 38. Princeton University Press. ISBN 978-1-4008-8391-2.
Porteous, Ian R. (1995). Clifford Algebras and the Classical Groups. Cambridge Studies in Advanced Mathematics. Vol. 50. Cambridge University Press. ISBN 978-0-521-55177-9. | Wikipedia/Classification_of_Clifford_algebras |
In physics, the screened Poisson equation is a Poisson equation, which arises in (for example) the Klein–Gordon equation, electric field screening in plasmas, and nonlocal granular fluidity in granular flow.
== Statement of the equation ==
The equation is
[
Δ
−
λ
2
]
u
(
r
)
=
−
f
(
r
)
,
{\displaystyle \left[\Delta -\lambda ^{2}\right]u(\mathbf {r} )=-f(\mathbf {r} ),}
where
Δ
{\displaystyle \Delta }
is the Laplace operator, λ is a constant that expresses the "screening", f is an arbitrary function of position (known as the "source function") and u is the function to be determined.
In the homogeneous case (f=0), the screened Poisson equation is the same as the time-independent Klein–Gordon equation. In the inhomogeneous case, the screened Poisson equation is very similar to the inhomogeneous Helmholtz equation, the only difference being the sign within the brackets.
=== Electrostatics ===
In electric-field screening, screened Poisson equation for the electric potential
ϕ
(
r
)
{\displaystyle \phi (\mathbf {r} )}
is usually written as (SI units)
[
Δ
−
k
0
2
]
ϕ
(
r
)
=
−
ρ
e
x
t
(
r
)
ϵ
0
,
{\displaystyle \left[\Delta -k_{0}^{2}\right]\phi (\mathbf {r} )=-{\frac {\rho _{\rm {ext}}(\mathbf {r} )}{\epsilon _{0}}},}
where
k
0
−
1
{\displaystyle k_{0}^{-1}}
is the screening length,
ρ
e
x
t
(
r
)
{\displaystyle \rho _{\rm {ext}}(\mathbf {r} )}
is the charge density produced by an external field in the absence of screening and
ϵ
0
{\displaystyle \epsilon _{0}}
is the vacuum permittivity. This equation can be derived in several screening models like Thomas–Fermi screening in solid-state physics and Debye screening in plasmas.
== Solutions ==
=== Three dimensions ===
Without loss of generality, we will take λ to be non-negative. When λ is zero, the equation reduces to Poisson's equation. Therefore, when λ is very small, the solution approaches that of the unscreened Poisson equation, which, in dimension
n
=
3
{\displaystyle n=3}
, is a superposition of 1/r functions weighted by the source function f:
u
(
r
)
(
Poisson
)
=
∭
d
3
r
′
f
(
r
′
)
4
π
|
r
−
r
′
|
.
{\displaystyle u(\mathbf {r} )_{({\text{Poisson}})}=\iiint \mathrm {d} ^{3}\mathbf {r} '{\frac {f(\mathbf {r} ')}{4\pi |\mathbf {r} -\mathbf {r} '|}}.}
On the other hand, when λ is extremely large, u approaches the value f/λ2, which goes to zero as λ goes to infinity. As we shall see, the solution for intermediate values of λ behaves as a superposition of screened (or damped) 1/r functions, with λ behaving as the strength of the screening.
The screened Poisson equation can be solved for general f using the method of Green's functions. The Green's function G is defined by
[
Δ
−
λ
2
]
G
(
r
)
=
−
δ
3
(
r
)
,
{\displaystyle \left[\Delta -\lambda ^{2}\right]G(\mathbf {r} )=-\delta ^{3}(\mathbf {r} ),}
where δ3 is a delta function with unit mass concentrated at the origin of R3.
Assuming u and its derivatives vanish at large r, we may perform a continuous Fourier transform in spatial coordinates:
G
~
(
k
)
=
∭
d
3
r
G
(
r
)
e
−
i
k
⋅
r
{\displaystyle {\tilde {G}}(\mathbf {k} )=\iiint \mathrm {d} ^{3}\mathbf {r} \;G(\mathbf {r} )e^{-i\mathbf {k} \cdot \mathbf {r} }}
where the integral is taken over all space. It is then straightforward to show that
[
k
2
+
λ
2
]
G
~
(
k
)
=
1.
{\displaystyle \left[k^{2}+\lambda ^{2}\right]{\tilde {G}}(\mathbf {k} )=1.}
The Green's function in r is therefore given by the inverse Fourier transform,
G
(
r
)
=
1
(
2
π
)
3
∭
d
3
k
e
i
k
⋅
r
k
2
+
λ
2
.
{\displaystyle G(\mathbf {r} )={\frac {1}{(2\pi )^{3}}}\;\iiint \mathrm {d} ^{3}\!\mathbf {k} \;{\frac {e^{i\mathbf {k} \cdot \mathbf {r} }}{k^{2}+\lambda ^{2}}}.}
This integral may be evaluated using spherical coordinates in k-space. The integration over the angular coordinates is straightforward, and the integral reduces to one over the radial wavenumber
k
r
{\displaystyle k_{r}}
:
G
(
r
)
=
1
2
π
2
r
∫
0
∞
d
k
r
k
r
sin
k
r
r
k
r
2
+
λ
2
.
{\displaystyle G(\mathbf {r} )={\frac {1}{2\pi ^{2}r}}\;\int _{0}^{\infty }\mathrm {d} k_{r}\;{\frac {k_{r}\,\sin k_{r}r}{k_{r}^{2}+\lambda ^{2}}}.}
This may be evaluated using contour integration. The result is:
G
(
r
)
=
e
−
λ
r
4
π
r
.
{\displaystyle G(\mathbf {r} )={\frac {e^{-\lambda r}}{4\pi r}}.}
The solution to the full problem is then given by
u
(
r
)
=
∫
d
3
r
′
G
(
r
−
r
′
)
f
(
r
′
)
=
∫
d
3
r
′
e
−
λ
|
r
−
r
′
|
4
π
|
r
−
r
′
|
f
(
r
′
)
.
{\displaystyle u(\mathbf {r} )=\int \mathrm {d} ^{3}\mathbf {r} 'G(\mathbf {r} -\mathbf {r} ')f(\mathbf {r} ')=\int \mathrm {d} ^{3}\mathbf {r} '{\frac {e^{-\lambda |\mathbf {r} -\mathbf {r} '|}}{4\pi |\mathbf {r} -\mathbf {r} '|}}f(\mathbf {r} ').}
As stated above, this is a superposition of screened 1/r functions, weighted by the source function f and with λ acting as the strength of the screening. The screened 1/r function is often encountered in physics as a screened Coulomb potential, also called a "Yukawa potential".
=== Two dimensions ===
In two dimensions:
In the case of a magnetized plasma, the screened Poisson equation is quasi-2D:
(
Δ
⊥
−
1
ρ
2
)
u
(
r
⊥
)
=
−
f
(
r
⊥
)
{\displaystyle \left(\Delta _{\perp }-{\frac {1}{\rho ^{2}}}\right)u(\mathbf {r} _{\perp })=-f(\mathbf {r} _{\perp })}
with
Δ
⊥
=
∇
⋅
∇
⊥
{\displaystyle \Delta _{\perp }=\nabla \cdot \nabla _{\perp }}
and
∇
⊥
=
∇
−
B
B
⋅
∇
{\displaystyle \nabla _{\perp }=\nabla -{\frac {\mathbf {B} }{B}}\cdot \nabla }
, with
B
{\displaystyle \mathbf {B} }
the magnetic field and
ρ
{\displaystyle \rho }
is the (ion) Larmor radius.
The two-dimensional Fourier Transform of the associated Green's function is:
G
~
(
k
⊥
)
=
∬
d
2
r
G
(
r
⊥
)
e
−
i
k
⊥
⋅
r
⊥
.
{\displaystyle {\tilde {G}}(\mathbf {k_{\perp }} )=\iint d^{2}\mathbf {r} ~G(\mathbf {r} _{\perp })e^{-i\mathbf {k} _{\perp }\cdot \mathbf {r} _{\perp }}.}
The 2D screened Poisson equation yields:
(
k
⊥
2
+
1
ρ
2
)
G
~
(
k
⊥
)
=
1.
{\displaystyle \left(k_{\perp }^{2}+{\frac {1}{\rho ^{2}}}\right){\tilde {G}}(\mathbf {k} _{\perp })=1.}
The Green's function is therefore given by the inverse Fourier transform:
G
(
r
⊥
)
=
1
4
π
2
∬
d
2
k
e
i
k
⊥
⋅
r
⊥
k
⊥
2
+
1
/
ρ
2
.
{\displaystyle G(\mathbf {r} _{\perp })={\frac {1}{4\pi ^{2}}}\;\iint \mathrm {d} ^{2}\!\mathbf {k} \;{\frac {e^{i\mathbf {k} _{\perp }\cdot \mathbf {r} _{\perp }}}{k_{\perp }^{2}+1/\rho ^{2}}}.}
This integral can be calculated using polar coordinates in k-space:
k
⊥
=
(
k
r
cos
(
θ
)
,
k
r
sin
(
θ
)
)
{\displaystyle \mathbf {k} _{\perp }=(k_{r}\cos(\theta ),k_{r}\sin(\theta ))}
The integration over the angular coordinate gives a Bessel function, and the integral reduces to one over the radial wavenumber
k
r
{\displaystyle k_{r}}
:
G
(
r
⊥
)
=
1
2
π
∫
0
∞
d
k
r
k
r
J
0
(
k
r
r
⊥
)
k
r
2
+
1
/
ρ
2
=
1
2
π
K
0
(
r
⊥
/
ρ
)
.
{\displaystyle G(\mathbf {r} _{\perp })={\frac {1}{2\pi }}\;\int _{0}^{\infty }\mathrm {d} k_{r}\;{\frac {k_{r}\,J_{0}(k_{r}r_{\perp })}{k_{r}^{2}+1/\rho ^{2}}}={\frac {1}{2\pi }}K_{0}(r_{\perp }\,/\,\rho ).}
== Connection to the Laplace distribution ==
The Green's functions in both 2D and 3D are identical to the probability density function of the multivariate Laplace distribution for two and three dimensions respectively.
== Application in differential geometry ==
The homogeneous case, studied in the context of differential geometry, involving Einstein warped product manifolds, explores cases where the warped function satisfies the homogeneous version of the screened Poisson equation. Under specific conditions, the manifold dimension, Ricci curvature, and screening parameter are interconnected via a quadratic relationship.
== See also ==
Yukawa interaction
== References == | Wikipedia/Screened_Poisson_equation |
The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, it is a characteristic of the system's total energy and momentum that is the same in all frames of reference related by Lorentz transformations. If a center-of-momentum frame exists for the system, then the invariant mass of a system is equal to its total mass in that "rest frame". In other reference frames, where the system's momentum is non-zero, the total mass (a.k.a. relativistic mass) of the system is greater than the invariant mass, but the invariant mass remains unchanged.
Because of mass–energy equivalence, the rest energy of the system is simply the invariant mass times the speed of light squared. Similarly, the total energy of the system is its total (relativistic) mass times the speed of light squared.
Systems whose four-momentum is a null vector, a light-like vector within the context of Minkowski space (for example, a single photon or many photons moving in exactly the same direction) have zero invariant mass and are referred to as massless. A physical object or particle moving faster than the speed of light would have space-like four-momenta (such as the hypothesized tachyon), and these do not appear to exist. Any time-like four-momentum possesses a reference frame where the momentum (3-dimensional) is zero, which is a center of momentum frame. In this case, invariant mass is positive and is referred to as the rest mass.
If objects within a system are in relative motion, then the invariant mass of the whole system will differ from the sum of the objects' rest masses. This is also equal to the total energy of the system divided by c2. See mass–energy equivalence for a discussion of definitions of mass. Since the mass of systems must be measured with a weight or mass scale in a center of momentum frame in which the entire system has zero momentum, such a scale always measures the system's invariant mass. For example, a scale would measure the kinetic energy of the molecules in a bottle of gas to be part of invariant mass of the bottle, and thus also its rest mass. The same is true for massless particles in such system, which add invariant mass and also rest mass to systems, according to their energy.
For an isolated massive system, the center of mass of the system moves in a straight line with a steady subluminal velocity (with a velocity depending on the reference frame used to view it). Thus, an observer can always be placed to move along with it. In this frame, which is the center-of-momentum frame, the total momentum is zero, and the system as a whole may be thought of as being "at rest" if it is a bound system (like a bottle of gas). In this frame, which exists under these assumptions, the invariant mass of the system is equal to the total system energy (in the zero-momentum frame) divided by c2. This total energy in the center of momentum frame, is the minimum energy which the system may be observed to have, when seen by various observers from various inertial frames.
Note that for reasons above, such a rest frame does not exist for single photons, or rays of light moving in one direction. When two or more photons move in different directions, however, a center of mass frame (or "rest frame" if the system is bound) exists. Thus, the mass of a system of several photons moving in different directions is positive, which means that an invariant mass exists for this system even though it does not exist for each photon.
== Sum of rest masses ==
The invariant mass of a system includes the mass of any kinetic energy of the system constituents that remains in the center of momentum frame, so the invariant mass of a system may be greater than sum of the invariant masses (rest masses) of its separate constituents. For example, rest mass and invariant mass are zero for individual photons even though they may add mass to the invariant mass of systems. For this reason, invariant mass is in general not an additive quantity (although there are a few rare situations where it may be, as is the case when massive particles in a system without potential or kinetic energy can be added to a total mass).
Consider the simple case of two-body system, where object A is moving towards another object B which is initially at rest (in any particular frame of reference). The magnitude of invariant mass of this two-body system (see definition below) is different from the sum of rest mass (i.e. their respective mass when stationary). Even if we consider the same system from center-of-momentum frame, where net momentum is zero, the magnitude of the system's invariant mass is not equal to the sum of the rest masses of the particles within it.
The kinetic energy of such particles and the potential energy of the force fields increase the total energy above the sum of the particle rest masses, and both terms contribute to the invariant mass of the system. The sum of the particle kinetic energies as calculated by an observer is smallest in the center of momentum frame (again, called the "rest frame" if the system is bound).
They will often also interact through one or more of the fundamental forces, giving them a potential energy of interaction, possibly negative.
== As defined in particle physics ==
In particle physics, the invariant mass m0 is equal to the mass in the rest frame of the particle, and can be calculated by the particle's energy E and its momentum p as measured in any frame, by the energy–momentum relation:
m
0
2
c
2
=
(
E
c
)
2
−
‖
p
‖
2
{\displaystyle m_{0}^{2}c^{2}=\left({\frac {E}{c}}\right)^{2}-\left\|\mathbf {p} \right\|^{2}}
or in natural units where c = 1,
m
0
2
=
E
2
−
‖
p
‖
2
.
{\displaystyle m_{0}^{2}=E^{2}-\left\|\mathbf {p} \right\|^{2}.}
This invariant mass is the same in all frames of reference (see also special relativity). This equation says that the invariant mass is the pseudo-Euclidean length of the four-vector (E, p), calculated using the relativistic version of the Pythagorean theorem which has a different sign for the space and time dimensions. This length is preserved under any Lorentz boost or rotation in four dimensions, just like the ordinary length of a vector is preserved under rotations. In quantum theory the invariant mass is a parameter in the relativistic Dirac equation for an elementary particle. The Dirac quantum operator corresponds to the particle four-momentum vector.
Since the invariant mass is determined from quantities which are conserved during a decay, the invariant mass calculated using the energy and momentum of the decay products of a single particle is equal to the mass of the particle that decayed.
The mass of a system of particles can be calculated from the general formula:
(
W
c
2
)
2
=
(
∑
E
)
2
−
‖
∑
p
c
‖
2
,
{\displaystyle \left(Wc^{2}\right)^{2}=\left(\sum E\right)^{2}-\left\|\sum \mathbf {p} c\right\|^{2},}
where
W
{\displaystyle W}
is the invariant mass of the system of particles, equal to the mass of the decay particle.
∑
E
{\textstyle \sum E}
is the sum of the energies of the particles
∑
p
{\textstyle \sum \mathbf {p} }
is the vector sum of the momentum of the particles (includes both magnitude and direction of the momenta)
The term invariant mass is also used in inelastic scattering experiments. Given an inelastic reaction with total incoming energy larger than the total detected energy (i.e. not all outgoing particles are detected in the experiment), the invariant mass (also known as the "missing mass") W of the reaction is defined as follows (in natural units):
W
2
=
(
∑
E
in
−
∑
E
out
)
2
−
‖
∑
p
in
−
∑
p
out
‖
2
.
{\displaystyle W^{2}=\left(\sum E_{\text{in}}-\sum E_{\text{out}}\right)^{2}-\left\|\sum \mathbf {p} _{\text{in}}-\sum \mathbf {p} _{\text{out}}\right\|^{2}.}
If there is one dominant particle which was not detected during an experiment, a plot of the invariant mass will show a sharp peak at the mass of the missing particle.
In those cases when the momentum along one direction cannot be measured (i.e. in the case of a neutrino, whose presence is only inferred from the missing energy) the transverse mass is used.
== Example: two-particle collision ==
In a two-particle collision (or a two-particle decay) the square of the invariant mass (in natural units) is
M
2
=
(
E
1
+
E
2
)
2
−
‖
p
1
+
p
2
‖
2
=
m
1
2
+
m
2
2
+
2
(
E
1
E
2
−
p
1
⋅
p
2
)
.
{\displaystyle {\begin{aligned}M^{2}&=(E_{1}+E_{2})^{2}-\left\|\mathbf {p} _{1}+\mathbf {p} _{2}\right\|^{2}\\&=m_{1}^{2}+m_{2}^{2}+2\left(E_{1}E_{2}-\mathbf {p} _{1}\cdot \mathbf {p} _{2}\right).\end{aligned}}}
=== Massless particles ===
The invariant mass of a system made of two massless particles whose momenta form an angle
θ
{\displaystyle \theta }
has a convenient expression:
M
2
=
(
E
1
+
E
2
)
2
−
‖
p
1
+
p
2
‖
2
=
[
(
p
1
,
0
,
0
,
p
1
)
+
(
p
2
,
0
,
p
2
sin
θ
,
p
2
cos
θ
)
]
2
=
(
p
1
+
p
2
)
2
−
p
2
2
sin
2
θ
−
(
p
1
+
p
2
cos
θ
)
2
=
2
p
1
p
2
(
1
−
cos
θ
)
.
{\displaystyle {\begin{aligned}M^{2}&=(E_{1}+E_{2})^{2}-\left\|{\textbf {p}}_{1}+{\textbf {p}}_{2}\right\|^{2}\\&=[(p_{1},0,0,p_{1})+(p_{2},0,p_{2}\sin \theta ,p_{2}\cos \theta )]^{2}\\&=(p_{1}+p_{2})^{2}-p_{2}^{2}\sin ^{2}\theta -(p_{1}+p_{2}\cos \theta )^{2}\\&=2p_{1}p_{2}(1-\cos \theta ).\end{aligned}}}
=== Collider experiments ===
In particle collider experiments, one often defines the angular position of a particle in terms of an azimuthal angle
ϕ
{\displaystyle \phi }
and pseudorapidity
η
{\displaystyle \eta }
. Additionally the transverse momentum,
p
T
{\displaystyle p_{T}}
, is usually measured. In this case if the particles are massless, or highly relativistic (
E
≫
m
{\displaystyle E\gg m}
) then the invariant mass becomes:
M
2
=
2
p
T
1
p
T
2
(
cosh
(
η
1
−
η
2
)
−
cos
(
ϕ
1
−
ϕ
2
)
)
.
{\displaystyle M^{2}=2p_{T1}p_{T2}(\cosh(\eta _{1}-\eta _{2})-\cos(\phi _{1}-\phi _{2})).}
== Rest energy ==
Rest energy (also called rest mass energy) is the energy associated with a particle's invariant mass.
The rest energy
E
0
{\displaystyle E_{0}}
of a particle is defined as:
E
0
=
m
0
c
2
,
{\displaystyle E_{0}=m_{0}c^{2},}
where
c
{\displaystyle c}
is the speed of light in vacuum. In general, only differences in energy have physical significance.
The concept of rest energy follows from the special theory of relativity that leads to Einstein's famous conclusion about equivalence of energy and mass. See Special relativity § Relativistic dynamics and invariance.
== See also ==
Mass in special relativity
Invariant (physics)
Transverse mass
== References ==
Landau, L.D.; Lifshitz, E.M. (1975). The Classical Theory of Fields: 4-th revised English Edition: Course of Theoretical Physics Vol. 2. Butterworth Heinemann. ISBN 0-7506-2768-9.
Halzen, Francis; Martin, Alan (1984). Quarks & Leptons: An Introductory Course in Modern Particle Physics. John Wiley & Sons. ISBN 0-471-88741-2.
== Citations == | Wikipedia/Rest_mass_energy |
In mathematics, orthogonal functions belong to a function space that is a vector space equipped with a bilinear form. When the function space has an interval as the domain, the bilinear form may be the integral of the product of functions over the interval:
⟨
f
,
g
⟩
=
∫
f
(
x
)
¯
g
(
x
)
d
x
.
{\displaystyle \langle f,g\rangle =\int {\overline {f(x)}}g(x)\,dx.}
The functions
f
{\displaystyle f}
and
g
{\displaystyle g}
are orthogonal when this integral is zero, i.e.
⟨
f
,
g
⟩
=
0
{\displaystyle \langle f,\,g\rangle =0}
whenever
f
≠
g
{\displaystyle f\neq g}
. As with a basis of vectors in a finite-dimensional space, orthogonal functions can form an infinite basis for a function space. Conceptually, the above integral is the equivalent of a vector dot product; two vectors are mutually independent (orthogonal) if their dot-product is zero.
Suppose
{
f
0
,
f
1
,
…
}
{\displaystyle \{f_{0},f_{1},\ldots \}}
is a sequence of orthogonal functions of nonzero L2-norms
‖
f
n
‖
2
=
⟨
f
n
,
f
n
⟩
=
(
∫
f
n
2
d
x
)
1
2
{\textstyle \left\|f_{n}\right\|_{2}={\sqrt {\langle f_{n},f_{n}\rangle }}=\left(\int f_{n}^{2}\ dx\right)^{\frac {1}{2}}}
. It follows that the sequence
{
f
n
/
‖
f
n
‖
2
}
{\displaystyle \left\{f_{n}/\left\|f_{n}\right\|_{2}\right\}}
is of functions of L2-norm one, forming an orthonormal sequence. To have a defined L2-norm, the integral must be bounded, which restricts the functions to being square-integrable.
== Trigonometric functions ==
Several sets of orthogonal functions have become standard bases for approximating functions. For example, the sine functions sin nx and sin mx are orthogonal on the interval
x
∈
(
−
π
,
π
)
{\displaystyle x\in (-\pi ,\pi )}
when
m
≠
n
{\displaystyle m\neq n}
and n and m are positive integers. For then
2
sin
(
m
x
)
sin
(
n
x
)
=
cos
(
(
m
−
n
)
x
)
−
cos
(
(
m
+
n
)
x
)
,
{\displaystyle 2\sin \left(mx\right)\sin \left(nx\right)=\cos \left(\left(m-n\right)x\right)-\cos \left(\left(m+n\right)x\right),}
and the integral of the product of the two sine functions vanishes. Together with cosine functions, these orthogonal functions may be assembled into a trigonometric polynomial to approximate a given function on the interval with its Fourier series.
== Polynomials ==
If one begins with the monomial sequence
{
1
,
x
,
x
2
,
…
}
{\displaystyle \left\{1,x,x^{2},\dots \right\}}
on the interval
[
−
1
,
1
]
{\displaystyle [-1,1]}
and applies the Gram–Schmidt process, then one obtains the Legendre polynomials. Another collection of orthogonal polynomials are the associated Legendre polynomials.
The study of orthogonal polynomials involves weight functions
w
(
x
)
{\displaystyle w(x)}
that are inserted in the bilinear form:
⟨
f
,
g
⟩
=
∫
w
(
x
)
f
(
x
)
g
(
x
)
d
x
.
{\displaystyle \langle f,g\rangle =\int w(x)f(x)g(x)\,dx.}
For Laguerre polynomials on
(
0
,
∞
)
{\displaystyle (0,\infty )}
the weight function is
w
(
x
)
=
e
−
x
{\displaystyle w(x)=e^{-x}}
.
Both physicists and probability theorists use Hermite polynomials on
(
−
∞
,
∞
)
{\displaystyle (-\infty ,\infty )}
, where the weight function is
w
(
x
)
=
e
−
x
2
{\displaystyle w(x)=e^{-x^{2}}}
or
w
(
x
)
=
e
−
x
2
/
2
{\displaystyle w(x)=e^{-x^{2}/2}}
.
Chebyshev polynomials are defined on
[
−
1
,
1
]
{\displaystyle [-1,1]}
and use weights
w
(
x
)
=
1
1
−
x
2
{\textstyle w(x)={\frac {1}{\sqrt {1-x^{2}}}}}
or
w
(
x
)
=
1
−
x
2
{\textstyle w(x)={\sqrt {1-x^{2}}}}
.
Zernike polynomials are defined on the unit disk and have orthogonality of both radial and angular parts.
== Binary-valued functions ==
Walsh functions and Haar wavelets are examples of orthogonal functions with discrete ranges.
== Rational functions ==
Legendre and Chebyshev polynomials provide orthogonal families for the interval [−1, 1] while occasionally orthogonal families are required on [0, ∞). In this case it is convenient to apply the Cayley transform first, to bring the argument into [−1, 1]. This procedure results in families of rational orthogonal functions called Legendre rational functions and Chebyshev rational functions.
== In differential equations ==
Solutions of linear differential equations with boundary conditions can often be written as a weighted sum of orthogonal solution functions (a.k.a. eigenfunctions), leading to generalized Fourier series.
== See also ==
Eigenvalues and eigenvectors
Hilbert space
Karhunen–Loève theorem
Lauricella's theorem
Wannier function
== References ==
George B. Arfken & Hans J. Weber (2005) Mathematical Methods for Physicists, 6th edition, chapter 10: Sturm-Liouville Theory — Orthogonal Functions, Academic Press.
Price, Justin J. (1975). "Topics in orthogonal functions". American Mathematical Monthly. 82: 594–609. doi:10.2307/2319690.
Giovanni Sansone (translated by Ainsley H. Diamond) (1959) Orthogonal Functions, Interscience Publishers.
== External links ==
Orthogonal Functions, on MathWorld. | Wikipedia/Orthogonal_functions |
The sine-Gordon equation is a second-order nonlinear partial differential equation for a function
φ
{\displaystyle \varphi }
dependent on two variables typically denoted
x
{\displaystyle x}
and
t
{\displaystyle t}
, involving the wave operator and the sine of
φ
{\displaystyle \varphi }
.
It was originally introduced by Edmond Bour (1862) in the course of study of surfaces of constant negative curvature as the Gauss–Codazzi equation for surfaces of constant Gaussian curvature −1 in 3-dimensional space. The equation was rediscovered by Frenkel and Kontorova (1939) in their study of crystal dislocations known as the Frenkel–Kontorova model.
This equation attracted a lot of attention in the 1970s due to the presence of soliton solutions, and is an example of an integrable PDE. Among well-known integrable PDEs, the sine-Gordon equation is the only relativistic system due to its Lorentz invariance.
== Realizations of the sine-Gordon equation ==
=== Differential geometry ===
This is the first derivation of the equation, by Bour (1862).
There are two equivalent forms of the sine-Gordon equation. In the (real) space-time coordinates, denoted
(
x
,
t
)
{\displaystyle (x,t)}
, the equation reads:
φ
t
t
−
φ
x
x
+
sin
φ
=
0
,
{\displaystyle \varphi _{tt}-\varphi _{xx}+\sin \varphi =0,}
where partial derivatives are denoted by subscripts. Passing to the light-cone coordinates (u, v), akin to asymptotic coordinates where
u
=
x
+
t
2
,
v
=
x
−
t
2
,
{\displaystyle u={\frac {x+t}{2}},\quad v={\frac {x-t}{2}},}
the equation takes the form
φ
u
v
=
sin
φ
.
{\displaystyle \varphi _{uv}=\sin \varphi .}
This is the original form of the sine-Gordon equation, as it was considered in the 19th century in the course of investigation of surfaces of constant Gaussian curvature K = −1, also called pseudospherical surfaces.
Consider an arbitrary pseudospherical surface. Across every point on the surface there are two asymptotic curves. This allows us to construct a distinguished coordinate system for such a surface, in which u = constant, v = constant are the asymptotic lines, and the coordinates are incremented by the arc length on the surface. At every point on the surface, let
φ
{\displaystyle \varphi }
be the angle between the asymptotic lines.
The first fundamental form of the surface is
d
s
2
=
d
u
2
+
2
cos
φ
d
u
d
v
+
d
v
2
,
{\displaystyle ds^{2}=du^{2}+2\cos \varphi \,du\,dv+dv^{2},}
and the second fundamental form is
L
=
N
=
0
,
M
=
sin
φ
{\displaystyle L=N=0,M=\sin \varphi }
and the Gauss–Codazzi equation is
φ
u
v
=
sin
φ
.
{\displaystyle \varphi _{uv}=\sin \varphi .}
Thus, any pseudospherical surface gives rise to a solution of the sine-Gordon equation, although with some caveats: if the surface is complete, it is necessarily singular due to the Hilbert embedding theorem. In the simplest case, the pseudosphere, also known as the tractroid, corresponds to a static one-soliton, but the tractroid has a singular cusp at its equator.
Conversely, one can start with a solution to the sine-Gordon equation to obtain a pseudosphere uniquely up to rigid transformations. There is a theorem, sometimes called the fundamental theorem of surfaces, that if a pair of matrix-valued bilinear forms satisfy the Gauss–Codazzi equations, then they are the first and second fundamental forms of an embedded surface in 3-dimensional space. Solutions to the sine-Gordon equation can be used to construct such matrices by using the forms obtained above.
=== New solutions from old ===
The study of this equation and of the associated transformations of pseudospherical surfaces in the 19th century by Bianchi and Bäcklund led to the discovery of Bäcklund transformations. Another transformation of pseudospherical surfaces is the Lie transform introduced by Sophus Lie in 1879, which corresponds to Lorentz boosts for solutions of the sine-Gordon equation.
There are also some more straightforward ways to construct new solutions but which do not give new surfaces. Since the sine-Gordon equation is odd, the negative of any solution is another solution. However this does not give a new surface, as the sign-change comes down to a choice of direction for the normal to the surface. New solutions can be found by translating the solution: if
φ
{\displaystyle \varphi }
is a solution, then so is
φ
+
2
n
π
{\displaystyle \varphi +2n\pi }
for
n
{\displaystyle n}
an integer.
=== Frenkel–Kontorova model ===
=== A mechanical model ===
Consider a line of pendula, hanging on a straight line, in constant gravity. Connect the bobs of the pendula together by a string in constant tension. Let the angle of the pendulum at location
x
{\displaystyle x}
be
φ
{\displaystyle \varphi }
, then schematically, the dynamics of the line of pendulum follows Newton's second law:
m
φ
t
t
⏟
mass times acceleration
=
T
φ
x
x
⏟
tension
−
m
g
sin
φ
⏟
gravity
{\displaystyle \underbrace {m\varphi _{tt}} _{\text{mass times acceleration}}=\underbrace {T\varphi _{xx}} _{\text{tension}}-\underbrace {mg\sin \varphi } _{\text{gravity}}}
and this is the sine-Gordon equation, after scaling time and distance appropriately.
Note that this is not exactly correct, since the net force on a pendulum due to the tension is not precisely
T
φ
x
x
{\displaystyle T\varphi _{xx}}
, but more accurately
T
φ
x
x
(
1
+
φ
x
2
)
−
3
/
2
{\displaystyle T\varphi _{xx}(1+\varphi _{x}^{2})^{-3/2}}
. However this does give an intuitive picture for the sine-gordon equation. One can produce exact mechanical realizations of the sine-gordon equation by more complex methods.
== Naming ==
The name "sine-Gordon equation" is a pun on the well-known Klein–Gordon equation in physics:
φ
t
t
−
φ
x
x
+
φ
=
0.
{\displaystyle \varphi _{tt}-\varphi _{xx}+\varphi =0.}
The sine-Gordon equation is the Euler–Lagrange equation of the field whose Lagrangian density is given by
L
SG
(
φ
)
=
1
2
(
φ
t
2
−
φ
x
2
)
−
1
+
cos
φ
.
{\displaystyle {\mathcal {L}}_{\text{SG}}(\varphi )={\frac {1}{2}}(\varphi _{t}^{2}-\varphi _{x}^{2})-1+\cos \varphi .}
Using the Taylor series expansion of the cosine in the Lagrangian,
cos
(
φ
)
=
∑
n
=
0
∞
(
−
φ
2
)
n
(
2
n
)
!
,
{\displaystyle \cos(\varphi )=\sum _{n=0}^{\infty }{\frac {(-\varphi ^{2})^{n}}{(2n)!}},}
it can be rewritten as the Klein–Gordon Lagrangian plus higher-order terms:
L
SG
(
φ
)
=
1
2
(
φ
t
2
−
φ
x
2
)
−
φ
2
2
+
∑
n
=
2
∞
(
−
φ
2
)
n
(
2
n
)
!
=
L
KG
(
φ
)
+
∑
n
=
2
∞
(
−
φ
2
)
n
(
2
n
)
!
.
{\displaystyle {\begin{aligned}{\mathcal {L}}_{\text{SG}}(\varphi )&={\frac {1}{2}}(\varphi _{t}^{2}-\varphi _{x}^{2})-{\frac {\varphi ^{2}}{2}}+\sum _{n=2}^{\infty }{\frac {(-\varphi ^{2})^{n}}{(2n)!}}\\&={\mathcal {L}}_{\text{KG}}(\varphi )+\sum _{n=2}^{\infty }{\frac {(-\varphi ^{2})^{n}}{(2n)!}}.\end{aligned}}}
== Soliton solutions ==
An interesting feature of the sine-Gordon equation is the existence of soliton and multisoliton solutions.
=== 1-soliton solutions ===
The sine-Gordon equation has the following 1-soliton solutions:
φ
soliton
(
x
,
t
)
:=
4
arctan
(
e
m
γ
(
x
−
v
t
)
+
δ
)
,
{\displaystyle \varphi _{\text{soliton}}(x,t):=4\arctan \left(e^{m\gamma (x-vt)+\delta }\right),}
where
γ
2
=
1
1
−
v
2
,
{\displaystyle \gamma ^{2}={\frac {1}{1-v^{2}}},}
and the slightly more general form of the equation is assumed:
φ
t
t
−
φ
x
x
+
m
2
sin
φ
=
0.
{\displaystyle \varphi _{tt}-\varphi _{xx}+m^{2}\sin \varphi =0.}
The 1-soliton solution for which we have chosen the positive root for
γ
{\displaystyle \gamma }
is called a kink and represents a twist in the variable
φ
{\displaystyle \varphi }
which takes the system from one constant solution
φ
=
0
{\displaystyle \varphi =0}
to an adjacent constant solution
φ
=
2
π
{\displaystyle \varphi =2\pi }
. The states
φ
≅
2
π
n
{\displaystyle \varphi \cong 2\pi n}
are known as vacuum states, as they are constant solutions of zero energy. The 1-soliton solution in which we take the negative root for
γ
{\displaystyle \gamma }
is called an antikink. The form of the 1-soliton solutions can be obtained through application of a Bäcklund transform to the trivial (vacuum) solution and the integration of the resulting first-order differentials:
φ
u
′
=
φ
u
+
2
β
sin
φ
′
+
φ
2
,
{\displaystyle \varphi '_{u}=\varphi _{u}+2\beta \sin {\frac {\varphi '+\varphi }{2}},}
φ
v
′
=
−
φ
v
+
2
β
sin
φ
′
−
φ
2
with
φ
=
φ
0
=
0
{\displaystyle \varphi '_{v}=-\varphi _{v}+{\frac {2}{\beta }}\sin {\frac {\varphi '-\varphi }{2}}{\text{ with }}\varphi =\varphi _{0}=0}
for all time.
The 1-soliton solutions can be visualized with the use of the elastic ribbon sine-Gordon model introduced by Julio Rubinstein in 1970. Here we take a clockwise (left-handed) twist of the elastic ribbon to be a kink with topological charge
θ
K
=
−
1
{\displaystyle \theta _{\text{K}}=-1}
. The alternative counterclockwise (right-handed) twist with topological charge
θ
AK
=
+
1
{\displaystyle \theta _{\text{AK}}=+1}
will be an antikink.
=== 2-soliton solutions ===
Multi-soliton solutions can be obtained through continued application of the Bäcklund transform to the 1-soliton solution, as prescribed by a Bianchi lattice relating the transformed results. The 2-soliton solutions of the sine-Gordon equation show some of the characteristic features of the solitons. The traveling sine-Gordon kinks and/or antikinks pass through each other as if perfectly permeable, and the only observed effect is a phase shift. Since the colliding solitons recover their velocity and shape, such an interaction is called an elastic collision.
The kink-kink solution is given by
φ
K
/
K
(
x
,
t
)
=
4
arctan
(
v
sinh
x
1
−
v
2
cosh
v
t
1
−
v
2
)
{\displaystyle \varphi _{K/K}(x,t)=4\arctan \left({\frac {v\sinh {\frac {x}{\sqrt {1-v^{2}}}}}{\cosh {\frac {vt}{\sqrt {1-v^{2}}}}}}\right)}
while the kink-antikink solution is given by
φ
K
/
A
K
(
x
,
t
)
=
4
arctan
(
v
cosh
x
1
−
v
2
sinh
v
t
1
−
v
2
)
{\displaystyle \varphi _{K/AK}(x,t)=4\arctan \left({\frac {v\cosh {\frac {x}{\sqrt {1-v^{2}}}}}{\sinh {\frac {vt}{\sqrt {1-v^{2}}}}}}\right)}
Another interesting 2-soliton solutions arise from the possibility of coupled kink-antikink behaviour known as a breather. There are known three types of breathers: standing breather, traveling large-amplitude breather, and traveling small-amplitude breather.
The standing breather solution is given by
φ
(
x
,
t
)
=
4
arctan
(
1
−
ω
2
cos
(
ω
t
)
ω
cosh
(
1
−
ω
2
x
)
)
.
{\displaystyle \varphi (x,t)=4\arctan \left({\frac {{\sqrt {1-\omega ^{2}}}\;\cos(\omega t)}{\omega \;\cosh({\sqrt {1-\omega ^{2}}}\;x)}}\right).}
=== 3-soliton solutions ===
3-soliton collisions between a traveling kink and a standing breather or a traveling antikink and a standing breather results in a phase shift of the standing breather. In the process of collision between a moving kink and a standing breather,
the shift of the breather
Δ
B
{\displaystyle \Delta _{\text{B}}}
is given by
Δ
B
=
2
artanh
(
1
−
ω
2
)
(
1
−
v
K
2
)
1
−
ω
2
,
{\displaystyle \Delta _{\text{B}}={\frac {2\operatorname {artanh} {\sqrt {(1-\omega ^{2})(1-v_{\text{K}}^{2})}}}{\sqrt {1-\omega ^{2}}}},}
where
v
K
{\displaystyle v_{\text{K}}}
is the velocity of the kink, and
ω
{\displaystyle \omega }
is the breather's frequency. If the old position of the standing breather is
x
0
{\displaystyle x_{0}}
, after the collision the new position will be
x
0
+
Δ
B
{\displaystyle x_{0}+\Delta _{\text{B}}}
.
== Bäcklund transformation ==
Suppose that
φ
{\displaystyle \varphi }
is a solution of the sine-Gordon equation
φ
u
v
=
sin
φ
.
{\displaystyle \varphi _{uv}=\sin \varphi .\,}
Then the system
ψ
u
=
φ
u
+
2
a
sin
(
ψ
+
φ
2
)
ψ
v
=
−
φ
v
+
2
a
sin
(
ψ
−
φ
2
)
{\displaystyle {\begin{aligned}\psi _{u}&=\varphi _{u}+2a\sin {\Bigl (}{\frac {\psi +\varphi }{2}}{\Bigr )}\\\psi _{v}&=-\varphi _{v}+{\frac {2}{a}}\sin {\Bigl (}{\frac {\psi -\varphi }{2}}{\Bigr )}\end{aligned}}\,\!}
where a is an arbitrary parameter, is solvable for a function
ψ
{\displaystyle \psi }
which will also satisfy the sine-Gordon equation. This is an example of an auto-Bäcklund transform, as both
φ
{\displaystyle \varphi }
and
ψ
{\displaystyle \psi }
are solutions to the same equation, that is, the sine-Gordon equation.
By using a matrix system, it is also possible to find a linear Bäcklund transform for solutions of sine-Gordon equation.
For example, if
φ
{\displaystyle \varphi }
is the trivial solution
φ
≡
0
{\displaystyle \varphi \equiv 0}
, then
ψ
{\displaystyle \psi }
is the one-soliton solution with
a
{\displaystyle a}
related to the boost applied to the soliton.
== Topological charge and energy ==
The topological charge or winding number of a solution
φ
{\displaystyle \varphi }
is
N
=
1
2
π
∫
R
d
φ
=
1
2
π
[
φ
(
x
=
∞
,
t
)
−
φ
(
x
=
−
∞
,
t
)
]
.
{\displaystyle N={\frac {1}{2\pi }}\int _{\mathbb {R} }d\varphi ={\frac {1}{2\pi }}\left[\varphi (x=\infty ,t)-\varphi (x=-\infty ,t)\right].}
The energy of a solution
φ
{\displaystyle \varphi }
is
E
=
∫
R
d
x
(
1
2
(
φ
t
2
+
φ
x
2
)
+
m
2
(
1
−
cos
φ
)
)
{\displaystyle E=\int _{\mathbb {R} }dx\left({\frac {1}{2}}(\varphi _{t}^{2}+\varphi _{x}^{2})+m^{2}(1-\cos \varphi )\right)}
where a constant energy density has been added so that the potential is non-negative. With it the first two terms in the Taylor expansion of the potential coincide with the potential of a massive scalar field, as mentioned in the naming section; the higher order terms can be thought of as interactions.
The topological charge is conserved if the energy is finite. The topological charge does not determine the solution, even up to Lorentz boosts. Both the trivial solution and the soliton-antisoliton pair solution have
N
=
0
{\displaystyle N=0}
.
== Zero-curvature formulation ==
The sine-Gordon equation is equivalent to the curvature of a particular
s
u
(
2
)
{\displaystyle {\mathfrak {su}}(2)}
-connection on
R
2
{\displaystyle \mathbb {R} ^{2}}
being equal to zero.
Explicitly, with coordinates
(
u
,
v
)
{\displaystyle (u,v)}
on
R
2
{\displaystyle \mathbb {R} ^{2}}
, the connection components
A
μ
{\displaystyle A_{\mu }}
are given by
A
u
=
(
i
λ
i
2
φ
u
i
2
φ
u
−
i
λ
)
=
1
2
φ
u
i
σ
1
+
λ
i
σ
3
,
{\displaystyle A_{u}={\begin{pmatrix}i\lambda &{\frac {i}{2}}\varphi _{u}\\{\frac {i}{2}}\varphi _{u}&-i\lambda \end{pmatrix}}={\frac {1}{2}}\varphi _{u}i\sigma _{1}+\lambda i\sigma _{3},}
A
v
=
(
−
i
4
λ
cos
φ
−
1
4
λ
sin
φ
1
4
λ
sin
φ
i
4
λ
cos
φ
)
=
−
1
4
λ
i
sin
φ
σ
2
−
1
4
λ
i
cos
φ
σ
3
,
{\displaystyle A_{v}={\begin{pmatrix}-{\frac {i}{4\lambda }}\cos \varphi &-{\frac {1}{4\lambda }}\sin \varphi \\{\frac {1}{4\lambda }}\sin \varphi &{\frac {i}{4\lambda }}\cos \varphi \end{pmatrix}}=-{\frac {1}{4\lambda }}i\sin \varphi \sigma _{2}-{\frac {1}{4\lambda }}i\cos \varphi \sigma _{3},}
where the
σ
i
{\displaystyle \sigma _{i}}
are the Pauli matrices.
Then the zero-curvature equation
∂
v
A
u
−
∂
u
A
v
+
[
A
u
,
A
v
]
=
0
{\displaystyle \partial _{v}A_{u}-\partial _{u}A_{v}+[A_{u},A_{v}]=0}
is equivalent to the sine-Gordon equation
φ
u
v
=
sin
φ
{\displaystyle \varphi _{uv}=\sin \varphi }
. The zero-curvature equation is so named as it corresponds to the curvature being equal to zero if it is defined
F
μ
ν
=
[
∂
μ
−
A
μ
,
∂
ν
−
A
ν
]
{\displaystyle F_{\mu \nu }=[\partial _{\mu }-A_{\mu },\partial _{\nu }-A_{\nu }]}
.
The pair of matrices
A
u
{\displaystyle A_{u}}
and
A
v
{\displaystyle A_{v}}
are also known as a Lax pair for the sine-Gordon equation, in the sense that the zero-curvature equation recovers the PDE rather than them satisfying Lax's equation.
== Related equations ==
The sinh-Gordon equation is given by
φ
x
x
−
φ
t
t
=
sinh
φ
.
{\displaystyle \varphi _{xx}-\varphi _{tt}=\sinh \varphi .}
This is the Euler–Lagrange equation of the Lagrangian
L
=
1
2
(
φ
t
2
−
φ
x
2
)
−
cosh
φ
.
{\displaystyle {\mathcal {L}}={\frac {1}{2}}(\varphi _{t}^{2}-\varphi _{x}^{2})-\cosh \varphi .}
Another closely related equation is the elliptic sine-Gordon equation or Euclidean sine-Gordon equation, given by
φ
x
x
+
φ
y
y
=
sin
φ
,
{\displaystyle \varphi _{xx}+\varphi _{yy}=\sin \varphi ,}
where
φ
{\displaystyle \varphi }
is now a function of the variables x and y. This is no longer a soliton equation, but it has many similar properties, as it is related to the sine-Gordon equation by the analytic continuation (or Wick rotation) y = it.
The elliptic sinh-Gordon equation may be defined in a similar way.
Another similar equation comes from the Euler–Lagrange equation for Liouville field theory
φ
x
x
−
φ
t
t
=
2
e
2
φ
.
{\displaystyle \varphi _{xx}-\varphi _{tt}=2e^{2\varphi }.}
A generalization is given by Toda field theory. More precisely, Liouville field theory is the Toda field theory for the finite Kac–Moody algebra
s
l
2
{\displaystyle {\mathfrak {sl}}_{2}}
, while sin(h)-Gordon is the Toda field theory for the affine Kac–Moody algebra
s
l
^
2
{\displaystyle {\hat {\mathfrak {sl}}}_{2}}
.
== Infinite volume and on a half line ==
One can also consider the sine-Gordon model on a circle, on a line segment, or on a half line. It is possible to find boundary conditions which preserve the integrability of the model. On a half line the spectrum contains boundary bound states in addition to the solitons and breathers.
== Quantum sine-Gordon model ==
In quantum field theory the sine-Gordon model contains a parameter that can be identified with the Planck constant. The particle spectrum consists of a soliton, an anti-soliton and a finite (possibly zero) number of breathers. The number of breathers depends on the value of the parameter. Multiparticle production cancels on mass shell.
Semi-classical quantization of the sine-Gordon model was done by Ludwig Faddeev and Vladimir Korepin. The exact quantum scattering matrix was discovered by Alexander Zamolodchikov.
This model is S-dual to the Thirring model, as discovered by Coleman. This is sometimes known as the Coleman correspondence and serves as an example of boson-fermion correspondence in the interacting case. This article also showed that the constants appearing in the model behave nicely under renormalization: there are three parameters
α
0
,
β
{\displaystyle \alpha _{0},\beta }
and
γ
0
{\displaystyle \gamma _{0}}
. Coleman showed
α
0
{\displaystyle \alpha _{0}}
receives only a multiplicative correction,
γ
0
{\displaystyle \gamma _{0}}
receives only an additive correction, and
β
{\displaystyle \beta }
is not renormalized. Further, for a critical, non-zero value
β
=
4
π
{\displaystyle \beta ={\sqrt {4\pi }}}
, the theory is in fact dual to a free massive Dirac field theory.
The quantum sine-Gordon equation should be modified so the exponentials become vertex operators
L
Q
s
G
=
1
2
∂
μ
φ
∂
μ
φ
+
1
2
m
0
2
φ
2
−
α
(
V
β
+
V
−
β
)
{\displaystyle {\mathcal {L}}_{QsG}={\frac {1}{2}}\partial _{\mu }\varphi \partial ^{\mu }\varphi +{\frac {1}{2}}m_{0}^{2}\varphi ^{2}-\alpha (V_{\beta }+V_{-\beta })}
with
V
β
=:
e
i
β
φ
:
{\displaystyle V_{\beta }=:e^{i\beta \varphi }:}
, where the semi-colons denote normal ordering. A possible mass term is included.
=== Regimes of renormalizability ===
For different values of the parameter
β
2
{\displaystyle \beta ^{2}}
, the renormalizability properties of the sine-Gordon theory change. The identification of these regimes is attributed to Jürg Fröhlich.
The finite regime is
β
2
<
4
π
{\displaystyle \beta ^{2}<4\pi }
, where no counterterms are needed to render the theory well-posed. The super-renormalizable regime is
4
π
<
β
2
<
8
π
{\displaystyle 4\pi <\beta ^{2}<8\pi }
, where a finite number of counterterms are needed to render the theory well-posed. More counterterms are needed for each threshold
n
n
+
1
8
π
{\displaystyle {\frac {n}{n+1}}8\pi }
passed. For
β
2
>
8
π
{\displaystyle \beta ^{2}>8\pi }
, the theory becomes ill-defined (Coleman 1975). The boundary values are
β
2
=
4
π
{\displaystyle \beta ^{2}=4\pi }
and
β
2
=
8
π
{\displaystyle \beta ^{2}=8\pi }
, which are respectively the free fermion point, as the theory is dual to a free fermion via the Coleman correspondence, and the self-dual point, where the vertex operators form an affine sl2 subalgebra, and the theory becomes strictly renormalizable (renormalizable, but not super-renormalizable).
== Stochastic sine-Gordon model ==
The stochastic or dynamical sine-Gordon model has been studied by Martin Hairer and Hao Shen
allowing heuristic results from the quantum sine-Gordon theory to be proven in a statistical setting.
The equation is
∂
t
u
=
1
2
Δ
u
+
c
sin
(
β
u
+
θ
)
+
ξ
,
{\displaystyle \partial _{t}u={\frac {1}{2}}\Delta u+c\sin(\beta u+\theta )+\xi ,}
where
c
,
β
,
θ
{\displaystyle c,\beta ,\theta }
are real-valued constants, and
ξ
{\displaystyle \xi }
is space-time white noise. The space dimension is fixed to 2. In the proof of existence of solutions, the thresholds
β
2
=
n
n
+
1
8
π
{\displaystyle \beta ^{2}={\frac {n}{n+1}}8\pi }
again play a role in determining convergence of certain terms.
== Supersymmetric sine-Gordon model ==
A supersymmetric extension of the sine-Gordon model also exists. Integrability preserving boundary conditions for this extension can be found as well.
== Physical applications ==
The sine-Gordon model arises as the continuum limit of the Frenkel–Kontorova model which models crystal dislocations.
Dynamics in long Josephson junctions are well-described by the sine-Gordon equations, and conversely provide a useful experimental system for studying the sine-Gordon model.
The sine-Gordon model is in the same universality class as the effective action for a Coulomb gas of vortices and anti-vortices in the continuous classical XY model, which is a model of magnetism. The Kosterlitz–Thouless transition for vortices can therefore be derived from a renormalization group analysis of the sine-Gordon field theory.
The sine-Gordon equation also arises as the formal continuum limit of a different model of magnetism, the quantum Heisenberg model, in particular the XXZ model.
== See also ==
Josephson effect
Fluxon
Shape waves
== References ==
== External links ==
sine-Gordon equation at EqWorld: The World of Mathematical Equations.
Sinh-Gordon Equation at EqWorld: The World of Mathematical Equations.
sine-Gordon equation Archived 2012-03-16 at the Wayback Machine at NEQwiki, the nonlinear equations encyclopedia. | Wikipedia/Sine–Gordon_equation |
In quantum mechanics, the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) equation (named after Vittorio Gorini, Andrzej Kossakowski, George Sudarshan and Göran Lindblad), master equation in Lindblad form, quantum Liouvillian, or Lindbladian is one of the general forms of Markovian master equations describing open quantum systems. It generalizes the Schrödinger equation to open quantum systems; that is, systems in contacts with their surroundings. The resulting dynamics are no longer unitary, but still satisfy the property of being trace-preserving and completely positive for any initial condition.
The Schrödinger equation or, actually, the von Neumann equation, is a special case of the GKSL equation, which has led to some speculation that quantum mechanics may be productively extended and expanded through further application and analysis of the Lindblad equation. The Schrödinger equation deals with state vectors, which can only describe pure quantum states and are thus less general than density matrices, which can describe mixed states as well.
== Motivation ==
Understanding the interaction of a quantum system with its environment is necessary for understanding many commonly observed phenomena like the spontaneous emission of light from excited atoms, or the performance of many quantum technological devices, like the laser.
In the canonical formulation of quantum mechanics, a system's time evolution is governed by unitary dynamics. This implies that there is no decay and phase coherence is maintained throughout the process, and is a consequence of the fact that all participating degrees of freedom are considered. However, any real physical system will interact with its environment, and is not absolutely isolated. The interaction with degrees of freedom that are external to the system results in dissipation of energy into the surroundings, causing decay and randomization of phase.
Certain mathematical techniques have been introduced to treat the interaction of a quantum system with its environment. One of these is the use of the density matrix, and its associated master equation. While in principle this approach to solving quantum dynamics is equivalent to the Schrödinger picture or Heisenberg picture, it allows more easily for the inclusion of incoherent processes, which represent environmental interactions. The density operator has the property that it can represent a classical mixture of quantum states, and is thus vital to accurately describe the dynamics of so-called open quantum systems.
== Definition ==
=== Diagonal form ===
The Lindblad master equation for system's density matrix ρ can be written as (for a pedagogical introduction you may refer to)
ρ
˙
=
−
i
ℏ
[
H
,
ρ
]
+
∑
i
γ
i
(
L
i
ρ
L
i
†
−
1
2
{
L
i
†
L
i
,
ρ
}
)
{\displaystyle {\dot {\rho }}=-{i \over \hbar }[H,\rho ]+\sum _{i}^{}\gamma _{i}\left(L_{i}\rho L_{i}^{\dagger }-{\frac {1}{2}}\left\{L_{i}^{\dagger }L_{i},\rho \right\}\right)}
where
{
a
,
b
}
=
a
b
+
b
a
{\displaystyle \{a,b\}=ab+ba}
is the anticommutator.
H
{\displaystyle H}
is the system Hamiltonian, describing the unitary aspects of the dynamics.
{
L
i
}
i
{\displaystyle \{L_{i}\}_{i}}
are a set of jump operators, describing the dissipative part of the dynamics. The shape of the jump operators describes how the environment acts on the system, and must either be determined from microscopic models of the system-environment dynamics, or phenomenologically modelled.
γ
i
≥
0
{\displaystyle \gamma _{i}\geq 0}
are a set of non-negative real coefficients called damping rates. If all
γ
i
=
0
{\displaystyle \gamma _{i}=0}
one recovers the von Neumann equation
ρ
˙
=
−
(
i
/
ℏ
)
[
H
,
ρ
]
{\displaystyle {\dot {\rho }}=-(i/\hbar )[H,\rho ]}
describing unitary dynamics, which is the quantum analog of the classical Liouville equation.
The entire equation can be written in superoperator form:
ρ
˙
=
L
(
ρ
)
{\displaystyle {\dot {\rho }}={\mathcal {L}}(\rho )}
which resembles the classical Liouville equation
ρ
˙
=
{
H
,
ρ
}
{\displaystyle {\dot {\rho }}=\{H,\rho \}}
. For this reason, the superoperator
L
{\displaystyle {\mathcal {L}}}
is called the Lindbladian superoperator or the Liouvillian superoperator.
=== General form ===
More generally, the GKSL equation has the form
ρ
˙
=
−
i
ℏ
[
H
,
ρ
]
+
∑
n
,
m
h
n
m
(
A
n
ρ
A
m
†
−
1
2
{
A
m
†
A
n
,
ρ
}
)
{\displaystyle {\dot {\rho }}=-{i \over \hbar }[H,\rho ]+\sum _{n,m}h_{nm}\left(A_{n}\rho A_{m}^{\dagger }-{\frac {1}{2}}\left\{A_{m}^{\dagger }A_{n},\rho \right\}\right)}
where
{
A
m
}
{\displaystyle \{A_{m}\}}
are arbitrary operators and h is a positive semidefinite matrix. The latter is a strict requirement to ensure the dynamics is trace-preserving and completely positive. The number of
A
m
{\displaystyle A_{m}}
operators is arbitrary, and they do not have to satisfy any special properties. But if the system is
N
{\displaystyle N}
-dimensional, it can be shown that the master equation can be fully described by a set of
N
2
−
1
{\displaystyle N^{2}-1}
operators, provided they form a basis for the space of operators.
The general form is not in fact more general, and can be reduced to the special form. Since the matrix h is positive semidefinite, it can be diagonalized with a unitary transformation u:
u
†
h
u
=
[
γ
1
0
⋯
0
0
γ
2
⋯
0
⋮
⋮
⋱
⋮
0
0
⋯
γ
N
2
−
1
]
{\displaystyle u^{\dagger }hu={\begin{bmatrix}\gamma _{1}&0&\cdots &0\\0&\gamma _{2}&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &\gamma _{N^{2}-1}\end{bmatrix}}}
where the eigenvalues γi are non-negative. If we define another orthonormal operator basis
L
i
=
∑
j
u
j
i
A
j
{\displaystyle L_{i}=\sum _{j}u_{ji}A_{j}}
This reduces the master equation to the same form as before:
ρ
˙
=
−
i
ℏ
[
H
,
ρ
]
+
∑
i
γ
i
(
L
i
ρ
L
i
†
−
1
2
{
L
i
†
L
i
,
ρ
}
)
{\displaystyle {\dot {\rho }}=-{i \over \hbar }[H,\rho ]+\sum _{i}^{}\gamma _{i}\left(L_{i}\rho L_{i}^{\dagger }-{\frac {1}{2}}\left\{L_{i}^{\dagger }L_{i},\rho \right\}\right)}
=== Quantum dynamical semigroup ===
The maps generated by a Lindbladian for various times are collectively referred to as a quantum dynamical semigroup—a family of quantum dynamical maps
ϕ
t
{\displaystyle \phi _{t}}
on the space of density matrices indexed by a single time parameter
t
≥
0
{\displaystyle t\geq 0}
that obey the semigroup property
ϕ
s
(
ϕ
t
(
ρ
)
)
=
ϕ
t
+
s
(
ρ
)
,
t
,
s
≥
0.
{\displaystyle \phi _{s}(\phi _{t}(\rho ))=\phi _{t+s}(\rho ),\qquad t,s\geq 0.}
The Lindblad equation can be obtained by
L
(
ρ
)
=
l
i
m
Δ
t
→
0
ϕ
Δ
t
(
ρ
)
−
ϕ
0
(
ρ
)
Δ
t
{\displaystyle {\mathcal {L}}(\rho )=\mathrm {lim} _{\Delta t\to 0}{\frac {\phi _{\Delta t}(\rho )-\phi _{0}(\rho )}{\Delta t}}}
which, by the linearity of
ϕ
t
{\displaystyle \phi _{t}}
, is a linear superoperator. The semigroup can be recovered as
ϕ
t
+
s
(
ρ
)
=
e
L
s
ϕ
t
(
ρ
)
.
{\displaystyle \phi _{t+s}(\rho )=e^{{\mathcal {L}}s}\phi _{t}(\rho ).}
=== Invariance properties ===
The Lindblad equation is invariant under any unitary transformation v of Lindblad operators and constants,
γ
i
L
i
→
γ
i
′
L
i
′
=
∑
j
v
i
j
γ
j
L
j
,
{\displaystyle {\sqrt {\gamma _{i}}}L_{i}\to {\sqrt {\gamma _{i}'}}L_{i}'=\sum _{j}v_{ij}{\sqrt {\gamma _{j}}}L_{j},}
and also under the inhomogeneous transformation
L
i
→
L
i
′
=
L
i
+
a
i
I
,
{\displaystyle L_{i}\to L_{i}'=L_{i}+a_{i}I,}
H
→
H
′
=
H
+
1
2
i
∑
j
γ
j
(
a
j
∗
L
j
−
a
j
L
j
†
)
+
b
I
,
{\displaystyle H\to H'=H+{\frac {1}{2i}}\sum _{j}\gamma _{j}\left(a_{j}^{*}L_{j}-a_{j}L_{j}^{\dagger }\right)+bI,}
where ai are complex numbers and b is a real number.
However, the first transformation destroys the orthonormality of the operators Li (unless all the γi are equal) and the second transformation destroys the tracelessness. Therefore, up to degeneracies among the γi, the Li of the diagonal form of the Lindblad equation are uniquely determined by the dynamics so long as we require them to be orthonormal and traceless.
=== Heisenberg picture ===
The Lindblad-type evolution of the density matrix in the Schrödinger picture can be equivalently described in the Heisenberg picture
using the following (diagonalized) equation of motion for each quantum observable X:
X
˙
=
i
ℏ
[
H
,
X
]
+
∑
i
γ
i
(
L
i
†
X
L
i
−
1
2
{
L
i
†
L
i
,
X
}
)
.
{\displaystyle {\dot {X}}={\frac {i}{\hbar }}[H,X]+\sum _{i}\gamma _{i}\left(L_{i}^{\dagger }XL_{i}-{\frac {1}{2}}\left\{L_{i}^{\dagger }L_{i},X\right\}\right).}
A similar equation describes the time evolution of the expectation values of observables, given by the Ehrenfest theorem.
Corresponding to the trace-preserving property of the Schrödinger picture Lindblad equation, the Heisenberg picture equation is unital, i.e. it preserves the identity operator.
== Physical derivation ==
The Lindblad master equation describes the evolution of various types of open quantum systems, e.g. a system weakly coupled to a Markovian reservoir.
Note that the H appearing in the equation is not necessarily equal to the bare system Hamiltonian, but may also incorporate effective unitary dynamics arising from the system-environment interaction.
A heuristic derivation, e.g., in the notes by Preskill, begins with a more general form of an open quantum system and converts it into Lindblad form by making the Markovian assumption and expanding in small time. A more physically motivated standard treatment covers three common types of derivations of the Lindbladian starting from a Hamiltonian acting on both the system and environment: the weak coupling limit (described in detail below), the low density approximation, and the singular coupling limit. Each of these relies on specific physical assumptions regarding, e.g., correlation functions of the environment. For example, in the weak coupling limit derivation, one typically assumes that (a) correlations of the system with the environment develop slowly, (b) excitations of the environment caused by system decay quickly, and (c) terms which are fast-oscillating when compared
to the system timescale of interest can be neglected. These three approximations are called Born,
Markov, and rotating wave, respectively.
The weak-coupling limit derivation assumes a quantum system with a finite number of degrees of freedom coupled to a bath containing an infinite number of degrees of freedom. The system and bath each possess a Hamiltonian written in terms of operators acting only on the respective subspace of the total Hilbert space. These Hamiltonians govern the internal dynamics of the uncoupled system and bath. There is a third Hamiltonian that contains products of system and bath operators, thus coupling the system and bath. The most general form of this Hamiltonian is
H
=
H
S
+
H
B
+
H
B
S
{\displaystyle H=H_{S}+H_{B}+H_{BS}\,}
The dynamics of the entire system can be described by the Liouville equation of motion,
χ
˙
=
−
i
[
H
,
χ
]
{\displaystyle {\dot {\chi }}=-i[H,\chi ]}
. This equation, containing an infinite number of degrees of freedom, is impossible to solve analytically except in very particular cases. What's more, under certain approximations, the bath degrees of freedom need not be considered, and an effective master equation can be derived in terms of the system density matrix,
ρ
=
tr
B
χ
{\displaystyle \rho =\operatorname {tr} _{B}\chi }
. The problem can be analyzed more easily by moving into the interaction picture, defined by the unitary transformation
M
~
=
U
0
M
U
0
†
{\displaystyle {\tilde {M}}=U_{0}MU_{0}^{\dagger }}
, where
M
{\displaystyle M}
is an arbitrary operator, and
U
0
=
e
i
(
H
S
+
H
B
)
t
{\displaystyle U_{0}=e^{i(H_{S}+H_{B})t}}
. Also note that
U
(
t
,
t
0
)
{\displaystyle U(t,t_{0})}
is the total unitary operator of the entire system. It is straightforward to confirm that the Liouville equation becomes
χ
~
˙
=
−
i
[
H
~
B
S
,
χ
~
]
{\displaystyle {\dot {\tilde {\chi }}}=-i[{\tilde {H}}_{BS},{\tilde {\chi }}]\,}
where the Hamiltonian
H
~
B
S
=
e
i
(
H
S
+
H
B
)
t
H
B
S
e
−
i
(
H
S
+
H
B
)
t
{\displaystyle {\tilde {H}}_{BS}=e^{i(H_{S}+H_{B})t}H_{BS}e^{-i(H_{S}+H_{B})t}}
is explicitly time dependent. Also, according to the interaction picture,
χ
~
=
U
B
S
(
t
,
t
0
)
χ
U
B
S
†
(
t
,
t
0
)
{\displaystyle {\tilde {\chi }}=U_{BS}(t,t_{0})\chi U_{BS}^{\dagger }(t,t_{0})}
, where
U
B
S
=
U
0
†
U
(
t
,
t
0
)
{\displaystyle U_{BS}=U_{0}^{\dagger }U(t,t_{0})}
. This equation can be integrated directly to give
χ
~
(
t
)
=
χ
~
(
0
)
−
i
∫
0
t
d
t
′
[
H
~
B
S
(
t
′
)
,
χ
~
(
t
′
)
]
{\displaystyle {\tilde {\chi }}(t)={\tilde {\chi }}(0)-i\int _{0}^{t}dt'[{\tilde {H}}_{BS}(t'),{\tilde {\chi }}(t')]}
This implicit equation for
χ
~
{\displaystyle {\tilde {\chi }}}
can be substituted back into the Liouville equation to obtain an exact differo-integral equation
χ
~
˙
=
−
i
[
H
~
B
S
(
t
)
,
χ
~
(
0
)
]
−
∫
0
t
d
t
′
[
H
~
B
S
(
t
)
,
[
H
~
B
S
(
t
′
)
,
χ
~
(
t
′
)
]
]
{\displaystyle {\dot {\tilde {\chi }}}=-i[{\tilde {H}}_{BS}(t),{\tilde {\chi }}(0)]-\int _{0}^{t}dt'[{\tilde {H}}_{BS}(t),[{\tilde {H}}_{BS}(t'),{\tilde {\chi }}(t')]]}
We proceed with the derivation by assuming the interaction is initiated at
t
=
0
{\displaystyle t=0}
, and at that time there are no correlations between the system and the bath. This implies that the initial condition is factorable as
χ
(
0
)
=
ρ
(
0
)
R
0
{\displaystyle \chi (0)=\rho (0)R_{0}}
, where
R
0
{\displaystyle R_{0}}
is the density operator of the bath initially.
Tracing over the bath degrees of freedom,
tr
R
χ
~
=
ρ
~
{\displaystyle \operatorname {tr} _{R}{\tilde {\chi }}={\tilde {\rho }}}
, of the aforementioned differo-integral equation yields
ρ
~
˙
=
−
∫
0
t
d
t
′
tr
R
{
[
H
~
B
S
(
t
)
,
[
H
~
B
S
(
t
′
)
,
χ
~
(
t
′
)
]
]
}
{\displaystyle {\dot {\tilde {\rho }}}=-\int _{0}^{t}dt'\operatorname {tr} _{R}\{[{\tilde {H}}_{BS}(t),[{\tilde {H}}_{BS}(t'),{\tilde {\chi }}(t')]]\}}
This equation is exact for the time dynamics of the system density matrix but requires full knowledge of the dynamics of the bath degrees of freedom. A simplifying assumption called the Born approximation rests on the largeness of the bath and the relative weakness of the coupling, which is to say the coupling of the system to the bath should not significantly alter the bath eigenstates. In this case the full density matrix is factorable for all times as
χ
~
(
t
)
=
ρ
~
(
t
)
R
0
{\displaystyle {\tilde {\chi }}(t)={\tilde {\rho }}(t)R_{0}}
. The master equation becomes
ρ
~
˙
=
−
∫
0
t
d
t
′
tr
R
{
[
H
~
B
S
(
t
)
,
[
H
~
B
S
(
t
′
)
,
ρ
~
(
t
′
)
R
0
]
]
}
{\displaystyle {\dot {\tilde {\rho }}}=-\int _{0}^{t}dt'\operatorname {tr} _{R}\{[{\tilde {H}}_{BS}(t),[{\tilde {H}}_{BS}(t'),{\tilde {\rho }}(t')R_{0}]]\}}
The equation is now explicit in the system degrees of freedom, but is very difficult to solve. A final assumption is the Born-Markov approximation that the time derivative of the density matrix depends only on its current state, and not on its past. This assumption is valid under fast bath dynamics, wherein correlations within the bath are lost extremely quickly, and amounts to replacing
ρ
(
t
′
)
→
ρ
(
t
)
{\displaystyle \rho (t')\rightarrow \rho (t)}
on the right hand side of the equation.
ρ
~
˙
=
−
∫
0
t
d
t
′
tr
R
{
[
H
~
B
S
(
t
)
,
[
H
~
B
S
(
t
′
)
,
ρ
~
(
t
)
R
0
]
]
}
{\displaystyle {\dot {\tilde {\rho }}}=-\int _{0}^{t}dt'\operatorname {tr} _{R}\{[{\tilde {H}}_{BS}(t),[{\tilde {H}}_{BS}(t'),{\tilde {\rho }}(t)R_{0}]]\}}
If the interaction Hamiltonian is assumed to have the form
H
B
S
=
∑
i
α
i
Γ
i
{\displaystyle H_{BS}=\sum _{i}\alpha _{i}\Gamma _{i}}
for system operators
α
i
{\displaystyle \alpha _{i}}
and bath operators
Γ
i
{\displaystyle \Gamma _{i}}
then
H
~
B
S
=
∑
i
α
~
i
Γ
~
i
{\displaystyle {\tilde {H}}_{BS}=\sum _{i}{\tilde {\alpha }}_{i}{\tilde {\Gamma }}_{i}}
. The master equation becomes
ρ
~
˙
=
−
∑
i
,
j
∫
0
t
d
t
′
tr
R
{
[
α
~
i
(
t
)
Γ
~
i
(
t
)
,
[
α
~
j
(
t
′
)
Γ
~
j
(
t
′
)
,
ρ
~
(
t
)
R
0
]
]
}
{\displaystyle {\dot {\tilde {\rho }}}=-\sum _{i,j}\int _{0}^{t}dt'\operatorname {tr} _{R}\{[{\tilde {\alpha }}_{i}(t){\tilde {\Gamma }}_{i}(t),[{\tilde {\alpha }}_{j}(t'){\tilde {\Gamma }}_{j}(t'),{\tilde {\rho }}(t)R_{0}]]\}}
which can be expanded as
ρ
~
˙
=
−
∑
i
,
j
∫
0
t
d
t
′
[
(
α
~
i
(
t
)
α
~
j
(
t
′
)
ρ
~
(
t
)
−
α
~
i
(
t
)
ρ
~
(
t
)
α
~
j
(
t
′
)
)
⟨
Γ
~
i
(
t
)
Γ
~
j
(
t
′
)
⟩
+
(
ρ
~
(
t
)
α
~
j
(
t
′
)
α
~
i
(
t
)
−
α
~
j
(
t
′
)
ρ
~
(
t
)
α
~
i
(
t
)
)
⟨
Γ
~
j
(
t
′
)
Γ
~
i
(
t
)
⟩
]
{\displaystyle {\dot {\tilde {\rho }}}=-\sum _{i,j}\int _{0}^{t}dt'\left[\left({\tilde {\alpha }}_{i}(t){\tilde {\alpha }}_{j}(t'){\tilde {\rho }}(t)-{\tilde {\alpha }}_{i}(t){\tilde {\rho }}(t){\tilde {\alpha }}_{j}(t')\right)\langle {\tilde {\Gamma }}_{i}(t){\tilde {\Gamma }}_{j}(t')\rangle +\left({\tilde {\rho }}(t){\tilde {\alpha }}_{j}(t'){\tilde {\alpha }}_{i}(t)-{\tilde {\alpha }}_{j}(t'){\tilde {\rho }}(t){\tilde {\alpha }}_{i}(t)\right)\langle {\tilde {\Gamma }}_{j}(t'){\tilde {\Gamma }}_{i}(t)\rangle \right]}
The expectation values
⟨
Γ
i
Γ
j
⟩
=
tr
{
Γ
i
Γ
j
R
0
}
{\displaystyle \langle \Gamma _{i}\Gamma _{j}\rangle =\operatorname {tr} \{\Gamma _{i}\Gamma _{j}R_{0}\}}
are with respect to the bath degrees of freedom.
By assuming rapid decay of these correlations (ideally
⟨
Γ
i
(
t
)
Γ
j
(
t
′
)
⟩
∝
δ
(
t
−
t
′
)
{\displaystyle \langle \Gamma _{i}(t)\Gamma _{j}(t')\rangle \propto \delta (t-t')}
), above form of the Lindblad superoperator L is achieved.
== Examples ==
In the simplest case, there is just one jump operator
F
{\displaystyle F}
and no unitary evolution. In this case, the Lindblad equation is
L
(
ρ
)
=
F
ρ
F
†
−
1
2
(
F
†
F
ρ
+
ρ
F
†
F
)
{\displaystyle {\mathcal {L}}(\rho )={F\rho F^{\dagger }}-{\frac {1}{2}}\left(F^{\dagger }F\rho +\rho F^{\dagger }F\right)}
This case is often used in quantum optics to model either absorption or emission of photons from a reservoir.
To model both absorption and emission, one would need a jump operator for each. This leads to the most common Lindblad equation describing the damping of a quantum harmonic oscillator (representing e.g. a Fabry–Perot cavity) coupled to a thermal bath, with jump operators:
F
1
=
a
,
γ
1
=
γ
2
(
n
¯
+
1
)
,
F
2
=
a
†
,
γ
2
=
γ
2
n
¯
.
{\displaystyle {\begin{aligned}F_{1}&=a,&\gamma _{1}&={\tfrac {\gamma }{2}}\left({\overline {n}}+1\right),\\F_{2}&=a^{\dagger },&\gamma _{2}&={\tfrac {\gamma }{2}}{\overline {n}}.\end{aligned}}}
Here
n
¯
{\displaystyle {\overline {n}}}
is the mean number of excitations in the reservoir damping the oscillator and γ is the decay rate.
To model the quantum harmonic oscillator Hamiltonian with frequency
ω
c
{\displaystyle \omega _{c}}
of the photons, we can add a further unitary evolution:
ρ
˙
=
−
i
[
ω
c
a
†
a
,
ρ
]
+
γ
1
D
[
F
1
]
(
ρ
)
+
γ
2
D
[
F
2
]
(
ρ
)
.
{\displaystyle {\dot {\rho }}=-i[\omega _{c}a^{\dagger }a,\rho ]+\gamma _{1}{\mathcal {D}}[F_{1}](\rho )+\gamma _{2}{\mathcal {D}}[F_{2}](\rho ).}
Additional Lindblad operators can be included to model various forms of dephasing and vibrational relaxation. These methods have been incorporated into grid-based density matrix propagation methods.
== See also ==
Quantum master equation
Redfield equation
Open quantum system
Quantum jump method
Sokhotski–Plemelj theorem § Heitler function
== References ==
Chruściński, Dariusz; Pascazio, Saverio (2017). "A Brief History of the GKLS Equation". Open Systems & Information Dynamics. 24 (3). arXiv:1710.05993. Bibcode:2017OSID...2440001C. doi:10.1142/S1230161217400017. S2CID 90357.
Kossakowski, A. (1972). "On quantum statistical mechanics of non-Hamiltonian systems". Rep. Math. Phys. 3 (4): 247. Bibcode:1972RpMP....3..247K. doi:10.1016/0034-4877(72)90010-9.
Belavin, A.A.; Zel'dovich, B. Ya.; Perelomov, A.M.; Popov, V.S. (1969). "Relaxation of Quantum Systems with Equidistant Spectra". JETP. 29: 145. Bibcode:1969JETP...29..145B.
Lindblad, G. (1976). "On the generators of quantum dynamical semigroups". Commun. Math. Phys. 48 (2): 119. Bibcode:1976CMaPh..48..119L. doi:10.1007/BF01608499. S2CID 55220796.
Gorini, V.; Kossakowski, A.; Sudarshan, E.C.G. (1976). "Completely positive dynamical semigroups of N-level systems". J. Math. Phys. 17 (5): 821. Bibcode:1976JMP....17..821G. doi:10.1063/1.522979.
Banks, T.; Susskind, L.; Peskin, M.E. (1984). "Difficulties for the evolution of pure states into mixed states". Nuclear Physics B. 244 (1): 125–134. Bibcode:1984NuPhB.244..125B. doi:10.1016/0550-3213(84)90184-6. OSTI 1447054.
Accardi, Luigi; Lu, Yun Gang; Volovich, I.V. (2002). Quantum Theory and Its Stochastic Limit. New York: Springer Verlag. ISBN 978-3-5404-1928-0.
Alicki, Robert (2002). "Invitation to quantum dynamical semigroups". Dynamics of Dissipation. Lecture Notes in Physics. 597: 239. arXiv:quant-ph/0205188. Bibcode:2002LNP...597..239A. doi:10.1007/3-540-46122-1_10. ISBN 978-3-540-44111-3. S2CID 118089738.
Alicki, Robert; Lendi, Karl (1987). Quantum Dynamical Semigroups and Applications. Berlin: Springer Verlag. ISBN 978-0-3871-8276-6.
Attal, Stéphane; Joye, Alain; Pillet, Claude-Alain (2006). Open Quantum Systems II: The Markovian Approach. Springer. ISBN 978-3-5403-0992-5.
Gardiner, C.W.; Zoller, Peter (2010). Quantum Noise. Springer Series in Synergetics (3rd ed.). Berlin Heidelberg: Springer-Verlag. ISBN 978-3-642-06094-6.
Ingarden, Roman S.; Kossakowski, A.; Ohya, M. (1997). Information Dynamics and Open Systems: Classical and Quantum Approach. New York: Springer Verlag. ISBN 978-0-7923-4473-5.
Tarasov, Vasily E. (2008). Quantum Mechanics of Non-Hamiltonian and Dissipative Systems. Amsterdam, Boston, London, New York: Elsevier Science. ISBN 978-0-0805-5971-1.
Pearle, P. (2012). "Simple derivation of the Lindblad equation". European Journal of Physics, 33(4), 805.
== External links ==
Quantum Optics Toolbox for Matlab
mcsolve Quantum jump (monte carlo) solver from QuTiP.
QuantumOptics.jl the quantum optics toolbox in Julia.
The Lindblad master equation | Wikipedia/Lindblad_equation |
The rotating-wave approximation is an approximation used in atom optics and magnetic resonance. In this approximation, terms in a Hamiltonian that oscillate rapidly are neglected. This is a valid approximation when the applied electromagnetic radiation is near resonance with an atomic transition, and the intensity is low. Explicitly, terms in the Hamiltonians that oscillate with frequencies
ω
L
+
ω
0
{\displaystyle \omega _{L}+\omega _{0}}
are neglected, while terms that oscillate with frequencies
ω
L
−
ω
0
{\displaystyle \omega _{L}-\omega _{0}}
are kept, where
ω
L
{\displaystyle \omega _{L}}
is the light frequency, and
ω
0
{\displaystyle \omega _{0}}
is a transition frequency.
The name of the approximation stems from the form of the Hamiltonian in the interaction picture, as shown below. By switching to this picture the evolution of an atom due to the corresponding atomic Hamiltonian is absorbed into the system ket, leaving only the evolution due to the interaction of the atom with the light field to consider. It is in this picture that the rapidly oscillating terms mentioned previously can be neglected. Since in some sense the interaction picture can be thought of as rotating with the system ket only that part of the electromagnetic wave that approximately co-rotates is kept; the counter-rotating component is discarded.
The rotating-wave approximation is closely related to, but different from, the secular approximation.
== Mathematical formulation ==
For simplicity consider a two-level atomic system with ground and excited states
|
g
⟩
{\displaystyle |{\text{g}}\rangle }
and
|
e
⟩
{\displaystyle |{\text{e}}\rangle }
, respectively (using the Dirac bracket notation). Let the energy difference between the states be
ℏ
ω
0
{\displaystyle \hbar \omega _{0}}
so that
ω
0
{\displaystyle \omega _{0}}
is the transition frequency of the system. Then the unperturbed Hamiltonian of the atom can be written as
H
0
=
ℏ
ω
0
2
|
e
⟩
⟨
e
|
−
ℏ
ω
0
2
|
g
⟩
⟨
g
|
{\displaystyle H_{0}={\frac {\hbar \omega _{0}}{2}}|{\text{e}}\rangle \langle {\text{e}}|-{\frac {\hbar \omega _{0}}{2}}|{\text{g}}\rangle \langle {\text{g}}|}
.
Suppose the atom experiences an external classical electric field of frequency
ω
L
{\displaystyle \omega _{L}}
, given by
E
→
(
t
)
=
E
→
0
e
−
i
ω
L
t
+
E
→
0
∗
e
i
ω
L
t
{\displaystyle {\vec {E}}(t)={\vec {E}}_{0}e^{-i\omega _{L}t}+{\vec {E}}_{0}^{*}e^{i\omega _{L}t}}
; e.g., a plane wave propagating in space. Then under the dipole approximation the interaction Hamiltonian between the atom and the electric field can be expressed as
H
1
=
−
d
→
⋅
E
→
{\displaystyle H_{1}=-{\vec {d}}\cdot {\vec {E}}}
,
where
d
→
{\displaystyle {\vec {d}}}
is the dipole moment operator of the atom. The total Hamiltonian for the atom-light system is therefore
H
=
H
0
+
H
1
.
{\displaystyle H=H_{0}+H_{1}.}
The atom does not have a dipole moment when it is in an energy eigenstate, so
⟨
e
|
d
→
|
e
⟩
=
⟨
g
|
d
→
|
g
⟩
=
0.
{\displaystyle \left\langle {\text{e}}\left|{\vec {d}}\right|{\text{e}}\right\rangle =\left\langle {\text{g}}\left|{\vec {d}}\right|{\text{g}}\right\rangle =0.}
This means that defining
d
→
eg
:=
⟨
e
|
d
→
|
g
⟩
{\displaystyle {\vec {d}}_{\text{eg}}\mathrel {:=} \left\langle {\text{e}}\left|{\vec {d}}\right|{\text{g}}\right\rangle }
allows the dipole operator to be written as
d
→
=
d
→
eg
|
e
⟩
⟨
g
|
+
d
→
eg
∗
|
g
⟩
⟨
e
|
{\displaystyle {\vec {d}}={\vec {d}}_{\text{eg}}|{\text{e}}\rangle \langle {\text{g}}|+{\vec {d}}_{\text{eg}}^{*}|{\text{g}}\rangle \langle {\text{e}}|}
(with
∗
{\displaystyle ^{*}}
denoting the complex conjugate). The interaction Hamiltonian can then be shown to be
H
1
=
−
ℏ
(
Ω
e
−
i
ω
L
t
+
Ω
~
e
i
ω
L
t
)
|
e
⟩
⟨
g
|
−
ℏ
(
Ω
~
∗
e
−
i
ω
L
t
+
Ω
∗
e
i
ω
L
t
)
|
g
⟩
⟨
e
|
{\displaystyle H_{1}=-\hbar \left(\Omega e^{-i\omega _{L}t}+{\tilde {\Omega }}e^{i\omega _{L}t}\right)|{\text{e}}\rangle \langle {\text{g}}|-\hbar \left({\tilde {\Omega }}^{*}e^{-i\omega _{L}t}+\Omega ^{*}e^{i\omega _{L}t}\right)|{\text{g}}\rangle \langle {\text{e}}|}
where
Ω
=
ℏ
−
1
d
→
eg
⋅
E
→
0
{\displaystyle \Omega =\hbar ^{-1}{\vec {d}}_{\text{eg}}\cdot {\vec {E}}_{0}}
is the Rabi frequency and
Ω
~
:=
ℏ
−
1
d
→
eg
⋅
E
→
0
∗
{\displaystyle {\tilde {\Omega }}\mathrel {:=} \hbar ^{-1}{\vec {d}}_{\text{eg}}\cdot {\vec {E}}_{0}^{*}}
is the counter-rotating frequency. To see why the
Ω
~
{\displaystyle {\tilde {\Omega }}}
terms are called counter-rotating consider a unitary transformation to the interaction or Dirac picture where the transformed Hamiltonian
H
1
,
I
{\displaystyle H_{1,I}}
is given by
H
1
,
I
=
−
ℏ
(
Ω
e
−
i
Δ
ω
t
+
Ω
~
e
i
(
ω
L
+
ω
0
)
t
)
|
e
⟩
⟨
g
|
−
ℏ
(
Ω
~
∗
e
−
i
(
ω
L
+
ω
0
)
t
+
Ω
∗
e
i
Δ
ω
t
)
|
g
⟩
⟨
e
|
,
{\displaystyle H_{1,I}=-\hbar \left(\Omega e^{-i\Delta \omega t}+{\tilde {\Omega }}e^{i(\omega _{L}+\omega _{0})t}\right)|{\text{e}}\rangle \langle {\text{g}}|-\hbar \left({\tilde {\Omega }}^{*}e^{-i(\omega _{L}+\omega _{0})t}+\Omega ^{*}e^{i\Delta \omega t}\right)|{\text{g}}\rangle \langle {\text{e}}|,}
where
Δ
ω
:=
ω
L
−
ω
0
{\displaystyle \Delta \omega \mathrel {:=} \omega _{L}-\omega _{0}}
is the detuning between the light field and the atom.
=== Making the approximation ===
This is the point at which the rotating wave approximation is made. The dipole approximation has been assumed, and for this to remain valid the electric field must be near resonance with the atomic transition. This means that
Δ
ω
≪
ω
L
+
ω
0
{\displaystyle \Delta \omega \ll \omega _{L}+\omega _{0}}
and the complex exponentials multiplying
Ω
~
{\displaystyle {\tilde {\Omega }}}
and
Ω
~
∗
{\displaystyle {\tilde {\Omega }}^{*}}
can be considered to be rapidly oscillating. Hence on any appreciable time scale, the oscillations will quickly average to 0. The rotating wave approximation is thus the claim that these terms may be neglected and thus the Hamiltonian can be written in the interaction picture as
H
1
,
I
RWA
=
−
ℏ
Ω
e
−
i
Δ
ω
t
|
e
⟩
⟨
g
|
−
ℏ
Ω
∗
e
i
Δ
ω
t
|
g
⟩
⟨
e
|
.
{\displaystyle H_{1,I}^{\text{RWA}}=-\hbar \Omega e^{-i\Delta \omega t}|{\text{e}}\rangle \langle {\text{g}}|-\hbar \Omega ^{*}e^{i\Delta \omega t}|{\text{g}}\rangle \langle {\text{e}}|.}
Finally, transforming back into the Schrödinger picture, the Hamiltonian is given by
H
RWA
=
ℏ
ω
0
2
|
e
⟩
⟨
e
|
−
ℏ
ω
0
2
|
g
⟩
⟨
g
|
−
ℏ
Ω
e
−
i
ω
L
t
|
e
⟩
⟨
g
|
−
ℏ
Ω
∗
e
i
ω
L
t
|
g
⟩
⟨
e
|
.
{\displaystyle H^{\text{RWA}}={\frac {\hbar \omega _{0}}{2}}|{\text{e}}\rangle \langle {\text{e}}|-{\frac {\hbar \omega _{0}}{2}}|{\text{g}}\rangle \langle {\text{g}}|-\hbar \Omega e^{-i\omega _{L}t}|{\text{e}}\rangle \langle {\text{g}}|-\hbar \Omega ^{*}e^{i\omega _{L}t}|{\text{g}}\rangle \langle {\text{e}}|.}
Another criterion for rotating wave approximation is the weak coupling condition, that is, the Rabi frequency should be much less than the transition frequency.
At this point the rotating wave approximation is complete. A common first step beyond this is to remove the remaining time dependence in the Hamiltonian via another unitary transformation.
== Derivation ==
Given the above definitions the interaction Hamiltonian is
H
1
=
−
d
→
⋅
E
→
=
−
(
d
→
eg
|
e
⟩
⟨
g
|
+
d
→
eg
∗
|
g
⟩
⟨
e
|
)
⋅
(
E
→
0
e
−
i
ω
L
t
+
E
→
0
∗
e
i
ω
L
t
)
=
−
(
d
→
eg
⋅
E
→
0
e
−
i
ω
L
t
+
d
→
eg
⋅
E
→
0
∗
e
i
ω
L
t
)
|
e
⟩
⟨
g
|
−
(
d
→
eg
∗
⋅
E
→
0
e
−
i
ω
L
t
+
d
→
eg
∗
⋅
E
→
0
∗
e
i
ω
L
t
)
|
g
⟩
⟨
e
|
=
−
ℏ
(
Ω
e
−
i
ω
L
t
+
Ω
~
e
i
ω
L
t
)
|
e
⟩
⟨
g
|
−
ℏ
(
Ω
~
∗
e
−
i
ω
L
t
+
Ω
∗
e
i
ω
L
t
)
|
g
⟩
⟨
e
|
,
{\displaystyle {\begin{aligned}H_{1}=-{\vec {d}}\cdot {\vec {E}}&=-\left({\vec {d}}_{\text{eg}}|{\text{e}}\rangle \langle {\text{g}}|+{\vec {d}}_{\text{eg}}^{*}|{\text{g}}\rangle \langle {\text{e}}|\right)\cdot \left({\vec {E}}_{0}e^{-i\omega _{L}t}+{\vec {E}}_{0}^{*}e^{i\omega _{L}t}\right)\\&=-\left({\vec {d}}_{\text{eg}}\cdot {\vec {E}}_{0}e^{-i\omega _{L}t}+{\vec {d}}_{\text{eg}}\cdot {\vec {E}}_{0}^{*}e^{i\omega _{L}t}\right)|{\text{e}}\rangle \langle {\text{g}}|-\left({\vec {d}}_{\text{eg}}^{*}\cdot {\vec {E}}_{0}e^{-i\omega _{L}t}+{\vec {d}}_{\text{eg}}^{*}\cdot {\vec {E}}_{0}^{*}e^{i\omega _{L}t}\right)|{\text{g}}\rangle \langle {\text{e}}|\\&=-\hbar \left(\Omega e^{-i\omega _{L}t}+{\tilde {\Omega }}e^{i\omega _{L}t}\right)|{\text{e}}\rangle \langle {\text{g}}|-\hbar \left({\tilde {\Omega }}^{*}e^{-i\omega _{L}t}+\Omega ^{*}e^{i\omega _{L}t}\right)|{\text{g}}\rangle \langle {\text{e}}|,\end{aligned}}}
as stated. The next step is to find the Hamiltonian in the interaction picture,
H
1
,
I
{\displaystyle H_{1,I}}
. The required unitary transformation is
U
=
e
i
H
0
t
/
ℏ
=
e
i
ω
0
t
/
2
(
|
e
⟩
⟨
e
|
−
|
g
⟩
⟨
g
|
)
=
cos
(
ω
0
t
2
)
(
|
e
⟩
⟨
e
|
+
|
g
⟩
⟨
g
|
)
+
i
sin
(
ω
0
t
2
)
(
|
e
⟩
⟨
e
|
−
|
g
⟩
⟨
g
|
)
=
e
−
i
ω
0
t
/
2
|
g
⟩
⟨
g
|
+
e
i
ω
0
t
/
2
|
e
⟩
⟨
e
|
=
e
−
i
ω
0
t
/
2
(
|
g
⟩
⟨
g
|
+
e
i
ω
0
t
|
e
⟩
⟨
e
|
)
{\displaystyle {\begin{aligned}U&=e^{iH_{0}t/\hbar }\\&=e^{i\omega _{0}t/2(|{\text{e}}\rangle \langle {\text{e}}|-|{\text{g}}\rangle \langle {\text{g}}|)}\\&=\cos({\frac {\omega _{0}t}{2}})\left(|{\text{e}}\rangle \langle {\text{e}}|+|{\text{g}}\rangle \langle {\text{g}}|\right)+i\sin({\frac {\omega _{0}t}{2}})\left(|{\text{e}}\rangle \langle {\text{e}}|-|{\text{g}}\rangle \langle {\text{g}}|\right)\\&=e^{-i\omega _{0}t/2}|{\text{g}}\rangle \langle {\text{g}}|+e^{i\omega _{0}t/2}|{\text{e}}\rangle \langle {\text{e}}|\\&=e^{-i\omega _{0}t/2}\left(|{\text{g}}\rangle \langle {\text{g}}|+e^{i\omega _{0}t}|{\text{e}}\rangle \langle {\text{e}}|\right)\end{aligned}}}
,
where the 3rd step can be proved by using a Taylor series expansion, and using the orthogonality of the states
|
g
⟩
{\displaystyle |{\text{g}}\rangle }
and
|
e
⟩
{\displaystyle |{\text{e}}\rangle }
. Note that a multiplication by an overall phase of
e
i
ω
0
t
/
2
{\displaystyle e^{i\omega _{0}t/2}}
on a unitary operator does not affect the underlying physics, so in the further usages of
U
{\displaystyle U}
we will neglect it. Applying
U
{\displaystyle U}
gives:
H
1
,
I
≡
U
H
1
U
†
=
−
ℏ
(
Ω
e
−
i
ω
L
t
+
Ω
~
e
i
ω
L
t
)
e
i
ω
0
t
|
e
⟩
⟨
g
|
−
ℏ
(
Ω
~
∗
e
−
i
ω
L
t
+
Ω
∗
e
i
ω
L
t
)
|
g
⟩
⟨
e
|
e
−
i
ω
0
t
=
−
ℏ
(
Ω
e
−
i
Δ
ω
t
+
Ω
~
e
i
(
ω
L
+
ω
0
)
t
)
|
e
⟩
⟨
g
|
−
ℏ
(
Ω
~
∗
e
−
i
(
ω
L
+
ω
0
)
t
+
Ω
∗
e
i
Δ
ω
t
)
|
g
⟩
⟨
e
|
.
{\displaystyle {\begin{aligned}H_{1,I}&\equiv UH_{1}U^{\dagger }\\&=-\hbar \left(\Omega e^{-i\omega _{L}t}+{\tilde {\Omega }}e^{i\omega _{L}t}\right)e^{i\omega _{0}t}|{\text{e}}\rangle \langle {\text{g}}|-\hbar \left({\tilde {\Omega }}^{*}e^{-i\omega _{L}t}+\Omega ^{*}e^{i\omega _{L}t}\right)|{\text{g}}\rangle \langle {\text{e}}|e^{-i\omega _{0}t}\\&=-\hbar \left(\Omega e^{-i\Delta \omega t}+{\tilde {\Omega }}e^{i(\omega _{L}+\omega _{0})t}\right)|{\text{e}}\rangle \langle {\text{g}}|-\hbar \left({\tilde {\Omega }}^{*}e^{-i(\omega _{L}+\omega _{0})t}+\Omega ^{*}e^{i\Delta \omega t}\right)|{\text{g}}\rangle \langle {\text{e}}|\ .\end{aligned}}}
Now we apply the RWA by eliminating the counter-rotating terms as explained in the previous section:
H
1
,
I
RWA
=
−
ℏ
Ω
e
−
i
Δ
ω
t
|
e
⟩
⟨
g
|
+
−
ℏ
Ω
∗
e
i
Δ
ω
t
|
g
⟩
⟨
e
|
{\displaystyle H_{1,I}^{\text{RWA}}=-\hbar \Omega e^{-i\Delta \omega t}|{\text{e}}\rangle \langle {\text{g}}|+-\hbar \Omega ^{*}e^{i\Delta \omega t}|{\text{g}}\rangle \langle {\text{e}}|}
Finally, we transform the approximate Hamiltonian
H
1
,
I
RWA
{\displaystyle H_{1,I}^{\text{RWA}}}
back to the Schrödinger picture:
H
1
RWA
=
U
†
H
1
,
I
RWA
U
=
−
ℏ
Ω
e
−
i
Δ
ω
t
e
−
i
ω
0
t
|
e
⟩
⟨
g
|
−
ℏ
Ω
∗
e
i
Δ
ω
t
|
g
⟩
⟨
e
|
e
i
ω
0
t
=
−
ℏ
Ω
e
−
i
ω
L
t
|
e
⟩
⟨
g
|
−
ℏ
Ω
∗
e
i
ω
L
t
|
g
⟩
⟨
e
|
.
{\displaystyle {\begin{aligned}H_{1}^{\text{RWA}}&=U^{\dagger }H_{1,I}^{\text{RWA}}U\\&=-\hbar \Omega e^{-i\Delta \omega t}e^{-i\omega _{0}t}|{\text{e}}\rangle \langle {\text{g}}|-\hbar \Omega ^{*}e^{i\Delta \omega t}|{\text{g}}\rangle \langle {\text{e}}|e^{i\omega _{0}t}\\&=-\hbar \Omega e^{-i\omega _{L}t}|{\text{e}}\rangle \langle {\text{g}}|-\hbar \Omega ^{*}e^{i\omega _{L}t}|{\text{g}}\rangle \langle {\text{e}}|.\end{aligned}}}
The atomic Hamiltonian was unaffected by the approximation, so the total Hamiltonian in the Schrödinger picture under the rotating wave approximation is
H
RWA
=
H
0
+
H
1
RWA
=
ℏ
ω
0
2
|
e
⟩
⟨
e
|
−
ℏ
ω
0
2
|
g
⟩
⟨
g
|
−
ℏ
Ω
e
−
i
ω
L
t
|
e
⟩
⟨
g
|
−
ℏ
Ω
∗
e
i
ω
L
t
|
g
⟩
⟨
e
|
.
{\displaystyle H^{\text{RWA}}=H_{0}+H_{1}^{\text{RWA}}={\frac {\hbar \omega _{0}}{2}}|{\text{e}}\rangle \langle {\text{e}}|-{\frac {\hbar \omega _{0}}{2}}|{\text{g}}\rangle \langle {\text{g}}|-\hbar \Omega e^{-i\omega _{L}t}|{\text{e}}\rangle \langle {\text{g}}|-\hbar \Omega ^{*}e^{i\omega _{L}t}|{\text{g}}\rangle \langle {\text{e}}|.}
== References == | Wikipedia/Rotating_wave_approximation |
The quantum jump method, also known as the Monte Carlo wave function (MCWF) is a technique in computational physics used for simulating open quantum systems and quantum dissipation. The quantum jump method was developed by Dalibard, Castin and Mølmer at a similar time to the similar method known as Quantum Trajectory Theory developed by Carmichael. Other contemporaneous works on wave-function-based Monte Carlo approaches to open quantum systems include those of Dum, Zoller and Ritsch and Hegerfeldt and Wilser.
== Method ==
The quantum jump method is an approach which is much like the master-equation treatment except that it operates on the wave function rather than using a density matrix approach. The main component of this method is evolving the system's wave function in time with a pseudo-Hamiltonian; where at each time step, a quantum jump (discontinuous change) may take place with some probability. The calculated system state as a function of time is known as a quantum trajectory, and the desired density matrix as a function of time may be calculated by averaging over many simulated trajectories. For a Hilbert space of dimension N, the number of wave function components is equal to N while the number of density matrix components is equal to N2. Consequently, for certain problems the quantum jump method offers a performance advantage over direct master-equation approaches.
== References ==
== Further reading ==
Plenio, M. B.; Knight, P. L. (1 January 1998). "The quantum-jump approach to dissipative dynamics in quantum optics". Reviews of Modern Physics. 70 (1): 101–144. arXiv:quant-ph/9702007. Bibcode:1998RvMP...70..101P. doi:10.1103/RevModPhys.70.101. S2CID 14721909.
== External links ==
mcsolve Quantum jump (Monte Carlo) solver from QuTiP for Python.
QuantumOptics.jl the quantum optics toolbox in Julia.
Quantum Optics Toolbox for Matlab | Wikipedia/Monte_Carlo_wave_function_method |
In condensed matter physics, scintillation ( SIN-til-ay-shun) is the physical process where a material, called a scintillator, emits ultraviolet or visible light under excitation from high energy photons (X-rays or gamma rays) or energetic particles (such as electrons, alpha particles, neutrons, or ions). See scintillator and scintillation counter for practical applications.
== Overview ==
Scintillation is an example of luminescence, whereby light of a characteristic spectrum is emitted following the absorption of radiation. The scintillation process can be summarized in three main stages: conversion, transport and energy transfer to the luminescence center, and luminescence. The emitted radiation is usually less energetic than the absorbed radiation, hence scintillation is generally a down-conversion process.
== Conversion processes ==
The first stage of scintillation, conversion, is the process where the energy from the incident radiation is absorbed by the scintillator and highly energetic electrons and holes are created in the material. The energy absorption mechanism by the scintillator depends on the type and energy of radiation involved. For highly energetic photons such as X-rays (0.1 keV <
E
γ
{\displaystyle E_{\gamma }}
< 100 keV) and γ-rays (
E
γ
{\displaystyle E_{\gamma }}
> 100 keV), three types of interactions are responsible for the energy conversion process in scintillation: photoelectric absorption, Compton scattering, and pair production, which only occurs when
E
γ
{\displaystyle E_{\gamma }}
> 1022 keV, i.e. the photon has enough energy to create an electron-positron pair.
These processes have different attenuation coefficients, which depend mainly on the energy of the incident radiation, the average atomic number of the material and the density of the material. Generally the absorption of high energy radiation is described by:
I
=
I
0
⋅
e
−
μ
d
{\displaystyle I=I_{0}\cdot e^{-\mu d}}
where
I
0
{\displaystyle I_{0}}
is the intensity of the incident radiation,
d
{\displaystyle d}
is the thickness of the material, and
μ
{\displaystyle \mu }
is the linear attenuation coefficient, which is the sum of the attenuation coefficients of the various contributions:
μ
=
μ
p
e
+
μ
c
s
+
μ
p
p
+
μ
o
c
{\displaystyle \mu =\mu _{pe}+\mu _{cs}+\mu _{pp}+\mu _{oc}}
At lower X-ray energies (
E
γ
≲
{\displaystyle E_{\gamma }\lesssim }
60 keV), the most dominant process is the photoelectric effect, where the photons are fully absorbed by bound electrons in the material, usually core electrons in the K- or L-shell of the atom, and then ejected, leading to the ionization of the host atom. The linear attenuation coefficient contribution for the photoelectric effect is given by:
μ
p
e
∝
ρ
Z
n
E
γ
3.5
{\displaystyle \mu _{pe}\propto {\rho Z^{n} \over E_{\gamma }^{3.5}}}
where
ρ
{\displaystyle \rho }
is the density of the scintillator,
Z
{\displaystyle Z}
is the average atomic number,
n
{\displaystyle n}
is a constant that varies between 3 and 4, and
E
γ
{\displaystyle E_{\gamma }}
is the energy of the photon. At low X-ray energies, scintillator materials with atoms with high atomic numbers and densities are favored for more efficient absorption of the incident radiation.
At higher energies (
E
γ
{\displaystyle E_{\gamma }}
≳
{\displaystyle \gtrsim }
60 keV) Compton scattering, the inelastic scattering of photons by bound electrons, often also leading to ionization of the host atom, becomes the more dominant conversion process. The linear attenuation coefficient contribution for Compton scattering is given by:
μ
c
s
∝
ρ
E
γ
{\displaystyle \mu _{cs}\propto {\rho \over {\sqrt {E_{\gamma }}}}}
Unlike the photoelectric effect, the absorption resulting from Compton scattering is independent of the atomic number of the atoms present in the crystal, but linearly on their density.
At γ-ray energies higher than
E
γ
{\displaystyle E_{\gamma }}
> 1022 keV, i.e. energies higher than twice the rest-mass energy of the electron, pair production starts to occur. Pair production is the relativistic phenomenon where the energy of a photon is converted into an electron-positron pair. The created electron and positron will then further interact with the scintillating material to generate energetic electron and holes. The attenuation coefficient contribution for pair production is given by:
μ
p
p
∝
ρ
Z
ln
(
2
E
γ
m
e
c
2
)
{\displaystyle \mu _{pp}\propto \rho Z\ln {\Bigl (}{2E_{\gamma } \over m_{e}c^{2}}{\Bigr )}}
where
m
e
{\displaystyle m_{e}}
is the rest mass of the electron and
c
{\displaystyle c}
is the speed of light. Hence, at high γ-ray energies, the energy absorption depends both on the density and average atomic number of the scintillator. In addition, unlike for the photoelectric effect and Compton scattering, pair production becomes more probable as the energy of the incident photons increases, and pair production becomes the most dominant conversion process above
E
γ
{\displaystyle E_{\gamma }}
~ 8 MeV.
The
μ
o
c
{\displaystyle \mu _{oc}}
term includes other (minor) contributions, such as Rayleigh (coherent) scattering at low energies and photonuclear reactions at very high energies, which also contribute to the conversion, however the contribution from Rayleigh scattering is almost negligible and photonuclear reactions become relevant only at very high energies.
After the energy of the incident radiation is absorbed and converted into so-called hot electrons and holes in the material, these energetic charge carriers will interact with other particles and quasi-particles in the scintillator (electrons, plasmons, phonons), leading to an "avalanche event", where a great number of secondary electron–hole pairs are produced until the hot electrons and holes have lost sufficient energy. The large number of electrons and holes that result from this process will then undergo thermalization, i.e. dissipation of part of their energy through interaction with phonons in the material
The resulting large number of energetic charge carriers will then undergo further energy dissipation called thermalization. This occurs via interaction with phonons for electrons and Auger processes for holes.
The average timescale for conversion, including energy absorption and thermalization has been estimated to be in the order of 1 ps, which is much faster than the average decay time in photoluminescence.
== Charge transport of excited carriers ==
The second stage of scintillation is the charge transport of thermalized electrons and holes towards luminescence centers and the energy transfer to the atoms involved in the luminescence process. In this stage, the large number of electrons and holes that have been generated during the conversion process, migrate inside the material. This is probably one of the most critical phases of scintillation, since it is generally in this stage where most loss of efficiency occur due to effects such as trapping or non-radiative recombination. These are mainly caused by the presence of defects in the scintillator crystal, such as impurities, ionic vacancies, and grain boundaries. The charge transport can also become a bottleneck for the timing of the scintillation process. The charge transport phase is also one of the least understood parts of scintillation and depends strongly on the type material involved and its intrinsic charge conduction properties.
== Luminescence ==
Once the electrons and holes reach the luminescence centers, the third and final stage of scintillation occurs: luminescence. In this stage the electrons and holes are captured potential paths by the luminescent center, and then the electrons and hole recombine radiatively. The exact details of the luminescence phase also depend on the type of material used for scintillation.
=== Inorganic crystals ===
For photons such as gamma rays, thallium activated NaI crystals (NaI(Tl)) are often used. For a faster response (but only 5% of the output) CsF crystals can be used.: 211
=== Organic scintillators ===
In organic molecules scintillation is a product of π-orbitals. Organic materials form molecular crystals where the molecules are loosely bound by Van der Waals forces. The ground state of 12C is 1s2 2s2 2p2. In valence bond theory, when carbon forms compounds, one of the 2s electrons is excited into the 2p state resulting in a configuration of 1s2 2s1 2p3. To describe the different valencies of carbon, the four valence electron orbitals, one 2s and three 2p, are considered to be mixed or hybridized in several alternative configurations. For example, in a tetrahedral configuration the s and p3 orbitals combine to produce four hybrid orbitals. In another configuration, known as trigonal configuration, one of the p-orbitals (say pz) remains unchanged and three hybrid orbitals are produced by mixing the s, px and py orbitals. The orbitals that are symmetrical about the bonding axes and plane of the molecule (sp2) are known as σ-electrons and the bonds are called σ-bonds. The pz orbital is called a π-orbital. A π-bond occurs when two π-orbitals interact. This occurs when their nodal planes are coplanar.
In certain organic molecules π-orbitals interact to produce a common nodal plane. These form delocalized π-electrons that can be excited by radiation. The de-excitation of the delocalized π-electrons results in luminescence.
The excited states of π-electron systems can be explained by the perimeter free-electron model (Platt 1949). This model is used for describing polycyclic hydrocarbons consisting of condensed systems of benzenoid rings in which no C atom belongs to more than two rings and every C atom is on the periphery.
The ring can be approximated as a circle with circumference l. The wave-function of the electron orbital must satisfy the condition of a plane rotator:
ψ
(
x
)
=
ψ
(
x
+
l
)
{\displaystyle \psi (x)=\psi (x+l)\,}
The corresponding solutions to the Schrödinger wave equation are:
ψ
0
=
(
1
l
)
1
2
ψ
q
1
=
(
2
l
)
1
2
cos
(
2
π
q
x
l
)
ψ
q
2
=
(
2
l
)
1
2
sin
(
2
π
q
x
l
)
E
q
=
q
2
ℏ
2
2
m
0
l
2
{\displaystyle {\begin{aligned}\psi _{0}&=\left({\frac {1}{l}}\right)^{\frac {1}{2}}\\\psi _{q1}&=\left({\frac {2}{l}}\right)^{\frac {1}{2}}\cos {\left({\frac {2\pi \ qx}{l}}\right)}\\\psi _{q2}&=\left({\frac {2}{l}}\right)^{\frac {1}{2}}\sin {\left({\frac {2\pi \ qx}{l}}\right)}\\E_{q}&={\frac {q^{2}\hbar ^{2}}{2m_{0}l^{2}}}\end{aligned}}}
where q is the orbital ring quantum number; the number of nodes of the wave-function. Since the electron can have spin up and spin down and can rotate about the circle in both directions all of the energy levels except the lowest are doubly degenerate.
The above shows the π-electronic energy levels of an organic molecule. Absorption of radiation is followed by molecular vibration to the S1 state. This is followed by a de-excitation to the S0 state called fluorescence. The population of triplet states is also possible by other means. The triplet states decay with a much longer decay time than singlet states, which results in what is called the slow component of the decay process (the fluorescence process is called the fast component). Depending on the particular energy loss of a certain particle (dE/dx), the "fast" and "slow" states are occupied in different proportions. The relative intensities in the light output of these states thus differs for different dE/dx. This property of scintillators allows for pulse shape discrimination: it is possible to identify which particle was detected by looking at the pulse shape. Of course, the difference in shape is visible in the trailing side of the pulse, since it is due to the decay of the excited states.
== See also ==
Positron emission tomography
== References == | Wikipedia/Scintillation_(physics) |
The Journal of Physics A: Mathematical and Theoretical is a peer-reviewed scientific journal published by IOP Publishing, the publishing branch of the Institute of Physics. It is part of the Journal of Physics series and covers theoretical physics focusing on sophisticated mathematical and computational techniques.
The journal is divided into six sections covering: statistical physics; chaotic and complex systems; mathematical physics; quantum mechanics and quantum information theory; classical and quantum field theory; fluid and plasma theory.
The editor in chief is Joseph A Minahan (Uppsala Universitet, Sweden). According to the Journal Citation Reports, the journal has a 2023 impact factor of 2.0.
== History ==
Journal of Physics A was established in 1968 as one of the subdivisions of the earlier title, Proceedings of the Physical Society, established in 1874, the flagship journal of the Physical Society of London. The Physical Society later became the Institute of Physics, the current publisher of the journal. Its papers began being made available electronically in 1991; by 2002, its entire back archive had been digitised, as the first step in a larger project to digitise all of the Institute's publishing archives.
== Indexing ==
The journal is indexed in:
Scopus
Inspec
Chemical Abstracts
GeoRef
INIS Atomindex
Astrophysics Data System
PASCAL
Referativny Zhurnal
Zentralblatt MATH
Science Citation Index and SciSearch
Current Contents/Physical, Chemical and Earth Sciences
== See also ==
Journal of Physics: Condensed Matter
== References ==
== External links ==
Official website | Wikipedia/Journal_of_Physics_A:_Mathematical_and_General |
In computer science, the controlled NOT gate (also C-NOT or CNOT), controlled-X gate, controlled-bit-flip gate, Feynman gate or controlled Pauli-X is a quantum logic gate that is an essential component in the construction of a gate-based quantum computer. It can be used to entangle and disentangle Bell states. Any quantum circuit can be simulated to an arbitrary degree of accuracy using a combination of CNOT gates and single qubit rotations. The gate is sometimes named after Richard Feynman who developed an early notation for quantum gate diagrams in 1986.
The CNOT can be expressed in the Pauli basis as:
CNOT
=
e
i
π
4
(
I
1
−
Z
1
)
(
I
2
−
X
2
)
=
e
−
i
π
4
(
I
1
−
Z
1
)
(
I
2
−
X
2
)
.
{\displaystyle {\mbox{CNOT}}=e^{i{\frac {\pi }{4}}(I_{1}-Z_{1})(I_{2}-X_{2})}=e^{-i{\frac {\pi }{4}}(I_{1}-Z_{1})(I_{2}-X_{2})}.}
Being both unitary and Hermitian, CNOT has the property
e
i
θ
U
=
(
cos
θ
)
I
+
(
i
sin
θ
)
U
{\displaystyle e^{i\theta U}=(\cos \theta )I+(i\sin \theta )U}
and
U
=
e
i
π
2
(
I
−
U
)
=
e
−
i
π
2
(
I
−
U
)
{\displaystyle U=e^{i{\frac {\pi }{2}}(I-U)}=e^{-i{\frac {\pi }{2}}(I-U)}}
, and is involutory.
The CNOT gate can be further decomposed as products of rotation operator gates and exactly one two qubit interaction gate, for example
CNOT
=
e
−
i
π
4
R
y
1
(
−
π
/
2
)
R
x
1
(
−
π
/
2
)
R
x
2
(
−
π
/
2
)
R
x
x
(
π
/
2
)
R
y
1
(
π
/
2
)
.
{\displaystyle {\mbox{CNOT}}=e^{-i{\frac {\pi }{4}}}R_{y_{1}}(-\pi /2)R_{x_{1}}(-\pi /2)R_{x_{2}}(-\pi /2)R_{xx}(\pi /2)R_{y_{1}}(\pi /2).}
In general, any single qubit unitary gate can be expressed as
U
=
e
i
H
{\displaystyle U=e^{iH}}
, where H is a Hermitian matrix, and then the controlled U is
C
U
=
e
i
1
2
(
I
1
−
Z
1
)
H
2
{\displaystyle CU=e^{i{\frac {1}{2}}(I_{1}-Z_{1})H_{2}}}
.
The CNOT gate is also used in classical reversible computing.
== Operation ==
The CNOT gate operates on a quantum register consisting of 2 qubits. The CNOT gate flips the second qubit (the target qubit) if and only if the first qubit (the control qubit) is
|
1
⟩
{\displaystyle |1\rangle }
.
If
{
|
0
⟩
,
|
1
⟩
}
{\displaystyle \{|0\rangle ,|1\rangle \}}
are the only allowed input values for both qubits, then the TARGET output of the CNOT gate corresponds to the result of a classical XOR gate. Fixing CONTROL as
|
1
⟩
{\displaystyle |1\rangle }
, the TARGET output of the CNOT gate yields the result of a classical NOT gate.
More generally, the inputs are allowed to be a linear superposition of
{
|
0
⟩
,
|
1
⟩
}
{\displaystyle \{|0\rangle ,|1\rangle \}}
. The CNOT gate transforms the quantum state:
a
|
00
⟩
+
b
|
01
⟩
+
c
|
10
⟩
+
d
|
11
⟩
{\displaystyle a|00\rangle +b|01\rangle +c|10\rangle +d|11\rangle }
into:
a
|
00
⟩
+
b
|
01
⟩
+
c
|
11
⟩
+
d
|
10
⟩
{\displaystyle a|00\rangle +b|01\rangle +c|11\rangle +d|10\rangle }
The action of the CNOT gate can be represented by the matrix (permutation matrix form):
CNOT
=
[
1
0
0
0
0
1
0
0
0
0
0
1
0
0
1
0
]
.
{\displaystyle \operatorname {CNOT} ={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&1&0\end{bmatrix}}.}
The first experimental realization of a CNOT gate was accomplished in 1995. Here, a single Beryllium ion in a trap was used. The two qubits were encoded into an optical state and into the vibrational state of the ion within the trap. At the time of the experiment, the reliability of the CNOT-operation was measured to be on the order of 90%.
In addition to a regular controlled NOT gate, one could construct a function-controlled NOT gate, which accepts an arbitrary number n+1 of qubits as input, where n+1 is greater than or equal to 2 (a quantum register). This gate flips the last qubit of the register if and only if a built-in function, with the first n qubits as input, returns a 1.
The function-controlled NOT gate is an essential element of the Deutsch–Jozsa algorithm.
== Behaviour in the Hadamard transformed basis ==
When viewed only in the computational basis
{
|
0
⟩
,
|
1
⟩
}
{\displaystyle \{|0\rangle ,|1\rangle \}}
, the behaviour of the CNOT appears to be like the equivalent classical gate. However, the simplicity of labelling one qubit the control and the other the target does not reflect the complexity of what happens for most input values of both qubits.
Insight can be gained by expressing the CNOT gate with respect to a Hadamard transformed basis
{
|
+
⟩
,
|
−
⟩
}
{\displaystyle \{|+\rangle ,|-\rangle \}}
. The Hadamard transformed basis of a one-qubit register is given by
|
+
⟩
=
1
2
(
|
0
⟩
+
|
1
⟩
)
,
|
−
⟩
=
1
2
(
|
0
⟩
−
|
1
⟩
)
,
{\displaystyle |+\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle +|1\rangle ),\qquad |-\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle -|1\rangle ),}
and the corresponding basis of a 2-qubit register is
|
+
+
⟩
=
|
+
⟩
⊗
|
+
⟩
=
1
2
(
|
0
⟩
+
|
1
⟩
)
⊗
(
|
0
⟩
+
|
1
⟩
)
=
1
2
(
|
00
⟩
+
|
01
⟩
+
|
10
⟩
+
|
11
⟩
)
{\displaystyle |++\rangle =|+\rangle \otimes |+\rangle ={\frac {1}{2}}(|0\rangle +|1\rangle )\otimes (|0\rangle +|1\rangle )={\frac {1}{2}}(|00\rangle +|01\rangle +|10\rangle +|11\rangle )}
,
etc. Viewing CNOT in this basis, the state of the second qubit remains unchanged, and the state of the first qubit is flipped, according to the state of the second bit. (For details see below.) "Thus, in this basis the sense of which bit is the control bit and which the target bit has reversed. But we have not changed the transformation at all, only the way we are thinking about it."
The "computational" basis
{
|
0
⟩
,
|
1
⟩
}
{\displaystyle \{|0\rangle ,|1\rangle \}}
is the eigenbasis for the spin in the Z-direction, whereas the Hadamard basis
{
|
+
⟩
,
|
−
⟩
}
{\displaystyle \{|+\rangle ,|-\rangle \}}
is the eigenbasis for spin in the X-direction. Switching X and Z and qubits 1 and 2, then, recovers the original transformation." This expresses a fundamental symmetry of the CNOT gate.
The observation that both qubits are (equally) affected in a CNOT interaction is of importance when considering information flow in entangled quantum systems.
=== Details of the computation ===
We now proceed to give the details of the computation. Working through each of the Hadamard basis states, the results on the right column show that the first qubit flips between
|
+
⟩
{\displaystyle |+\rangle }
and
|
−
⟩
{\displaystyle |-\rangle }
when the second qubit is
|
−
⟩
{\displaystyle |-\rangle }
:
A quantum circuit that performs a Hadamard transform followed by CNOT then another Hadamard transform, can be described as performing the CNOT gate in the Hadamard basis (i.e. a change of basis):
(H1 ⊗ H1)−1 . CNOT . (H1 ⊗ H1)
The single-qubit Hadamard transform, H1, is Hermitian and therefore its own inverse. The tensor product of two Hadamard transforms operating (independently) on two qubits is labelled H2. We can therefore write the matrices as:
H2 . CNOT . H2
When multiplied out, this yields a matrix that swaps the
|
01
⟩
{\displaystyle |01\rangle }
and
|
11
⟩
{\displaystyle |11\rangle }
terms over, while leaving the
|
00
⟩
{\displaystyle |00\rangle }
and
|
10
⟩
{\displaystyle |10\rangle }
terms alone. This is equivalent to a CNOT gate where qubit 2 is the control qubit and qubit 1 is the target qubit:
1
2
[
1
1
1
1
1
−
1
1
−
1
1
1
−
1
−
1
1
−
1
−
1
1
]
.
[
1
0
0
0
0
1
0
0
0
0
0
1
0
0
1
0
]
.
1
2
[
1
1
1
1
1
−
1
1
−
1
1
1
−
1
−
1
1
−
1
−
1
1
]
=
[
1
0
0
0
0
0
0
1
0
0
1
0
0
1
0
0
]
{\displaystyle {\frac {1}{2}}{\begin{bmatrix}{\begin{array}{rrrr}1&1&1&1\\1&-1&1&-1\\1&1&-1&-1\\1&-1&-1&1\end{array}}\end{bmatrix}}.{\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&1&0\end{bmatrix}}.{\frac {1}{2}}{\begin{bmatrix}{\begin{array}{rrrr}1&1&1&1\\1&-1&1&-1\\1&1&-1&-1\\1&-1&-1&1\end{array}}\end{bmatrix}}={\begin{bmatrix}1&0&0&0\\0&0&0&1\\0&0&1&0\\0&1&0&0\end{bmatrix}}}
== Constructing a Bell state ==
A common application of the CNOT gate is to maximally entangle two qubits into the
|
Φ
+
⟩
{\displaystyle |\Phi ^{+}\rangle }
Bell state; this forms part of the setup of the superdense coding, quantum teleportation, and entangled quantum cryptography algorithms.
To construct
|
Φ
+
⟩
{\displaystyle |\Phi ^{+}\rangle }
, the inputs A (control) and B (target) to the CNOT gate are
1
2
(
|
0
⟩
+
|
1
⟩
)
A
{\displaystyle {\frac {1}{\sqrt {2}}}(|0\rangle +|1\rangle )_{A}}
and
|
0
⟩
B
{\displaystyle |0\rangle _{B}}
.
After applying CNOT, the resulting Bell state
1
2
(
|
00
⟩
+
|
11
⟩
)
{\textstyle {\frac {1}{\sqrt {2}}}(|00\rangle +|11\rangle )}
has the property that the individual qubits can be measured using any basis and will always present a 50/50 chance of resolving to each state. In effect, the individual qubits are in an undefined state. The correlation between the two qubits is the complete description of the state of the two qubits; if we both choose the same basis to measure both qubits and compare notes, the measurements will perfectly correlate.
When viewed in the computational basis, it appears that qubit A is affecting qubit B. Changing our viewpoint to the Hadamard basis demonstrates that, in a symmetrical way, qubit B is affecting qubit A.
The input state can alternately be viewed as
|
+
⟩
A
{\displaystyle |+\rangle _{A}}
and
1
2
(
|
+
⟩
+
|
−
⟩
)
B
{\displaystyle {\frac {1}{\sqrt {2}}}(|+\rangle +|-\rangle )_{B}}
.
In the Hadamard view, the control and target qubits have conceptually swapped and qubit A is inverted when qubit B is
|
−
⟩
B
{\displaystyle |-\rangle _{B}}
. The output state after applying the CNOT gate is
1
2
(
|
+
+
⟩
+
|
−
−
⟩
)
,
{\displaystyle {\tfrac {1}{\sqrt {2}}}(|++\rangle +|--\rangle ),}
which can be shown as follows:
=
1
2
(
|
+
⟩
A
|
+
⟩
B
+
|
−
⟩
A
|
−
⟩
B
)
{\displaystyle ={\frac {1}{\sqrt {2}}}(|+\rangle _{A}|+\rangle _{B}+|-\rangle _{A}|-\rangle _{B})}
=
1
2
2
(
(
|
0
⟩
A
+
|
1
⟩
A
)
(
|
0
⟩
B
+
|
1
⟩
B
)
+
(
|
0
⟩
A
−
|
1
⟩
A
)
(
|
0
⟩
B
−
|
1
⟩
B
)
)
{\displaystyle ={\frac {1}{2{\sqrt {2}}}}((|0\rangle _{A}+|1\rangle _{A})(|0\rangle _{B}+|1\rangle _{B})+(|0\rangle _{A}-|1\rangle _{A})(|0\rangle _{B}-|1\rangle _{B}))}
=
1
2
2
(
(
|
00
⟩
+
|
01
⟩
+
|
10
⟩
+
|
11
⟩
)
+
(
|
00
⟩
−
|
01
⟩
−
|
10
⟩
+
|
11
⟩
)
)
{\displaystyle ={\frac {1}{2{\sqrt {2}}}}((|00\rangle +|01\rangle +|10\rangle +|11\rangle )+(|00\rangle -|01\rangle -|10\rangle +|11\rangle ))}
=
1
2
(
|
00
⟩
+
|
11
⟩
)
.
{\displaystyle ={\frac {1}{\sqrt {2}}}(|00\rangle +|11\rangle ).}
== C-ROT gate ==
The C-ROT gate (controlled Rabi rotation) is equivalent to a C-NOT gate except for a
π
/
2
{\displaystyle \pi /2}
rotation of the nuclear spin around the z axis.
== Implementations ==
Trapped ion quantum computers:
Cirac–Zoller controlled-NOT gate
Mølmer–Sørensen gate
== Regulation ==
In May, 2024, Canada implemented export restrictions on the sale of quantum computers containing more than 34 qubits and error rates below a certain CNOT error threshold, along with restrictions for quantum computers with more qubits and higher error rates. The same restrictions quickly popped up in the UK, France, Spain and the Netherlands. They offered few explanations for this action, but all of them are Wassenaar Arrangement states, and the restrictions seem related to national security concerns potentially including quantum cryptography or protection from competition.
== See also ==
Toffoli gate (controlled-controlled-NOT gate)
== Notes ==
== References ==
== External links ==
Michael Westmoreland: "Isolation and information flow in quantum dynamics" - discussion around the Cnot gate | Wikipedia/Controlled_NOT_gate |
Quantum tomography or quantum state tomography is the process by which a quantum state is reconstructed using measurements on an ensemble of identical quantum states. The source of these states may be any device or system which prepares quantum states either consistently into quantum pure states or otherwise into general mixed states. To be able to uniquely identify the state, the measurements must be tomographically complete. That is, the measured operators must form an operator basis on the Hilbert space of the system, providing all the information about the state. Such a set of observations is sometimes called a quorum. The term tomography was first used in the quantum physics literature in a 1993 paper introducing experimental optical homodyne tomography.
In quantum process tomography on the other hand, known quantum states are used to probe a quantum process to find out how the process can be described. Similarly, quantum measurement tomography works to find out what measurement is being performed. Whereas, randomized benchmarking scalably obtains a figure of merit of the overlap between the error prone physical quantum process and its ideal counterpart.
The general principle behind quantum state tomography is that by repeatedly performing many different measurements on quantum systems described by identical density matrices, frequency counts can be used to infer probabilities, and these probabilities are combined with Born's rule to determine a density matrix which fits the best with the observations.
This can be easily understood by making a classical analogy. Consider a harmonic oscillator (e.g. a pendulum). The position and momentum of the oscillator at any given point can be measured and therefore the motion can be completely described by the phase space. This is shown in figure 1. By performing this measurement for a large number of identical oscillators we get a probability distribution in the phase space (figure 2). This distribution can be normalized (the oscillator at a given time has to be somewhere) and the distribution must be non-negative. So we have retrieved a function
W
(
x
,
p
)
{\displaystyle W(x,p)}
which gives a description of the chance of finding the particle at a given point with a given momentum.
For quantum mechanical particles the same can be done. The only difference is that the Heisenberg's uncertainty principle mustn't be violated, meaning that we cannot measure the particle's momentum and position at the same time. The particle's momentum and its position are called quadratures (see Optical phase space for more information) in quantum related states. By measuring one of the quadratures of a large number of identical quantum states will give us a probability density corresponding to that particular quadrature. This is called the marginal distribution,
p
r
(
X
)
{\displaystyle \mathrm {pr} (X)}
or
p
r
(
P
)
{\displaystyle \mathrm {pr} (P)}
(see figure 3). In the following text we will see that this probability density is needed to characterize the particle's quantum state, which is the whole point of quantum tomography.
== What quantum state tomography is used for ==
Quantum tomography is applied on a source of systems, to determine the quantum state of the output of that source. Unlike a measurement on a single system, which determines the system's current state after the measurement (in general, the act of making a measurement alters the quantum state), quantum tomography works to determine the state(s) prior to the measurements.
Quantum tomography can be used for characterizing optical signals, including measuring the signal gain and loss of optical devices, as well as in quantum computing and quantum information theory to reliably determine the actual states of the qubits. One can imagine a situation in which a person Bob prepares many identical objects (particles or fields) in the same quantum states and then gives them to Alice to measure. Not confident with Bob's description of the state, Alice may wish to do quantum tomography to classify the state herself.
== Methods of quantum state tomography ==
=== Linear inversion ===
Using Born's rule, one can derive the simplest form of quantum tomography. Generally, being in a pure state is not known in advance, and a state may be mixed. In this case, many different types of measurements will have to be performed, many times each. To fully reconstruct the density matrix for a mixed state in a finite-dimensional Hilbert space, the following technique may be used.
Born's rule states
P
(
E
i
|
ρ
)
=
T
r
a
c
e
(
E
i
ρ
)
{\displaystyle \mathrm {P} (E_{i}|\rho )=\mathrm {Trace} (E_{i}\rho )}
, where
E
i
{\displaystyle E_{i}}
is a particular measurement outcome projector and
ρ
{\displaystyle \rho }
is the density matrix of the system.
Given a histogram of observations for each measurement, one has an approximation
p
i
{\displaystyle p_{i}}
to
P
(
E
i
|
ρ
)
{\displaystyle \mathrm {P} (E_{i}|\rho )}
for each
E
i
{\displaystyle E_{i}}
.
Given linear operators
S
{\displaystyle S}
and
T
{\displaystyle T}
, define the inner product
S
⋅
T
=
T
r
[
S
†
T
]
=
S
→
†
T
→
{\displaystyle S\cdot T=\mathrm {Tr} [S^{\dagger }T]={\vec {S}}^{\dagger }{\vec {T}}}
where
T
→
{\displaystyle {\vec {T}}}
is representation of the
T
{\displaystyle T}
operator as a column vector and
S
→
†
{\displaystyle {\vec {S}}^{\dagger }}
a row vector such that
S
→
†
T
→
{\displaystyle {\vec {S}}^{\dagger }{\vec {T}}}
is the inner product in
C
d
{\displaystyle \mathbb {C} ^{d}}
of the two.
Define the matrix
A
{\displaystyle A}
as
A
=
(
E
→
1
†
E
→
2
†
E
→
3
†
⋮
)
{\displaystyle A={\begin{pmatrix}{\vec {E}}_{1}^{\dagger }\\{\vec {E}}_{2}^{\dagger }\\{\vec {E}}_{3}^{\dagger }\\\vdots \end{pmatrix}}}
.
Here Ei is some fixed list of individual measurements (with binary outcomes), and A does all the measurements at once.
Then applying this to
ρ
→
{\displaystyle {\vec {\rho }}}
yields the probabilities:
A
ρ
→
=
(
E
→
1
†
ρ
→
E
→
2
†
ρ
→
E
→
3
†
ρ
→
⋮
)
=
(
E
1
⋅
ρ
E
2
⋅
ρ
E
3
⋅
ρ
⋮
)
=
(
P
(
E
1
|
ρ
)
P
(
E
2
|
ρ
)
P
(
E
3
|
ρ
)
⋮
)
≈
(
p
1
p
2
p
3
⋮
)
=
p
→
{\displaystyle A{\vec {\rho }}={\begin{pmatrix}{\vec {E}}_{1}^{\dagger }{\vec {\rho }}\\{\vec {E}}_{2}^{\dagger }{\vec {\rho }}\\{\vec {E}}_{3}^{\dagger }{\vec {\rho }}\\\vdots \end{pmatrix}}={\begin{pmatrix}E_{1}\cdot \rho \\E_{2}\cdot \rho \\E_{3}\cdot \rho \\\vdots \end{pmatrix}}={\begin{pmatrix}\mathrm {P} (E_{1}|\rho )\\\mathrm {P} (E_{2}|\rho )\\\mathrm {P} (E_{3}|\rho )\\\vdots \end{pmatrix}}\approx {\begin{pmatrix}p_{1}\\p_{2}\\p_{3}\\\vdots \end{pmatrix}}={\vec {p}}}
.
Linear inversion corresponds to inverting this system using the observed relative frequencies
p
→
{\displaystyle {\vec {p}}}
to derive
ρ
→
{\displaystyle {\vec {\rho }}}
(which is isomorphic to
ρ
{\displaystyle \displaystyle \rho }
).
This system is not going to be square in general, as for each measurement being made there will generally be multiple measurement outcome projectors
E
i
{\displaystyle E_{i}}
. For example, in a 2-D Hilbert space with 3 measurements
σ
x
,
σ
y
,
σ
z
{\displaystyle \sigma _{x},\sigma _{y},\sigma _{z}}
, each measurement has 2 outcomes, each of which has a projector Ei, for 6 projectors, whereas the real dimension of the space of density matrices is (2⋅22)/2=4, leaving
A
{\displaystyle A}
to be 6 x 4. To solve the system, multiply on the left by
A
T
{\displaystyle A^{T}}
:
A
T
A
ρ
→
=
A
T
p
→
{\displaystyle A^{T}A{\vec {\rho }}=A^{T}{\vec {p}}}
.
Now solving for
ρ
→
{\displaystyle {\vec {\rho }}}
yields the pseudoinverse:
ρ
→
=
(
A
T
A
)
−
1
A
T
p
→
{\displaystyle {\vec {\rho }}=(A^{T}A)^{-1}A^{T}{\vec {p}}}
.
This works in general only if the measurement list Ei is tomographically complete. Otherwise, the matrix
A
T
A
{\displaystyle A^{T}A}
will not be invertible.
==== Continuous variables and quantum homodyne tomography ====
In infinite dimensional Hilbert spaces, e.g. in measurements of continuous variables such as position, the methodology is somewhat more complex. One notable example is in the tomography of light, known as optical homodyne tomography. Using balanced homodyne measurements, one can derive the Wigner function and a density matrix for the state of the light.
One approach involves measurements along different rotated directions in phase space. For each direction
θ
{\displaystyle \theta }
, one can find a probability distribution
w
(
q
,
θ
)
{\displaystyle w(q,\theta )}
for the probability density of measurements in the
θ
{\displaystyle \theta }
direction of phase space yielding the value
q
{\displaystyle q}
. Using an inverse Radon transformation (the filtered back projection) on
w
(
q
,
θ
)
{\displaystyle w(q,\theta )}
leads to the Wigner function,
W
(
x
,
p
)
{\displaystyle \mathrm {W} (x,p)}
, which can be converted by an inverse Fourier transform into the density matrix for the state in any basis. A similar technique is often used in medical tomography.
==== Example: single-qubit state tomography ====
The density matrix of a single qubit can be expressed in terms of its Bloch vector
r
→
{\displaystyle {\vec {r}}}
and the Pauli vector
σ
→
{\displaystyle {\vec {\sigma }}}
:
ρ
=
1
2
(
I
+
r
→
⋅
σ
→
)
=
1
2
(
1
+
r
z
r
x
−
i
r
y
r
x
+
i
r
y
1
−
r
z
)
{\displaystyle \rho ={\frac {1}{2}}\left(I+{\vec {r}}\cdot {\vec {\sigma }}\right)={\frac {1}{2}}{\begin{pmatrix}1+r_{z}&r_{x}-\mathrm {i} r_{y}\\r_{x}+\mathrm {i} r_{y}&1-r_{z}\end{pmatrix}}}
.
The single-qubit state tomography can be performed by means of single-qubit Pauli measurements:
First, create a list of three quantum circuits, with the first one measuring the qubit in the computational basis (Z-basis), the second one performing a Hadamard gate before measurement (which makes the measurement in X-basis), and the third one performing the appropriate phase shift gate (that is
Z
†
=
|
0
⟩
⟨
0
|
+
exp
(
−
i
π
/
2
)
|
1
⟩
⟨
1
|
{\displaystyle {\sqrt {Z}}^{\dagger }=|0\rangle \langle 0|+\exp(-\mathrm {i} \pi /2)|1\rangle \langle 1|}
) followed by a Hadamard gate before measurement (which makes the measurement in Y-basis);
Then, run these circuits (typically thousands of times), and the counts in the measurement results of the first circuit produces
z
¯
=
(
n
z
,
+
−
n
z
,
−
)
/
(
n
z
,
+
+
n
z
,
−
)
{\displaystyle {\bar {z}}=(n_{z,+}-n_{z,-})/(n_{z,+}+n_{z,-})}
, the second circuit
x
¯
{\displaystyle {\bar {x}}}
, and the third circuit
y
¯
{\displaystyle {\bar {y}}}
;
Finally, if
x
¯
2
+
y
¯
2
+
z
¯
2
≤
1
{\displaystyle {\bar {x}}^{2}+{\bar {y}}^{2}+{\bar {z}}^{2}\leq 1}
, then a measured Bloch vector is produced as
r
→
m
=
(
x
¯
,
y
¯
,
z
¯
)
{\displaystyle {\vec {r}}_{m}=({\bar {x}},{\bar {y}},{\bar {z}})}
, and the measured density matrix is
ρ
m
=
1
2
(
I
+
r
→
m
⋅
σ
→
)
{\displaystyle \rho _{m}={\frac {1}{2}}\left(I+{\vec {r}}_{m}\cdot {\vec {\sigma }}\right)}
; If
x
¯
2
+
y
¯
2
+
z
¯
2
>
1
{\displaystyle {\bar {x}}^{2}+{\bar {y}}^{2}+{\bar {z}}^{2}>1}
, it'll be necessary to renormalize the measured Bloch vector as
r
→
m
=
(
x
¯
,
y
¯
,
z
¯
)
/
x
¯
2
+
y
¯
2
+
z
¯
2
{\displaystyle {\vec {r}}_{m}=({\bar {x}},{\bar {y}},{\bar {z}})/{\sqrt {{\bar {x}}^{2}+{\bar {y}}^{2}+{\bar {z}}^{2}}}}
before using it to calculate the measured density matrix.
This algorithm is the foundation for qubit tomography and is used in some quantum programming routines, like that of Qiskit.
==== Example: homodyne tomography. ====
Electromagnetic field amplitudes (quadratures) can be measured with high efficiency using photodetectors together with temporal mode selectivity. Balanced homodyne tomography is a reliable technique of reconstructing quantum states in the optical domain. This technique combines the advantages of the high efficiencies of photodiodes in measuring the intensity or photon number of light, together with measuring the quantum features of light by a clever set-up called the homodyne tomography detector.
Quantum homodyne tomography is understood by the following example.
A laser is directed onto a 50-50% beamsplitter, splitting the laser beam into two beams. One is used as a local oscillator (LO) and the other is used to generate photons with a particular quantum state. The generation of quantum states can be realized, e.g. by directing the laser beam through a frequency doubling crystal and then onto a parametric down-conversion crystal. This crystal generates two photons in a certain quantum state. One of the photons is used as a trigger signal used to trigger (start) the readout event of the homodyne tomography detector. The other photon is directed into the homodyne tomography detector, in order to reconstruct its quantum state. Since the trigger and signal photons are entangled (this is explained by the spontaneous parametric down-conversion article), it is important to realize that the optical mode of the signal state is created nonlocal only when the trigger photon impinges the photodetector (of the trigger event readout module) and is actually measured. More simply said, it is only when the trigger photon is measured, that the signal photon can be measured by the homodyne detector.
Now consider the homodyne tomography detector as depicted in figure 4 (figure missing). The signal photon (this is the quantum state we want to reconstruct) interferes with the local oscillator, when they are directed onto a 50-50% beamsplitter. Since the two beams originate from the same so called master laser, they have the same fixed phase relation. The local oscillator must be intense, compared to the signal so it provides a precise phase reference. The local oscillator is so intense, that we can treat it classically (a = α) and neglect the quantum fluctuations.
The signal field is spatially and temporally controlled by the local oscillator, which has a controlled shape. Where the local oscillator is zero, the signal is rejected. Therefore, we have temporal-spatial mode selectivity of the signal.
The beamsplitter redirects the two beams to two photodetectors. The photodetectors generate an electric current proportional to the photon number. The two detector currents are subtracted and the resulting current is proportional to the electric field operator in the signal mode, depended on relative optical phase of signal and local oscillator.
Since the electric field amplitude of the local oscillator is much higher than that of the signal the intensity or fluctuations in the signal field can be seen. The homodyne tomography system functions as an amplifier. The system can be seen as an interferometer with such a high intensity reference beam (the local oscillator) that unbalancing the interference by a single photon in the signal is measurable. This amplification is well above the photodetectors noise floor.
The measurement is reproduced a large number of times. Then the phase difference between the signal and local oscillator is changed in order to ‘scan’ a different angle in the phase space. This can be seen from figure 4. The measurement is repeated again a large number of times and a marginal distribution is retrieved from the current difference. The marginal distribution can be transformed into the density matrix and/or the Wigner function. Since the density matrix and the Wigner function give information about the quantum state of the photon, we have reconstructed the quantum state of the photon.
The advantage of this balanced detection method is that this arrangement is insensitive to fluctuations in the intensity of the laser.
The quantum computations for retrieving the quadrature component from the current difference are performed as follows.
The photon number operator for the beams striking the photodetectors after the beamsplitter is given by:
n
^
i
=
a
^
i
†
a
^
i
{\displaystyle {\hat {n}}_{i}={\hat {a}}_{i}^{\dagger }{\hat {a}}_{i}}
,
where i is 1 and 2, for respectively beam one and two.
The mode operators of the field emerging the beamsplitters are given by:
a
^
1
=
2
−
1
/
2
(
a
^
−
α
L
O
)
{\displaystyle {\hat {a}}_{1}=2^{-1/2}({\hat {a}}-\alpha _{LO})}
a
^
2
=
2
−
1
/
2
(
a
^
+
α
L
O
)
{\displaystyle {\hat {a}}_{2}=2^{-1/2}({\hat {a}}+\alpha _{LO})}
The
a
^
{\displaystyle {\hat {a}}}
denotes the annihilation operator of the signal and alpha the complex amplitude of the local oscillator.
The number of photon difference is eventually proportional to the quadrature and given by:
n
^
21
=
n
^
2
−
n
^
1
=
α
L
O
∗
a
^
+
α
L
O
a
^
†
{\displaystyle {\hat {n}}_{21}={\hat {n}}_{2}-{\hat {n}}_{1}=\alpha _{LO}^{*}{\hat {a}}+\alpha _{LO}{\hat {a}}^{\dagger }}
,
Rewriting this with the relation:
q
^
=
2
−
1
/
2
(
a
^
†
+
a
^
)
{\displaystyle {\hat {q}}=2^{-1/2}({\hat {a}}^{\dagger }+{\hat {a}})}
Results in the following relation:
n
^
21
=
2
1
/
2
|
α
^
L
O
|
q
^
θ
{\displaystyle {\hat {n}}_{21}=2^{1/2}|{\hat {\alpha }}_{LO}|{\hat {q}}_{\theta }}
,
where we see clear relation between the photon number difference and the quadrature component
q
^
θ
{\displaystyle {\hat {q}}_{\theta }}
. By keeping track of the sum current, one can recover information about the local oscillator's intensity, since this is usually an unknown quantity, but an important quantity for calculating the quadrature component
q
^
θ
{\displaystyle {\hat {q}}_{\theta }}
.
==== Problems with linear inversion ====
One of the primary problems with using linear inversion to solve for the density matrix is that in general the computed solution will not be a valid density matrix. For example, it could give negative probabilities or probabilities greater than 1 to certain measurement outcomes. This is particularly an issue when fewer measurements are made.
Another issue is that in infinite dimensional Hilbert spaces, an infinite number of measurement outcomes would be required. Making assumptions about the structure and using a finite measurement basis leads to artifacts in the phase space density.
=== Maximum likelihood estimation ===
Maximum likelihood estimation (also known as MLE or MaxLik) is a popular technique for dealing with the problems of linear inversion. By restricting the domain of density matrices to the proper space, and searching for the density matrix which maximizes the likelihood of giving the experimental results, it guarantees the state to be theoretically valid while giving a close fit to the data. The likelihood of a state is the probability that would be assigned to the observed results had the system been in that state.
Suppose the measurements
{
|
y
j
⟩
⟨
y
j
|
}
{\displaystyle \{|y_{j}\rangle \langle y_{j}|\}}
have been observed with frequencies
f
j
{\displaystyle f_{j}}
. Then the likelihood associated with a state
ρ
^
{\displaystyle {\hat {\rho }}}
is
L
(
ρ
^
)
=
∏
j
⟨
y
j
|
ρ
^
|
y
j
⟩
f
j
{\displaystyle L({\hat {\rho }})=\prod _{j}\langle y_{j}|{\hat {\rho }}|y_{j}\rangle ^{f_{j}}}
where
⟨
y
j
|
ρ
^
|
y
j
⟩
{\displaystyle \langle y_{j}|{\hat {\rho }}|y_{j}\rangle }
is the probability of outcome
y
j
{\displaystyle y_{j}}
for the state
ρ
^
{\displaystyle {\hat {\rho }}}
.
Finding the maximum of this function is non-trivial and generally involves iterative methods. The methods are an active topic of research.
==== Problems with maximum likelihood estimation ====
Maximum likelihood estimation suffers from some less obvious problems than linear inversion. One problem is that it makes predictions about probabilities that cannot be justified by the data. This is seen most easily by looking at the problem of zero eigenvalues. The computed solution using MLE often contains eigenvalues which are 0, i.e. it is rank deficient. In these cases, the solution then lies on the boundary of the n-dimensional Bloch sphere. This can be seen as related to linear inversion giving states which lie outside the valid space (the Bloch sphere). MLE in these cases picks a nearby point that is valid, and the nearest points are generally on the boundary.
This is not physically a problem, the real state might have zero eigenvalues. However, since no value may be less than 0, an estimate of an eigenvalue being 0 implies that the estimator is certain the value is 0, otherwise they would have estimated some
ϵ
{\displaystyle \epsilon }
greater than 0 with a small degree of uncertainty as the best estimate. This is where the problem arises, in that it is not logical to conclude with absolute certainty after a finite number of measurements that any eigenvalue (that is, the probability of a particular outcome) is 0. For example, if a coin is flipped 5 times and each time heads was observed, it does not mean there is 0 probability of getting tails, despite that being the most likely description of the coin.
=== Bayesian methods ===
Bayesian mean estimation (BME) is a relatively new approach which addresses the problems of maximum likelihood estimation. It focuses on finding optimal solutions which are also honest in that they include error bars in the estimate. The general idea is to start with a likelihood function and a function describing the experimenter's prior knowledge (which might be a constant function), then integrate over all density matrices using the product of the likelihood function and prior knowledge function as a weight.
Given a reasonable prior knowledge function, BME will yield a state strictly within the n-dimensional Bloch sphere. In the case of a coin flipped N times to get N heads described above, with a constant prior knowledge function, BME would assign
1
N
+
2
{\displaystyle \scriptstyle {\frac {1}{N+2}}}
as the probability for tails.
BME provides a high degree of accuracy in that it minimizes the operational divergences of the estimate from the actual state.
=== Methods for incomplete data ===
The number of measurements needed for a full quantum state tomography for a multi-particle system scales exponentially with the number of particles, which
makes such a procedure impossible even for modest system sizes. Hence, several methods have been developed to
realize quantum tomography with fewer measurements.
The concept of matrix completion and compressed sensing have been applied to reconstruct density matrices from an incomplete set of measurements (that is, a set of measurements which is not a quorum). In general, this is impossible, but under assumptions (for example, if the density matrix is a pure state, or a combination of just a few pure states) then the density matrix has fewer degrees of freedom, and it may be possible to reconstruct the state from the incomplete measurements.
Permutationally Invariant Quantum Tomography
is a procedure that has been developed mostly for states that are close to being
permutationally symmetric, which is typical in nowadays experiments. For two-state particles, the number of measurements needed scales only quadratically with the number of particles.
Besides the modest measurement effort, the processing of the measured data can also be done efficiently:
It is possible to carry out the fitting of a physical density matrix on the measured data even for large systems.
Permutationally Invariant Quantum Tomography has been combined with compressed sensing in a six-qubit
photonic experiment.
== Quantum measurement tomography ==
One can imagine a situation in which an apparatus performs some measurement on quantum systems, and determining what particular measurement is desired. The strategy is to send in systems of various known states, and use these states to estimate the outcomes of the unknown measurement. Also known as "quantum estimation", tomography techniques are increasingly important including those for quantum measurement tomography and the very similar quantum state tomography. Since a measurement can always be characterized by a set of POVM's, the goal is to reconstruct the characterizing POVM's
Π
l
{\displaystyle \Pi _{l}}
. The simplest approach is linear inversion. As in quantum state observation, use
T
r
[
Π
l
ρ
m
]
=
P
(
l
|
ρ
m
)
{\displaystyle \displaystyle \mathrm {Tr} [\Pi _{l}\rho _{m}]=\mathrm {P} (l|\rho _{m})}
.
Exploiting linearity as above, this can be inverted to solve for the
Π
l
{\displaystyle \Pi _{l}}
.
Not surprisingly, this suffers from the same pitfalls as in quantum state tomography: namely, non-physical results, in particular negative probabilities. Here the
Π
l
{\displaystyle \Pi _{l}}
will not be valid POVM's, as they will not be positive. Bayesian methods as well as Maximum likelihood estimation of the density matrix can be used to restrict the operators to valid physical results.
== Quantum process tomography ==
Quantum process tomography (QPT) deals with identifying an unknown quantum dynamical process. The first approach, introduced in 1996 and sometimes known as standard quantum process tomography (SQPT) involves preparing an ensemble of quantum states and sending them through the process, then using quantum state tomography to identify the resultant states. Other techniques include ancilla-assisted process tomography (AAPT) and entanglement-assisted process tomography (EAPT) which require an extra copy of the system.
Each of the techniques listed above are known as indirect methods for characterization of quantum dynamics, since they require the use of quantum state tomography to reconstruct the process. In contrast, there are direct methods such as direct characterization of quantum dynamics (DCQD) which provide a full characterization of quantum systems without any state tomography.
The number of experimental configurations (state preparations and measurements) required for full quantum process tomography grows exponentially with the number of constituent particles of a system. Consequently, in general, QPT is an impossible task for large-scale quantum systems. However, under weak decoherence assumption, a quantum dynamical map can find a sparse representation. The method of compressed quantum process tomography (CQPT) uses the compressed sensing technique and applies the sparsity assumption to reconstruct a quantum dynamical map from an incomplete set of measurements or test state preparations.
=== Quantum dynamical maps ===
A quantum process, also known as a quantum dynamical map,
E
(
ρ
)
{\displaystyle {\mathcal {E}}(\rho )}
, can be described by a completely positive map
E
(
ρ
)
=
∑
i
A
i
ρ
A
i
†
{\displaystyle {\mathcal {E}}(\rho )=\sum _{i}A_{i}\rho A_{i}^{\dagger }}
,
where
ρ
∈
B
(
H
)
{\displaystyle \rho \in {\mathcal {B(H)}}}
, the bounded operators on Hilbert space; with operation elements
A
i
{\displaystyle \displaystyle A_{i}}
satisfying
∑
i
A
i
†
A
i
≤
I
{\displaystyle \textstyle \sum _{i}A_{i}^{\dagger }A_{i}\leq I}
so that
T
r
[
E
(
ρ
)
]
≤
1
{\displaystyle \mathrm {Tr} [{\mathcal {E}}(\rho )]\leq 1}
.
Let
{
E
i
}
{\displaystyle \displaystyle \{E_{i}\}}
be an orthogonal basis for
B
(
H
)
{\displaystyle {\mathcal {B(H)}}}
. Write the
A
i
{\displaystyle \displaystyle A_{i}}
operators in this basis
A
i
=
∑
m
a
i
m
E
m
{\displaystyle \displaystyle A_{i}=\sum _{m}a_{im}E_{m}}
.
This leads to
E
(
ρ
)
=
∑
m
,
n
χ
m
n
E
m
ρ
E
n
†
{\displaystyle {\mathcal {E}}(\rho )=\sum _{m,n}\chi _{mn}E_{m}\rho E_{n}^{\dagger }}
,
where
χ
m
n
=
∑
i
a
i
m
a
i
n
∗
{\displaystyle \chi _{mn}=\sum _{i}a_{im}a_{in}^{*}}
.
The goal is then to solve for
χ
{\displaystyle \displaystyle \chi }
, which is a positive superoperator and completely characterizes
E
{\displaystyle {\mathcal {E}}}
with respect to the
{
E
i
}
{\displaystyle \displaystyle \{E_{i}\}}
basis.
=== Standard quantum process tomography ===
SQPT approaches this using
d
2
{\displaystyle d^{2}}
linearly independent inputs
ρ
j
{\displaystyle \rho _{j}}
, where
d
{\displaystyle d}
is the dimension of the Hilbert space
H
{\displaystyle {\mathcal {H}}}
. For each of these input states
ρ
j
{\displaystyle \rho _{j}}
, sending it through the process gives an output state
E
(
ρ
)
{\displaystyle {\mathcal {E}}(\rho )}
which can be written as a linear combination of the
ρ
k
{\displaystyle \rho _{k}}
, i.e.
E
(
ρ
j
)
=
∑
k
c
j
k
ρ
k
{\displaystyle \textstyle {\mathcal {E}}(\rho _{j})=\sum _{k}c_{jk}\rho _{k}}
. By sending each
ρ
j
{\displaystyle \rho _{j}}
through many times, quantum state tomography can be used to determine the coefficients
c
j
k
{\displaystyle c_{jk}}
experimentally.
Write
E
m
ρ
j
E
n
†
=
∑
k
B
m
,
n
,
j
,
k
ρ
k
{\displaystyle E_{m}\rho _{j}E_{n}^{\dagger }=\sum _{k}B_{m,n,j,k}\rho _{k}}
,
where
B
{\displaystyle B}
is a matrix of coefficients.
Then
∑
k
c
j
k
ρ
k
=
E
(
ρ
j
)
=
∑
m
,
n
χ
m
,
n
E
m
ρ
j
E
n
†
=
∑
m
,
n
∑
k
χ
m
,
n
B
m
,
n
,
j
,
k
ρ
k
{\displaystyle \sum _{k}c_{jk}\rho _{k}={\mathcal {E}}(\rho _{j})=\sum _{m,n}\chi _{m,n}E_{m}\rho _{j}E_{n}^{\dagger }=\sum _{m,n}\sum _{k}\chi _{m,n}B_{m,n,j,k}\rho _{k}}
.
Since
ρ
k
{\displaystyle \rho _{k}}
form a linearly independent basis,
c
j
k
=
∑
m
,
n
χ
m
,
n
B
m
,
n
,
j
,
k
{\displaystyle \displaystyle c_{jk}=\sum _{m,n}\chi _{m,n}B_{m,n,j,k}}
.
Inverting
B
{\displaystyle B}
gives
χ
{\displaystyle \chi }
:
χ
m
,
n
=
∑
j
,
k
B
m
,
n
,
j
,
k
−
1
c
j
k
{\displaystyle \chi _{m,n}=\sum _{j,k}B_{m,n,j,k}^{-1}c_{jk}}
.
== See also ==
Quantum discord
Quantum process
Quantum entanglement § Entanglement of top quarks
== References == | Wikipedia/Quantum_tomography |
In physics, an observable is a physical property or physical quantity that can be measured. In classical mechanics, an observable is a real-valued "function" on the set of all possible system states, e.g., position and momentum. In quantum mechanics, an observable is an operator, or gauge, where the property of the quantum state can be determined by some sequence of operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value.
Physically meaningful observables must also satisfy transformation laws that relate observations performed by different observers in different frames of reference. These transformation laws are automorphisms of the state space, that is bijective transformations that preserve certain mathematical properties of the space in question.
== Quantum mechanics ==
In quantum mechanics, observables manifest as self-adjoint operators on a separable complex Hilbert space representing the quantum state space. Observables assign values to outcomes of particular measurements, corresponding to the eigenvalue of the operator. If these outcomes represent physically allowable states (i.e. those that belong to the Hilbert space) the eigenvalues are real; however, the converse is not necessarily true. As a consequence, only certain measurements can determine the value of an observable for some state of a quantum system. In classical mechanics, any measurement can be made to determine the value of an observable.
The relation between the state of a quantum system and the value of an observable requires some linear algebra for its description. In the mathematical formulation of quantum mechanics, up to a phase constant, pure states are given by non-zero vectors in a Hilbert space V. Two vectors v and w are considered to specify the same state if and only if
w
=
c
v
{\displaystyle \mathbf {w} =c\mathbf {v} }
for some non-zero
c
∈
C
{\displaystyle c\in \mathbb {C} }
. Observables are given by self-adjoint operators on V. Not every self-adjoint operator corresponds to a physically meaningful observable. Also, not all physical observables are associated with non-trivial self-adjoint operators. For example, in quantum theory, mass appears as a parameter in the Hamiltonian, not as a non-trivial operator.
In the case of transformation laws in quantum mechanics, the requisite automorphisms are unitary (or antiunitary) linear transformations of the Hilbert space V. Under Galilean relativity or special relativity, the mathematics of frames of reference is particularly simple, considerably restricting the set of physically meaningful observables.
In quantum mechanics, measurement of observables exhibits some seemingly unintuitive properties. Specifically, if a system is in a state described by a vector in a Hilbert space, the measurement process affects the state in a non-deterministic but statistically predictable way. In particular, after a measurement is applied, the state description by a single vector may be destroyed, being replaced by a statistical ensemble. The irreversible nature of measurement operations in quantum physics is sometimes referred to as the measurement problem and is described mathematically by quantum operations. By the structure of quantum operations, this description is mathematically equivalent to that offered by the relative state interpretation where the original system is regarded as a subsystem of a larger system and the state of the original system is given by the partial trace of the state of the larger system.
In quantum mechanics, dynamical variables
A
{\displaystyle A}
such as position, translational (linear) momentum, orbital angular momentum, spin, and total angular momentum are each associated with a self-adjoint operator
A
^
{\displaystyle {\hat {A}}}
that acts on the state of the quantum system. The eigenvalues of operator
A
^
{\displaystyle {\hat {A}}}
correspond to the possible values that the dynamical variable can be observed as having. For example, suppose
|
ψ
a
⟩
{\displaystyle |\psi _{a}\rangle }
is an eigenket (eigenvector) of the observable
A
^
{\displaystyle {\hat {A}}}
, with eigenvalue
a
{\displaystyle a}
, and exists in a Hilbert space. Then
A
^
|
ψ
a
⟩
=
a
|
ψ
a
⟩
.
{\displaystyle {\hat {A}}|\psi _{a}\rangle =a|\psi _{a}\rangle .}
This eigenket equation says that if a measurement of the observable
A
^
{\displaystyle {\hat {A}}}
is made while the system of interest is in the state
|
ψ
a
⟩
{\displaystyle |\psi _{a}\rangle }
, then the observed value of that particular measurement must return the eigenvalue
a
{\displaystyle a}
with certainty. However, if the system of interest is in the general state
|
ϕ
⟩
∈
H
{\displaystyle |\phi \rangle \in {\mathcal {H}}}
(and
|
ϕ
⟩
{\displaystyle |\phi \rangle }
and
|
ψ
a
⟩
{\displaystyle |\psi _{a}\rangle }
are unit vectors, and the eigenspace of
a
{\displaystyle a}
is one-dimensional), then the eigenvalue
a
{\displaystyle a}
is returned with probability
|
⟨
ψ
a
|
ϕ
⟩
|
2
{\displaystyle |\langle \psi _{a}|\phi \rangle |^{2}}
, by the Born rule.
=== Compatible and incompatible observables in quantum mechanics ===
A crucial difference between classical quantities and quantum mechanical observables is that some pairs of quantum observables may not be simultaneously measurable, a property referred to as complementarity. This is mathematically expressed by non-commutativity of their corresponding operators, to the effect that the commutator
[
A
^
,
B
^
]
:=
A
^
B
^
−
B
^
A
^
≠
0
^
.
{\displaystyle \left[{\hat {A}},{\hat {B}}\right]:={\hat {A}}{\hat {B}}-{\hat {B}}{\hat {A}}\neq {\hat {0}}.}
This inequality expresses a dependence of measurement results on the order in which measurements of observables
A
^
{\displaystyle {\hat {A}}}
and
B
^
{\displaystyle {\hat {B}}}
are performed. A measurement of
A
^
{\displaystyle {\hat {A}}}
alters the quantum state in a way that is incompatible with the subsequent measurement of
B
^
{\displaystyle {\hat {B}}}
and vice versa.
Observables corresponding to commuting operators are called compatible observables. For example, momentum along say the
x
{\displaystyle x}
and
y
{\displaystyle y}
axes are compatible. Observables corresponding to non-commuting operators are called incompatible observables or complementary variables. For example, the position and momentum along the same axis are incompatible.: 155
Incompatible observables cannot have a complete set of common eigenfunctions. Note that there can be some simultaneous eigenvectors of
A
^
{\displaystyle {\hat {A}}}
and
B
^
{\displaystyle {\hat {B}}}
, but not enough in number to constitute a complete basis.
== See also ==
== References ==
== Further reading ==
Auyang, Sunny Y. (1995). How is quantum field theory possible?. New York, N.Y.: Oxford University Press. ISBN 978-0195093452.
Cohen-Tannoudji, Claude; Diu, Bernard; Laloë, Franck (2019). Quantum Mechanics, Volume 1. Weinheim: John Wiley & Sons. ISBN 978-3-527-34553-3.
de la Madrid Modino, R. (2001). Quantum mechanics in rigged Hilbert space language (PhD thesis). Universidad de Valladolid.
Teschl, G. (2014). Mathematical Methods in Quantum Mechanics. Providence (R.I): American Mathematical Soc. ISBN 978-1-4704-1704-8.
von Neumann, John (1996). Mathematical foundations of quantum mechanics. Translated by Robert T. Beyer (12. print., 1. paperback print. ed.). Princeton, N.J.: Princeton Univ. Press. ISBN 978-0691028934.
Varadarajan, V.S. (2007). Geometry of quantum theory (2nd ed.). New York: Springer. ISBN 9780387493862.
Weyl, Hermann (2009). "Appendix C: Quantum physics and causality". Philosophy of mathematics and natural science. Revised and augmented English edition based on a translation by Olaf Helmer. Princeton, N.J.: Princeton University Press. pp. 253–265. ISBN 9780691141206.
Moretti, Valter (2017). Spectral Theory and Quantum Mechanics: Mathematical Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation (2 ed.). Springer. ISBN 978-3319707068.
Moretti, Valter (2019). Fundamental Mathematical Structures of Quantum Theory: Spectral Theory, Foundational Issues, Symmetries, Algebraic Formulation. Springer. ISBN 978-3030183462. | Wikipedia/Observable_(quantum_mechanics) |
The Ghirardi–Rimini–Weber theory (GRW) is a spontaneous collapse theory in quantum mechanics, proposed in 1986 by Giancarlo Ghirardi, Alberto Rimini, and Tullio Weber.
== Measurement problem and spontaneous collapses ==
Quantum mechanics has two fundamentally different dynamical principles: the linear and deterministic Schrödinger equation, and the nonlinear and stochastic wave packet reduction postulate. The orthodox interpretation, or Copenhagen interpretation of quantum mechanics, posits a wave function collapse every time an observer performs a measurement. One thus faces the problem of defining what an “observer” and a “measurement” are. Another issue of quantum mechanics is that it forecasts superpositions of macroscopic objects, which are not observed in nature (see Schrödinger's cat paradox). The theory does not tell where the threshold between the microscopic and macroscopic worlds is, that is when quantum mechanics should leave space to classical mechanics. The aforementioned issues constitute the measurement problem in quantum mechanics.
Collapse theories avoid the measurement problem by merging the two dynamical principles of quantum mechanics in a unique dynamical description. The physical idea that underlies collapse theories is that particles undergo spontaneous wave-function collapses, which occur randomly both in time (at a given average rate), and in space (according to the Born rule). The imprecise “observer” and “measurement” that plague the orthodox interpretation are thus avoided because the wave function collapses spontaneously. Furthermore, thanks to a so-called “amplification mechanism” (later discussed), collapse theories recover both quantum mechanics for microscopic objects, and classical mechanics for macroscopic ones.
The GRW is the first spontaneous collapse theory that was devised. In the following years several different models were proposed. Among these are
the continuous spontaneous localization model (CSL model), which is formulated in terms of identical particles;
the Diósi–Penrose model, which relates the spontaneous collapse to gravity;
the quantum mechanics with universal position localization (QMUPL) model, which proves important mathematical results on collapse theories; and the coloured QMUPL model, which is the only collapse model involving coloured stochastic processes for which the exact solution is known.
== Description ==
The first assumption of the GRW theory is that the wave function (or state vector) represents the most accurate possible specification of the state of a physical system. This is a feature that the GRW theory shares with the standard Interpretations of quantum mechanics, and distinguishes it from hidden variable theories, like the de Broglie–Bohm theory, according to which the wave function does not give a complete description of a physical system. The GRW theory differs from standard quantum mechanics for the dynamical principles according to which the wave function evolves. More philosophical issues related to the GRW theory and to collapse theories in general one have been discussed by Ghirardi and Bassi.
=== Working principles ===
Each particle of a system described by the multi-particle wave function
|
ψ
⟩
{\displaystyle |\psi \rangle }
independently undergoes a spontaneous localization process (or jump):
|
ψ
⟩
→
|
ψ
x
i
⟩
⟨
ψ
x
i
|
ψ
x
i
⟩
{\displaystyle |\psi \rangle \rightarrow {\frac {|\psi _{x}^{i}\rangle }{\sqrt {\langle \psi _{x}^{i}|\psi _{x}^{i}\rangle }}}}
,
where
|
ψ
x
i
⟩
=
L
^
x
i
|
ψ
⟩
{\displaystyle |\psi _{x}^{i}\rangle ={\hat {L}}_{x}^{i}|\psi \rangle }
is the state after the operator
L
^
x
i
{\displaystyle {\hat {L}}_{x}^{i}}
has localized the
i
{\displaystyle i}
-th particle around the position
x
{\displaystyle x}
.
The localization process is random both in space and time. The jumps are Poisson distributed in time, with mean rate
λ
{\displaystyle \lambda }
; the probability density for a jump to occur at position
x
{\displaystyle x}
is
P
i
(
x
)
=
⟨
ψ
x
i
|
ψ
x
i
⟩
{\displaystyle P_{i}(x)=\langle \psi _{x}^{i}|\psi _{x}^{i}\rangle }
.
The localization operator has a Gaussian form:
L
^
x
i
=
(
1
π
r
C
2
)
3
4
e
−
(
q
^
i
−
x
)
2
2
r
C
2
{\displaystyle {\hat {L}}_{x}^{i}=\left({\frac {1}{\pi r_{C}^{2}}}\right)^{\frac {3}{4}}e^{-{\frac {({\hat {q}}_{i}-x)^{2}}{2r_{C}^{2}}}}}
,
where
q
^
i
{\displaystyle {\hat {q}}_{i}}
is the position operator of the
i
{\displaystyle i}
-th particle, and
r
C
{\displaystyle r_{C}}
is the localization distance.
In between two localization processes, the wave function evolves according to the Schrödinger equation.
These principles can be expressed in a more compact way with the statistical operator formalism. Since the localization process is Poissonian, in a time interval
d
t
{\displaystyle dt}
there is a probability
λ
d
t
{\displaystyle \lambda dt}
that a collapse occurs, i.e. that the pure state
ρ
=
|
ψ
⟩
⟨
ψ
|
{\displaystyle \rho =|\psi \rangle \langle \psi |}
is transformed into the statistical mixture
T
^
i
[
ρ
]
≡
∫
d
x
L
^
x
i
|
ψ
⟩
⟨
ψ
|
L
^
x
i
{\displaystyle {\hat {T}}_{i}[\rho ]\equiv \int dx\,{\hat {L}}_{x}^{i}|\psi \rangle \langle \psi |{\hat {L}}_{x}^{i}}
.
In the same time interval, there is a probability
1
−
λ
d
t
{\displaystyle 1-\lambda dt}
that the system keeps evolving according to the Schrödinger equation. Accordingly, the GRW master equation for
N
{\displaystyle N}
particles reads
d
d
t
ρ
(
t
)
=
−
i
ℏ
[
H
^
,
ρ
(
t
)
]
−
∑
i
=
1
N
λ
i
(
ρ
(
t
)
−
T
^
i
[
ρ
]
)
{\displaystyle {\frac {d}{dt}}\rho (t)=-{\frac {i}{\hbar }}[{\hat {H}},\rho (t)]-\sum _{i=1}^{N}\lambda _{i}\left(\rho (t)-{\hat {T}}_{i}[\rho ]\right)}
,
where
H
^
{\displaystyle {\hat {H}}}
is the Hamiltonian of the system, and the square brackets denote a commutator.
Two new parameters are introduced by the GRW theory, namely the collapse rate
λ
{\displaystyle \lambda }
and the localization distance
r
C
{\displaystyle r_{C}}
. These are phenomenological parameters, whose values are not fixed by any principle and should be understood as new constants of Nature. Comparison of the model's predictions with experimental data permits bounding of the values of the parameters (see CSL model). The collapse rate should be such that microscopic object are almost never localized, thus effectively recovering standard quantum mechanics. The value originally proposed was
λ
=
10
−
16
s
−
1
{\displaystyle \lambda =10^{-16}\mathrm {s} ^{-1}}
, while more recently Stephen L. Adler proposed that the value
λ
=
10
−
8
s
−
1
{\displaystyle \lambda =10^{-8}\mathrm {s} ^{-1}}
(with an uncertainty of two orders of magnitude) is more adequate. There is a general consensus on the value
r
C
=
10
−
7
m
{\displaystyle r_{C}=10^{-7}\mathrm {m} }
for the localization distance. This is a mesoscopic distance, such that microscopic superpositions are left unaltered, while macroscopic ones are collapsed.
== Examples ==
When the wave function is hit by a sudden jump, the action of the localization operator essentially results in the multiplication of the wave function by the collapse Gaussian.
Let us consider a Gaussian wave function with spread
σ
{\displaystyle \sigma }
, centered at
x
=
a
{\displaystyle x=a}
, and let us assume that this undergoes a localization process at the position
x
=
a
{\displaystyle x=a}
. One thus has (in one dimension)
ψ
(
x
)
=
1
(
π
σ
)
1
4
e
−
(
x
−
a
)
2
2
σ
2
⟶
ψ
a
(
x
)
=
L
^
x
=
a
ψ
(
x
)
=
N
e
−
(
x
−
a
)
2
2
r
C
2
e
−
(
x
−
a
)
2
2
σ
2
{\displaystyle \psi (x)={\frac {1}{(\pi \sigma )^{\frac {1}{4}}}}\,e^{-{\frac {(x-a)^{2}}{2\sigma ^{2}}}}\quad \longrightarrow \quad \psi _{a}(x)={\hat {L}}_{x=a}\,\psi (x)={\cal {N}}e^{-{\frac {(x-a)^{2}}{2r_{C}^{2}}}}\,e^{-{\frac {(x-a)^{2}}{2\sigma ^{2}}}}}
,
where
N
{\displaystyle {\cal {N}}}
is a normalization factor. Let us further assume that the initial state is delocalised, i.e. that
σ
≫
r
C
{\displaystyle \sigma \gg r_{C}}
. In this case one has
ψ
a
(
x
)
≃
N
′
e
−
(
x
−
a
)
2
2
r
C
2
{\displaystyle \psi _{a}(x)\simeq {\cal {N}}'e^{-{\frac {(x-a)^{2}}{2r_{C}^{2}}}}}
,
where
N
′
{\displaystyle {\cal {N}}'}
is another normalization factor. One thus finds that after the sudden jump has occurred, the initially delocalised wave function has become localized.
Another interesting case is when the initial state is the superposition of two Gaussian states, centered at
x
=
−
a
{\displaystyle x=-a}
and
x
=
a
{\displaystyle x=a}
respectively:
ψ
(
x
)
=
1
2
(
π
σ
)
1
4
[
e
−
(
x
+
a
)
2
2
σ
2
+
e
−
(
x
−
a
)
2
2
σ
2
]
{\displaystyle \psi (x)={\frac {1}{2(\pi \sigma )^{\frac {1}{4}}}}\,\left[e^{-{\frac {(x+a)^{2}}{2\sigma ^{2}}}}+e^{-{\frac {(x-a)^{2}}{2\sigma ^{2}}}}\right]}
. If the localization occurs e.g. around
x
=
a
{\displaystyle x=a}
one has
ψ
a
(
x
)
=
N
e
−
(
x
−
a
)
2
2
r
C
2
[
e
−
(
x
+
a
)
2
2
σ
2
+
e
−
(
x
−
a
)
2
2
σ
2
]
=
N
[
e
−
σ
2
+
r
C
2
2
σ
2
r
C
2
(
x
+
σ
2
−
r
C
2
σ
2
+
r
C
2
a
)
2
−
2
a
2
σ
2
+
r
C
2
+
e
−
σ
2
+
r
C
2
2
σ
2
r
C
2
(
x
−
a
)
2
]
{\displaystyle \psi _{a}(x)={\cal {N}}e^{-{\frac {(x-a)^{2}}{2r_{C}^{2}}}}\left[e^{-{\frac {(x+a)^{2}}{2\sigma ^{2}}}}+e^{-{\frac {(x-a)^{2}}{2\sigma ^{2}}}}\right]={\cal {N}}\left[e^{-{\frac {\sigma ^{2}+r_{C}^{2}}{2\sigma ^{2}r_{C}^{2}}}\,\left(x+{\frac {\sigma ^{2}-r_{C}^{2}}{\sigma ^{2}+r_{C}^{2}}}a\right)^{2}-{\frac {2a^{2}}{\sigma ^{2}+r_{C}^{2}}}}+e^{-{\frac {\sigma ^{2}+r_{C}^{2}}{2\sigma ^{2}r_{C}^{2}}}\,(x-a)^{2}}\right]}
.
If one assumes that each Gaussian is localized (
σ
≪
r
C
{\displaystyle \sigma \ll r_{C}}
) and that the overall superposition is delocalised (
2
a
≫
r
C
{\displaystyle 2a\gg r_{C}}
), one finds
ψ
a
(
x
)
≃
N
′
[
e
−
(
x
+
a
)
2
2
σ
2
−
2
a
2
r
C
2
+
e
−
(
x
−
a
)
2
2
σ
2
]
{\displaystyle \psi _{a}(x)\simeq {\cal {N}}'\left[e^{-{\frac {\left(x+a\right)^{2}}{2\sigma ^{2}}}-{\frac {2a^{2}}{r_{C}^{2}}}}+e^{-{\frac {(x-a)^{2}}{2\sigma ^{2}}}}\right]}
.
We thus see that the Gaussian that is hit by the localization is left unchanged, while the other is exponentially suppressed.
== Amplification mechanism ==
This is one of the most important features of the GRW theory, because it allows us to recover classical mechanics for macroscopic objects. Let us consider a rigid body of
N
{\displaystyle N}
particles whose statistical operator evolves according to the master equation described above. We introduce the center of mass (
Q
^
{\displaystyle {\hat {Q}}}
) and relative (
r
^
i
{\displaystyle {\hat {r}}_{i}}
) position operators, which allow us to rewrite each particle's position operator as follows:
q
^
i
=
Q
^
+
r
^
i
{\displaystyle {\hat {q}}_{i}={\hat {Q}}+{\hat {r}}_{i}}
. One can show that, when the system Hamiltonian can be split into a center of mass Hamiltonian
H
C
M
{\displaystyle H_{\mathrm {CM} }}
and a relative Hamiltonian
H
r
{\displaystyle H_{r}}
, the center of mass statistical operator
ρ
C
M
{\displaystyle \rho _{\mathrm {CM} }}
evolves according to the following master equation:
d
d
t
ρ
C
M
(
t
)
=
−
i
ℏ
[
H
^
C
M
,
ρ
C
M
(
t
)
]
−
∑
i
=
1
N
λ
i
(
ρ
C
M
(
t
)
−
T
^
C
M
[
ρ
C
M
(
t
)
]
)
{\displaystyle {\frac {d}{dt}}\rho _{\mathrm {CM} }(t)=-{\frac {i}{\hbar }}[{\hat {H}}_{\mathrm {CM} },\rho _{\mathrm {CM} }(t)]-\sum _{i=1}^{N}\lambda _{i}\left(\rho _{\mathrm {CM} }(t)-{\hat {T}}_{\mathrm {CM} }[\rho _{\mathrm {CM} }(t)]\right)}
,
where
T
^
C
M
[
ρ
C
M
(
t
)
]
=
(
1
π
r
C
2
)
3
2
∫
∞
∞
d
3
x
e
−
(
Q
^
−
x
)
2
2
r
C
2
ρ
C
M
(
t
)
e
−
(
Q
^
−
x
)
2
2
r
C
2
{\displaystyle {\hat {T}}_{\mathrm {CM} }[\rho _{\mathrm {CM} }(t)]=\left({\frac {1}{\pi r_{C}^{2}}}\right)^{\frac {3}{2}}\int _{\infty }^{\infty }d^{3}x\,e^{-{\frac {({\hat {Q}}-x)^{2}}{2r_{C}^{2}}}}\,\rho _{\mathrm {CM} }(t)\,e^{-{\frac {({\hat {Q}}-x)^{2}}{2r_{C}^{2}}}}}
.
One thus sees that the center of mass collapses with a rate
Λ
{\displaystyle \Lambda }
that is the sum of the rates of its constituents: this is the amplification mechanism. If for simplicity one assumes that all particles collapse with the same rate
λ
{\displaystyle \lambda }
, one simply gets
Λ
=
N
λ
{\displaystyle \Lambda =N\,\lambda }
.
An object that consists of in the order of the Avogadro number of nucleons (
N
≃
10
23
{\displaystyle N\simeq 10^{23}}
) collapses almost instantly: GRW's and Adler's values of
λ
{\displaystyle \lambda }
give respectively
Λ
=
10
7
s
−
1
{\displaystyle \Lambda =10^{7}\,\mathrm {s} ^{-1}}
and
Λ
=
10
15
s
−
1
{\displaystyle \Lambda =10^{15}\,\mathrm {s} ^{-1}}
. Fast reduction of macroscopic object superpositions is thus guaranteed, and the GRW theory effectively recovers classical mechanics for macroscopic objects.
== Other features ==
The GRW theory makes different predictions than standard quantum mechanics, and as such can be tested against it (see CSL model).
The collapse noise repeatedly kicks the particles, thus inducing a diffusion process (Brownian motion). This introduces a steady amount of energy in the system, thus leading to a violation of the energy conservation principle. For the GRW model, one can show that energy grows linearly in time with rate
λ
ℏ
2
/
4
m
r
C
2
{\displaystyle \lambda \hbar ^{2}/4mr_{C}^{2}}
, which for a macroscopic object amounts to
≃
10
−
14
e
r
g
s
−
1
{\displaystyle \simeq 10^{-14}\mathrm {erg\,\,s} ^{-1}}
. Although such an energy increase is negligible, this feature of the model is not appealing. For this reason, a dissipative extension of the GRW theory has been investigated.
The GRW theory does not allow for identical particles. An extension of the theory with identical particles has been proposed by Tumulka.
GRW is a non relativistic theory, its relativistic extension for non-interacting particles has been investigated by Tumulka, while interacting models are still under investigation.
The master equation of the GRW theory describes a decoherence process according to which the off-diagonal elements of the statistical operator are suppressed exponentially. This is a feature that the GRW theory shares with other collapse theories: those involving white noises are associated to Lindblad master equations, while the coloured QMUPL model follows a non-Markovian Gaussian master equation.
== See also ==
Quantum decoherence
Penrose interpretation
Interpretations of quantum mechanics
== References == | Wikipedia/Ghirardi–Rimini–Weber_theory |
The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system.: 1–2 Its discovery was a significant landmark in the development of quantum mechanics. It is named after Erwin Schrödinger, an Austrian physicist, who postulated the equation in 1925 and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933.
Conceptually, the Schrödinger equation is the quantum counterpart of Newton's second law in classical mechanics. Given a set of known initial conditions, Newton's second law makes a mathematical prediction as to what path a given physical system will take over time. The Schrödinger equation gives the evolution over time of the wave function, the quantum-mechanical characterization of an isolated physical system. The equation was postulated by Schrödinger based on a postulate of Louis de Broglie that all matter has an associated matter wave. The equation predicted bound states of the atom in agreement with experimental observations.: II:268
The Schrödinger equation is not the only way to study quantum mechanical systems and make predictions. Other formulations of quantum mechanics include matrix mechanics, introduced by Werner Heisenberg, and the path integral formulation, developed chiefly by Richard Feynman. When these approaches are compared, the use of the Schrödinger equation is sometimes called "wave mechanics".
The equation given by Schrödinger is nonrelativistic because it contains a first derivative in time and a second derivative in space, and therefore space and time are not on equal footing. Paul Dirac incorporated special relativity and quantum mechanics into a single formulation that simplifies to the Schrödinger equation in the non-relativistic limit. This is the Dirac equation, which contains a single derivative in both space and time. Another partial differential equation, the Klein–Gordon equation, led to a problem with probability density even though it was a relativistic wave equation. The probability density could be negative, which is physically unviable. This was fixed by Dirac by taking the so-called square root of the Klein–Gordon operator and in turn introducing Dirac matrices. In a modern context, the Klein–Gordon equation describes spin-less particles, while the Dirac equation describes spin-1/2 particles.
== Definition ==
=== Preliminaries ===
Introductory courses on physics or chemistry typically introduce the Schrödinger equation in a way that can be appreciated knowing only the concepts and notations of basic calculus, particularly derivatives with respect to space and time. A special case of the Schrödinger equation that admits a statement in those terms is the position-space Schrödinger equation for a single nonrelativistic particle in one dimension:
i
ℏ
∂
∂
t
Ψ
(
x
,
t
)
=
[
−
ℏ
2
2
m
∂
2
∂
x
2
+
V
(
x
,
t
)
]
Ψ
(
x
,
t
)
.
{\displaystyle i\hbar {\frac {\partial }{\partial t}}\Psi (x,t)=\left[-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}+V(x,t)\right]\Psi (x,t).}
Here,
Ψ
(
x
,
t
)
{\displaystyle \Psi (x,t)}
is a wave function, a function that assigns a complex number to each point
x
{\displaystyle x}
at each time
t
{\displaystyle t}
. The parameter
m
{\displaystyle m}
is the mass of the particle, and
V
(
x
,
t
)
{\displaystyle V(x,t)}
is the potential that represents the environment in which the particle exists.: 74 The constant
i
{\displaystyle i}
is the imaginary unit, and
ℏ
{\displaystyle \hbar }
is the reduced Planck constant, which has units of action (energy multiplied by time).: 10
Broadening beyond this simple case, the mathematical formulation of quantum mechanics developed by Paul Dirac, David Hilbert, John von Neumann, and Hermann Weyl defines the state of a quantum mechanical system to be a vector
|
ψ
⟩
{\displaystyle |\psi \rangle }
belonging to a separable complex Hilbert space
H
{\displaystyle {\mathcal {H}}}
. This vector is postulated to be normalized under the Hilbert space's inner product, that is, in Dirac notation it obeys
⟨
ψ
|
ψ
⟩
=
1
{\displaystyle \langle \psi |\psi \rangle =1}
. The exact nature of this Hilbert space is dependent on the system – for example, for describing position and momentum the Hilbert space is the space of square-integrable functions
L
2
{\displaystyle L^{2}}
, while the Hilbert space for the spin of a single proton is the two-dimensional complex vector space
C
2
{\displaystyle \mathbb {C} ^{2}}
with the usual inner product.: 322
Physical quantities of interest – position, momentum, energy, spin – are represented by observables, which are self-adjoint operators acting on the Hilbert space. A wave function can be an eigenvector of an observable, in which case it is called an eigenstate, and the associated eigenvalue corresponds to the value of the observable in that eigenstate. More generally, a quantum state will be a linear combination of the eigenstates, known as a quantum superposition. When an observable is measured, the result will be one of its eigenvalues with probability given by the Born rule: in the simplest case the eigenvalue
λ
{\displaystyle \lambda }
is non-degenerate and the probability is given by
|
⟨
λ
|
ψ
⟩
|
2
{\displaystyle |\langle \lambda |\psi \rangle |^{2}}
, where
|
λ
⟩
{\displaystyle |\lambda \rangle }
is its associated eigenvector. More generally, the eigenvalue is degenerate and the probability is given by
⟨
ψ
|
P
λ
|
ψ
⟩
{\displaystyle \langle \psi |P_{\lambda }|\psi \rangle }
, where
P
λ
{\displaystyle P_{\lambda }}
is the projector onto its associated eigenspace.
A momentum eigenstate would be a perfectly monochromatic wave of infinite extent, which is not square-integrable. Likewise a position eigenstate would be a Dirac delta distribution, not square-integrable and technically not a function at all. Consequently, neither can belong to the particle's Hilbert space. Physicists sometimes regard these eigenstates, composed of elements outside the Hilbert space, as "generalized eigenvectors". These are used for calculational convenience and do not represent physical states.: 100–105 Thus, a position-space wave function
Ψ
(
x
,
t
)
{\displaystyle \Psi (x,t)}
as used above can be written as the inner product of a time-dependent state vector
|
Ψ
(
t
)
⟩
{\displaystyle |\Psi (t)\rangle }
with unphysical but convenient "position eigenstates"
|
x
⟩
{\displaystyle |x\rangle }
:
Ψ
(
x
,
t
)
=
⟨
x
|
Ψ
(
t
)
⟩
.
{\displaystyle \Psi (x,t)=\langle x|\Psi (t)\rangle .}
=== Time-dependent equation ===
The form of the Schrödinger equation depends on the physical situation. The most general form is the time-dependent Schrödinger equation, which gives a description of a system evolving with time:: 143
where
t
{\displaystyle t}
is time,
|
Ψ
(
t
)
⟩
{\displaystyle \vert \Psi (t)\rangle }
is the state vector of the quantum system (
Ψ
{\displaystyle \Psi }
being the Greek letter psi), and
H
^
{\displaystyle {\hat {H}}}
is an observable, the Hamiltonian operator.
The term "Schrödinger equation" can refer to both the general equation, or the specific nonrelativistic version. The general equation is indeed quite general, used throughout quantum mechanics, for everything from the Dirac equation to quantum field theory, by plugging in diverse expressions for the Hamiltonian. The specific nonrelativistic version is an approximation that yields accurate results in many situations, but only to a certain extent (see relativistic quantum mechanics and relativistic quantum field theory).
To apply the Schrödinger equation, write down the Hamiltonian for the system, accounting for the kinetic and potential energies of the particles constituting the system, then insert it into the Schrödinger equation. The resulting partial differential equation is solved for the wave function, which contains information about the system. In practice, the square of the absolute value of the wave function at each point is taken to define a probability density function.: 78 For example, given a wave function in position space
Ψ
(
x
,
t
)
{\displaystyle \Psi (x,t)}
as above, we have
Pr
(
x
,
t
)
=
|
Ψ
(
x
,
t
)
|
2
.
{\displaystyle \Pr(x,t)=|\Psi (x,t)|^{2}.}
=== Time-independent equation ===
The time-dependent Schrödinger equation described above predicts that wave functions can form standing waves, called stationary states. These states are particularly important as their individual study later simplifies the task of solving the time-dependent Schrödinger equation for any state. Stationary states can also be described by a simpler form of the Schrödinger equation, the time-independent Schrödinger equation.
where
E
{\displaystyle E}
is the energy of the system.: 134 This is only used when the Hamiltonian itself is not dependent on time explicitly. However, even in this case the total wave function is dependent on time as explained in the section on linearity below. In the language of linear algebra, this equation is an eigenvalue equation. Therefore, the wave function is an eigenfunction of the Hamiltonian operator with corresponding eigenvalue(s)
E
{\displaystyle E}
.
== Properties ==
=== Linearity ===
The Schrödinger equation is a linear differential equation, meaning that if two state vectors
|
ψ
1
⟩
{\displaystyle |\psi _{1}\rangle }
and
|
ψ
2
⟩
{\displaystyle |\psi _{2}\rangle }
are solutions, then so is any linear combination
|
ψ
⟩
=
a
|
ψ
1
⟩
+
b
|
ψ
2
⟩
{\displaystyle |\psi \rangle =a|\psi _{1}\rangle +b|\psi _{2}\rangle }
of the two state vectors where a and b are any complex numbers.: 25 Moreover, the sum can be extended for any number of state vectors. This property allows superpositions of quantum states to be solutions of the Schrödinger equation. Even more generally, it holds that a general solution to the Schrödinger equation can be found by taking a weighted sum over a basis of states. A choice often employed is the basis of energy eigenstates, which are solutions of the time-independent Schrödinger equation. In this basis, a time-dependent state vector
|
Ψ
(
t
)
⟩
{\displaystyle |\Psi (t)\rangle }
can be written as the linear combination
|
Ψ
(
t
)
⟩
=
∑
n
A
n
e
−
i
E
n
t
/
ℏ
|
ψ
E
n
⟩
,
{\displaystyle |\Psi (t)\rangle =\sum _{n}A_{n}e^{{-iE_{n}t}/\hbar }|\psi _{E_{n}}\rangle ,}
where
A
n
{\displaystyle A_{n}}
are complex numbers and the vectors
|
ψ
E
n
⟩
{\displaystyle |\psi _{E_{n}}\rangle }
are solutions of the time-independent equation
H
^
|
ψ
E
n
⟩
=
E
n
|
ψ
E
n
⟩
{\displaystyle {\hat {H}}|\psi _{E_{n}}\rangle =E_{n}|\psi _{E_{n}}\rangle }
.
=== Unitarity ===
Holding the Hamiltonian
H
^
{\displaystyle {\hat {H}}}
constant, the Schrödinger equation has the solution
|
Ψ
(
t
)
⟩
=
e
−
i
H
^
t
/
ℏ
|
Ψ
(
0
)
⟩
.
{\displaystyle |\Psi (t)\rangle =e^{-i{\hat {H}}t/\hbar }|\Psi (0)\rangle .}
The operator
U
^
(
t
)
=
e
−
i
H
^
t
/
ℏ
{\displaystyle {\hat {U}}(t)=e^{-i{\hat {H}}t/\hbar }}
is known as the time-evolution operator, and it is unitary: it preserves the inner product between vectors in the Hilbert space. Unitarity is a general feature of time evolution under the Schrödinger equation. If the initial state is
|
Ψ
(
0
)
⟩
{\displaystyle |\Psi (0)\rangle }
, then the state at a later time
t
{\displaystyle t}
will be given by
|
Ψ
(
t
)
⟩
=
U
^
(
t
)
|
Ψ
(
0
)
⟩
{\displaystyle |\Psi (t)\rangle ={\hat {U}}(t)|\Psi (0)\rangle }
for some unitary operator
U
^
(
t
)
{\displaystyle {\hat {U}}(t)}
. Conversely, suppose that
U
^
(
t
)
{\displaystyle {\hat {U}}(t)}
is a continuous family of unitary operators parameterized by
t
{\displaystyle t}
. Without loss of generality, the parameterization can be chosen so that
U
^
(
0
)
{\displaystyle {\hat {U}}(0)}
is the identity operator and that
U
^
(
t
/
N
)
N
=
U
^
(
t
)
{\displaystyle {\hat {U}}(t/N)^{N}={\hat {U}}(t)}
for any
N
>
0
{\displaystyle N>0}
. Then
U
^
(
t
)
{\displaystyle {\hat {U}}(t)}
depends upon the parameter
t
{\displaystyle t}
in such a way that
U
^
(
t
)
=
e
−
i
G
^
t
{\displaystyle {\hat {U}}(t)=e^{-i{\hat {G}}t}}
for some self-adjoint operator
G
^
{\displaystyle {\hat {G}}}
, called the generator of the family
U
^
(
t
)
{\displaystyle {\hat {U}}(t)}
. A Hamiltonian is just such a generator (up to the factor of the Planck constant that would be set to 1 in natural units).
To see that the generator is Hermitian, note that with
U
^
(
δ
t
)
≈
U
^
(
0
)
−
i
G
^
δ
t
{\displaystyle {\hat {U}}(\delta t)\approx {\hat {U}}(0)-i{\hat {G}}\delta t}
, we have
U
^
(
δ
t
)
†
U
^
(
δ
t
)
≈
(
U
^
(
0
)
†
+
i
G
^
†
δ
t
)
(
U
^
(
0
)
−
i
G
^
δ
t
)
=
I
+
i
δ
t
(
G
^
†
−
G
^
)
+
O
(
δ
t
2
)
,
{\displaystyle {\hat {U}}(\delta t)^{\dagger }{\hat {U}}(\delta t)\approx ({\hat {U}}(0)^{\dagger }+i{\hat {G}}^{\dagger }\delta t)({\hat {U}}(0)-i{\hat {G}}\delta t)=I+i\delta t({\hat {G}}^{\dagger }-{\hat {G}})+O(\delta t^{2}),}
so
U
^
(
t
)
{\displaystyle {\hat {U}}(t)}
is unitary only if, to first order, its derivative is Hermitian.
=== Changes of basis ===
The Schrödinger equation is often presented using quantities varying as functions of position, but as a vector-operator equation it has a valid representation in any arbitrary complete basis of kets in Hilbert space. As mentioned above, "bases" that lie outside the physical Hilbert space are also employed for calculational purposes. This is illustrated by the position-space and momentum-space Schrödinger equations for a nonrelativistic, spinless particle.: 182 The Hilbert space for such a particle is the space of complex square-integrable functions on three-dimensional Euclidean space, and its Hamiltonian is the sum of a kinetic-energy term that is quadratic in the momentum operator and a potential-energy term:
i
ℏ
d
d
t
|
Ψ
(
t
)
⟩
=
(
1
2
m
p
^
2
+
V
^
)
|
Ψ
(
t
)
⟩
.
{\displaystyle i\hbar {\frac {d}{dt}}|\Psi (t)\rangle =\left({\frac {1}{2m}}{\hat {p}}^{2}+{\hat {V}}\right)|\Psi (t)\rangle .}
Writing
r
{\displaystyle \mathbf {r} }
for a three-dimensional position vector and
p
{\displaystyle \mathbf {p} }
for a three-dimensional momentum vector, the position-space Schrödinger equation is
i
ℏ
∂
∂
t
Ψ
(
r
,
t
)
=
−
ℏ
2
2
m
∇
2
Ψ
(
r
,
t
)
+
V
(
r
)
Ψ
(
r
,
t
)
.
{\displaystyle i\hbar {\frac {\partial }{\partial t}}\Psi (\mathbf {r} ,t)=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\Psi (\mathbf {r} ,t)+V(\mathbf {r} )\Psi (\mathbf {r} ,t).}
The momentum-space counterpart involves the Fourier transforms of the wave function and the potential:
i
ℏ
∂
∂
t
Ψ
~
(
p
,
t
)
=
p
2
2
m
Ψ
~
(
p
,
t
)
+
(
2
π
ℏ
)
−
3
/
2
∫
d
3
p
′
V
~
(
p
−
p
′
)
Ψ
~
(
p
′
,
t
)
.
{\displaystyle i\hbar {\frac {\partial }{\partial t}}{\tilde {\Psi }}(\mathbf {p} ,t)={\frac {\mathbf {p} ^{2}}{2m}}{\tilde {\Psi }}(\mathbf {p} ,t)+(2\pi \hbar )^{-3/2}\int d^{3}\mathbf {p} '\,{\tilde {V}}(\mathbf {p} -\mathbf {p} '){\tilde {\Psi }}(\mathbf {p} ',t).}
The functions
Ψ
(
r
,
t
)
{\displaystyle \Psi (\mathbf {r} ,t)}
and
Ψ
~
(
p
,
t
)
{\displaystyle {\tilde {\Psi }}(\mathbf {p} ,t)}
are derived from
|
Ψ
(
t
)
⟩
{\displaystyle |\Psi (t)\rangle }
by
Ψ
(
r
,
t
)
=
⟨
r
|
Ψ
(
t
)
⟩
,
{\displaystyle \Psi (\mathbf {r} ,t)=\langle \mathbf {r} |\Psi (t)\rangle ,}
Ψ
~
(
p
,
t
)
=
⟨
p
|
Ψ
(
t
)
⟩
,
{\displaystyle {\tilde {\Psi }}(\mathbf {p} ,t)=\langle \mathbf {p} |\Psi (t)\rangle ,}
where
|
r
⟩
{\displaystyle |\mathbf {r} \rangle }
and
|
p
⟩
{\displaystyle |\mathbf {p} \rangle }
do not belong to the Hilbert space itself, but have well-defined inner products with all elements of that space.
When restricted from three dimensions to one, the position-space equation is just the first form of the Schrödinger equation given above. The relation between position and momentum in quantum mechanics can be appreciated in a single dimension. In canonical quantization, the classical variables
x
{\displaystyle x}
and
p
{\displaystyle p}
are promoted to self-adjoint operators
x
^
{\displaystyle {\hat {x}}}
and
p
^
{\displaystyle {\hat {p}}}
that satisfy the canonical commutation relation
[
x
^
,
p
^
]
=
i
ℏ
.
{\displaystyle [{\hat {x}},{\hat {p}}]=i\hbar .}
This implies that: 190
⟨
x
|
p
^
|
Ψ
⟩
=
−
i
ℏ
d
d
x
Ψ
(
x
)
,
{\displaystyle \langle x|{\hat {p}}|\Psi \rangle =-i\hbar {\frac {d}{dx}}\Psi (x),}
so the action of the momentum operator
p
^
{\displaystyle {\hat {p}}}
in the position-space representation is
−
i
ℏ
d
d
x
{\textstyle -i\hbar {\frac {d}{dx}}}
. Thus,
p
^
2
{\displaystyle {\hat {p}}^{2}}
becomes a second derivative, and in three dimensions, the second derivative becomes the Laplacian
∇
2
{\displaystyle \nabla ^{2}}
.
The canonical commutation relation also implies that the position and momentum operators are Fourier conjugates of each other. Consequently, functions originally defined in terms of their position dependence can be converted to functions of momentum using the Fourier transform.: 103–104 In solid-state physics, the Schrödinger equation is often written for functions of momentum, as Bloch's theorem ensures the periodic crystal lattice potential couples
Ψ
~
(
p
)
{\displaystyle {\tilde {\Psi }}(p)}
with
Ψ
~
(
p
+
ℏ
K
)
{\displaystyle {\tilde {\Psi }}(p+\hbar K)}
for only discrete reciprocal lattice vectors
K
{\displaystyle K}
. This makes it convenient to solve the momentum-space Schrödinger equation at each point in the Brillouin zone independently of the other points in the Brillouin zone.: 138
=== Probability current ===
The Schrödinger equation is consistent with local probability conservation.: 238 It also ensures that a normalized wavefunction remains normalized after time evolution. In matrix mechanics, this means that the time evolution operator is a unitary operator. In contrast to, for example, the Klein Gordon equation, although a redefined inner product of a wavefunction can be time independent, the total volume integral of modulus square of the wavefunction need not be time independent.
The continuity equation for probability in non relativistic quantum mechanics is stated as:
∂
∂
t
ρ
(
r
,
t
)
+
∇
⋅
j
=
0
,
{\displaystyle {\frac {\partial }{\partial t}}\rho \left(\mathbf {r} ,t\right)+\nabla \cdot \mathbf {j} =0,}
where
j
=
1
2
m
(
Ψ
∗
p
^
Ψ
−
Ψ
p
^
Ψ
∗
)
=
−
i
ℏ
2
m
(
ψ
∗
∇
ψ
−
ψ
∇
ψ
∗
)
=
ℏ
m
Im
(
ψ
∗
∇
ψ
)
{\displaystyle \mathbf {j} ={\frac {1}{2m}}\left(\Psi ^{*}{\hat {\mathbf {p} }}\Psi -\Psi {\hat {\mathbf {p} }}\Psi ^{*}\right)=-{\frac {i\hbar }{2m}}(\psi ^{*}\nabla \psi -\psi \nabla \psi ^{*})={\frac {\hbar }{m}}\operatorname {Im} (\psi ^{*}\nabla \psi )}
is the probability current or probability flux (flow per unit area).
If the wavefunction is represented as
ψ
(
x
,
t
)
=
ρ
(
x
,
t
)
exp
(
i
S
(
x
,
t
)
ℏ
)
,
{\textstyle \psi ({\bf {x}},t)={\sqrt {\rho ({\bf {x}},t)}}\exp \left({\frac {iS({\bf {x}},t)}{\hbar }}\right),}
where
S
(
x
,
t
)
{\displaystyle S(\mathbf {x} ,t)}
is a real function which represents the complex phase of the wavefunction, then the probability flux is calculated as:
j
=
ρ
∇
S
m
{\displaystyle \mathbf {j} ={\frac {\rho \nabla S}{m}}}
Hence, the spatial variation of the phase of a wavefunction is said to characterize the probability flux of the wavefunction. Although the
∇
S
m
{\textstyle {\frac {\nabla S}{m}}}
term appears to play the role of velocity, it does not represent velocity at a point since simultaneous measurement of position and velocity violates uncertainty principle.
=== Separation of variables ===
If the Hamiltonian is not an explicit function of time, Schrödinger's equation reads:
i
ℏ
∂
∂
t
Ψ
(
r
,
t
)
=
[
−
ℏ
2
2
m
∇
2
+
V
(
r
)
]
Ψ
(
r
,
t
)
.
{\displaystyle i\hbar {\frac {\partial }{\partial t}}\Psi (\mathbf {r} ,t)=\left[-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V(\mathbf {r} )\right]\Psi (\mathbf {r} ,t).}
The operator on the left side depends only on time; the one on the right side depends only on space.
Solving the equation by separation of variables means seeking a solution of the form of a product of spatial and temporal parts
Ψ
(
r
,
t
)
=
ψ
(
r
)
τ
(
t
)
,
{\displaystyle \Psi (\mathbf {r} ,t)=\psi (\mathbf {r} )\tau (t),}
where
ψ
(
r
)
{\displaystyle \psi (\mathbf {r} )}
is a function of all the spatial coordinate(s) of the particle(s) constituting the system only, and
τ
(
t
)
{\displaystyle \tau (t)}
is a function of time only. Substituting this expression for
Ψ
{\displaystyle \Psi }
into the time dependent left hand side shows that
τ
(
t
)
{\displaystyle \tau (t)}
is a phase factor:
Ψ
(
r
,
t
)
=
ψ
(
r
)
e
−
i
E
t
/
ℏ
.
{\displaystyle \Psi (\mathbf {r} ,t)=\psi (\mathbf {r} )e^{-i{Et/\hbar }}.}
A solution of this type is called stationary, since the only time dependence is a phase factor that cancels when the probability density is calculated via the Born rule.: 143ff
The spatial part of the full wave function solves the equation
∇
2
ψ
(
r
)
+
2
m
ℏ
2
[
E
−
V
(
r
)
]
ψ
(
r
)
=
0
,
{\displaystyle \nabla ^{2}\psi (\mathbf {r} )+{\frac {2m}{\hbar ^{2}}}\left[E-V(\mathbf {r} )\right]\psi (\mathbf {r} )=0,}
where the energy
E
{\displaystyle E}
appears in the phase factor.
This generalizes to any number of particles in any number of dimensions (in a time-independent potential): the standing wave solutions of the time-independent equation are the states with definite energy, instead of a probability distribution of different energies. In physics, these standing waves are called "stationary states" or "energy eigenstates"; in chemistry they are called "atomic orbitals" or "molecular orbitals". Superpositions of energy eigenstates change their properties according to the relative phases between the energy levels. The energy eigenstates form a basis: any wave function may be written as a sum over the discrete energy states or an integral over continuous energy states, or more generally as an integral over a measure. This is an example of the spectral theorem, and in a finite-dimensional state space it is just a statement of the completeness of the eigenvectors of a Hermitian matrix.
Separation of variables can also be a useful method for the time-independent Schrödinger equation. For example, depending on the symmetry of the problem, the Cartesian axes might be separated, as in
ψ
(
r
)
=
ψ
x
(
x
)
ψ
y
(
y
)
ψ
z
(
z
)
,
{\displaystyle \psi (\mathbf {r} )=\psi _{x}(x)\psi _{y}(y)\psi _{z}(z),}
or radial and angular coordinates might be separated:
ψ
(
r
)
=
ψ
r
(
r
)
ψ
θ
(
θ
)
ψ
ϕ
(
ϕ
)
.
{\displaystyle \psi (\mathbf {r} )=\psi _{r}(r)\psi _{\theta }(\theta )\psi _{\phi }(\phi ).}
== Examples ==
=== Particle in a box ===
The particle in a one-dimensional potential energy box is the most mathematically simple example where restraints lead to the quantization of energy levels. The box is defined as having zero potential energy inside a certain region and infinite potential energy outside.: 77–78 For the one-dimensional case in the
x
{\displaystyle x}
direction, the time-independent Schrödinger equation may be written
−
ℏ
2
2
m
d
2
ψ
d
x
2
=
E
ψ
.
{\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}\psi }{dx^{2}}}=E\psi .}
With the differential operator defined by
p
^
x
=
−
i
ℏ
d
d
x
{\displaystyle {\hat {p}}_{x}=-i\hbar {\frac {d}{dx}}}
the previous equation is evocative of the classic kinetic energy analogue,
1
2
m
p
^
x
2
=
E
,
{\displaystyle {\frac {1}{2m}}{\hat {p}}_{x}^{2}=E,}
with state
ψ
{\displaystyle \psi }
in this case having energy
E
{\displaystyle E}
coincident with the kinetic energy of the particle.
The general solutions of the Schrödinger equation for the particle in a box are
ψ
(
x
)
=
A
e
i
k
x
+
B
e
−
i
k
x
E
=
ℏ
2
k
2
2
m
{\displaystyle \psi (x)=Ae^{ikx}+Be^{-ikx}\qquad \qquad E={\frac {\hbar ^{2}k^{2}}{2m}}}
or, from Euler's formula,
ψ
(
x
)
=
C
sin
(
k
x
)
+
D
cos
(
k
x
)
.
{\displaystyle \psi (x)=C\sin(kx)+D\cos(kx).}
The infinite potential walls of the box determine the values of
C
,
D
,
{\displaystyle C,D,}
and
k
{\displaystyle k}
at
x
=
0
{\displaystyle x=0}
and
x
=
L
{\displaystyle x=L}
where
ψ
{\displaystyle \psi }
must be zero. Thus, at
x
=
0
{\displaystyle x=0}
,
ψ
(
0
)
=
0
=
C
sin
(
0
)
+
D
cos
(
0
)
=
D
{\displaystyle \psi (0)=0=C\sin(0)+D\cos(0)=D}
and
D
=
0
{\displaystyle D=0}
. At
x
=
L
{\displaystyle x=L}
,
ψ
(
L
)
=
0
=
C
sin
(
k
L
)
,
{\displaystyle \psi (L)=0=C\sin(kL),}
in which
C
{\displaystyle C}
cannot be zero as this would conflict with the postulate that
ψ
{\displaystyle \psi }
has norm 1. Therefore, since
sin
(
k
L
)
=
0
{\displaystyle \sin(kL)=0}
,
k
L
{\displaystyle kL}
must be an integer multiple of
π
{\displaystyle \pi }
,
k
=
n
π
L
n
=
1
,
2
,
3
,
…
.
{\displaystyle k={\frac {n\pi }{L}}\qquad \qquad n=1,2,3,\ldots .}
This constraint on
k
{\displaystyle k}
implies a constraint on the energy levels, yielding
E
n
=
ℏ
2
π
2
n
2
2
m
L
2
=
n
2
h
2
8
m
L
2
.
{\displaystyle E_{n}={\frac {\hbar ^{2}\pi ^{2}n^{2}}{2mL^{2}}}={\frac {n^{2}h^{2}}{8mL^{2}}}.}
A finite potential well is the generalization of the infinite potential well problem to potential wells having finite depth. The finite potential well problem is mathematically more complicated than the infinite particle-in-a-box problem as the wave function is not pinned to zero at the walls of the well. Instead, the wave function must satisfy more complicated mathematical boundary conditions as it is nonzero in regions outside the well. Another related problem is that of the rectangular potential barrier, which furnishes a model for the quantum tunneling effect that plays an important role in the performance of modern technologies such as flash memory and scanning tunneling microscopy.
=== Harmonic oscillator ===
The Schrödinger equation for this situation is
E
ψ
=
−
ℏ
2
2
m
d
2
d
x
2
ψ
+
1
2
m
ω
2
x
2
ψ
,
{\displaystyle E\psi =-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}\psi +{\frac {1}{2}}m\omega ^{2}x^{2}\psi ,}
where
x
{\displaystyle x}
is the displacement and
ω
{\displaystyle \omega }
the angular frequency. Furthermore, it can be used to describe approximately a wide variety of other systems, including vibrating atoms, molecules, and atoms or ions in lattices, and approximating other potentials near equilibrium points. It is also the basis of perturbation methods in quantum mechanics.
The solutions in position space are
ψ
n
(
x
)
=
1
2
n
n
!
(
m
ω
π
ℏ
)
1
/
4
e
−
m
ω
x
2
2
ℏ
H
n
(
m
ω
ℏ
x
)
,
{\displaystyle \psi _{n}(x)={\sqrt {\frac {1}{2^{n}\,n!}}}\ \left({\frac {m\omega }{\pi \hbar }}\right)^{1/4}\ e^{-{\frac {m\omega x^{2}}{2\hbar }}}\ {\mathcal {H}}_{n}\left({\sqrt {\frac {m\omega }{\hbar }}}x\right),}
where
n
∈
{
0
,
1
,
2
,
…
}
{\displaystyle n\in \{0,1,2,\ldots \}}
, and the functions
H
n
{\displaystyle {\mathcal {H}}_{n}}
are the Hermite polynomials of order
n
{\displaystyle n}
. The solution set may be generated by
ψ
n
(
x
)
=
1
n
!
(
m
ω
2
ℏ
)
n
(
x
−
ℏ
m
ω
d
d
x
)
n
(
m
ω
π
ℏ
)
1
4
e
−
m
ω
x
2
2
ℏ
.
{\displaystyle \psi _{n}(x)={\frac {1}{\sqrt {n!}}}\left({\sqrt {\frac {m\omega }{2\hbar }}}\right)^{n}\left(x-{\frac {\hbar }{m\omega }}{\frac {d}{dx}}\right)^{n}\left({\frac {m\omega }{\pi \hbar }}\right)^{\frac {1}{4}}e^{\frac {-m\omega x^{2}}{2\hbar }}.}
The eigenvalues are
E
n
=
(
n
+
1
2
)
ℏ
ω
.
{\displaystyle E_{n}=\left(n+{\frac {1}{2}}\right)\hbar \omega .}
The case
n
=
0
{\displaystyle n=0}
is called the ground state, its energy is called the zero-point energy, and the wave function is a Gaussian.
The harmonic oscillator, like the particle in a box, illustrates the generic feature of the Schrödinger equation that the energies of bound eigenstates are discretized.: 352
=== Hydrogen atom ===
The Schrödinger equation for the electron in a hydrogen atom (or a hydrogen-like atom) is
E
ψ
=
−
ℏ
2
2
μ
∇
2
ψ
−
q
2
4
π
ε
0
r
ψ
{\displaystyle E\psi =-{\frac {\hbar ^{2}}{2\mu }}\nabla ^{2}\psi -{\frac {q^{2}}{4\pi \varepsilon _{0}r}}\psi }
where
q
{\displaystyle q}
is the electron charge,
r
{\displaystyle \mathbf {r} }
is the position of the electron relative to the nucleus,
r
=
|
r
|
{\displaystyle r=|\mathbf {r} |}
is the magnitude of the relative position, the potential term is due to the Coulomb interaction, wherein
ε
0
{\displaystyle \varepsilon _{0}}
is the permittivity of free space and
μ
=
m
q
m
p
m
q
+
m
p
{\displaystyle \mu ={\frac {m_{q}m_{p}}{m_{q}+m_{p}}}}
is the 2-body reduced mass of the hydrogen nucleus (just a proton) of mass
m
p
{\displaystyle m_{p}}
and the electron of mass
m
q
{\displaystyle m_{q}}
. The negative sign arises in the potential term since the proton and electron are oppositely charged. The reduced mass in place of the electron mass is used since the electron and proton together orbit each other about a common center of mass, and constitute a two-body problem to solve. The motion of the electron is of principal interest here, so the equivalent one-body problem is the motion of the electron using the reduced mass.
The Schrödinger equation for a hydrogen atom can be solved by separation of variables. In this case, spherical polar coordinates are the most convenient. Thus,
ψ
(
r
,
θ
,
φ
)
=
R
(
r
)
Y
ℓ
m
(
θ
,
φ
)
=
R
(
r
)
Θ
(
θ
)
Φ
(
φ
)
,
{\displaystyle \psi (r,\theta ,\varphi )=R(r)Y_{\ell }^{m}(\theta ,\varphi )=R(r)\Theta (\theta )\Phi (\varphi ),}
where R are radial functions and
Y
l
m
(
θ
,
φ
)
{\displaystyle Y_{l}^{m}(\theta ,\varphi )}
are spherical harmonics of degree
ℓ
{\displaystyle \ell }
and order
m
{\displaystyle m}
. This is the only atom for which the Schrödinger equation has been solved for exactly. Multi-electron atoms require approximate methods. The family of solutions are:
ψ
n
ℓ
m
(
r
,
θ
,
φ
)
=
(
2
n
a
0
)
3
(
n
−
ℓ
−
1
)
!
2
n
[
(
n
+
ℓ
)
!
]
e
−
r
/
n
a
0
(
2
r
n
a
0
)
ℓ
L
n
−
ℓ
−
1
2
ℓ
+
1
(
2
r
n
a
0
)
⋅
Y
ℓ
m
(
θ
,
φ
)
{\displaystyle \psi _{n\ell m}(r,\theta ,\varphi )={\sqrt {\left({\frac {2}{na_{0}}}\right)^{3}{\frac {(n-\ell -1)!}{2n[(n+\ell )!]}}}}e^{-r/na_{0}}\left({\frac {2r}{na_{0}}}\right)^{\ell }L_{n-\ell -1}^{2\ell +1}\left({\frac {2r}{na_{0}}}\right)\cdot Y_{\ell }^{m}(\theta ,\varphi )}
where
a
0
=
4
π
ε
0
ℏ
2
m
q
q
2
{\displaystyle a_{0}={\frac {4\pi \varepsilon _{0}\hbar ^{2}}{m_{q}q^{2}}}}
is the Bohr radius,
L
n
−
ℓ
−
1
2
ℓ
+
1
(
⋯
)
{\displaystyle L_{n-\ell -1}^{2\ell +1}(\cdots )}
are the generalized Laguerre polynomials of degree
n
−
ℓ
−
1
{\displaystyle n-\ell -1}
,
n
,
ℓ
,
m
{\displaystyle n,\ell ,m}
are the principal, azimuthal, and magnetic quantum numbers respectively, which take the values
n
=
1
,
2
,
3
,
…
,
{\displaystyle n=1,2,3,\dots ,}
ℓ
=
0
,
1
,
2
,
…
,
n
−
1
,
{\displaystyle \ell =0,1,2,\dots ,n-1,}
m
=
−
ℓ
,
…
,
ℓ
.
{\displaystyle m=-\ell ,\dots ,\ell .}
=== Approximate solutions ===
It is typically not possible to solve the Schrödinger equation exactly for situations of physical interest. Accordingly, approximate solutions are obtained using techniques like variational methods and WKB approximation. It is also common to treat a problem of interest as a small modification to a problem that can be solved exactly, a method known as perturbation theory.
== Semiclassical limit ==
One simple way to compare classical to quantum mechanics is to consider the time-evolution of the expected position and expected momentum, which can then be compared to the time-evolution of the ordinary position and momentum in classical mechanics.: 302 The quantum expectation values satisfy the Ehrenfest theorem. For a one-dimensional quantum particle moving in a potential
V
{\displaystyle V}
, the Ehrenfest theorem says
m
d
d
t
⟨
x
⟩
=
⟨
p
⟩
;
d
d
t
⟨
p
⟩
=
−
⟨
V
′
(
X
)
⟩
.
{\displaystyle m{\frac {d}{dt}}\langle x\rangle =\langle p\rangle ;\quad {\frac {d}{dt}}\langle p\rangle =-\left\langle V'(X)\right\rangle .}
Although the first of these equations is consistent with the classical behavior, the second is not: If the pair
(
⟨
X
⟩
,
⟨
P
⟩
)
{\displaystyle (\langle X\rangle ,\langle P\rangle )}
were to satisfy Newton's second law, the right-hand side of the second equation would have to be
−
V
′
(
⟨
X
⟩
)
{\displaystyle -V'\left(\left\langle X\right\rangle \right)}
which is typically not the same as
−
⟨
V
′
(
X
)
⟩
{\displaystyle -\left\langle V'(X)\right\rangle }
. For a general
V
′
{\displaystyle V'}
, therefore, quantum mechanics can lead to predictions where expectation values do not mimic the classical behavior. In the case of the quantum harmonic oscillator, however,
V
′
{\displaystyle V'}
is linear and this distinction disappears, so that in this very special case, the expected position and expected momentum do exactly follow the classical trajectories.
For general systems, the best we can hope for is that the expected position and momentum will approximately follow the classical trajectories. If the wave function is highly concentrated around a point
x
0
{\displaystyle x_{0}}
, then
V
′
(
⟨
X
⟩
)
{\displaystyle V'\left(\left\langle X\right\rangle \right)}
and
⟨
V
′
(
X
)
⟩
{\displaystyle \left\langle V'(X)\right\rangle }
will be almost the same, since both will be approximately equal to
V
′
(
x
0
)
{\displaystyle V'(x_{0})}
. In that case, the expected position and expected momentum will remain very close to the classical trajectories, at least for as long as the wave function remains highly localized in position.
The Schrödinger equation in its general form
i
ℏ
∂
∂
t
Ψ
(
r
,
t
)
=
H
^
Ψ
(
r
,
t
)
{\displaystyle i\hbar {\frac {\partial }{\partial t}}\Psi \left(\mathbf {r} ,t\right)={\hat {H}}\Psi \left(\mathbf {r} ,t\right)}
is closely related to the Hamilton–Jacobi equation (HJE)
−
∂
∂
t
S
(
q
i
,
t
)
=
H
(
q
i
,
∂
S
∂
q
i
,
t
)
{\displaystyle -{\frac {\partial }{\partial t}}S(q_{i},t)=H\left(q_{i},{\frac {\partial S}{\partial q_{i}}},t\right)}
where
S
{\displaystyle S}
is the classical action and
H
{\displaystyle H}
is the Hamiltonian function (not operator).: 308 Here the generalized coordinates
q
i
{\displaystyle q_{i}}
for
i
=
1
,
2
,
3
{\displaystyle i=1,2,3}
(used in the context of the HJE) can be set to the position in Cartesian coordinates as
r
=
(
q
1
,
q
2
,
q
3
)
=
(
x
,
y
,
z
)
{\displaystyle \mathbf {r} =(q_{1},q_{2},q_{3})=(x,y,z)}
.
Substituting
Ψ
=
ρ
(
r
,
t
)
e
i
S
(
r
,
t
)
/
ℏ
{\displaystyle \Psi ={\sqrt {\rho (\mathbf {r} ,t)}}e^{iS(\mathbf {r} ,t)/\hbar }}
where
ρ
{\displaystyle \rho }
is the probability density, into the Schrödinger equation and then taking the limit
ℏ
→
0
{\displaystyle \hbar \to 0}
in the resulting equation yield the Hamilton–Jacobi equation.
== Density matrices ==
Wave functions are not always the most convenient way to describe quantum systems and their behavior. When the preparation of a system is only imperfectly known, or when the system under investigation is a part of a larger whole, density matrices may be used instead.: 74 A density matrix is a positive semi-definite operator whose trace is equal to 1. (The term "density operator" is also used, particularly when the underlying Hilbert space is infinite-dimensional.) The set of all density matrices is convex, and the extreme points are the operators that project onto vectors in the Hilbert space. These are the density-matrix representations of wave functions; in Dirac notation, they are written
ρ
^
=
|
Ψ
⟩
⟨
Ψ
|
.
{\displaystyle {\hat {\rho }}=|\Psi \rangle \langle \Psi |.}
The density-matrix analogue of the Schrödinger equation for wave functions is
i
ℏ
∂
ρ
^
∂
t
=
[
H
^
,
ρ
^
]
,
{\displaystyle i\hbar {\frac {\partial {\hat {\rho }}}{\partial t}}=[{\hat {H}},{\hat {\rho }}],}
where the brackets denote a commutator. This is variously known as the von Neumann equation, the Liouville–von Neumann equation, or just the Schrödinger equation for density matrices.: 312 If the Hamiltonian is time-independent, this equation can be easily solved to yield
ρ
^
(
t
)
=
e
−
i
H
^
t
/
ℏ
ρ
^
(
0
)
e
i
H
^
t
/
ℏ
.
{\displaystyle {\hat {\rho }}(t)=e^{-i{\hat {H}}t/\hbar }{\hat {\rho }}(0)e^{i{\hat {H}}t/\hbar }.}
More generally, if the unitary operator
U
^
(
t
)
{\displaystyle {\hat {U}}(t)}
describes wave function evolution over some time interval, then the time evolution of a density matrix over that same interval is given by
ρ
^
(
t
)
=
U
^
(
t
)
ρ
^
(
0
)
U
^
(
t
)
†
.
{\displaystyle {\hat {\rho }}(t)={\hat {U}}(t){\hat {\rho }}(0){\hat {U}}(t)^{\dagger }.}
Unitary evolution of a density matrix conserves its von Neumann entropy.: 267
== Relativistic quantum physics and quantum field theory ==
The one-particle Schrödinger equation described above is valid essentially in the nonrelativistic domain. For one reason, it is essentially invariant under Galilean transformations, which form the symmetry group of Newtonian dynamics. Moreover, processes that change particle number are natural in relativity, and so an equation for one particle (or any fixed number thereof) can only be of limited use. A more general form of the Schrödinger equation that also applies in relativistic situations can be formulated within quantum field theory (QFT), a framework that allows the combination of quantum mechanics with special relativity. The region in which both simultaneously apply may be described by relativistic quantum mechanics. Such descriptions may use time evolution generated by a Hamiltonian operator, as in the Schrödinger functional method.
=== Klein–Gordon and Dirac equations ===
Attempts to combine quantum physics with special relativity began with building relativistic wave equations from the relativistic energy–momentum relation
E
2
=
(
p
c
)
2
+
(
m
0
c
2
)
2
,
{\displaystyle E^{2}=(pc)^{2}+\left(m_{0}c^{2}\right)^{2},}
instead of nonrelativistic energy equations. The Klein–Gordon equation and the Dirac equation are two such equations. The Klein–Gordon equation,
−
1
c
2
∂
2
∂
t
2
ψ
+
∇
2
ψ
=
m
2
c
2
ℏ
2
ψ
,
{\displaystyle -{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\psi +\nabla ^{2}\psi ={\frac {m^{2}c^{2}}{\hbar ^{2}}}\psi ,}
was the first such equation to be obtained, even before the nonrelativistic one-particle Schrödinger equation, and applies to massive spinless particles. Historically, Dirac obtained the Dirac equation by seeking a differential equation that would be first-order in both time and space, a desirable property for a relativistic theory. Taking the "square root" of the left-hand side of the Klein–Gordon equation in this way required factorizing it into a product of two operators, which Dirac wrote using 4 × 4 matrices
α
1
,
α
2
,
α
3
,
β
{\displaystyle \alpha _{1},\alpha _{2},\alpha _{3},\beta }
. Consequently, the wave function also became a four-component function, governed by the Dirac equation that, in free space, read
(
β
m
c
2
+
c
(
∑
n
=
1
3
α
n
p
n
)
)
ψ
=
i
ℏ
∂
ψ
∂
t
.
{\displaystyle \left(\beta mc^{2}+c\left(\sum _{n\mathop {=} 1}^{3}\alpha _{n}p_{n}\right)\right)\psi =i\hbar {\frac {\partial \psi }{\partial t}}.}
This has again the form of the Schrödinger equation, with the time derivative of the wave function being given by a Hamiltonian operator acting upon the wave function. Including influences upon the particle requires modifying the Hamiltonian operator. For example, the Dirac Hamiltonian for a particle of mass m and electric charge q in an electromagnetic field (described by the electromagnetic potentials φ and A) is:
H
^
Dirac
=
γ
0
[
c
γ
⋅
(
p
^
−
q
A
)
+
m
c
2
+
γ
0
q
φ
]
,
{\displaystyle {\hat {H}}_{\text{Dirac}}=\gamma ^{0}\left[c{\boldsymbol {\gamma }}\cdot \left({\hat {\mathbf {p} }}-q\mathbf {A} \right)+mc^{2}+\gamma ^{0}q\varphi \right],}
in which the γ = (γ1, γ2, γ3) and γ0 are the Dirac gamma matrices related to the spin of the particle. The Dirac equation is true for all spin-1⁄2 particles, and the solutions to the equation are 4-component spinor fields with two components corresponding to the particle and the other two for the antiparticle.
For the Klein–Gordon equation, the general form of the Schrödinger equation is inconvenient to use, and in practice the Hamiltonian is not expressed in an analogous way to the Dirac Hamiltonian. The equations for relativistic quantum fields, of which the Klein–Gordon and Dirac equations are two examples, can be obtained in other ways, such as starting from a Lagrangian density and using the Euler–Lagrange equations for fields, or using the representation theory of the Lorentz group in which certain representations can be used to fix the equation for a free particle of given spin (and mass).
In general, the Hamiltonian to be substituted in the general Schrödinger equation is not just a function of the position and momentum operators (and possibly time), but also of spin matrices. Also, the solutions to a relativistic wave equation, for a massive particle of spin s, are complex-valued 2(2s + 1)-component spinor fields.
=== Fock space ===
As originally formulated, the Dirac equation is an equation for a single quantum particle, just like the single-particle Schrödinger equation with wave function
Ψ
(
x
,
t
)
{\displaystyle \Psi (x,t)}
. This is of limited use in relativistic quantum mechanics, where particle number is not fixed. Heuristically, this complication can be motivated by noting that mass–energy equivalence implies material particles can be created from energy. A common way to address this in QFT is to introduce a Hilbert space where the basis states are labeled by particle number, a so-called Fock space. The Schrödinger equation can then be formulated for quantum states on this Hilbert space. However, because the Schrödinger equation picks out a preferred time axis, the Lorentz invariance of the theory is no longer manifest, and accordingly, the theory is often formulated in other ways.
== History ==
Following Max Planck's quantization of light (see black-body radiation), Albert Einstein interpreted Planck's quanta to be photons, particles of light, and proposed that the energy of a photon is proportional to its frequency, one of the first signs of wave–particle duality. Since energy and momentum are related in the same way as frequency and wave number in special relativity, it followed that the momentum
p
{\displaystyle p}
of a photon is inversely proportional to its wavelength
λ
{\displaystyle \lambda }
, or proportional to its wave number
k
{\displaystyle k}
:
p
=
h
λ
=
ℏ
k
,
{\displaystyle p={\frac {h}{\lambda }}=\hbar k,}
where
h
{\displaystyle h}
is the Planck constant and
ℏ
=
h
/
2
π
{\displaystyle \hbar ={h}/{2\pi }}
is the reduced Planck constant. Louis de Broglie hypothesized that this is true for all particles, even particles which have mass such as electrons. He showed that, assuming that the matter waves propagate along with their particle counterparts, electrons form standing waves, meaning that only certain discrete rotational frequencies about the nucleus of an atom are allowed.
These quantized orbits correspond to discrete energy levels, and de Broglie reproduced the Bohr model formula for the energy levels. The Bohr model was based on the assumed quantization of angular momentum
L
{\displaystyle L}
according to
L
=
n
h
2
π
=
n
ℏ
.
{\displaystyle L=n{\frac {h}{2\pi }}=n\hbar .}
According to de Broglie, the electron is described by a wave, and a whole number of wavelengths must fit along the circumference of the electron's orbit:
n
λ
=
2
π
r
.
{\displaystyle n\lambda =2\pi r.}
This approach essentially confined the electron wave in one dimension, along a circular orbit of radius
r
{\displaystyle r}
.
In 1921, prior to de Broglie, Arthur C. Lunn at the University of Chicago had used the same argument based on the completion of the relativistic energy–momentum 4-vector to derive what we now call the de Broglie relation. Unlike de Broglie, Lunn went on to formulate the differential equation now known as the Schrödinger equation and solve for its energy eigenvalues for the hydrogen atom; the paper was rejected by the Physical Review, according to Kamen.
Following up on de Broglie's ideas, physicist Peter Debye made an offhand comment that if particles behaved as waves, they should satisfy some sort of wave equation. Inspired by Debye's remark, Schrödinger decided to find a proper 3-dimensional wave equation for the electron. He was guided by William Rowan Hamilton's analogy between mechanics and optics, encoded in the observation that the zero-wavelength limit of optics resembles a mechanical system—the trajectories of light rays become sharp tracks that obey Fermat's principle, an analog of the principle of least action.
The equation he found is
i
ℏ
∂
∂
t
Ψ
(
r
,
t
)
=
−
ℏ
2
2
m
∇
2
Ψ
(
r
,
t
)
+
V
(
r
)
Ψ
(
r
,
t
)
.
{\displaystyle i\hbar {\frac {\partial }{\partial t}}\Psi (\mathbf {r} ,t)=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\Psi (\mathbf {r} ,t)+V(\mathbf {r} )\Psi (\mathbf {r} ,t).}
By that time Arnold Sommerfeld had refined the Bohr model with relativistic corrections. Schrödinger used the relativistic energy–momentum relation to find what is now known as the Klein–Gordon equation in a Coulomb potential (in natural units):
(
E
+
e
2
r
)
2
ψ
(
x
)
=
−
∇
2
ψ
(
x
)
+
m
2
ψ
(
x
)
.
{\displaystyle \left(E+{\frac {e^{2}}{r}}\right)^{2}\psi (x)=-\nabla ^{2}\psi (x)+m^{2}\psi (x).}
He found the standing waves of this relativistic equation, but the relativistic corrections disagreed with Sommerfeld's formula. Discouraged, he put away his calculations and secluded himself with a mistress in a mountain cabin in December 1925.
While at the cabin, Schrödinger decided that his earlier nonrelativistic calculations were novel enough to publish and decided to leave off the problem of relativistic corrections for the future. Despite the difficulties in solving the differential equation for hydrogen (he had sought help from his friend the mathematician Hermann Weyl: 3 ) Schrödinger showed that his nonrelativistic version of the wave equation produced the correct spectral energies of hydrogen in a paper published in 1926.: 1 Schrödinger computed the hydrogen spectral series by treating a hydrogen atom's electron as a wave
Ψ
(
x
,
t
)
{\displaystyle \Psi (\mathbf {x} ,t)}
, moving in a potential well
V
{\displaystyle V}
, created by the proton. This computation accurately reproduced the energy levels of the Bohr model.
The Schrödinger equation details the behavior of
Ψ
{\displaystyle \Psi }
but says nothing of its nature. Schrödinger tried to interpret the real part of
Ψ
∂
Ψ
∗
∂
t
{\displaystyle \Psi {\frac {\partial \Psi ^{*}}{\partial t}}}
as a charge density, and then revised this proposal, saying in his next paper that the modulus squared of
Ψ
{\displaystyle \Psi }
is a charge density. This approach was, however, unsuccessful. In 1926, just a few days after this paper was published, Max Born successfully interpreted
Ψ
{\displaystyle \Psi }
as the probability amplitude, whose modulus squared is equal to probability density.: 220 Later, Schrödinger himself explained this interpretation as follows:
The already ... mentioned psi-function.... is now the means for predicting probability of measurement results. In it is embodied the momentarily attained sum of theoretically based future expectation, somewhat as laid down in a catalog.
== Interpretation ==
The Schrödinger equation provides a way to calculate the wave function of a system and how it changes dynamically in time. However, the Schrödinger equation does not directly say what, exactly, the wave function is. The meaning of the Schrödinger equation and how the mathematical entities in it relate to physical reality depends upon the interpretation of quantum mechanics that one adopts.
In the views often grouped together as the Copenhagen interpretation, a system's wave function is a collection of statistical information about that system. The Schrödinger equation relates information about the system at one time to information about it at another. While the time-evolution process represented by the Schrödinger equation is continuous and deterministic, in that knowing the wave function at one instant is in principle sufficient to calculate it for all future times, wave functions can also change discontinuously and stochastically during a measurement. The wave function changes, according to this school of thought, because new information is available. The post-measurement wave function generally cannot be known prior to the measurement, but the probabilities for the different possibilities can be calculated using the Born rule. Other, more recent interpretations of quantum mechanics, such as relational quantum mechanics and QBism also give the Schrödinger equation a status of this sort.
Schrödinger himself suggested in 1952 that the different terms of a superposition evolving under the Schrödinger equation are "not alternatives but all really happen simultaneously". This has been interpreted as an early version of Everett's many-worlds interpretation. This interpretation, formulated independently in 1956, holds that all the possibilities described by quantum theory simultaneously occur in a multiverse composed of mostly independent parallel universes. This interpretation removes the axiom of wave function collapse, leaving only continuous evolution under the Schrödinger equation, and so all possible states of the measured system and the measuring apparatus, together with the observer, are present in a real physical quantum superposition. While the multiverse is deterministic, we perceive non-deterministic behavior governed by probabilities, because we do not observe the multiverse as a whole, but only one parallel universe at a time. Exactly how this is supposed to work has been the subject of much debate. Why should we assign probabilities at all to outcomes that are certain to occur in some worlds, and why should the probabilities be given by the Born rule? Several ways to answer these questions in the many-worlds framework have been proposed, but there is no consensus on whether they are successful.
Bohmian mechanics reformulates quantum mechanics to make it deterministic, at the price of adding a force due to a "quantum potential". It attributes to each physical system not only a wave function but in addition a real position that evolves deterministically under a nonlocal guiding equation. The evolution of a physical system is given at all times by the Schrödinger equation together with the guiding equation.
== See also ==
== Notes ==
== References ==
== External links ==
"Schrödinger equation". Encyclopedia of Mathematics. EMS Press. 2001 [1994].
Quantum Cook Book (PDF) and PHYS 201: Fundamentals of Physics II by Ramamurti Shankar, Yale OpenCourseware
The Modern Revolution in Physics – an online textbook.
Quantum Physics I at MIT OpenCourseWare | Wikipedia/Schrödinger_wave_equation |
There is a diversity of views that propose interpretations of quantum mechanics. They vary in how many physicists accept or reject them. An interpretation of quantum mechanics is a conceptual scheme that proposes to relate the mathematical formalism to the physical phenomena of interest. The present article is about those interpretations which, independently of their intrinsic value, remain today less known, or are simply less debated by the scientific community, for different reasons.
== History ==
The historical dichotomy between the "orthodox" Copenhagen interpretation and "unorthodox" minority views developed in the 1950s debate surrounding Bohmian mechanics.
During most of the 20th century, collapse theories were clearly the mainstream view, and the question of interpretation of quantum mechanics mostly revolved around how to interpret "collapse". Proponents of either "pilot-wave" (de Broglie-Bohm-like) or "many-worlds" (Everettian) interpretations tend to emphasize how their respective camps were intellectually marginalized throughout 1950s to 1980s. In this (historical) sense, all non-collapse theories are (historically) "minority" interpretations.
The term 'Copenhagen interpretation' suggests some definite set of rules for interpreting the mathematical formalism of quantum mechanics. However, no such text exists, apart from some informal popular lectures by Bohr and Heisenberg, which contradict each other on several important issues.
It appears that the term "Copenhagen interpretation", with its more definite sense, was coined by Heisenberg in the 1950s, while criticizing "unorthodox" interpretations such as that of David Bohm.
Before the book was released for sale, Heisenberg privately expressed regret for having used the term, due to its suggestion of the existence of other interpretations, that he considered to be "nonsense".
Since the 1990s, there has been a resurgence of interest in non-collapse theories. Interpretations of quantum mechanics now mostly fall into the categories of collapse theories (including the Copenhagen interpretation), hidden variables ("Bohm-like"), many-worlds ("Everettian") and quantum information approaches. While collapse theories continue to be seen as the default or mainstream position, there is no longer any clear dichotomy between "orthodox" and "unorthodox" views. The Stanford Encyclopedia as of 2015 groups interpretations of quantum mechanics into five classes (all of which contain further divisions): "Bohmian mechanics" (pilot-wave theories), "collapse theories", "many-worlds interpretations", "modal interpretations"
and "relational interpretations".
Some of the historically relevant approaches to quantum mechanics have now themselves become "minority interpretations", or widely seen as obsolete. In this sense, there is a variety of reasons for why a specific approach may be considered marginal: because it is a very specialized sub-variant of a more widely known class of interpretations, because it is seen as obsolete (in spite of possible historical significance), because it is a very recent suggestion that has not received wide attention, or because it is rejected as flawed.
As a rough guide to a picture of what are the relevant "minority" views, consider the "snapshot" of opinions collected in a poll by Schlosshauer et al. at the 2011 "Quantum Physics and the Nature of Reality" conference of July 2011.
The authors reference a similarly informal poll carried out by Max Tegmark at the "Fundamental Problems
in Quantum Theory" conference in August 1997.
In both polls, the Copenhagen interpretation received the largest number of votes. In Tegmark's poll, many-worlds interpretations came in second place, while in the 2011 poll, many-worlds was at third place (18%), behind quantum information approaches in second place (24%).
Other options given as "interpretation of quantum mechanics" in the 2011 poll were:
objective collapse theories (9% support), Quantum Bayesianism (6% support) and Relational quantum mechanics (6% support),
besides consistent histories, de Broglie–Bohm theory, modal interpretation, ensemble interpretation and transactional interpretation which received no votes.
== List of interpretations ==
=== Many-worlds ===
"Everettian" (many-worlds) interpretations as a whole were long a "minority" field in general, but they have grown in popularity. Multiple variants and offshoots of Everett's original proposal exist, which have sometimes developed the basic ideas in contradictory ways. Interpretations of an Everettian type include the following.
Many-minds interpretation
"Cosmological interpretations", such as that proposed by Anthony Aguirre and Max Tegmark, in which the wavefunction for a quantum system describes not an imaginary ensemble of possibilities for what the system might be doing, but rather the actual spatial collection of identical copies of the system that exist in the infinite space that is, hypothetically, generated by eternal inflation.
"Self-locating uncertainty" interpretation
Relative state interpretation
=== Quantum information ===
QBism and other variants of Quantum Bayesianism
Relational quantum mechanics treats the state of a quantum system as being observer-dependent, that is, the state is the relation between the observer and the system. While a relational conception of quantum states dates back at least to Grete Hermann in 1935, in modern usage "relational quantum mechanics" refers to an interpretation delineated by Carlo Rovelli in 1996. It uses some ideas from Wheeler about quantum information.
=== Hidden variables ===
"Bohm-like" (hidden variable) theories as a whole are a "minority view" as compared to Copenhagen-type or many-worlds (Everettian) interpretations.
Popper's propensity-based interpretation
Stochastic interpretation, the most well-known variant of which was due to Edward Nelson, further elaborated upon by a conjecture of Francesco Calogero
Time-symmetric interpretations
Transactional interpretation
Zitterbewegung interpretation
Sutherland Interpretation
=== Collapse theories ===
Consciousness causes collapse, mostly of historical interest
Objective-collapse theories: these are extensions of quantum mechanics rather than "interpretations" in the narrow sense.
Penrose interpretation
Ghirardi-Rimini-Weber theory
=== Other ===
The ensemble interpretation, or statistical interpretation can be viewed as a minimalist approach; The wave function in this interpretation is not a property of any individual system, it is by its nature a statistical description of a hypothetical "ensemble" of similar systems. This is the interpretation historically advocated by Albert Einstein.
Modal interpretation (van Fraassen 1972) Van Fraassen's proposal is "modal" because it leads to a modal logic of quantum propositions. Since the 1980s, a number of authors have developed other "realist" proposals which can in retrospect be classed with van Fraassen's "modal" proposal.
Superdeterminism (Bell 1977), the idea that the universe is completely deterministic, and thus Bell's theorem does not apply, as observers are not free to make independent choices in their measurements, rather everything is predetermined from the Big Bang.
Consistent histories (Dowker and Kent 1995), based on a consistency criterion that then allows probabilities to be assigned to various alternative histories of a system.
"Montevideo interpretation" (Gambini and Pullin 2009), suggesting that quantum gravity makes for fundamental limitations on the accuracy of clocks, which imply a type of decoherence.
"Pondicherry interpretation" (Mohrhoff 2000–2005), based on the idea of objective probability and "supervenience of the microscopic on the macroscopic".
"Psychophysical interpretation"(Pradhan 2012) based on Psychophysical parallelism takes the conjugate quantities to represent psychic counterparts to the corresponding physical aspects such as states and observables.
The interpretation from a bundle-theoretic view of objective idealism (Korth 2022), based on the idea that quantum 'weirdness' follows from objects being bundles of universals.
== Quantum mysticism ==
Quantum mysticism is a set of metaphysical beliefs and associated practices that seek to relate consciousness, intelligence, spirituality, or mystical worldviews to the ideas of quantum mechanics and its interpretations. Quantum mysticism is considered by most scientists to be pseudoscience or quackery.
== References == | Wikipedia/Minority_interpretations_of_quantum_mechanics |
The continuous spontaneous localization (CSL) model is a spontaneous collapse model in quantum mechanics, proposed in 1989 by Philip Pearle. and finalized in 1990 Gian Carlo Ghirardi, Philip Pearle and Alberto Rimini.
== Introduction ==
The most widely studied among the dynamical reduction (also known as collapse) models is the CSL model. Building on the Ghirardi-Rimini-Weber model, the CSL model describes the collapse of the wave function as occurring continuously in time, in contrast to the Ghirardi-Rimini-Weber model.
Some of the key features of the model are:
The localization takes place in position, which is the preferred basis in this model.
The model does not significantly alter the dynamics of microscopic systems, while it becomes strong for macroscopic objects: the amplification mechanism ensures this scaling.
It preserves the symmetry properties of identical particles.
It is characterized by two parameters:
λ
{\displaystyle \lambda }
and
r
C
{\displaystyle r_{C}}
, which are respectively the collapse rate and the correlation length of the model.
== Dynamical equation ==
The CSL dynamical equation for the wave function is stochastic and non-linear:
d
|
ψ
t
⟩
=
[
−
i
ℏ
H
^
d
t
+
λ
m
0
∫
d
x
N
^
t
(
x
)
d
W
t
(
x
)
−
λ
2
m
0
2
∫
d
x
∫
d
y
g
(
x
−
y
)
N
^
t
(
x
)
N
^
t
(
y
)
d
t
]
|
ψ
t
⟩
.
{\displaystyle \operatorname {d} \!|\psi _{t}\rangle =\left[-{\frac {i}{\hbar }}{\hat {H}}\operatorname {d} \!t+{\frac {\sqrt {\lambda }}{m_{0}}}\int \operatorname {d} \!{\bf {x}}\,{\hat {N}}_{t}({\bf {x}})\operatorname {d} \!W_{t}({\bf {x}})\right.\left.-{\frac {\lambda }{2m_{0}^{2}}}\int \operatorname {d} \!{\bf {x}}\int \operatorname {d} \!{\bf {y}}\,g({\bf {x}}-{\bf {y}}){\hat {N}}_{t}({\bf {x}}){\hat {N}}_{t}({\bf {y}})\operatorname {d} \!t\right]|\psi _{t}\rangle .}
Here
H
^
{\displaystyle {\hat {H}}}
is the Hamiltonian describing the quantum mechanical dynamics,
m
0
{\displaystyle m_{0}}
is a reference mass taken equal to that of a nucleon,
g
(
x
−
y
)
=
e
−
(
x
−
y
)
2
/
4
r
C
2
{\displaystyle g({\bf {x}}-{\bf {y}})=e^{-{({\bf {x}}-{\bf {y}})^{2}}/{4r_{C}^{2}}}}
, and the noise field
w
t
(
x
)
=
d
W
t
(
x
)
/
d
t
{\displaystyle w_{t}({\bf {x}})=\operatorname {d} \!W_{t}({\bf {x}})/\operatorname {d} \!t}
has zero average and correlation equal to
E
[
w
t
(
x
)
w
s
(
y
)
]
=
g
(
x
−
y
)
δ
(
t
−
s
)
,
{\displaystyle \mathbb {E} [w_{t}({\bf {x}})w_{s}({\bf {y}})]=g({\bf {x}}-{\bf {y}})\delta (t-s),}
where
E
[
⋅
]
{\displaystyle \mathbb {E} [\ \cdot \ ]}
denotes the stochastic average over the noise. Finally, we write
N
^
t
(
x
)
=
M
^
(
x
)
−
⟨
ψ
t
|
M
^
(
x
)
|
ψ
t
⟩
,
{\displaystyle {\hat {N}}_{t}({\bf {x}})={\hat {M}}({\bf {x}})-\langle \psi _{t}|{\hat {M}}({\bf {x}})|\psi _{t}\rangle ,}
where
M
^
(
x
)
{\displaystyle {\hat {M}}({\bf {x}})}
is the mass density operator, which reads
M
^
(
x
)
=
∑
j
m
j
∑
s
a
^
j
†
(
x
,
s
)
a
^
j
(
x
,
s
)
,
{\displaystyle {\hat {M}}({\bf {x}})=\sum _{j}m_{j}\sum _{s}{\hat {a}}_{j}^{\dagger }({\bf {x}},s){\hat {a}}_{j}({\bf {x}},s),}
where
a
^
j
†
(
y
,
s
)
{\displaystyle {\hat {a}}_{j}^{\dagger }({\bf {y}},s)}
and
a
^
j
(
y
,
s
)
{\displaystyle {\hat {a}}_{j}({\bf {y}},s)}
are, respectively, the second quantized creation and annihilation operators of a particle of type
j
{\displaystyle j}
with spin
s
{\displaystyle s}
at the point
y
{\displaystyle {\bf {y}}}
of mass
m
j
{\displaystyle m_{j}}
. The use of these operators satisfies the conservation of the symmetry properties of identical particles. Moreover, the mass proportionality implements automatically the amplification mechanism. The choice of the form of
M
^
(
x
)
{\displaystyle {\hat {M}}({\bf {x}})}
ensures the collapse in the position basis.
The action of the CSL model is quantified by the values of the two phenomenological parameters
λ
{\displaystyle \lambda }
and
r
C
{\displaystyle r_{C}}
. Originally, the Ghirardi-Rimini-Weber model proposed
λ
=
10
−
17
{\displaystyle \lambda =10^{-17}\,}
s
−
1
{\displaystyle ^{-1}}
at
r
C
=
10
−
7
{\displaystyle r_{C}=10^{-7}\,}
m, while later Adler considered larger values:
λ
=
10
−
8
±
2
{\displaystyle \lambda =10^{-8\pm 2}\,}
s
−
1
{\displaystyle ^{-1}}
for
r
C
=
10
−
7
{\displaystyle r_{C}=10^{-7}\,}
m, and
λ
=
10
−
6
±
2
{\displaystyle \lambda =10^{-6\pm 2}\,}
s
−
1
{\displaystyle ^{-1}}
for
r
C
=
10
−
6
{\displaystyle r_{C}=10^{-6}\,}
m. Eventually, these values have to be bounded by experiments.
From the dynamics of the wave function one can obtain the corresponding master equation for the statistical operator
ρ
^
t
{\displaystyle {\hat {\rho }}_{t}}
:
d
ρ
^
t
d
t
=
−
i
ℏ
[
H
^
,
ρ
^
t
]
−
λ
2
m
0
2
∫
d
x
∫
d
y
g
(
x
−
y
)
[
M
^
(
x
)
,
[
M
^
(
y
)
,
ρ
^
t
]
]
.
{\displaystyle {\frac {\operatorname {d} \!{\hat {\rho }}_{t}}{\operatorname {d} \!t}}=-{\frac {i}{\hbar }}\left[{\hat {H}},{{\hat {\rho }}_{t}}\right]-{\frac {\lambda }{2m_{0}^{2}}}\int \operatorname {d} \!{\bf {x}}\int \operatorname {d} \!{\bf {y}}\,g({\bf {x}}-{\bf {y}})\left[{{\hat {M}}({\bf {x}})},\left[{{{\hat {M}}({\bf {y}})},{{\hat {\rho }}_{t}}}\right]\right].}
Once the master equation is represented in the position basis, it becomes clear that its direct action is to diagonalize the density matrix in position. For a single point-like particle of mass
m
{\displaystyle m}
, it reads
∂
⟨
x
|
ρ
^
t
|
y
⟩
∂
t
=
−
i
ℏ
⟨
x
|
[
H
^
,
ρ
^
t
]
|
y
⟩
−
λ
m
2
m
0
2
(
1
−
e
−
(
x
−
y
)
2
4
r
C
2
)
⟨
x
|
ρ
^
t
|
y
⟩
,
{\displaystyle {\frac {\partial \langle {{\bf {x}}|{\hat {\rho }}_{t}|{\bf {y}}}\rangle }{\partial t}}=-{\frac {i}{\hbar }}\langle {{\bf {x}}|\left[{\hat {H}},{{\hat {\rho }}_{t}}\right]|{\bf {y}}}\rangle -\lambda {\frac {m^{2}}{m_{0}^{2}}}\left(1-e^{-{\tfrac {({\bf {x}}-{\bf {y}})^{2}}{4r_{C}^{2}}}}\right)\langle {{\bf {x}}|{\hat {\rho }}_{t}|{\bf {y}}}\rangle ,}
where the off-diagonal terms, which have
x
≠
y
{\displaystyle {\bf {x}}\neq {\bf {y}}}
, decay exponentially. Conversely, the diagonal terms, characterized by
x
=
y
{\displaystyle {\bf {x}}={\bf {y}}}
, are preserved. For a composite system, the single-particle collapse rate
λ
{\displaystyle \lambda }
should be replaced with that of the composite system
λ
m
2
m
0
2
→
λ
r
C
3
π
3
/
2
m
0
2
∫
d
k
|
μ
~
(
k
)
|
2
e
−
k
2
r
C
2
,
{\displaystyle \lambda {\frac {m^{2}}{m_{0}^{2}}}\to \lambda {\frac {r_{C}^{3}}{\pi ^{3/2}m_{0}^{2}}}\int \operatorname {d} \!{\bf {k}}|{\tilde {\mu }}({\bf {k}})|^{2}e^{-k^{2}r_{C}^{2}},}
where
μ
~
(
k
)
{\displaystyle {\tilde {\mu }}(k)}
is the Fourier transform of the mass density of the system.
== Experimental tests ==
Contrary to most other proposed solutions of the measurement problem, collapse models are experimentally testable. Experiments testing the CSL model can be divided in two classes: interferometric and non-interferometric experiments, which respectively probe direct and indirect effects of the collapse mechanism.
=== Interferometric experiments ===
Interferometric experiments can detect the direct action of the collapse, which is to localize the wavefunction in space. They include all experiments where a superposition is generated and, after some time, its interference pattern is probed. The action of CSL is a reduction of the interference contrast, which is quantified by the reduction of the off-diagonal terms of the statistical operator
ρ
(
x
,
x
′
,
t
)
=
1
2
π
ℏ
∫
−
∞
+
∞
d
k
∫
−
∞
+
∞
d
w
e
−
i
k
w
/
ℏ
F
C
S
L
(
k
,
x
−
x
′
,
t
)
ρ
Q
M
(
x
+
w
,
x
′
+
w
,
t
)
,
{\displaystyle \rho (x,x',t)={\frac {1}{2\pi \hbar }}\int _{-\infty }^{+\infty }\operatorname {d} \!k\int _{-\infty }^{+\infty }\operatorname {d} \!w\,e^{-ikw/\hbar }F_{CSL}(k,x-x',t)\rho ^{QM}(x+w,x'+w,t),}
where
ρ
Q
M
{\textstyle \rho ^{QM}}
denotes the statistical operator described by quantum mechanics, and we define
F
C
S
L
(
k
,
q
,
t
)
=
exp
[
−
λ
m
2
m
0
2
t
(
1
−
1
t
∫
0
t
d
τ
e
−
(
q
−
k
τ
m
)
2
/
4
r
C
2
)
]
.
{\displaystyle F_{CSL}(k,q,t)=\exp {\bigg [}-\lambda {\frac {m^{2}}{m_{0}^{2}}}t\left(1-{\frac {1}{t}}\int _{0}^{t}\operatorname {d} \!\tau \,e^{-{(q-{\frac {k\tau }{m}})^{2}}/{4r_{C}^{2}}}\right){\bigg ]}.}
Experiments testing such a reduction of the interference contrast are carried out with cold-atoms, molecules, entangled diamonds and mechanical oscillators .
Similarly, one can also quantify the minimum collapse strength to solve the measurement problem at the macroscopic level. Specifically, an estimate can be obtained by requiring that a superposition of a single-layered graphene disk of radius
≃
10
−
5
{\displaystyle \simeq 10^{-5}}
m collapses in less than
≃
10
−
2
{\displaystyle \simeq 10^{-2}}
s.
=== Non-interferometric experiments ===
Non-interferometric experiments consist in CSL tests, which are not based on the preparation of a superposition. They exploit an indirect effect of the collapse, which consists in a Brownian motion induced by the interaction with the collapse noise. The effect of this noise amounts to an effective stochastic force acting on the system, and several experiments can be designed to quantify such a force. They include:
Radiation emission from charged particles. If a particle is electrically charged, the action of the coupling with the collapse noise will induce the emission of radiation. This result is in net contrast with the predictions of quantum mechanics, where no radiation is expected from a free particle. The predicted CSL-induced emission rate at frequency
ω
{\displaystyle \omega }
for a particle of charge
Q
{\displaystyle Q}
is given by:
d
Γ
(
ω
)
d
ω
=
ℏ
Q
2
λ
2
π
2
ϵ
0
c
3
m
0
2
r
C
2
ω
,
{\displaystyle {\frac {\operatorname {d} \!\Gamma (\omega )}{\operatorname {d} \!\omega }}={\frac {\hbar Q^{2}\lambda }{2\pi ^{2}\epsilon _{0}c^{3}m_{0}^{2}r_{C}^{2}\omega }},}
where
ϵ
0
{\displaystyle \epsilon _{0}}
is the vacuum dielectric constant and
c
{\displaystyle c}
is the light speed. This prediction of CSL can be tested by analyzing the X-ray emission spectrum from a bulk Germanium test mass.
Heating in bulk materials. A prediction of CSL is the increase of the total energy of a system. For example, the total energy
E
{\displaystyle E}
of a free particle of mass
m
{\displaystyle m}
in three dimensions grows linearly in time according to
E
(
t
)
=
E
(
0
)
+
3
m
λ
ℏ
2
4
m
0
2
r
C
2
t
,
{\displaystyle E(t)=E(0)+{\frac {3m\lambda \hbar ^{2}}{4m_{0}^{2}r_{C}^{2}}}t,}
where
E
(
0
)
{\displaystyle E(0)}
is the initial energy of the system. This increase is effectively small; for example, the temperature of a hydrogen atom increases by
≃
10
−
14
{\displaystyle \simeq 10^{-14}}
K per year considering the values
λ
=
10
−
16
{\displaystyle \lambda =10^{-16}}
s
−
1
{\displaystyle ^{-1}}
and
r
C
=
10
−
7
{\displaystyle r_{C}=10^{-7}}
m. Although small, such an energy increase can be tested by monitoring cold atoms. and bulk materials, as Bravais lattices, low temperature experiments, neutron stars and planets
Diffusive effects. Another prediction of the CSL model is the increase of the spread in position of center-of-mass of a system. For a free particle, the position spread in one dimension reads
⟨
x
^
2
⟩
t
=
⟨
x
^
2
⟩
t
(
Q
M
)
+
ℏ
2
η
t
3
3
m
2
,
{\displaystyle \langle {{\hat {x}}^{2}}\rangle _{t}=\langle {{\hat {x}}^{2}}\rangle _{t}^{(QM)}+{\frac {\hbar ^{2}\eta t^{3}}{3m^{2}}},}
where
⟨
x
^
2
⟩
t
(
Q
M
)
{\displaystyle \langle {{\hat {x}}^{2}}\rangle _{t}^{(QM)}}
is the free quantum mechanical spread and
η
{\displaystyle \eta }
is the CSL diffusion constant, defined as
η
=
λ
r
C
3
2
π
3
/
2
m
0
2
∫
d
k
e
−
k
2
r
C
2
k
x
2
|
μ
~
(
k
)
|
2
,
{\displaystyle \eta ={\frac {\lambda r_{C}^{3}}{2\pi ^{3/2}m_{0}^{2}}}\int \operatorname {d} \!{\bf {k}}\,e^{-{\bf {k}}^{2}r_{C}^{2}}k_{x}^{2}|{\tilde {\mu }}({\bf {k}})|^{2},}
where the motion is assumed to occur along the
x
{\displaystyle x}
axis;
μ
~
(
k
)
{\displaystyle {\tilde {\mu }}({\bf {k}})}
is the Fourier transform of the mass density
μ
(
r
)
{\displaystyle \mu ({\bf {r}})}
. In experiments, such an increase is limited by the dissipation rate
γ
{\displaystyle \gamma }
. Assuming that the experiment is performed at temperature
T
{\displaystyle T}
, a particle of mass
m
{\displaystyle m}
, harmonically trapped at frequency
ω
0
{\displaystyle \omega _{0}}
, at equilibrium reaches a spread in position given by
⟨
x
^
2
⟩
e
q
=
k
B
T
m
ω
0
2
+
ℏ
2
η
2
m
2
ω
0
2
γ
,
{\displaystyle \langle {{\hat {x}}^{2}}\rangle _{eq}={\frac {k_{B}T}{m\omega _{0}^{2}}}+{\frac {\hbar ^{2}\eta }{2m^{2}\omega _{0}^{2}\gamma }},}
where
k
B
{\displaystyle k_{B}}
is the Boltzmann constant. Several experiments can test such a spread. They range from cold atom free expansion, nano-cantilevers cooled to millikelvin temperatures, gravitational wave detectors, levitated optomechanics, torsion pendula.
== Dissipative and colored extensions ==
The CSL model consistently describes the collapse mechanism as a dynamical process. It has, however, two weak points.
CSL does not conserve the energy of isolated systems. Although this increase is small, it is an unpleasant feature for a phenomenological model. The dissipative extensions of the CSL model gives a remedy. One associates to the collapse noise a finite temperature
T
C
S
L
{\displaystyle T_{CSL}}
at which the system eventually thermalizes. Thus, as an example, for a free point-like particle of mass
m
{\displaystyle m}
in three dimensions, the energy evolution in Ref. is described by
E
(
t
)
=
e
−
β
t
(
E
(
0
)
−
E
a
s
)
+
E
a
s
,
{\displaystyle E(t)=e^{-\beta t}(E(0)-E_{as})+E_{as},}
where
E
a
s
=
3
2
k
B
T
C
S
L
{\displaystyle E_{as}={\tfrac {3}{2}}k_{B}T_{CSL}}
,
β
=
4
χ
λ
/
(
1
+
χ
)
5
{\displaystyle \beta =4\chi \lambda /(1+\chi )^{5}}
and
χ
=
ℏ
2
/
(
8
m
0
k
B
T
C
S
L
r
C
2
)
{\displaystyle \chi =\hbar ^{2}/(8m_{0}k_{B}T_{CSL}r_{C}^{2})}
. Assuming that the CSL noise has a cosmological origin (which is reasonable due to its supposed universality), a plausible value such a temperature is
T
C
S
L
=
1
{\displaystyle T_{CSL}=1}
K, although only experiments can indicate a definite value. Several interferometric and non-interferometric tests bound the CSL parameter space for different choices of
T
C
S
L
{\displaystyle T_{CSL}}
.
The CSL noise spectrum is white. If one attributes a physical origin to the CSL noise, then its spectrum cannot be white, but colored. In particular, in place of the white noise
w
t
(
x
)
{\displaystyle w_{t}({\bf {x}})}
, whose correlation is proportional to a Dirac delta in time, a non-white noise is considered, which is characterized by a non-trivial temporal correlation function
f
(
t
)
{\displaystyle f(t)}
. The effect can be quantified by a rescaling of
F
C
S
L
(
k
,
q
,
t
)
{\displaystyle F_{CSL}(k,q,t)}
, which becomes
F
c
C
S
L
(
k
,
q
,
t
)
=
F
C
S
L
(
k
,
q
,
t
)
exp
[
λ
τ
¯
2
(
e
−
(
q
−
k
t
/
m
)
2
/
4
r
C
2
−
e
−
q
2
/
4
r
C
2
)
]
,
{\displaystyle F_{cCSL}(k,q,t)=F_{CSL}(k,q,t)\exp \left[{\frac {\lambda {\bar {\tau }}}{2}}\left(e^{-(q-kt/m)^{2}/4r_{C}^{2}}-e^{-q^{2}/4r_{C}^{2}}\right)\right],}
where
τ
¯
=
∫
0
t
d
s
f
(
s
)
{\displaystyle {\bar {\tau }}=\int _{0}^{t}\operatorname {d} \!s\,f(s)}
. As an example, one can consider an exponentially decaying noise, whose time correlation function can be of the form
f
(
t
)
=
1
2
Ω
C
e
−
Ω
C
|
t
|
{\displaystyle f(t)={\tfrac {1}{2}}\Omega _{C}e^{-\Omega _{C}|t|}}
. In such a way, one introduces a frequency cutoff
Ω
C
{\displaystyle \Omega _{C}}
, whose inverse describes the time scale of the noise correlations. The parameter
Ω
C
{\displaystyle \Omega _{C}}
works now as the third parameter of the colored CSL model together with
λ
{\displaystyle \lambda }
and
r
C
{\displaystyle r_{C}}
. Assuming a cosmological origin of the noise, a reasonable guess is
Ω
C
=
10
12
{\displaystyle \Omega _{C}=10^{12}\,}
Hz. As for the dissipative extension, experimental bounds were obtained for different values of
Ω
C
{\displaystyle \Omega _{C}}
: they include interferometric and non-interferometric tests.
== References == | Wikipedia/Continuous_spontaneous_localization_model |
Integrated quantum photonics, uses photonic integrated circuits to control photonic quantum states for applications in quantum technologies. As such, integrated quantum photonics provides a promising approach to the miniaturisation and scaling up of optical quantum circuits. The major application of integrated quantum photonics is Quantum technology:, for example quantum computing, quantum communication, quantum simulation, quantum walks and quantum metrology.
== History ==
Linear optics was not seen as a potential technology platform for quantum computation until the seminal work of Knill, Laflamme, and Milburn, which demonstrated the feasibility of linear optical quantum computers using detection and feed-forward to produce deterministic two-qubit gates. Following this there were several experimental proof-of-principle demonstrations of two-qubit gates performed in bulk optics. It soon became clear that integrated optics could provide a powerful enabling technology for this emerging field. Early experiments in integrated optics demonstrated the feasibility of the field via demonstrations of high-visibility non-classical and classical interference. Typically, linear optical components such as directional couplers (which act as beamsplitters between waveguide modes), and phase shifters to form nested Mach–Zehnder interferometers are used to encode a qubit in the spatial degree of freedom. That is, a single photon is in superposition between two waveguides, where the zero and one states of the qubit correspond to the photon's presence in one or the other waveguide. These basic components are combined to produce more complex structures, such as entangling gates and reconfigurable quantum circuits. Reconfigurability is achieved by tuning the phase shifters, which are manipulated by using thermo- or electro-optical elements.
Another area of research in which integrated optics will prove pivotal is Quantum communication and has been marked by extensive experimental development demonstrating, for example, quantum key distribution (QKD), quantum relays based on entanglement swapping, and quantum repeaters.
Since the birth of integrated quantum optics experiments have ranged from technological demonstrations, for example integrated single photon sources and integrated single photon detectors, to fundamental tests of nature, new methods for quantum key distribution, and the generation of new quantum states of light. It has also been demonstrated that a single reconfigurable integrated device is sufficient to implement the full field of linear optics, by using a reconfigurable universal interferometer.
As the field has progressed new quantum algorithms have been developed which provide short and long term routes towards the demonstration of the superiority of quantum computers over their classical counterparts. Cluster state quantum computation is now generally accepted as the approach that will be used to develop a fully fledged quantum computer. Whilst development of quantum computer will require the synthesis of many aspects of integrated optics, boson sampling seeks to demonstrate the power of quantum information processing via readily available technologies and is therefore a very promising near term algorithm for doing so. In fact, shortly after its introduction, there were several small scale experimental demonstrations of the effectiveness of the boson sampling algorithm
== Introduction ==
Quantum photonics is the science of generating, manipulating and detecting light in regimes where it's possible to coherently control individual quanta of the light field (photons). Historically, quantum photonics has been fundamental to exploring quantum phenomena, for example with the EPR paradox and Bell test experiments,. Quantum photonics is also expected to play a central role in advancing future technologies, such as Quantum computing, Quantum key distribution and Quantum metrology. Photons are particularly attractive carriers of quantum information due to their low decoherence properties, light-speed transmission and ease of manipulation. Quantum photonics experiments traditionally involved 'bulk optics' technology—individual optical components (lenses, beamsplitters, etc.) mounted on a large optical table, with a combined mass of hundreds of kilograms.
The application of Integrated quantum photonic circuits to quantum photonics, is seen as an important step in developing useful quantum technology. Single die photonic circuits offer the following advantages over bulk optics:
Miniaturisation - Size, weight, and power consumption are reduced by orders of magnitude by virtue of smaller system size.
Stability - Miniaturised components produced with advanced lithographic techniques produce waveguides and components which are inherently phase stable (coherent) and do not require optical alignment
Experiment size - Large numbers of optical components can be integrated into a device measuring a few square centimeters.
Manufacturability - Devices can be manufactured in large volumes at much lower cost.
Being based on well-developed fabrication techniques, the elements employed in Integrated Quantum Photonics are more readily miniaturisable, and products based on this approach can be manufactured using existing production processes and methods.
== Materials ==
Control over photons can be achieved with integrated devices that can be realised in diverse material substrates such as silica, silicon, gallium arsenide, lithium niobate and indium phosphide and silicon nitride.
=== Silica ===
Three methods for using silica:
Flame hydrolysis.
Photolithography.
Direct write - uses a single material and laser (a computer controlled laser "damages" the glass by manipulating the laser focus and path to create circuit lines by altering the refractive index of the material along that path, thereby producing waveguides). This method has the benefit of not needing a clean room and is the most common method now for making silica waveguides. It's also excellent for rapid prototyping and has been used to advantage in several demonstrations of topological photonics.
The main challenges of the silica platform are the low refractive index contrast, the lack of active tunability post-fabrication (as opposed to all the other substrates) and the difficulty of mass production with reproducibility and high yield due to the serial nature of the inscription process.
=== Silicon ===
A big advantage of using silicon is that the circuits can be tuned actively using integrated thermal microheaters or p-i-n modulators, after the devices have been fabricated. The other big benefit of silicon is its compatibility with CMOS technology, which allows leveraging the mature fabrication infrastructure of the semiconductor electronics industry. The structures differ from modern electronic ones, however, as they are readily scalable. Silicon has a really high refractive index of ~3.5 at the 1550 nm wavelength commonly used in optical telecommunications. It therefore offers one of the highest component densities in integrated photonics. The large contrast in refractive index with glass (1.44) allows waveguides formed of silicon surrounded by glass to have very tight bends, which allows for a high component density and reduced system size. Large silicon-on-insulator (SOI) wafers up to 300 mm in diameter can be obtained commercially, making the technology both available and reproducible. Many of the largest systems (up to several hundred components) have been demonstrated on the silicon photonics platform, with up to eight simultaneous photons, generation of graph states (cluster states), and up to 15 dimensional qubits). Photon sources in silicon waveguide circuits leverage silicon's third-order nonlinearity to produce pairs of photons in spontaneous four-wave mixing. Silicon is opaque for wavelengths of light below ~1200 nm, limiting applicability to infrared photons. Phase modulators based on thermo-optic and electro-optic phases are characteristically slow (KHz) and lossy (several dB) respectively, limiting applications and the ability to perform feed-forward measurements for quantum computation.
=== Lithium Niobate ===
Lithium niobate offers a large second-order optical nonlinearity, enabling generation of photon pairs via spontaneous parametric down-conversion. This can also be leveraged to manipulate phase and perform mode conversion at high speeds, and offers a promising route to feed-forward for quantum computation, multiplexed (deterministic) single photons sources). Historically, waveguides are defined using titanium indiffusion, resulting in large waveguides (large bend radius).
=== III-V Materials on Insulator ===
Photonic waveguides made from group III-V materials on insulator, such as (Al)GaAs and InP, provide some of the largest second and third order nonlinearities, large refractive index contrast providing large modal confinement, and wide optical bandgaps resulting in negligible two-photon absorption at telecommunications wavelengths. III-V materials are capable of low-loss passive and high-speed active components, such as active gain for on-chip lasers, high-speed electro-optic modulators (Pockels and Kerr effects), and on-chip detectors. Compared to other materials such as silica, silicon, and silicon nitride, the large optical nonlinearity, simultaneously with low waveguide loss and tight modal confinement, has resulted in ultrabright entangled-photon pair generation from microring resonators.
== Fabrication ==
Conventional fabrication technologies are based on photolithographic processes, which enable strong miniaturisation and mass production. In quantum optics applications a relevant role has also been played by the direct inscription of the circuits by femtosecond lasers or UV lasers; these are low-volume fabrication technologies, which are particularly convenient for research purposes where novel designs have to be tested with rapid fabrication turnaround.
However, laser-written waveguides are not suitable for mass production and miniaturisation due to the serial nature of the inscription technique, and due to the very low refractive index contrast allowed by these materials, as opposed to silicon photonic circuits. Femtosecond laser-written quantum circuits have proven particularly suited for the manipulation of the polarisation degree of freedom and for building circuits with innovative three-dimensional designs. Quantum information is encoded on-chip in either the path, polarisation, time bin, or frequency state of the photon and manipulated using active integrated components in a compact and stable manner.
== Components ==
Though the same fundamental components are used in quantum as classical photonic integrated circuits, there are also some practical differences. Since amplification of single photon quantum states is not possible (no-cloning theorem), loss is the top priority in components in quantum photonics.
Single photon sources are built from building blocks (waveguides, directional couplers, phase shifters). Typically, optical ring resonators, and long waveguide sections provide increased nonlinear interaction for photon pair generation, though progress is also being made to integrate solid state systems single photon sources based on quantum dots, and nitrogen-vacancy centers with waveguide photonic circuits.
== See also ==
Linear optical quantum computing
Quantum information
Quantum key distribution
List of companies involved in quantum computing or communication
List of quantum processors
== References ==
== External links ==
QUCHIP Project
3D-QUEST Project
Center for Quantum Photonics, University of Bristol
Fast Group, Istituto di Fotonica e Nanotecnologie, Consiglio Nazionale delle Ricerche
Integrated Quantum Optics, Paderborn University
Integrated Quantum Technologies, Griffith University | Wikipedia/Integrated_quantum_photonics |
An optical neural network is a physical implementation of an artificial neural network with optical components. Early optical neural networks used a photorefractive Volume hologram to interconnect arrays of input neurons to arrays of output with synaptic weights in proportion to the multiplexed hologram's strength. Volume holograms were further multiplexed using spectral hole burning to add one dimension of wavelength to space to achieve four dimensional interconnects of two dimensional arrays of neural inputs and outputs. This research led to extensive research on alternative methods using the strength of the optical interconnect for implementing neuronal communications.
Some artificial neural networks that have been implemented as optical neural networks include the Hopfield neural network and the Kohonen self-organizing map with liquid crystal spatial light modulators Optical neural networks can also be based on the principles of neuromorphic engineering, creating neuromorphic photonic systems. Typically, these systems encode information in the networks using spikes, mimicking the functionality of spiking neural networks in optical and photonic hardware. Photonic devices that have demonstrated neuromorphic functionalities include (among others) vertical-cavity surface-emitting lasers, integrated photonic modulators, optoelectronic systems based on superconducting Josephson junctions or systems based on resonant tunnelling diodes.
== Electrochemical vs. optical neural networks ==
Biological neural networks function on an electrochemical basis, while optical neural networks use electromagnetic waves. Optical interfaces to biological neural networks can be created with optogenetics, but is not the same as an optical neural networks. In biological neural networks there exist a lot of different mechanisms for dynamically changing the state of the neurons, these include short-term and long-term synaptic plasticity. Synaptic plasticity is among the electrophysiological phenomena used to control the efficiency of synaptic transmission, long-term for learning and memory, and short-term for short transient changes in synaptic transmission efficiency. Implementing this with optical components is difficult, and ideally requires advanced photonic materials. Properties that might be desirable in photonic materials for optical neural networks include the ability to change their efficiency of transmitting light, based on the intensity of incoming light.
== Rising Era of Optical Neural Networks ==
With the increasing significance of computer vision in various domains, the computational cost of these tasks has increased, making it more important to develop the new approaches of the processing acceleration. Optical computing has emerged as a potential alternative to GPU acceleration for modern neural networks, particularly considering the looming obsolescence of Moore's Law. Consequently, optical neural networks have garnered increased attention in the research community. Presently, two primary methods of optical neural computing are under research: silicon photonics-based and free-space optics. Each approach has its benefits and drawbacks; while silicon photonics may offer superior speed, it lacks the massive parallelism that free-space optics can deliver.
Given the substantial parallelism capabilities of free-space optics, researchers have focused on taking advantage of it. One implementation, proposed by Lin et al., involves the training and fabrication of phase masks for a handwritten digit classifier. By stacking 3D-printed phase masks, light passing through the fabricated network can be read by a photodetector array of ten detectors, each representing a digit class ranging from 1 to 10. Although this network can achieve terahertz-range classification, it lacks flexibility, as the phase masks are fabricated for a specific task and cannot be retrained.
An alternative method for classification in free-space optics, introduced by Cahng et al., employs a 4F system that is based on the convolution theorem to perform convolution operations. This system uses two lenses to execute the Fourier transforms of the convolution operation, enabling passive conversion into the Fourier domain without power consumption or latency. However, the convolution operation kernels in this implementation are also fabricated phase masks, limiting the device's functionality to specific convolutional layers of the network only.
In contrast, Li et al. proposed a technique involving kernel tiling to use the parallelism of the 4F system while using a Digital Micromirror Device (DMD) instead of a phase mask. This approach allows users to upload various kernels into the 4F system and execute the entire network's inference on a single device. Unfortunately, modern neural networks are not designed for the 4F systems, as they were primarily developed during the CPU/GPU era. Mostly because they tend to use a lower resolution and a high number of channels in their feature maps.
== Other Implementations ==
In 2007 there was one model of Optical Neural Network: the Programmable Optical Array/Analogic Computer (POAC). It had been implemented in the year 2000 and reported based on modified Joint Fourier Transform Correlator (JTC) and Bacteriorhodopsin (BR) as a holographic optical memory. Full parallelism, large array size and the speed of light are three promises offered by POAC to implement an optical CNN. They had been investigated during the last years with their practical limitations and considerations yielding the design of the first portable POAC version.
The practical details – hardware (optical setups) and software (optical templates) – were published. However, POAC is a general purpose and programmable array computer that has a wide range of applications including:
image processing
pattern recognition
target tracking
real-time video processing
document security
optical switching
== Progress in the 2020s ==
Taichi from Tsinghua University in Beijing is a hybrid ONN that combines the power efficiency and parallelism of optical diffraction and the configurability of optical interference. Taichi offers 13.96 million parameters. Taichi avoids the high error rates that afflict deep (multi-layer) networks by combining clusters of fewer-layer diffractive units with arrays of interferometers for reconfigurable computation. Its encoding protocol divides large network models into sub-models that can be distributed across multiple chiplets in parallel.
Taichi achieved 91.89% accuracy in tests with the Omniglot database. It was also used to generate music Bach and generate images the styles of Van Gogh and Munch.
The developers claimed energy efficiency of up to 160 trillion operations second−1 watt−1 and an area efficiency of 880 trillion multiply-accumulate operations mm−2 or 103 more energy efficient than the NVIDIA H100, and 102 times more energy efficient and 10 times more area efficient than previous ONNs.
Time dimension has recently been introduced into diffrative nueral network by fs laser lithography of perovskite hydration. The temporal behaviour of the neuron can be modulated by the fs laser at the nanoscale, enabling a programmable holographic neural network with temporal evolution functionality, i.e., the functionality can change with time under the hydration stimuli. An in-memory temporal inference functionality was demonstrated to mimic the function evolution of the human brain,i.e.,the functionality can change from simple digit image classification to more complicated digit and clothing product image classification with time. This is the first time of introducting time dimension into the optical neural network, laying a foundation for future brain-like photonic chip development.
== See also ==
Optical computing
Quantum neural network
== References == | Wikipedia/Optical_neural_network |
For holographic data storage, holographic associative memory (HAM) is an information storage and retrieval system based on the principles of holography. Holograms are made by using two beams of light, called a "reference beam" and an "object beam". They produce a pattern on the film that contains them both. Afterwards, by reproducing the reference beam, the hologram recreates a visual image of the original object. In theory, one could use the object beam to do the same thing: reproduce the original reference beam. In HAM, the pieces of information act like the two beams. Each can be used to retrieve the other from the pattern. It can be thought of as an artificial neural network which mimics the way the brain uses information. The information is presented in abstract form by a complex vector which may be expressed directly by a waveform possessing frequency and magnitude. This waveform is analogous to electrochemical impulses believed to transmit information between biological neuron cells.
== Definition ==
HAM is part of the family of analog, correlation-based, associative, stimulus-response memories, where information is mapped onto the phase orientation of complex numbers. It can be considered as a complex valued artificial neural network. The holographic associative memory exhibits some remarkable characteristics. Holographs have been shown to be effective for associative memory tasks, generalization, and pattern recognition with changeable attention. Ability of dynamic search localization is central to natural memory. For example, in visual perception, humans always tend to focus on some specific objects in a pattern. Humans can effortlessly change the focus from object to object without requiring relearning. HAM provides a computational model which can mimic this ability by creating representation for focus. At the heart of this new memory lies a novel bi-modal representation of pattern and a hologram-like complex spherical weight state-space. Besides the usual advantages of associative computing, this technique also has excellent potential for fast optical realization because the underlying hyper-spherical computations can be naturally implemented on optical computations.
It is based on principle of information storage in the form of stimulus-response patterns where information is presented by phase angle orientations of complex numbers on a Riemann surface. A very large number of stimulus-response patterns may be superimposed or "enfolded" on a single neural element. Stimulus-response associations may be both encoded and decoded in one non-iterative transformation. The mathematical basis requires no optimization of parameters or error backpropagation, unlike connectionist neural networks. The principal requirement is for stimulus patterns to be made symmetric or orthogonal in the complex domain. HAM typically employs sigmoid pre-processing where raw inputs are orthogonalized and converted to Gaussian distributions.
=== Principles of operation ===
Stimulus-response associations are both learned and expressed in one non-iterative transformation. No backpropagation of error terms or iterative processing required.
The method forms a non-connectionist model in which the ability to superimpose a very large set of analog stimulus-response patterns or complex associations exists within the individual neuron cell.
The generated phase angle communicates response information, and magnitude communicates a measure of recognition (or confidence in the result).
The process permits a capability with neural system to establish dominance profile of stored information, thus exhibiting a memory profile of any range - from short-term to long-term memory.
The process follows the non-disturbance rule, that is prior stimulus-response associations are minimally influenced by subsequent learning.
The information is presented in abstract form by a complex vector which may be expressed directly by a waveform possessing frequency and magnitude. This waveform is analogous to electrochemical impulses believed to transmit information between biological neuron cells.
== See also ==
AND Corporation – Canadian technology company
Holonomic brain theory – Quantum interpretation of neuroscience
Self-organizing map – Machine learning technique useful for dimensionality reduction
Sparse distributed memory – Mathematical model of memory
== References ==
== Further reading ==
Gopalan, R. P.; Lee, G (2002). McKay, R. I.; Slaney, J. (eds.). Indexing of Image Databases Using Untrained 4D Holographic Memory Model. 15th Australian Joint Conference on Artificial Intelligence. Springer. pp. 237–248.
Hendra, Y.; Gopalan, R. P.; Nair, M. G. (1999). A method for dynamic indexing of large image databases. IEEE SMC'99. Systems, Man, and Cybernetics.
Khan, J. I. (August 1995). Attention Modulated Associative Computing and Content-Associative Search in Image Archive (PDF) (PhD thesis). University of Hawaii.
Michel, H. E.; Awwal, A. A. S. (1999). Enhanced artificial neural networks using complex numbers. IJCNN'99. International Joint Conference on Neural Networks. Proceedings (Cat. No.99CH36339. Vol. 1. Washington, DC, USA. pp. 456–461. doi:10.1109/IJCNN.1999.831538.
Michel, H. E.; Kunjithapatham, S. (2002). Dasarathy, Belur V. (ed.). "Processing Landsat TM data using complex-valued neural networks" (PDF). Proceedings of SPIE. Data Mining and Knowledge Discovery: Theory, Tools, and Technology IV. 4730. International Society for Optical: 43–51. doi:10.1117/12.460209. S2CID 7664487. Archived from the original (PDF) on 2017-09-11.
Stoop, R.; Buchli, J.; Keller, G.; Steeb, W. H. (2003). "Stochastic resonance in pattern recognition by a holographic neuron model" (PDF). Physical Review E. 67. | Wikipedia/Holographic_associative_memory |
An artificial neuron is a mathematical function conceived as a model of a biological neuron in a neural network. The artificial neuron is the elementary unit of an artificial neural network.
The design of the artificial neuron was inspired by biological neural circuitry. Its inputs are analogous to excitatory postsynaptic potentials and inhibitory postsynaptic potentials at neural dendrites, or activation. Its weights are analogous to synaptic weights, and its output is analogous to a neuron's action potential which is transmitted along its axon.
Usually, each input is separately weighted, and the sum is often added to a term known as a bias (loosely corresponding to the threshold potential), before being passed through a nonlinear function known as an activation function. Depending on the task, these functions could have a sigmoid shape (e.g. for binary classification), but they may also take the form of other nonlinear functions, piecewise linear functions, or step functions. They are also often monotonically increasing, continuous, differentiable, and bounded. Non-monotonic, unbounded, and oscillating activation functions with multiple zeros that outperform sigmoidal and ReLU-like activation functions on many tasks have also been recently explored. The threshold function has inspired building logic gates referred to as threshold logic; applicable to building logic circuits resembling brain processing. For example, new devices such as memristors have been extensively used to develop such logic.
The artificial neuron activation function should not be confused with a linear system's transfer function.
An artificial neuron may be referred to as a semi-linear unit, Nv neuron, binary neuron, linear threshold function, or McCulloch–Pitts (MCP) neuron, depending on the structure used.
Simple artificial neurons, such as the McCulloch–Pitts model, are sometimes described as "caricature models", since they are intended to reflect one or more neurophysiological observations, but without regard to realism. Artificial neurons can also refer to artificial cells in neuromorphic engineering that are similar to natural physical neurons.
== Basic structure ==
For a given artificial neuron
k
{\displaystyle k}
, let there be
m
+
1
{\displaystyle m+1}
inputs with signals
x
0
{\displaystyle x_{0}}
through
x
m
{\displaystyle x_{m}}
and weights
w
k
0
{\displaystyle w_{k0}}
through
w
k
m
{\displaystyle w_{km}}
. Usually, the input
x
0
{\displaystyle x_{0}}
is assigned the value +1, which makes it a bias input with
w
k
0
=
b
k
{\displaystyle w_{k0}=b_{k}}
. This leaves only
m
{\displaystyle m}
actual inputs to the neuron:
x
1
{\displaystyle x_{1}}
to
x
m
{\displaystyle x_{m}}
.
The output of the
k
{\displaystyle k}
-th neuron is:
y
k
=
φ
(
∑
j
=
0
m
w
k
j
x
j
)
{\displaystyle y_{k}=\varphi \left(\sum _{j=0}^{m}w_{kj}x_{j}\right)}
,
where
φ
{\displaystyle \varphi }
(phi) is the activation function.
The output is analogous to the axon of a biological neuron, and its value propagates to the input of the next layer, through a synapse. It may also exit the system, possibly as part of an output vector.
It has no learning process as such. Its activation function weights are calculated, and its threshold value is predetermined.
== McCulloch–Pitts (MCP) neuron ==
An MCP neuron is a kind of restricted artificial neuron which operates in discrete time-steps. Each has zero or more inputs, and are written as
x
1
,
.
.
.
,
x
n
{\displaystyle x_{1},...,x_{n}}
. It has one output, written as
y
{\displaystyle y}
. Each input can be either excitatory or inhibitory. The output can either be quiet or firing. An MCP neuron also has a threshold
b
∈
{
0
,
1
,
2
,
.
.
.
}
{\displaystyle b\in \{0,1,2,...\}}
.
In an MCP neural network, all the neurons operate in synchronous discrete time-steps of
t
=
0
,
1
,
2
,
3
,
.
.
.
{\displaystyle t=0,1,2,3,...}
. At time
t
+
1
{\displaystyle t+1}
, the output of the neuron is
y
(
t
+
1
)
=
1
{\displaystyle y(t+1)=1}
if the number of firing excitatory inputs is at least equal to the threshold, and no inhibitory inputs are firing;
y
(
t
+
1
)
=
0
{\displaystyle y(t+1)=0}
otherwise.
Each output can be the input to an arbitrary number of neurons, including itself (i.e., self-loops are possible). However, an output cannot connect more than once with a single neuron. Self-loops do not cause contradictions, since the network operates in synchronous discrete time-steps.
As a simple example, consider a single neuron with threshold 0, and a single inhibitory self-loop. Its output would oscillate between 0 and 1 at every step, acting as a "clock".
Any finite state machine can be simulated by a MCP neural network. Furnished with an infinite tape, MCP neural networks can simulate any Turing machine.
== Biological models ==
Artificial neurons are designed to mimic aspects of their biological counterparts. However a significant performance gap exists between biological and artificial neural networks. In particular single biological neurons in the human brain with oscillating activation function capable of learning the XOR function have been discovered.
Dendrites – in biological neurons, dendrites act as the input vector. These dendrites allow the cell to receive signals from a large (>1000) number of neighboring neurons. As in the above mathematical treatment, each dendrite is able to perform "multiplication" by that dendrite's "weight value." The multiplication is accomplished by increasing or decreasing the ratio of synaptic neurotransmitters to signal chemicals introduced into the dendrite in response to the synaptic neurotransmitter. A negative multiplication effect can be achieved by transmitting signal inhibitors (i.e. oppositely charged ions) along the dendrite in response to the reception of synaptic neurotransmitters.
Soma – in biological neurons, the soma acts as the summation function, seen in the above mathematical description. As positive and negative signals (exciting and inhibiting, respectively) arrive in the soma from the dendrites, the positive and negative ions are effectively added in summation, by simple virtue of being mixed together in the solution inside the cell's body.
Axon – the axon gets its signal from the summation behavior which occurs inside the soma. The opening to the axon essentially samples the electrical potential of the solution inside the soma. Once the soma reaches a certain potential, the axon will transmit an all-in signal pulse down its length. In this regard, the axon behaves as the ability for us to connect our artificial neuron to other artificial neurons.
Unlike most artificial neurons, however, biological neurons fire in discrete pulses. Each time the electrical potential inside the soma reaches a certain threshold, a pulse is transmitted down the axon. This pulsing can be translated into continuous values. The rate (activations per second, etc.) at which an axon fires converts directly into the rate at which neighboring cells get signal ions introduced into them. The faster a biological neuron fires, the faster nearby neurons accumulate electrical potential (or lose electrical potential, depending on the "weighting" of the dendrite that connects to the neuron that fired). It is this conversion that allows computer scientists and mathematicians to simulate biological neural networks using artificial neurons which can output distinct values (often from −1 to 1).
=== Encoding ===
Research has shown that unary coding is used in the neural circuits responsible for birdsong production. The use of unary in biological networks is presumably due to the inherent simplicity of the coding. Another contributing factor could be that unary coding provides a certain degree of error correction.
== Physical artificial cells ==
There is research and development into physical artificial neurons – organic and inorganic.
For example, some artificial neurons can receive and release dopamine (chemical signals rather than electrical signals) and communicate with natural rat muscle and brain cells, with potential for use in BCIs/prosthetics.
Low-power biocompatible memristors may enable construction of artificial neurons which function at voltages of biological action potentials and could be used to directly process biosensing signals, for neuromorphic computing and/or direct communication with biological neurons.
Organic neuromorphic circuits made out of polymers, coated with an ion-rich gel to enable a material to carry an electric charge like real neurons, have been built into a robot, enabling it to learn sensorimotorically within the real world, rather than via simulations or virtually. Moreover, artificial spiking neurons made of soft matter (polymers) can operate in biologically relevant environments and enable the synergetic communication between the artificial and biological domains.
== History ==
The first artificial neuron was the Threshold Logic Unit (TLU), or Linear Threshold Unit, first proposed by Warren McCulloch and Walter Pitts in 1943 in A logical calculus of the ideas immanent in nervous activity. The model was specifically targeted as a computational model of the "nerve net" in the brain. As an activation function, it employed a threshold, equivalent to using the Heaviside step function. Initially, only a simple model was considered, with binary inputs and outputs, some restrictions on the possible weights, and a more flexible threshold value. Since the beginning it was already noticed that any Boolean function could be implemented by networks of such devices, what is easily seen from the fact that one can implement the AND and OR functions, and use them in the disjunctive or the conjunctive normal form.
Researchers also soon realized that cyclic networks, with feedbacks through neurons, could define dynamical systems with memory, but most of the research concentrated (and still does) on strictly feed-forward networks because of the smaller difficulty they present.
One important and pioneering artificial neural network that used the linear threshold function was the perceptron, developed by Frank Rosenblatt. This model already considered more flexible weight values in the neurons, and was used in machines with adaptive capabilities. The representation of the threshold values as a bias term was introduced by Bernard Widrow in 1960 – see ADALINE.
In the late 1980s, when research on neural networks regained strength, neurons with more continuous shapes started to be considered. The possibility of differentiating the activation function allows the direct use of the gradient descent and other optimization algorithms for the adjustment of the weights. Neural networks also started to be used as a general function approximation model. The best known training algorithm called backpropagation has been rediscovered several times but its first development goes back to the work of Paul Werbos.
== Types of activation function ==
The activation function of a neuron is chosen to have a number of properties which either enhance or simplify the network containing the neuron. Crucially, for instance, any multilayer perceptron using a linear activation function has an equivalent single-layer network; a non-linear function is therefore necessary to gain the advantages of a multi-layer network.
Below,
u
{\displaystyle u}
refers in all cases to the weighted sum of all the inputs to the neuron, i.e. for
n
{\displaystyle n}
inputs,
u
=
∑
i
=
1
n
w
i
x
i
{\displaystyle u=\sum _{i=1}^{n}w_{i}x_{i}}
where
w
{\displaystyle w}
is a vector of synaptic weights and
x
{\displaystyle x}
is a vector of inputs.
=== Step function ===
The output
y
{\displaystyle y}
of this activation function is binary, depending on whether the input meets a specified threshold,
θ
{\displaystyle \theta }
(theta). The "signal" is sent, i.e. the output is set to 1, if the activation meets or exceeds the threshold.
y
=
{
1
if
u
≥
θ
0
if
u
<
θ
{\displaystyle y={\begin{cases}1&{\text{if }}u\geq \theta \\0&{\text{if }}u<\theta \end{cases}}}
This function is used in perceptrons, and appears in many other models. It performs a division of the space of inputs by a hyperplane. It is specially useful in the last layer of a network, intended for example to perform binary classification of the inputs.
=== Linear combination ===
In this case, the output unit is simply the weighted sum of its inputs, plus a bias term. A number of such linear neurons perform a linear transformation of the input vector. This is usually more useful in the early layers of a network. A number of analysis tools exist based on linear models, such as harmonic analysis, and they can all be used in neural networks with this linear neuron. The bias term allows us to make affine transformations to the data.
=== Sigmoid ===
A fairly simple nonlinear function, the sigmoid function such as the logistic function also has an easily calculated derivative, which can be important when calculating the weight updates in the network. It thus makes the network more easily manipulable mathematically, and was attractive to early computer scientists who needed to minimize the computational load of their simulations. It was previously commonly seen in multilayer perceptrons. However, recent work has shown sigmoid neurons to be less effective than rectified linear neurons. The reason is that the gradients computed by the backpropagation algorithm tend to diminish towards zero as activations propagate through layers of sigmoidal neurons, making it difficult to optimize neural networks using multiple layers of sigmoidal neurons.
=== Rectifier ===
In the context of artificial neural networks, the rectifier or ReLU (Rectified Linear Unit) is an activation function defined as the positive part of its argument:
f
(
x
)
=
x
+
=
max
(
0
,
x
)
,
{\displaystyle f(x)=x^{+}=\max(0,x),}
where
x
{\displaystyle x}
is the input to a neuron. This is also known as a ramp function and is analogous to half-wave rectification in electrical engineering. This activation function was first introduced to a dynamical network by Hahnloser et al. in a 2000 paper in Nature with strong biological motivations and mathematical justifications. It has been demonstrated for the first time in 2011 to enable better training of deeper networks, compared to the widely used activation functions prior to 2011, i.e., the logistic sigmoid (which is inspired by probability theory; see logistic regression) and its more practical counterpart, the hyperbolic tangent.
A commonly used variant of the ReLU activation function is the Leaky ReLU which allows a small, positive gradient when the unit is not active:
f
(
x
)
=
{
x
if
x
>
0
,
a
x
otherwise
.
{\displaystyle f(x)={\begin{cases}x&{\text{if }}x>0,\\ax&{\text{otherwise}}.\end{cases}}}
where
x
{\displaystyle x}
is the input to the neuron and
a
{\displaystyle a}
is a small positive constant (set to 0.01 in the original paper).
== Pseudocode algorithm ==
The following is a simple pseudocode implementation of a single Threshold Logic Unit (TLU) which takes Boolean inputs (true or false), and returns a single Boolean output when activated. An object-oriented model is used. No method of training is defined, since several exist. If a purely functional model were used, the class TLU below would be replaced with a function TLU with input parameters threshold, weights, and inputs that returned a Boolean value.
class TLU defined as:
data member threshold : number
data member weights : list of numbers of size X
function member fire(inputs : list of booleans of size X) : boolean defined as:
variable T : number
T ← 0
for each i in 1 to X do
if inputs(i) is true then
T ← T + weights(i)
end if
end for each
if T > threshold then
return true
else:
return false
end if
end function
end class
== See also ==
Binding neuron
Connectionism
== References ==
== Further reading ==
== External links ==
Artifical [sic] neuron mimicks function of human cells
McCulloch-Pitts Neurons (Overview) | Wikipedia/McCulloch-Pitts_neuron |
In quantum computing, Grover's algorithm, also known as the quantum search algorithm, is a quantum algorithm for unstructured search that finds with high probability the unique input to a black box function that produces a particular output value, using just
O
(
N
)
{\displaystyle O({\sqrt {N}})}
evaluations of the function, where
N
{\displaystyle N}
is the size of the function's domain. It was devised by Lov Grover in 1996.
The analogous problem in classical computation would have a query complexity
O
(
N
)
{\displaystyle O(N)}
(i.e., the function would have to be evaluated
O
(
N
)
{\displaystyle O(N)}
times: there is no better approach than trying out all input values one after the other, which, on average, takes
N
/
2
{\displaystyle N/2}
steps).
Charles H. Bennett, Ethan Bernstein, Gilles Brassard, and Umesh Vazirani proved that any quantum solution to the problem needs to evaluate the function
Ω
(
N
)
{\displaystyle \Omega ({\sqrt {N}})}
times, so Grover's algorithm is asymptotically optimal. Since classical algorithms for NP-complete problems require exponentially many steps, and Grover's algorithm provides at most a quadratic speedup over the classical solution for unstructured search, this suggests that Grover's algorithm by itself will not provide polynomial-time solutions for NP-complete problems (as the square root of an exponential function is still an exponential, not a polynomial function).
Unlike other quantum algorithms, which may provide exponential speedup over their classical counterparts, Grover's algorithm provides only a quadratic speedup. However, even quadratic speedup is considerable when
N
{\displaystyle N}
is large, and Grover's algorithm can be applied to speed up broad classes of algorithms. Grover's algorithm could brute-force a 128-bit symmetric cryptographic key in roughly 264 iterations, or a 256-bit key in roughly 2128 iterations. It may not be the case that Grover's algorithm poses a significantly increased risk to encryption over existing classical algorithms, however.
== Applications and limitations ==
Grover's algorithm, along with variants like amplitude amplification, can be used to speed up a broad range of algorithms. In particular, algorithms for NP-complete problems which contain exhaustive search as a subroutine can be sped up by Grover's algorithm. The current theoretical best algorithm, in terms of worst-case complexity, for 3SAT is one such example. Generic constraint satisfaction problems also see quadratic speedups with Grover. These algorithms do not require that the input be given in the form of an oracle, since Grover's algorithm is being applied with an explicit function, e.g. the function checking that a set of bits satisfies a 3SAT instance. However, it is unclear whether Grover's algorithm could speed up best practical algorithms for these problems.
Grover's algorithm can also give provable speedups for black-box problems in quantum query complexity, including element distinctness and the collision problem (solved with the Brassard–Høyer–Tapp algorithm). In these types of problems, one treats the oracle function f as a database, and the goal is to use the quantum query to this function as few times as possible.
=== Cryptography ===
Grover's algorithm essentially solves the task of function inversion. Roughly speaking, if we have a function
y
=
f
(
x
)
{\displaystyle y=f(x)}
that can be evaluated on a quantum computer, Grover's algorithm allows us to calculate
x
{\displaystyle x}
when given
y
{\displaystyle y}
. Consequently, Grover's algorithm gives broad asymptotic speed-ups to many kinds of brute-force attacks on symmetric-key cryptography, including collision attacks and pre-image attacks. However, this may not necessarily be the most efficient algorithm since, for example, the Pollard's rho algorithm is able to find a collision in SHA-2 more efficiently than Grover's algorithm.
=== Limitations ===
Grover's original paper described the algorithm as a database search algorithm, and this description is still common. The database in this analogy is a table of all of the function's outputs, indexed by the corresponding input. However, this database is not represented explicitly. Instead, an oracle is invoked to evaluate an item by its index. Reading a full database item by item and converting it into such a representation may take a lot longer than Grover's search. To account for such effects, Grover's algorithm can be viewed as solving an equation or satisfying a constraint. In such applications, the oracle is a way to check the constraint and is not related to the search algorithm. This separation usually prevents algorithmic optimizations, whereas conventional search algorithms often rely on such optimizations and avoid exhaustive search. Fortunately, fast Grover's oracle implementation is possible for many constraint satisfaction and optimization problems.
The major barrier to instantiating a speedup from Grover's algorithm is that the quadratic speedup achieved is too modest to overcome the large overhead of near-term quantum computers. However, later generations of fault-tolerant quantum computers with better hardware performance may be able to realize these speedups for practical instances of data.
== Problem description ==
As input for Grover's algorithm, suppose we have a function
f
:
{
0
,
1
,
…
,
N
−
1
}
→
{
0
,
1
}
{\displaystyle f\colon \{0,1,\ldots ,N-1\}\to \{0,1\}}
. In the "unstructured database" analogy, the domain represent indices to a database, and f(x) = 1 if and only if the data that x points to satisfies the search criterion. We additionally assume that only one index satisfies f(x) = 1, and we call this index ω. Our goal is to identify ω.
We can access f with a subroutine (sometimes called an oracle) in the form of a unitary operator Uω that acts as follows:
{
U
ω
|
x
⟩
=
−
|
x
⟩
for
x
=
ω
, that is,
f
(
x
)
=
1
,
U
ω
|
x
⟩
=
|
x
⟩
for
x
≠
ω
, that is,
f
(
x
)
=
0.
{\displaystyle {\begin{cases}U_{\omega }|x\rangle =-|x\rangle &{\text{for }}x=\omega {\text{, that is, }}f(x)=1,\\U_{\omega }|x\rangle =|x\rangle &{\text{for }}x\neq \omega {\text{, that is, }}f(x)=0.\end{cases}}}
This uses the
N
{\displaystyle N}
-dimensional state space
H
{\displaystyle {\mathcal {H}}}
, which is supplied by a register with
n
=
⌈
log
2
N
⌉
{\displaystyle n=\lceil \log _{2}N\rceil }
qubits.
This is often written as
U
ω
|
x
⟩
=
(
−
1
)
f
(
x
)
|
x
⟩
.
{\displaystyle U_{\omega }|x\rangle =(-1)^{f(x)}|x\rangle .}
Grover's algorithm outputs ω with probability at least 1/2 using
O
(
N
)
{\displaystyle O({\sqrt {N}})}
applications of Uω. This probability can be made arbitrarily large by running Grover's algorithm multiple times. If one runs Grover's algorithm until ω is found, the expected number of applications is still
O
(
N
)
{\displaystyle O({\sqrt {N}})}
, since it will only be run twice on average.
=== Alternative oracle definition ===
This section compares the above oracle
U
ω
{\displaystyle U_{\omega }}
with an oracle
U
f
{\displaystyle U_{f}}
.
Uω is different from the standard quantum oracle for a function f. This standard oracle, denoted here as Uf, uses an ancillary qubit system. The operation then represents an inversion (NOT gate) on the main system conditioned by the value of f(x) from the ancillary system:
{
U
f
|
x
⟩
|
y
⟩
=
|
x
⟩
|
¬
y
⟩
for
x
=
ω
, that is,
f
(
x
)
=
1
,
U
f
|
x
⟩
|
y
⟩
=
|
x
⟩
|
y
⟩
for
x
≠
ω
, that is,
f
(
x
)
=
0
,
{\displaystyle {\begin{cases}U_{f}|x\rangle |y\rangle =|x\rangle |\neg y\rangle &{\text{for }}x=\omega {\text{, that is, }}f(x)=1,\\U_{f}|x\rangle |y\rangle =|x\rangle |y\rangle &{\text{for }}x\neq \omega {\text{, that is, }}f(x)=0,\end{cases}}}
or briefly,
U
f
|
x
⟩
|
y
⟩
=
|
x
⟩
|
y
⊕
f
(
x
)
⟩
.
{\displaystyle U_{f}|x\rangle |y\rangle =|x\rangle |y\oplus f(x)\rangle .}
These oracles are typically realized using uncomputation.
If we are given Uf as our oracle, then we can also implement Uω, since Uω is Uf when the ancillary qubit is in the state
|
−
⟩
=
1
2
(
|
0
⟩
−
|
1
⟩
)
=
H
|
1
⟩
{\displaystyle |-\rangle ={\frac {1}{\sqrt {2}}}{\big (}|0\rangle -|1\rangle {\big )}=H|1\rangle }
:
U
f
(
|
x
⟩
⊗
|
−
⟩
)
=
1
2
(
U
f
|
x
⟩
|
0
⟩
−
U
f
|
x
⟩
|
1
⟩
)
=
1
2
(
|
x
⟩
|
0
⊕
f
(
x
)
⟩
−
|
x
⟩
|
1
⊕
f
(
x
)
⟩
)
=
{
1
2
(
−
|
x
⟩
|
0
⟩
+
|
x
⟩
|
1
⟩
)
if
f
(
x
)
=
1
,
1
2
(
|
x
⟩
|
0
⟩
−
|
x
⟩
|
1
⟩
)
if
f
(
x
)
=
0
=
(
U
ω
|
x
⟩
)
⊗
|
−
⟩
{\displaystyle {\begin{aligned}U_{f}{\big (}|x\rangle \otimes |-\rangle {\big )}&={\frac {1}{\sqrt {2}}}\left(U_{f}|x\rangle |0\rangle -U_{f}|x\rangle |1\rangle \right)\\&={\frac {1}{\sqrt {2}}}\left(|x\rangle |0\oplus f(x)\rangle -|x\rangle |1\oplus f(x)\rangle \right)\\&={\begin{cases}{\frac {1}{\sqrt {2}}}\left(-|x\rangle |0\rangle +|x\rangle |1\rangle \right)&{\text{if }}f(x)=1,\\{\frac {1}{\sqrt {2}}}\left(|x\rangle |0\rangle -|x\rangle |1\rangle \right)&{\text{if }}f(x)=0\end{cases}}\\&=(U_{\omega }|x\rangle )\otimes |-\rangle \end{aligned}}}
So, Grover's algorithm can be run regardless of which oracle is given. If Uf is given, then we must maintain an additional qubit in the state
|
−
⟩
{\displaystyle |-\rangle }
and apply Uf in place of Uω.
== Algorithm ==
The steps of Grover's algorithm are given as follows:
Initialize the system to the uniform superposition over all states
|
s
⟩
=
1
N
∑
x
=
0
N
−
1
|
x
⟩
.
{\displaystyle |s\rangle ={\frac {1}{\sqrt {N}}}\sum _{x=0}^{N-1}|x\rangle .}
Perform the following "Grover iteration"
r
(
N
)
{\displaystyle r(N)}
times:
Apply the operator
U
ω
{\displaystyle U_{\omega }}
Apply the Grover diffusion operator
U
s
=
2
|
s
⟩
⟨
s
|
−
I
{\displaystyle U_{s}=2\left|s\right\rangle \!\!\left\langle s\right|-I}
Measure the resulting quantum state in the computational basis.
For the correctly chosen value of
r
{\displaystyle r}
, the output will be
|
ω
⟩
{\displaystyle |\omega \rangle }
with probability approaching 1 for N ≫ 1. Analysis shows that this eventual value for
r
(
N
)
{\displaystyle r(N)}
satisfies
r
(
N
)
≤
⌈
π
4
N
⌉
{\displaystyle r(N)\leq {\Big \lceil }{\frac {\pi }{4}}{\sqrt {N}}{\Big \rceil }}
.
Implementing the steps for this algorithm can be done using a number of gates linear in the number of qubits. Thus, the gate complexity of this algorithm is
O
(
log
(
N
)
r
(
N
)
)
{\displaystyle O(\log(N)r(N))}
, or
O
(
log
(
N
)
)
{\displaystyle O(\log(N))}
per iteration.
== Geometric proof of correctness ==
There is a geometric interpretation of Grover's algorithm, following from the observation that the quantum state of Grover's algorithm stays in a two-dimensional subspace after each step. Consider the plane spanned by
|
s
⟩
{\displaystyle |s\rangle }
and
|
ω
⟩
{\displaystyle |\omega \rangle }
; equivalently, the plane spanned by
|
ω
⟩
{\displaystyle |\omega \rangle }
and the perpendicular ket
|
s
′
⟩
=
1
N
−
1
∑
x
≠
ω
|
x
⟩
{\displaystyle \textstyle |s'\rangle ={\frac {1}{\sqrt {N-1}}}\sum _{x\neq \omega }|x\rangle }
.
Grover's algorithm begins with the initial ket
|
s
⟩
{\displaystyle |s\rangle }
, which lies in the subspace. The operator
U
ω
{\displaystyle U_{\omega }}
is a reflection at the hyperplane orthogonal to
|
ω
⟩
{\displaystyle |\omega \rangle }
for vectors in the plane spanned by
|
s
′
⟩
{\displaystyle |s'\rangle }
and
|
ω
⟩
{\displaystyle |\omega \rangle }
, i.e. it acts as a reflection across
|
s
′
⟩
{\displaystyle |s'\rangle }
. This can be seen by writing
U
ω
{\displaystyle U_{\omega }}
in the form of a Householder reflection:
U
ω
=
I
−
2
|
ω
⟩
⟨
ω
|
.
{\displaystyle U_{\omega }=I-2|\omega \rangle \langle \omega |.}
The operator
U
s
=
2
|
s
⟩
⟨
s
|
−
I
{\displaystyle U_{s}=2|s\rangle \langle s|-I}
is a reflection through
|
s
⟩
{\displaystyle |s\rangle }
. Both operators
U
s
{\displaystyle U_{s}}
and
U
ω
{\displaystyle U_{\omega }}
take states in the plane spanned by
|
s
′
⟩
{\displaystyle |s'\rangle }
and
|
ω
⟩
{\displaystyle |\omega \rangle }
to states in the plane. Therefore, Grover's algorithm stays in this plane for the entire algorithm.
It is straightforward to check that the operator
U
s
U
ω
{\displaystyle U_{s}U_{\omega }}
of each Grover iteration step rotates the state vector by an angle of
θ
=
2
arcsin
1
N
{\displaystyle \theta =2\arcsin {\tfrac {1}{\sqrt {N}}}}
. So, with enough iterations, one can rotate from the initial state
|
s
⟩
{\displaystyle |s\rangle }
to the desired output state
|
ω
⟩
{\displaystyle |\omega \rangle }
. The initial ket is close to the state orthogonal to
|
ω
⟩
{\displaystyle |\omega \rangle }
:
⟨
s
′
|
s
⟩
=
N
−
1
N
.
{\displaystyle \langle s'|s\rangle ={\sqrt {\frac {N-1}{N}}}.}
In geometric terms, the angle
θ
/
2
{\displaystyle \theta /2}
between
|
s
⟩
{\displaystyle |s\rangle }
and
|
s
′
⟩
{\displaystyle |s'\rangle }
is given by
sin
θ
2
=
1
N
.
{\displaystyle \sin {\frac {\theta }{2}}={\frac {1}{\sqrt {N}}}.}
We need to stop when the state vector passes close to
|
ω
⟩
{\displaystyle |\omega \rangle }
; after this, subsequent iterations rotate the state vector away from
|
ω
⟩
{\displaystyle |\omega \rangle }
, reducing the probability of obtaining the correct answer. The exact probability of measuring the correct answer is
sin
2
(
(
r
+
1
2
)
θ
)
,
{\displaystyle \sin ^{2}\left({\Big (}r+{\frac {1}{2}}{\Big )}\theta \right),}
where r is the (integer) number of Grover iterations. The earliest time that we get a near-optimal measurement is therefore
r
≈
π
N
/
4
{\displaystyle r\approx \pi {\sqrt {N}}/4}
.
== Algebraic proof of correctness ==
To complete the algebraic analysis, we need to find out what happens when we repeatedly apply
U
s
U
ω
{\displaystyle U_{s}U_{\omega }}
. A natural way to do this is by eigenvalue analysis of a matrix. Notice that during the entire computation, the state of the algorithm is a linear combination of
s
{\displaystyle s}
and
ω
{\displaystyle \omega }
. We can write the action of
U
s
{\displaystyle U_{s}}
and
U
ω
{\displaystyle U_{\omega }}
in the space spanned by
{
|
s
⟩
,
|
ω
⟩
}
{\displaystyle \{|s\rangle ,|\omega \rangle \}}
as:
U
s
:
a
|
ω
⟩
+
b
|
s
⟩
↦
[
|
ω
⟩
|
s
⟩
]
[
−
1
0
2
/
N
1
]
[
a
b
]
.
U
ω
:
a
|
ω
⟩
+
b
|
s
⟩
↦
[
|
ω
⟩
|
s
⟩
]
[
−
1
−
2
/
N
0
1
]
[
a
b
]
.
{\displaystyle {\begin{aligned}U_{s}:a|\omega \rangle +b|s\rangle &\mapsto [|\omega \rangle \,|s\rangle ]{\begin{bmatrix}-1&0\\2/{\sqrt {N}}&1\end{bmatrix}}{\begin{bmatrix}a\\b\end{bmatrix}}.\\U_{\omega }:a|\omega \rangle +b|s\rangle &\mapsto [|\omega \rangle \,|s\rangle ]{\begin{bmatrix}-1&-2/{\sqrt {N}}\\0&1\end{bmatrix}}{\begin{bmatrix}a\\b\end{bmatrix}}.\end{aligned}}}
So in the basis
{
|
ω
⟩
,
|
s
⟩
}
{\displaystyle \{|\omega \rangle ,|s\rangle \}}
(which is neither orthogonal nor a basis of the whole space) the action
U
s
U
ω
{\displaystyle U_{s}U_{\omega }}
of applying
U
ω
{\displaystyle U_{\omega }}
followed by
U
s
{\displaystyle U_{s}}
is given by the matrix
U
s
U
ω
=
[
−
1
0
2
/
N
1
]
[
−
1
−
2
/
N
0
1
]
=
[
1
2
/
N
−
2
/
N
1
−
4
/
N
]
.
{\displaystyle U_{s}U_{\omega }={\begin{bmatrix}-1&0\\2/{\sqrt {N}}&1\end{bmatrix}}{\begin{bmatrix}-1&-2/{\sqrt {N}}\\0&1\end{bmatrix}}={\begin{bmatrix}1&2/{\sqrt {N}}\\-2/{\sqrt {N}}&1-4/N\end{bmatrix}}.}
This matrix happens to have a very convenient Jordan form. If we define
t
=
arcsin
(
1
/
N
)
{\displaystyle t=\arcsin(1/{\sqrt {N}})}
, it is
U
s
U
ω
=
M
[
e
2
i
t
0
0
e
−
2
i
t
]
M
−
1
{\displaystyle U_{s}U_{\omega }=M{\begin{bmatrix}e^{2it}&0\\0&e^{-2it}\end{bmatrix}}M^{-1}}
where
M
=
[
−
i
i
e
i
t
e
−
i
t
]
.
{\displaystyle M={\begin{bmatrix}-i&i\\e^{it}&e^{-it}\end{bmatrix}}.}
It follows that r-th power of the matrix (corresponding to r iterations) is
(
U
s
U
ω
)
r
=
M
[
e
2
r
i
t
0
0
e
−
2
r
i
t
]
M
−
1
.
{\displaystyle (U_{s}U_{\omega })^{r}=M{\begin{bmatrix}e^{2rit}&0\\0&e^{-2rit}\end{bmatrix}}M^{-1}.}
Using this form, we can use trigonometric identities to compute the probability of observing ω after r iterations mentioned in the previous section,
|
[
⟨
ω
|
ω
⟩
⟨
ω
|
s
⟩
]
(
U
s
U
ω
)
r
[
0
1
]
|
2
=
sin
2
(
(
2
r
+
1
)
t
)
.
{\displaystyle \left|{\begin{bmatrix}\langle \omega |\omega \rangle &\langle \omega |s\rangle \end{bmatrix}}(U_{s}U_{\omega })^{r}{\begin{bmatrix}0\\1\end{bmatrix}}\right|^{2}=\sin ^{2}\left((2r+1)t\right).}
Alternatively, one might reasonably imagine that a near-optimal time to distinguish would be when the angles 2rt and −2rt are as far apart as possible, which corresponds to
2
r
t
≈
π
/
2
{\displaystyle 2rt\approx \pi /2}
, or
r
=
π
/
4
t
=
π
/
4
arcsin
(
1
/
N
)
≈
π
N
/
4
{\displaystyle r=\pi /4t=\pi /4\arcsin(1/{\sqrt {N}})\approx \pi {\sqrt {N}}/4}
. Then the system is in state
[
|
ω
⟩
|
s
⟩
]
(
U
s
U
ω
)
r
[
0
1
]
≈
[
|
ω
⟩
|
s
⟩
]
M
[
i
0
0
−
i
]
M
−
1
[
0
1
]
=
|
ω
⟩
1
cos
(
t
)
−
|
s
⟩
sin
(
t
)
cos
(
t
)
.
{\displaystyle [|\omega \rangle \,|s\rangle ](U_{s}U_{\omega })^{r}{\begin{bmatrix}0\\1\end{bmatrix}}\approx [|\omega \rangle \,|s\rangle ]M{\begin{bmatrix}i&0\\0&-i\end{bmatrix}}M^{-1}{\begin{bmatrix}0\\1\end{bmatrix}}=|\omega \rangle {\frac {1}{\cos(t)}}-|s\rangle {\frac {\sin(t)}{\cos(t)}}.}
A short calculation now shows that the observation yields the correct answer ω with error
O
(
1
N
)
{\displaystyle O\left({\frac {1}{N}}\right)}
.
== Extensions and variants ==
=== Multiple matching entries ===
If, instead of 1 matching entry, there are k matching entries, the same algorithm works, but the number of iterations must be
π
4
(
N
k
)
1
/
2
{\textstyle {\frac {\pi }{4}}{\left({\frac {N}{k}}\right)^{1/2}}}
instead of
π
4
N
1
/
2
.
{\textstyle {\frac {\pi }{4}}{N^{1/2}}.}
There are several ways to handle the case if k is unknown. A simple solution performs optimally up to a constant factor: run Grover's algorithm repeatedly for increasingly small values of k, e.g., taking k = N, N/2, N/4, ..., and so on, taking
k
=
N
/
2
t
{\displaystyle k=N/2^{t}}
for iteration t until a matching entry is found.
With sufficiently high probability, a marked entry will be found by iteration
t
=
log
2
(
N
/
k
)
+
c
{\displaystyle t=\log _{2}(N/k)+c}
for some constant c. Thus, the total number of iterations taken is at most
π
4
(
1
+
2
+
4
+
⋯
+
N
k
2
c
)
=
O
(
N
/
k
)
.
{\displaystyle {\frac {\pi }{4}}{\Big (}1+{\sqrt {2}}+{\sqrt {4}}+\cdots +{\sqrt {\frac {N}{k2^{c}}}}{\Big )}=O{\big (}{\sqrt {N/k}}{\big )}.}
Another approach if k is unknown is to derive it via the quantum counting algorithm prior.
If
k
=
N
/
2
{\displaystyle k=N/2}
(or the traditional one marked state Grover's Algorithm if run with
N
=
2
{\displaystyle N=2}
), the algorithm will provide no amplification. If
k
>
N
/
2
{\displaystyle k>N/2}
, increasing k will begin to increase the number of iterations necessary to obtain a solution. On the other hand, if
k
≥
N
/
2
{\displaystyle k\geq N/2}
, a classical running of the checking oracle on a single random choice of input will more likely than not give a correct solution.
A version of this algorithm is used in order to solve the collision problem.
=== Quantum partial search ===
A modification of Grover's algorithm called quantum partial search was described by Grover and Radhakrishnan in 2004. In partial search, one is not interested in finding the exact address of the target item, only the first few digits of the address. Equivalently, we can think of "chunking" the search space into blocks, and then asking "in which block is the target item?". In many applications, such a search yields enough information if the target address contains the information wanted. For instance, to use the example given by L. K. Grover, if one has a list of students organized by class rank, we may only be interested in whether a student is in the lower 25%, 25–50%, 50–75% or 75–100% percentile.
To describe partial search, we consider a database separated into
K
{\displaystyle K}
blocks, each of size
b
=
N
/
K
{\displaystyle b=N/K}
. The partial search problem is easier. Consider the approach we would take classically – we pick one block at random, and then perform a normal search through the rest of the blocks (in set theory language, the complement). If we do not find the target, then we know it is in the block we did not search. The average number of iterations drops from
N
/
2
{\displaystyle N/2}
to
(
N
−
b
)
/
2
{\displaystyle (N-b)/2}
.
Grover's algorithm requires
π
4
N
{\textstyle {\frac {\pi }{4}}{\sqrt {N}}}
iterations. Partial search will be faster by a numerical factor that depends on the number of blocks
K
{\displaystyle K}
. Partial search uses
n
1
{\displaystyle n_{1}}
global iterations and
n
2
{\displaystyle n_{2}}
local iterations. The global Grover operator is designated
G
1
{\displaystyle G_{1}}
and the local Grover operator is designated
G
2
{\displaystyle G_{2}}
.
The global Grover operator acts on the blocks. Essentially, it is given as follows:
Perform
j
1
{\displaystyle j_{1}}
standard Grover iterations on the entire database.
Perform
j
2
{\displaystyle j_{2}}
local Grover iterations. A local Grover iteration is a direct sum of Grover iterations over each block.
Perform one standard Grover iteration.
The optimal values of
j
1
{\displaystyle j_{1}}
and
j
2
{\displaystyle j_{2}}
are discussed in the paper by Grover and Radhakrishnan. One might also wonder what happens if one applies successive partial searches at different levels of "resolution". This idea was studied in detail by Vladimir Korepin and Xu, who called it binary quantum search. They proved that it is not in fact any faster than performing a single partial search.
== Optimality ==
Grover's algorithm is optimal up to sub-constant factors. That is, any algorithm that accesses the database only by using the operator Uω must apply Uω at least a
1
−
o
(
1
)
{\displaystyle 1-o(1)}
fraction as many times as Grover's algorithm. The extension of Grover's algorithm to k matching entries, π(N/k)1/2/4, is also optimal. This result is important in understanding the limits of quantum computation.
If the Grover's search problem was solvable with logc N applications of Uω, that would imply that NP is contained in BQP, by transforming problems in NP into Grover-type search problems. The optimality of Grover's algorithm suggests that quantum computers cannot solve NP-Complete problems in polynomial time, and thus NP is not contained in BQP.
It has been shown that a class of non-local hidden variable quantum computers could implement a search of an
N
{\displaystyle N}
-item database in at most
O
(
N
3
)
{\displaystyle O({\sqrt[{3}]{N}})}
steps. This is faster than the
O
(
N
)
{\displaystyle O({\sqrt {N}})}
steps taken by Grover's algorithm.
== See also ==
Amplitude amplification
Brassard–Høyer–Tapp algorithm (for solving the collision problem)
Shor's algorithm (for factorization)
Quantum walk search
== Notes ==
== References ==
Grover L.K.: A fast quantum mechanical algorithm for database search, Proceedings, 28th Annual ACM Symposium on the Theory of Computing, (May 1996) p. 212
Grover L.K.: From Schrödinger's equation to quantum search algorithm, American Journal of Physics, 69(7): 769–777, 2001. Pedagogical review of the algorithm and its history.
Grover L.K.: QUANTUM COMPUTING: How the weird logic of the subatomic world could make it possible for machines to calculate millions of times faster than they do today The Sciences, July/August 1999, pp. 24–30.
Nielsen, M.A. and Chuang, I.L. Quantum computation and quantum information. Cambridge University Press, 2000. Chapter 6.
What's a Quantum Phone Book?, Lov Grover, Lucent Technologies
== External links ==
Davy Wybiral. "Quantum Circuit Simulator". Archived from the original on 2017-01-16. Retrieved 2017-01-13.
Craig Gidney (2013-03-05). "Grover's Quantum Search Algorithm". Archived from the original on 2020-11-17. Retrieved 2013-03-08.
François Schwarzentruber (2013-05-18). "Grover's algorithm".
Alexander Prokopenya. "Quantum Circuit Implementing Grover's Search Algorithm". Wolfram Alpha.
"Quantum computation, theory of", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Roberto Maestre (2018-05-11). "Grover's Algorithm implemented in R and C". GitHub.
Bernhard Ömer. "QCL - A Programming Language for Quantum Computers". Retrieved 2022-04-30. Implemented in /qcl-0.6.4/lib/grover.qcl | Wikipedia/Grover_search_algorithm |
A Hopfield network (or associative memory) is a form of recurrent neural network, or a spin glass system, that can serve as a content-addressable memory. The Hopfield network, named for John Hopfield, consists of a single layer of neurons, where each neuron is connected to every other neuron except itself. These connections are bidirectional and symmetric, meaning the weight of the connection from neuron i to neuron j is the same as the weight from neuron j to neuron i. Patterns are associatively recalled by fixing certain inputs, and dynamically evolve the network to minimize an energy function, towards local energy minimum states that correspond to stored patterns. Patterns are associatively learned (or "stored") by a Hebbian learning algorithm.
One of the key features of Hopfield networks is their ability to recover complete patterns from partial or noisy inputs, making them robust in the face of incomplete or corrupted data. Their connection to statistical mechanics, recurrent networks, and human cognitive psychology has led to their application in various fields, including physics, psychology, neuroscience, and machine learning theory and practice.
== History ==
One origin of associative memory is human cognitive psychology, specifically the associative memory. Frank Rosenblatt studied "close-loop cross-coupled perceptrons", which are 3-layered perceptron networks whose middle layer contains recurrent connections that change by a Hebbian learning rule.: 73–75 : Chapter 19, 21
Another model of associative memory is where the output does not loop back to the input. W. K. Taylor proposed such a model trained by Hebbian learning in 1956. Karl Steinbuch, who wanted to understand learning, and inspired by watching his children learn, published the Lernmatrix in 1961. It was translated to English in 1963. Similar research was done with the correlogram of D. J. Willshaw et al. in 1969. Teuvo Kohonen trained an associative memory by gradient descent in 1974.
Another origin of associative memory was statistical mechanics. The Ising model was published in 1920s as a model of magnetism, however it studied the thermal equilibrium, which does not change with time. Roy J. Glauber in 1963 studied the Ising model evolving in time, as a process towards thermal equilibrium (Glauber dynamics), adding in the component of time.
The second component to be added was adaptation to stimulus. Described independently by Kaoru Nakano in 1971 and Shun'ichi Amari in 1972, they proposed to modify the weights of an Ising model by Hebbian learning rule as a model of associative memory. The same idea was published by William A. Little in 1974, who was acknowledged by Hopfield in his 1982 paper.
See Carpenter (1989) and Cowan (1990) for a technical description of some of these early works in associative memory.
The Sherrington–Kirkpatrick model of spin glass, published in 1975, is the Hopfield network with random initialization. Sherrington and Kirkpatrick found that it is highly likely for the energy function of the SK model to have many local minima. In the 1982 paper, Hopfield applied this recently developed theory to study the Hopfield network with binary activation functions. In a 1984 paper he extended this to continuous activation functions. It became a standard model for the study of neural networks through statistical mechanics.
A major advance in memory storage capacity was developed by Dimitry Krotov and Hopfield in 2016 through a change in network dynamics and energy function. This idea was further extended by Demircigil and collaborators in 2017. The continuous dynamics of large memory capacity models was developed in a series of papers between 2016 and 2020. Large memory storage capacity Hopfield Networks are now called Dense Associative Memories or modern Hopfield networks.
In 2024, John J. Hopfield and Geoffrey E. Hinton were awarded the Nobel Prize in Physics for their foundational contributions to machine learning, such as the Hopfield network.
== Structure ==
The units in Hopfield nets are binary threshold units, i.e. the units only take on two different values for their states, and the value is determined by whether or not the unit's input exceeds its threshold
U
i
{\displaystyle U_{i}}
. Discrete Hopfield nets describe relationships between binary (firing or not-firing) neurons
1
,
2
,
…
,
i
,
j
,
…
,
N
{\displaystyle 1,2,\ldots ,i,j,\ldots ,N}
. At a certain time, the state of the neural net is described by a vector
V
{\displaystyle V}
, which records which neurons are firing in a binary word of
N
{\displaystyle N}
bits.
The interactions
w
i
j
{\displaystyle w_{ij}}
between neurons have units that usually take on values of 1 or −1, and this convention will be used throughout this article. However, other literature might use units that take values of 0 and 1. These interactions are "learned" via Hebb's law of association, such that, for a certain state
V
s
{\displaystyle V^{s}}
and distinct nodes
i
,
j
{\displaystyle i,j}
w
i
j
=
V
i
s
V
j
s
{\displaystyle w_{ij}=V_{i}^{s}V_{j}^{s}}
but
w
i
i
=
0
{\displaystyle w_{ii}=0}
.
(Note that the Hebbian learning rule takes the form
w
i
j
=
(
2
V
i
s
−
1
)
(
2
V
j
s
−
1
)
{\displaystyle w_{ij}=(2V_{i}^{s}-1)(2V_{j}^{s}-1)}
when the units assume values in
{
0
,
1
}
{\displaystyle \{0,1\}}
.)
Once the network is trained,
w
i
j
{\displaystyle w_{ij}}
no longer evolve. If a new state of neurons
V
s
′
{\displaystyle V^{s'}}
is introduced to the neural network, the net acts on neurons such that
V
i
s
′
→
1
{\displaystyle V_{i}^{s'}\rightarrow 1}
if
∑
j
w
i
j
V
j
s
′
≥
U
i
{\displaystyle \sum _{j}w_{ij}V_{j}^{s'}\geq U_{i}}
V
i
s
′
→
−
1
{\displaystyle V_{i}^{s'}\rightarrow -1}
if
∑
j
w
i
j
V
j
s
′
<
U
i
{\displaystyle \sum _{j}w_{ij}V_{j}^{s'}<U_{i}}
where
U
i
{\displaystyle U_{i}}
is the threshold value of the i'th neuron (often taken to be 0). In this way, Hopfield networks have the ability to "remember" states stored in the interaction matrix, because if a new state
V
s
′
{\displaystyle V^{s'}}
is subjected to the interaction matrix, each neuron will change until it matches the original state
V
s
{\displaystyle V^{s}}
(see the Updates section below).
The connections in a Hopfield net typically have the following restrictions:
w
i
i
=
0
,
∀
i
{\displaystyle w_{ii}=0,\forall i}
(no unit has a connection with itself)
w
i
j
=
w
j
i
,
∀
i
,
j
{\displaystyle w_{ij}=w_{ji},\forall i,j}
(connections are symmetric)
The constraint that weights are symmetric guarantees that the energy function decreases monotonically while following the activation rules. A network with asymmetric weights may exhibit some periodic or chaotic behaviour; however, Hopfield found that this behavior is confined to relatively small parts of the phase space and does not impair the network's ability to act as a content-addressable associative memory system.
Hopfield also modeled neural nets for continuous values, in which the electric output of each neuron is not binary but some value between 0 and 1. He found that this type of network was also able to store and reproduce memorized states.
Notice that every pair of units i and j in a Hopfield network has a connection that is described by the connectivity weight
w
i
j
{\displaystyle w_{ij}}
. In this sense, the Hopfield network can be formally described as a complete undirected graph
G
=
⟨
V
,
f
⟩
{\displaystyle G=\langle V,f\rangle }
, where
V
{\displaystyle V}
is a set of McCulloch–Pitts neurons and
f
:
V
2
→
R
{\displaystyle f:V^{2}\rightarrow \mathbb {R} }
is a function that links pairs of units to a real value, the connectivity weight.
== Updating ==
Updating one unit (node in the graph simulating the artificial neuron) in the Hopfield network is performed using the following rule:
s
i
←
{
+
1
if
∑
j
w
i
j
s
j
≥
θ
i
,
−
1
otherwise.
{\displaystyle s_{i}\leftarrow \left\{{\begin{array}{ll}+1&{\text{if }}\sum _{j}{w_{ij}s_{j}}\geq \theta _{i},\\-1&{\text{otherwise.}}\end{array}}\right.}
where:
w
i
j
{\displaystyle w_{ij}}
is the strength of the connection weight from unit j to unit i (the weight of the connection).
s
i
{\displaystyle s_{i}}
is the state of unit i.
θ
i
{\displaystyle \theta _{i}}
is the threshold of unit i.
Updates in the Hopfield network can be performed in two different ways:
Asynchronous: Only one unit is updated at a time. This unit can be picked at random, or a pre-defined order can be imposed from the very beginning.
Synchronous: All units are updated at the same time. This requires a central clock to the system in order to maintain synchronization. This method is viewed by some as less realistic, based on an absence of observed global clock influencing analogous biological or physical systems of interest.
=== Neurons "attract or repel each other" in state space ===
The weight between two units has a powerful impact upon the values of the neurons. Consider the connection weight
w
i
j
{\displaystyle w_{ij}}
between two neurons i and j. If
w
i
j
>
0
{\displaystyle w_{ij}>0}
, the updating rule implies that:
when
s
j
=
1
{\displaystyle s_{j}=1}
, the contribution of j in the weighted sum is positive. Thus,
s
i
{\displaystyle s_{i}}
is pulled by j towards its value
s
i
=
1
{\displaystyle s_{i}=1}
when
s
j
=
−
1
{\displaystyle s_{j}=-1}
, the contribution of j in the weighted sum is negative. Then again,
s
i
{\displaystyle s_{i}}
is pushed by j towards its value
s
i
=
−
1
{\displaystyle s_{i}=-1}
Thus, the values of neurons i and j will converge if the weight between them is positive. Similarly, they will diverge if the weight is negative.
== Convergence properties of discrete and continuous Hopfield networks ==
Bruck in his paper in 1990 studied discrete Hopfield networks and proved a generalized convergence theorem that is based on the connection between the network's dynamics and cuts in the associated graph. This generalization covered both asynchronous as well as synchronous dynamics and presented elementary proofs based on greedy algorithms for max-cut in graphs. A subsequent paper further investigated the behavior of any neuron in both discrete-time and continuous-time Hopfield networks when the corresponding energy function is minimized during an optimization process. Bruck showed that neuron j changes its state if and only if it further decreases the following biased pseudo-cut. The discrete Hopfield network minimizes the following biased pseudo-cut for the synaptic weight matrix of the Hopfield net.
J
p
s
e
u
d
o
−
c
u
t
(
k
)
=
∑
i
∈
C
1
(
k
)
∑
j
∈
C
2
(
k
)
w
i
j
+
∑
j
∈
C
1
(
k
)
θ
j
{\displaystyle J_{pseudo-cut}(k)=\sum _{i\in C_{1}(k)}\sum _{j\in C_{2}(k)}w_{ij}+\sum _{j\in C_{1}(k)}{\theta _{j}}}
where
C
1
(
k
)
{\displaystyle C_{1}(k)}
and
C
2
(
k
)
{\displaystyle C_{2}(k)}
represents the set of neurons which are −1 and +1, respectively, at time
k
{\displaystyle k}
. For further details, see the recent paper.
The discrete-time Hopfield Network always minimizes exactly the following pseudo-cut
U
(
k
)
=
∑
i
=
1
N
∑
j
=
1
N
w
i
j
(
s
i
(
k
)
−
s
j
(
k
)
)
2
+
2
∑
j
=
1
N
θ
j
s
j
(
k
)
{\displaystyle U(k)=\sum _{i=1}^{N}\sum _{j=1}^{N}w_{ij}(s_{i}(k)-s_{j}(k))^{2}+2\sum _{j=1}^{N}\theta _{j}s_{j}(k)}
The continuous-time Hopfield network always minimizes an upper bound to the following weighted cut
V
(
t
)
=
∑
i
=
1
N
∑
j
=
1
N
w
i
j
(
f
(
s
i
(
t
)
)
−
f
(
s
j
(
t
)
)
2
+
2
∑
j
=
1
N
θ
j
f
(
s
j
(
t
)
)
{\displaystyle V(t)=\sum _{i=1}^{N}\sum _{j=1}^{N}w_{ij}(f(s_{i}(t))-f(s_{j}(t))^{2}+2\sum _{j=1}^{N}\theta _{j}f(s_{j}(t))}
where
f
(
⋅
)
{\displaystyle f(\cdot )}
is a zero-centered sigmoid function.
The complex Hopfield network, on the other hand, generally tends to minimize the so-called shadow-cut of the complex weight matrix of the net.
== Energy ==
Hopfield nets have a scalar value associated with each state of the network, referred to as the "energy", E, of the network, where:
E
=
−
1
2
∑
i
,
j
w
i
j
s
i
s
j
−
∑
i
θ
i
s
i
{\displaystyle E=-{\frac {1}{2}}\sum _{i,j}w_{ij}s_{i}s_{j}-\sum _{i}\theta _{i}s_{i}}
This quantity is called "energy" because it either decreases or stays the same upon network units being updated. Furthermore, under repeated updating the network will eventually converge to a state which is a local minimum in the energy function (which is considered to be a Lyapunov function). Thus, if a state is a local minimum in the energy function it is a stable state for the network. Note that this energy function belongs to a general class of models in physics under the name of Ising models; these in turn are a special case of Markov networks, since the associated probability measure, the Gibbs measure, has the Markov property.
== Hopfield network in optimization ==
Hopfield and Tank presented the Hopfield network application in solving the classical traveling-salesman problem in 1985. Since then, the Hopfield network has been widely used for optimization. The idea of using the Hopfield network in optimization problems is straightforward: If a constrained/unconstrained cost function can be written in the form of the Hopfield energy function E, then there exists a Hopfield network whose equilibrium points represent solutions to the constrained/unconstrained optimization problem. Minimizing the Hopfield energy function both minimizes the objective function and satisfies the constraints also as the constraints are "embedded" into the synaptic weights of the network. Although including the optimization constraints into the synaptic weights in the best possible way is a challenging task, many difficult optimization problems with constraints in different disciplines have been converted to the Hopfield energy function: Associative memory systems, Analog-to-Digital conversion, job-shop scheduling problem, quadratic assignment and other related NP-complete problems, channel allocation problem in wireless networks, mobile ad-hoc network routing problem, image restoration, system identification, combinatorial optimization, etc, just to name a few. However, while it is possible to convert hard optimization problems to Hopfield energy functions, it does not guarantee convergence to a solution (even in exponential time).
== Initialization and running ==
Initialization of the Hopfield networks is done by setting the values of the units to the desired start pattern. Repeated updates are then performed until the network converges to an attractor pattern. Convergence is generally assured, as Hopfield proved that the attractors of this nonlinear dynamical system are stable, not periodic or chaotic as in some other systems. Therefore, in the context of Hopfield networks, an attractor pattern is a final stable state, a pattern that cannot change any value within it under updating.
== Training ==
Training a Hopfield net involves lowering the energy of states that the net should "remember". This allows the net to serve as a content addressable memory system, that is to say, the network will converge to a "remembered" state if it is given only part of the state. The net can be used to recover from a distorted input to the trained state that is most similar to that input. This is called associative memory because it recovers memories on the basis of similarity. For example, if we train a Hopfield net with five units so that the state (1, −1, 1, −1, 1) is an energy minimum, and we give the network the state (1, −1, −1, −1, 1) it will converge to (1, −1, 1, −1, 1). Thus, the network is properly trained when the energy of states which the network should remember are local minima. Note that, in contrast to Perceptron training, the thresholds of the neurons are never updated.
=== Learning rules ===
There are various different learning rules that can be used to store information in the memory of the Hopfield network. It is desirable for a learning rule to have both of the following two properties:
Local: A learning rule is local if each weight is updated using information available to neurons on either side of the connection that is associated with that particular weight.
Incremental: New patterns can be learned without using information from the old patterns that have been also used for training. That is, when a new pattern is used for training, the new values for the weights only depend on the old values and on the new pattern.
These properties are desirable, since a learning rule satisfying them is more biologically plausible. For example, since the human brain is always learning new concepts, one can reason that human learning is incremental. A learning system that was not incremental would generally be trained only once, with a huge batch of training data.
=== Hebbian learning rule for Hopfield networks ===
Hebbian theory was introduced by Donald Hebb in 1949 in order to explain "associative learning", in which simultaneous activation of neuron cells leads to pronounced increases in synaptic strength between those cells. It is often summarized as "Neurons that fire together wire together. Neurons that fire out of sync fail to link".
The Hebbian rule is both local and incremental. For the Hopfield networks, it is implemented in the following manner when learning
n
{\displaystyle n}
binary patterns:
w
i
j
=
1
n
∑
μ
=
1
n
ϵ
i
μ
ϵ
j
μ
{\displaystyle w_{ij}={\frac {1}{n}}\sum _{\mu =1}^{n}\epsilon _{i}^{\mu }\epsilon _{j}^{\mu }}
where
ϵ
i
μ
{\displaystyle \epsilon _{i}^{\mu }}
represents bit i from pattern
μ
{\displaystyle \mu }
.
If the bits corresponding to neurons i and j are equal in pattern
μ
{\displaystyle \mu }
, then the product
ϵ
i
μ
ϵ
j
μ
{\displaystyle \epsilon _{i}^{\mu }\epsilon _{j}^{\mu }}
will be positive. This would, in turn, have a positive effect on the weight
w
i
j
{\displaystyle w_{ij}}
and the values of i and j will tend to become equal. The opposite happens if the bits corresponding to neurons i and j are different.
=== Storkey learning rule ===
This rule was introduced by Amos Storkey in 1997 and is both local and incremental. Storkey also showed that a Hopfield network trained using this rule has a greater capacity than a corresponding network trained using the Hebbian rule. The weight matrix of an attractor neural network is said to follow the Storkey learning rule if it obeys:
w
i
j
ν
=
w
i
j
ν
−
1
+
1
n
ϵ
i
ν
ϵ
j
ν
−
1
n
ϵ
i
ν
h
j
i
ν
−
1
n
ϵ
j
ν
h
i
j
ν
{\displaystyle w_{ij}^{\nu }=w_{ij}^{\nu -1}+{\frac {1}{n}}\epsilon _{i}^{\nu }\epsilon _{j}^{\nu }-{\frac {1}{n}}\epsilon _{i}^{\nu }h_{ji}^{\nu }-{\frac {1}{n}}\epsilon _{j}^{\nu }h_{ij}^{\nu }}
where
h
i
j
ν
=
∑
k
=
1
:
i
≠
k
≠
j
n
w
i
k
ν
−
1
ϵ
k
ν
{\displaystyle h_{ij}^{\nu }=\sum _{k=1~:~i\neq k\neq j}^{n}w_{ik}^{\nu -1}\epsilon _{k}^{\nu }}
is a form of local field at neuron i.
This learning rule is local, since the synapses take into account only neurons at their sides. The rule makes use of more information from the patterns and weights than the generalized Hebbian rule, due to the effect of the local field.
== Spurious patterns ==
Patterns that the network uses for training (called retrieval states) become attractors of the system. Repeated updates would eventually lead to convergence to one of the retrieval states. However, sometimes the network will converge to spurious patterns (different from the training patterns). In fact, the number of spurious patterns can be exponential in the number of stored patterns, even if the stored patterns are orthogonal. The energy in these spurious patterns is also a local minimum. For each stored pattern x, the negation -x is also a spurious pattern.
A spurious state can also be a linear combination of an odd number of retrieval states. For example, when using 3 patterns
μ
1
,
μ
2
,
μ
3
{\displaystyle \mu _{1},\mu _{2},\mu _{3}}
, one can get the following spurious state:
ϵ
i
m
i
x
=
±
sgn
(
±
ϵ
i
μ
1
±
ϵ
i
μ
2
±
ϵ
i
μ
3
)
{\displaystyle \epsilon _{i}^{\rm {mix}}=\pm \operatorname {sgn}(\pm \epsilon _{i}^{\mu _{1}}\pm \epsilon _{i}^{\mu _{2}}\pm \epsilon _{i}^{\mu _{3}})}
Spurious patterns that have an even number of states cannot exist, since they might sum up to zero
== Capacity ==
The Network capacity of the Hopfield network model is determined by neuron amounts and connections within a given network. Therefore, the number of memories that are able to be stored is dependent on neurons and connections. Furthermore, it was shown that the recall accuracy between vectors and nodes was 0.138 (approximately 138 vectors can be recalled from storage for every 1000 nodes) (Hertz et al., 1991). Therefore, it is evident that many mistakes will occur if one tries to store a large number of vectors. When the Hopfield model does not recall the right pattern, it is possible that an intrusion has taken place, since semantically related items tend to confuse the individual, and recollection of the wrong pattern occurs. Therefore, the Hopfield network model is shown to confuse one stored item with that of another upon retrieval. Perfect recalls and high capacity, >0.14, can be loaded in the network by Storkey learning method; ETAM, ETAM experiments also in. Ulterior models inspired by the Hopfield network were later devised to raise the storage limit and reduce the retrieval error rate, with some being capable of one-shot learning.
The storage capacity can be given as
C
≅
n
2
log
2
n
{\displaystyle C\cong {\frac {n}{2\log _{2}n}}}
where
n
{\displaystyle n}
is the number of neurons in the net.
== Human memory ==
The Hopfield network is a model for human associative learning and recall. It accounts for associative memory through the incorporation of memory vectors. Memory vectors can be slightly used, and this would spark the retrieval of the most similar vector in the network. However, we will find out that due to this process, intrusions can occur. In associative memory for the Hopfield network, there are two types of operations: auto-association and hetero-association. The first being when a vector is associated with itself, and the latter being when two different vectors are associated in storage. Furthermore, both types of operations are possible to store within a single memory matrix, but only if that given representation matrix is not one or the other of the operations, but rather the combination (auto-associative and hetero-associative) of the two.
Hopfield's network model utilizes the same learning rule as Hebb's (1949) learning rule, which characterised learning as being a result of the strengthening of the weights in cases of neuronal activity.
Rizzuto and Kahana (2001) were able to show that the neural network model can account for repetition on recall accuracy by incorporating a probabilistic-learning algorithm. During the retrieval process, no learning occurs. As a result, the weights of the network remain fixed, showing that the model is able to switch from a learning stage to a recall stage. By adding contextual drift they were able to show the rapid forgetting that occurs in a Hopfield model during a cued-recall task. The entire network contributes to the change in the activation of any single node.
McCulloch and Pitts' (1943) dynamical rule, which describes the behavior of neurons, does so in a way that shows how the activations of multiple neurons map onto the activation of a new neuron's firing rate, and how the weights of the neurons strengthen the synaptic connections between the new activated neuron (and those that activated it). Hopfield would use McCulloch–Pitts's dynamical rule in order to show how retrieval is possible in the Hopfield network. However, Hopfield would do so in a repetitious fashion. Hopfield would use a nonlinear activation function, instead of using a linear function. This would therefore create the Hopfield dynamical rule and with this, Hopfield was able to show that with the nonlinear activation function, the dynamical rule will always modify the values of the state vector in the direction of one of the stored patterns.
== Dense associative memory or modern Hopfield network ==
Hopfield networks are recurrent neural networks with dynamical trajectories converging to fixed point attractor states and described by an energy function. The state of each model neuron
i
{\textstyle i}
is defined by a time-dependent variable
V
i
{\displaystyle V_{i}}
, which can be chosen to be either discrete or continuous. A complete model describes the mathematics of how the future state of activity of each neuron depends on the known present or previous activity of all the neurons.
In the original Hopfield model of associative memory, the variables were binary, and the dynamics were described by a one-at-a-time update of the state of the neurons. An energy function quadratic in the
V
i
{\displaystyle V_{i}}
was defined, and the dynamics consisted of changing the activity of each single neuron
i
{\displaystyle i}
only if doing so would lower the total energy of the system. This same idea was extended to the case of
V
i
{\displaystyle V_{i}}
being a continuous variable representing the output of neuron
i
{\displaystyle i}
, and
V
i
{\displaystyle V_{i}}
being a monotonic function of an input current. The dynamics became expressed as a set of first-order differential equations for which the "energy" of the system always decreased. The energy in the continuous case has one term which is quadratic in the
V
i
{\displaystyle V_{i}}
(as in the binary model), and a second term which depends on the gain function (neuron's activation function). While having many desirable properties of associative memory, both of these classical systems suffer from a small memory storage capacity, which scales linearly with the number of input features. In contrast, by increasing the number of parameters in the model so that there are not just pair-wise but also higher-order interactions between the neurons, one can increase the memory storage capacity.
Dense Associative Memories (also known as the modern Hopfield networks) are generalizations of the classical Hopfield Networks that break the linear scaling relationship between the number of input features and the number of stored memories. This is achieved by introducing stronger non-linearities (either in the energy function or neurons' activation functions) leading to super-linear (even an exponential) memory storage capacity as a function of the number of feature neurons, in effect increasing the order of interactions between the neurons. The network still requires a sufficient number of hidden neurons.
The key theoretical idea behind dense associative memory networks is to use an energy function and an update rule that is more sharply peaked around the stored memories in the space of neuron's configurations compared to the classical model, as demonstrated when the higher-order interactions and subsequent energy landscapes are explicitly modelled.
=== Discrete variables ===
A simple example of the modern Hopfield network can be written in terms of binary variables
V
i
{\displaystyle V_{i}}
that represent the active
V
i
=
+
1
{\displaystyle V_{i}=+1}
and inactive
V
i
=
−
1
{\displaystyle V_{i}=-1}
state of the model neuron
i
{\displaystyle i}
.
E
=
−
∑
μ
=
1
N
mem
F
(
∑
i
=
1
N
f
ξ
μ
i
V
i
)
{\displaystyle E=-\sum \limits _{\mu =1}^{N_{\text{mem}}}F{\Big (}\sum \limits _{i=1}^{N_{f}}\xi _{\mu i}V_{i}{\Big )}}
In this formula the weights
ξ
μ
i
{\textstyle \xi _{\mu i}}
represent the matrix of memory vectors (index
μ
=
1...
N
mem
{\displaystyle \mu =1...N_{\text{mem}}}
enumerates different memories, and index
i
=
1...
N
f
{\displaystyle i=1...N_{f}}
enumerates the content of each memory corresponding to the
i
{\displaystyle i}
-th feature neuron), and the function
F
(
x
)
{\displaystyle F(x)}
is a rapidly growing non-linear function. The update rule for individual neurons (in the asynchronous case) can be written in the following form
V
i
(
t
+
1
)
=
S
i
g
n
[
∑
μ
=
1
N
mem
(
F
(
ξ
μ
i
+
∑
j
≠
i
ξ
μ
j
V
j
(
t
)
)
−
F
(
−
ξ
μ
i
+
∑
j
≠
i
ξ
μ
j
V
j
(
t
)
)
)
]
{\displaystyle V_{i}^{(t+1)}=Sign{\bigg [}\sum \limits _{\mu =1}^{N_{\text{mem}}}{\bigg (}F{\Big (}\xi _{\mu i}+\sum \limits _{j\neq i}\xi _{\mu j}V_{j}^{(t)}{\Big )}-F{\Big (}-\xi _{\mu i}+\sum \limits _{j\neq i}\xi _{\mu j}V_{j}^{(t)}{\Big )}{\bigg )}{\bigg ]}}
which states that in order to calculate the updated state of the
i
{\textstyle i}
-th neuron the network compares two energies: the energy of the network with the
i
{\displaystyle i}
-th neuron in the ON state and the energy of the network with the
i
{\displaystyle i}
-th neuron in the OFF state, given the states of the remaining neuron. The updated state of the
i
{\displaystyle i}
-th neuron selects the state that has the lowest of the two energies.
In the limiting case when the non-linear energy function is quadratic
F
(
x
)
=
x
2
{\displaystyle F(x)=x^{2}}
these equations reduce to the familiar energy function and the update rule for the classical binary Hopfield Network.
The memory storage capacity of these networks can be calculated for random binary patterns. For the power energy function
F
(
x
)
=
x
n
{\displaystyle F(x)=x^{n}}
the maximal number of memories that can be stored and retrieved from this network without errors is given by
N
mem
m
a
x
≈
1
2
(
2
n
−
3
)
!
!
N
f
n
−
1
ln
(
N
f
)
{\displaystyle N_{\text{mem}}^{max}\approx {\frac {1}{2(2n-3)!!}}{\frac {N_{f}^{n-1}}{\ln(N_{f})}}}
For an exponential energy function
F
(
x
)
=
e
x
{\textstyle F(x)=e^{x}}
the memory storage capacity is exponential in the number of feature neurons
N
mem
m
a
x
≈
2
N
f
/
2
{\displaystyle N_{\text{mem}}^{max}\approx 2^{N_{f}/2}}
=== Continuous variables ===
Modern Hopfield networks or dense associative memories can be best understood in continuous variables and continuous time. Consider the network architecture, shown in Fig.1, and the equations for neuron's states evolutionwhere the currents of the feature neurons are denoted by
x
i
{\textstyle x_{i}}
, and the currents of the memory neurons are denoted by
h
μ
{\displaystyle h_{\mu }}
(
h
{\displaystyle h}
stands for hidden neurons). There are no synaptic connections among the feature neurons or the memory neurons. A matrix
ξ
μ
i
{\displaystyle \xi _{\mu i}}
denotes the strength of synapses from a feature neuron
i
{\displaystyle i}
to the memory neuron
μ
{\displaystyle \mu }
. The synapses are assumed to be symmetric, so that the same value characterizes a different physical synapse from the memory neuron
μ
{\displaystyle \mu }
to the feature neuron
i
{\displaystyle i}
. The outputs of the memory neurons and the feature neurons are denoted by
f
μ
{\displaystyle f_{\mu }}
and
g
i
{\displaystyle g_{i}}
, which are non-linear functions of the corresponding currents. In general these outputs can depend on the currents of all the neurons in that layer so that
f
μ
=
f
(
{
h
μ
}
)
{\displaystyle f_{\mu }=f(\{h_{\mu }\})}
and
g
i
=
g
(
{
x
i
}
)
{\textstyle g_{i}=g(\{x_{i}\})}
. It is convenient to define these activation functions as derivatives of the Lagrangian functions for the two groups of neuronsThis way the specific form of the equations for neuron's states is completely defined once the Lagrangian functions are specified. Finally, the time constants for the two groups of neurons are denoted by
τ
f
{\displaystyle \tau _{f}}
and
τ
h
{\displaystyle \tau _{h}}
,
I
i
{\displaystyle I_{i}}
is the input current to the network that can be driven by the presented data.
General systems of non-linear differential equations can have many complicated behaviors that can depend on the choice of the non-linearities and the initial conditions. For Hopfield Networks, however, this is not the case - the dynamical trajectories always converge to a fixed point attractor state. This property is achieved because these equations are specifically engineered so that they have an underlying energy function The terms grouped into square brackets represent a Legendre transform of the Lagrangian function with respect to the states of the neurons. If the Hessian matrices of the Lagrangian functions are positive semi-definite, the energy function is guaranteed to decrease on the dynamical trajectory This property makes it possible to prove that the system of dynamical equations describing temporal evolution of neurons' activities will eventually reach a fixed point attractor state.
In certain situations one can assume that the dynamics of hidden neurons equilibrates at a much faster time scale compared to the feature neurons,
τ
h
≪
τ
f
{\textstyle \tau _{h}\ll \tau _{f}}
. In this case the steady state solution of the second equation in the system (1) can be used to express the currents of the hidden units through the outputs of the feature neurons. This makes it possible to reduce the general theory (1) to an effective theory for feature neurons only. The resulting effective update rules and the energies for various common choices of the Lagrangian functions are shown in Fig.2. In the case of log-sum-exponential Lagrangian function the update rule (if applied once) for the states of the feature neurons is the attention mechanism commonly used in many modern AI systems (see Ref. for the derivation of this result from the continuous time formulation).
=== Relationship to classical Hopfield network with continuous variables ===
Classical formulation of continuous Hopfield Networks can be understood as a special limiting case of the modern Hopfield networks with one hidden layer. Continuous Hopfield Networks for neurons with graded response are typically described by the dynamical equations and the energy function where
V
i
=
g
(
x
i
)
{\textstyle V_{i}=g(x_{i})}
, and
g
−
1
(
z
)
{\displaystyle g^{-1}(z)}
is the inverse of the activation function
g
(
x
)
{\displaystyle g(x)}
. This model is a special limit of the class of models that is called models A, with the following choice of the Lagrangian functions that, according to the definition (2), leads to the activation functions If we integrate out the hidden neurons the system of equations (1) reduces to the equations on the feature neurons (5) with
T
i
j
=
∑
μ
=
1
N
h
ξ
μ
i
ξ
μ
j
{\displaystyle T_{ij}=\sum \limits _{\mu =1}^{N_{h}}\xi _{\mu i}\xi _{\mu j}}
, and the general expression for the energy (3) reduces to the effective energy While the first two terms in equation (6) are the same as those in equation (9), the third terms look superficially different. In equation (9) it is a Legendre transform of the Lagrangian for the feature neurons, while in (6) the third term is an integral of the inverse activation function. Nevertheless, these two expressions are in fact equivalent, since the derivatives of a function and its Legendre transform are inverse functions of each other. The easiest way to see that these two terms are equal explicitly is to differentiate each one with respect to
x
i
{\displaystyle x_{i}}
. The results of these differentiations for both expressions are equal to
x
i
g
(
x
i
)
′
{\displaystyle x_{i}g(x_{i})'}
. Thus, the two expressions are equal up to an additive constant. This completes the proof that the classical Hopfield Network with continuous states is a special limiting case of the modern Hopfield network (1) with energy (3).
=== General formulation of the modern Hopfield network ===
Biological neural networks have a large degree of heterogeneity in terms of different cell types. This section describes a mathematical model of a fully connected modern Hopfield network assuming the extreme degree of heterogeneity: every single neuron is different. Specifically, an energy function and the corresponding dynamical equations are described assuming that each neuron has its own activation function and kinetic time scale. The network is assumed to be fully connected, so that every neuron is connected to every other neuron using a symmetric matrix of weights
W
I
J
{\displaystyle W_{IJ}}
, indices
I
{\displaystyle I}
and
J
{\displaystyle J}
enumerate different neurons in the network, see Fig.3. The easiest way to mathematically formulate this problem is to define the architecture through a Lagrangian function
L
(
{
x
I
}
)
{\displaystyle L(\{x_{I}\})}
that depends on the activities of all the neurons in the network. The activation function for each neuron is defined as a partial derivative of the Lagrangian with respect to that neuron's activity From the biological perspective one can think about
g
I
{\displaystyle g_{I}}
as an axonal output of the neuron
I
{\displaystyle I}
. In the simplest case, when the Lagrangian is additive for different neurons, this definition results in the activation that is a non-linear function of that neuron's activity. For non-additive Lagrangians this activation function can depend on the activities of a group of neurons. For instance, it can contain contrastive (softmax) or divisive normalization. The dynamical equations describing temporal evolution of a given neuron are given by This equation belongs to the class of models called firing rate models in neuroscience. Each neuron
I
{\displaystyle I}
collects the axonal outputs
g
J
{\displaystyle g_{J}}
from all the neurons, weights them with the synaptic coefficients
W
I
J
{\displaystyle W_{IJ}}
and produces its own time-dependent activity
x
I
{\displaystyle x_{I}}
. The temporal evolution has a time constant
τ
I
{\displaystyle \tau _{I}}
, which in general can be different for every neuron. This network has a global energy function where the first two terms represent the Legendre transform of the Lagrangian function with respect to the neurons' currents
x
I
{\displaystyle x_{I}}
. The temporal derivative of this energy function can be computed on the dynamical trajectories leading to (see for details) The last inequality sign holds provided that the matrix
M
I
K
{\displaystyle M_{IK}}
(or its symmetric part) is positive semi-definite. If, in addition to this, the energy function is bounded from below the non-linear dynamical equations are guaranteed to converge to a fixed point attractor state. The advantage of formulating this network in terms of the Lagrangian functions is that it makes it possible to easily experiment with different choices of the activation functions and different architectural arrangements of neurons. For all those flexible choices the conditions of convergence are determined by the properties of the matrix
M
I
J
{\displaystyle M_{IJ}}
and the existence of the lower bound on the energy function.
=== Hierarchical associative memory network ===
The neurons can be organized in layers so that every neuron in a given layer has the same activation function and the same dynamic time scale. If we assume that there are no horizontal connections between the neurons within the layer (lateral connections) and there are no skip-layer connections, the general fully connected network (11), (12) reduces to the architecture shown in Fig.4. It has
N
layer
{\displaystyle N_{\text{layer}}}
layers of recurrently connected neurons with the states described by continuous variables
x
i
A
{\displaystyle x_{i}^{A}}
and the activation functions
g
i
A
{\displaystyle g_{i}^{A}}
, index
A
{\displaystyle A}
enumerates the layers of the network, and index
i
{\displaystyle i}
enumerates individual neurons in that layer. The activation functions can depend on the activities of all the neurons in the layer. Every layer can have a different number of neurons
N
A
{\displaystyle N_{A}}
. These neurons are recurrently connected with the neurons in the preceding and the subsequent layers. The matrices of weights that connect neurons in layers
A
{\displaystyle A}
and
B
{\displaystyle B}
are denoted by
ξ
i
j
(
A
,
B
)
{\displaystyle \xi _{ij}^{(A,B)}}
(the order of the upper indices for weights is the same as the order of the lower indices, in the example above this means that the index
i
{\displaystyle i}
enumerates neurons in the layer
A
{\displaystyle A}
, and index
j
{\displaystyle j}
enumerates neurons in the layer
B
{\displaystyle B}
). The feedforward weights and the feedback weights are equal. The dynamical equations for the neurons' states can be written as with boundary conditions The main difference between these equations and those from the conventional feedforward networks is the presence of the second term, which is responsible for the feedback from higher layers. These top-down signals help neurons in lower layers to decide on their response to the presented stimuli. Following the general recipe it is convenient to introduce a Lagrangian function
L
A
(
{
x
i
A
}
)
{\displaystyle L^{A}(\{x_{i}^{A}\})}
for the
A
{\displaystyle A}
-th hidden layer, which depends on the activities of all the neurons in that layer. The activation functions in that layer can be defined as partial derivatives of the Lagrangian With these definitions the energy (Lyapunov) function is given by If the Lagrangian functions, or equivalently the activation functions, are chosen in such a way that the Hessians for each layer are positive semi-definite and the overall energy is bounded from below, this system is guaranteed to converge to a fixed point attractor state. The temporal derivative of this energy function is given by Thus, the hierarchical layered network is indeed an attractor network with the global energy function. This network is described by a hierarchical set of synaptic weights that can be learned for each specific problem.
== See also ==
Associative memory (disambiguation)
Autoassociative memory
Boltzmann machine – like a Hopfield net but uses annealed Gibbs sampling instead of gradient descent
Dynamical systems model of cognition
Ising model
Hebbian theory
== References ==
== External links ==
Rojas, Raul (12 July 1996). "13. The Hopfield model" (PDF). Neural Networks – A Systematic Introduction. Springer. ISBN 978-3-540-60505-8.
Hopfield Network Javascript
The Travelling Salesman Problem Archived 2015-05-30 at the Wayback Machine – Hopfield Neural Network JAVA Applet
Hopfield, John (2007). "Hopfield network". Scholarpedia. 2 (5): 1977. Bibcode:2007SchpJ...2.1977H. doi:10.4249/scholarpedia.1977.
"Don't Forget About Associative Memories". The Gradient. November 7, 2020. Retrieved September 27, 2024.
Fletcher, Tristan. "Hopfield Network Learning Using Deterministic Latent Variables" (PDF) (Tutorial). Archived from the original (PDF) on 2011-10-05. | Wikipedia/Hopfield_neural_network |
The classical limit or correspondence limit is the ability of a physical theory to approximate or "recover" classical mechanics when considered over special values of its parameters. The classical limit is used with physical theories that predict non-classical behavior.
== Quantum theory ==
A heuristic postulate called the correspondence principle was introduced to quantum theory by Niels Bohr: in effect it states that some kind of continuity argument should apply to the classical limit of quantum systems as the value of the Planck constant normalized by the action of these systems becomes very small. Often, this is approached through "quasi-classical" techniques (cf. WKB approximation).
More rigorously, the mathematical operation involved in classical limits is a group contraction, approximating physical systems where the relevant action is much larger than the reduced Planck constant ħ, so the "deformation parameter" ħ/S can be effectively taken to be zero (cf. Weyl quantization.) Thus typically, quantum commutators (equivalently, Moyal brackets) reduce to Poisson brackets, in a group contraction.
In quantum mechanics, due to Heisenberg's uncertainty principle, an electron can never be at rest; it must always have a non-zero kinetic energy, a result not found in classical mechanics. For example, if we consider something very large relative to an electron, like a baseball, the uncertainty principle predicts that it cannot really have zero kinetic energy, but the uncertainty in kinetic energy is so small that the baseball can effectively appear to be at rest, and hence it appears to obey classical mechanics. In general, if large energies and large objects (relative to the size and energy levels of an electron) are considered in quantum mechanics, the result will appear to obey classical mechanics. The typical occupation numbers involved are huge: a macroscopic harmonic oscillator with ω = 2 Hz, m = 10 g, and maximum amplitude x0 = 10 cm, has S ≈ E/ω ≈ mωx20/2 ≈ 10−4 kg·m2/s = ħn, so that n ≃ 1030. Further see coherent states. It is less clear, however, how the classical limit applies to chaotic systems, a field known as quantum chaos.
Quantum mechanics and classical mechanics are usually treated with entirely different formalisms: quantum theory using Hilbert space, and classical mechanics using a representation in phase space. One can bring the two into a common mathematical framework in various ways. In the phase space formulation of quantum mechanics, which is statistical in nature, logical connections between quantum mechanics and classical statistical mechanics are made, enabling natural comparisons between them, including the violations of Liouville's theorem (Hamiltonian) upon quantization.
In a crucial paper (1933), Dirac explained how classical mechanics is an emergent phenomenon of quantum mechanics: destructive interference among paths with non-extremal macroscopic actions S » ħ obliterate amplitude contributions in the path integral he introduced, leaving the extremal action Sclass, thus the classical action path as the dominant contribution, an observation further elaborated by Feynman in his 1942 PhD dissertation. (Further see quantum decoherence.)
== Time-evolution of expectation values ==
One simple way to compare classical to quantum mechanics is to consider the time-evolution of the expected position and expected momentum, which can then be compared to the time-evolution of the ordinary position and momentum in classical mechanics. The quantum expectation values satisfy the Ehrenfest theorem. For a one-dimensional quantum particle moving in a potential
V
{\displaystyle V}
, the Ehrenfest theorem says
m
d
d
t
⟨
x
⟩
=
⟨
p
⟩
;
d
d
t
⟨
p
⟩
=
−
⟨
V
′
(
X
)
⟩
.
{\displaystyle m{\frac {d}{dt}}\langle x\rangle =\langle p\rangle ;\quad {\frac {d}{dt}}\langle p\rangle =-\left\langle V'(X)\right\rangle .}
Although the first of these equations is consistent with the classical mechanics, the second is not: If the pair
(
⟨
X
⟩
,
⟨
P
⟩
)
{\displaystyle (\langle X\rangle ,\langle P\rangle )}
were to satisfy Newton's second law, the right-hand side of the second equation would have read
d
d
t
⟨
p
⟩
=
−
V
′
(
⟨
X
⟩
)
{\displaystyle {\frac {d}{dt}}\langle p\rangle =-V'\left(\left\langle X\right\rangle \right)}
.
But in most cases,
⟨
V
′
(
X
)
⟩
≠
V
′
(
⟨
X
⟩
)
{\displaystyle \left\langle V'(X)\right\rangle \neq V'(\left\langle X\right\rangle )}
.
If for example, the potential
V
{\displaystyle V}
is cubic, then
V
′
{\displaystyle V'}
is quadratic, in which case, we are talking about the distinction between
⟨
X
2
⟩
{\displaystyle \langle X^{2}\rangle }
and
⟨
X
⟩
2
{\displaystyle \langle X\rangle ^{2}}
, which differ by
(
Δ
X
)
2
{\displaystyle (\Delta X)^{2}}
.
An exception occurs in case when the classical equations of motion are linear, that is, when
V
{\displaystyle V}
is quadratic and
V
′
{\displaystyle V'}
is linear. In that special case,
V
′
(
⟨
X
⟩
)
{\displaystyle V'\left(\left\langle X\right\rangle \right)}
and
⟨
V
′
(
X
)
⟩
{\displaystyle \left\langle V'(X)\right\rangle }
do agree. In particular, for a free particle or a quantum harmonic oscillator, the expected position and expected momentum exactly follows solutions of Newton's equations.
For general systems, the best we can hope for is that the expected position and momentum will approximately follow the classical trajectories. If the wave function is highly concentrated around a point
x
0
{\displaystyle x_{0}}
, then
V
′
(
⟨
X
⟩
)
{\displaystyle V'\left(\left\langle X\right\rangle \right)}
and
⟨
V
′
(
X
)
⟩
{\displaystyle \left\langle V'(X)\right\rangle }
will be almost the same, since both will be approximately equal to
V
′
(
x
0
)
{\displaystyle V'(x_{0})}
. In that case, the expected position and expected momentum will remain very close to the classical trajectories, at least for as long as the wave function remains highly localized in position.
Now, if the initial state is very localized in position, it will be very spread out in momentum, and thus we expect that the wave function will rapidly spread out, and the connection with the classical trajectories will be lost. When the Planck constant is small, however, it is possible to have a state that is well localized in both position and momentum. The small uncertainty in momentum ensures that the particle remains well localized in position for a long time, so that expected position and momentum continue to closely track the classical trajectories for a long time.
== Relativity and other deformations ==
Other familiar deformations in physics involve:
The deformation of classical Newtonian into relativistic mechanics (special relativity), with deformation parameter v/c; the classical limit involves small speeds, so v/c → 0, and the systems appear to obey Newtonian mechanics.
Similarly for the deformation of Newtonian gravity into general relativity, with deformation parameter Schwarzschild-radius/characteristic-dimension, we find that objects once again appear to obey classical mechanics (flat space), when the mass of an object times the square of the Planck length is much smaller than its size and the sizes of the problem addressed. See Newtonian limit.
Wave optics might also be regarded as a deformation of ray optics for deformation parameter λ/a.
Likewise, thermodynamics deforms to statistical mechanics with deformation parameter 1/N.
== See also ==
Classical probability density
Ehrenfest theorem
Madelung equations
Fresnel integral
Mathematical formulation of quantum mechanics
Quantum chaos
Quantum decoherence
Quantum limit
Semiclassical physics
Wigner–Weyl transform
WKB approximation
== References ==
Hall, Brian C. (2013), Quantum Theory for Mathematicians, Graduate Texts in Mathematics, vol. 267, Springer, Bibcode:2013qtm..book.....H, ISBN 978-1461471158 | Wikipedia/Classical_limit_of_quantum_mechanics |
Algebraic quantum field theory (AQFT) is an application to local quantum physics of C*-algebra theory. Also referred to as the Haag–Kastler axiomatic framework for quantum field theory, because it was introduced by Rudolf Haag and Daniel Kastler (1964). The axioms are stated in terms of an algebra given for every open set in Minkowski space, and mappings between those.
== Haag–Kastler axioms ==
Let
O
{\displaystyle {\mathcal {O}}}
be the set of all open and bounded subsets of Minkowski space. An algebraic quantum field theory is defined via a set
{
A
(
O
)
}
O
∈
O
{\displaystyle \{{\mathcal {A}}(O)\}_{O\in {\mathcal {O}}}}
of von Neumann algebras
A
(
O
)
{\displaystyle {\mathcal {A}}(O)}
on a common Hilbert space
H
{\displaystyle {\mathcal {H}}}
satisfying the following axioms:
Isotony:
O
1
⊂
O
2
{\displaystyle O_{1}\subset O_{2}}
implies
A
(
O
1
)
⊂
A
(
O
2
)
{\displaystyle {\mathcal {A}}(O_{1})\subset {\mathcal {A}}(O_{2})}
.
Causality: If
O
1
{\displaystyle O_{1}}
is space-like separated from
O
2
{\displaystyle O_{2}}
, then
[
A
(
O
1
)
,
A
(
O
2
)
]
=
0
{\displaystyle [{\mathcal {A}}(O_{1}),{\mathcal {A}}(O_{2})]=0}
.
Poincaré covariance: A strongly continuous unitary representation
U
(
P
)
{\displaystyle U({\mathcal {P}})}
of the Poincaré group
P
{\displaystyle {\mathcal {P}}}
on
H
{\displaystyle {\mathcal {H}}}
exists such that
A
(
g
O
)
=
U
(
g
)
A
(
O
)
U
(
g
)
∗
,
g
∈
P
.
{\displaystyle {\mathcal {A}}(gO)=U(g){\mathcal {A}}(O)U(g)^{*},\,\,g\in {\mathcal {P}}.}
Spectrum condition: The joint spectrum
S
p
(
P
)
{\displaystyle \mathrm {Sp} (P)}
of the energy-momentum operator
P
{\displaystyle P}
(i.e. the generator of space-time translations) is contained in the closed forward lightcone.
Existence of a vacuum vector: A cyclic and Poincaré-invariant vector
Ω
∈
H
{\displaystyle \Omega \in {\mathcal {H}}}
exists.
The net algebras
A
(
O
)
{\displaystyle {\mathcal {A}}(O)}
are called local algebras and the C* algebra
A
:=
⋃
O
∈
O
A
(
O
)
¯
{\displaystyle {\mathcal {A}}:={\overline {\bigcup _{O\in {\mathcal {O}}}{\mathcal {A}}(O)}}}
is called the quasilocal algebra.
== Category-theoretic formulation ==
Let Mink be the category of open subsets of Minkowski space M with inclusion maps as morphisms. We are given a covariant functor
A
{\displaystyle {\mathcal {A}}}
from Mink to uC*alg, the category of unital C* algebras, such that every morphism in Mink maps to a monomorphism in uC*alg (isotony).
The Poincaré group acts continuously on Mink. There exists a pullback of this action, which is continuous in the norm topology of
A
(
M
)
{\displaystyle {\mathcal {A}}(M)}
(Poincaré covariance).
Minkowski space has a causal structure. If an open set V lies in the causal complement of an open set U, then the image of the maps
A
(
i
U
,
U
∪
V
)
{\displaystyle {\mathcal {A}}(i_{U,U\cup V})}
and
A
(
i
V
,
U
∪
V
)
{\displaystyle {\mathcal {A}}(i_{V,U\cup V})}
commute (spacelike commutativity). If
U
¯
{\displaystyle {\bar {U}}}
is the causal completion of an open set U, then
A
(
i
U
,
U
¯
)
{\displaystyle {\mathcal {A}}(i_{U,{\bar {U}}})}
is an isomorphism (primitive causality).
A state with respect to a C*-algebra is a positive linear functional over it with unit norm. If we have a state over
A
(
M
)
{\displaystyle {\mathcal {A}}(M)}
, we can take the "partial trace" to get states associated with
A
(
U
)
{\displaystyle {\mathcal {A}}(U)}
for each open set via the net monomorphism. The states over the open sets form a presheaf structure.
According to the GNS construction, for each state, we can associate a Hilbert space representation of
A
(
M
)
.
{\displaystyle {\mathcal {A}}(M).}
Pure states correspond to irreducible representations and mixed states correspond to reducible representations. Each irreducible representation (up to equivalence) is called a superselection sector. We assume there is a pure state called the vacuum such that the Hilbert space associated with it is a unitary representation of the Poincaré group compatible with the Poincaré covariance of the net such that if we look at the Poincaré algebra, the spectrum with respect to energy-momentum (corresponding to spacetime translations) lies on and in the positive light cone. This is the vacuum sector.
== QFT in curved spacetime ==
More recently, the approach has been further implemented to include an algebraic version of quantum field theory in curved spacetime. Indeed, the viewpoint of local quantum physics is in particular suitable to generalize the renormalization procedure to the theory of quantum fields developed on curved backgrounds. Several rigorous results concerning QFT in presence of a black hole have been obtained.
== References ==
== Further reading ==
Haag, Rudolf; Kastler, Daniel (1964), "An Algebraic Approach to Quantum Field Theory", Journal of Mathematical Physics, 5 (7): 848–861, Bibcode:1964JMP.....5..848H, doi:10.1063/1.1704187, ISSN 0022-2488, MR 0165864
Haag, Rudolf (1996) [1992], Local Quantum Physics: Fields, Particles, Algebras, Theoretical and Mathematical Physics (2nd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-3-642-61458-3, ISBN 978-3-540-61451-7, MR 1405610
Brunetti, Romeo; Fredenhagen, Klaus; Verch, Rainer (2003). "The Generally Covariant Locality Principle – A New Paradigm for Local Quantum Field Theory". Communications in Mathematical Physics. 237 (1–2): 31–68. arXiv:math-ph/0112041. Bibcode:2003CMaPh.237...31B. doi:10.1007/s00220-003-0815-7. S2CID 13950246.
Brunetti, Romeo; Dütsch, Michael; Fredenhagen, Klaus (2009). "Perturbative Algebraic Quantum Field Theory and the Renormalization Groups". Advances in Theoretical and Mathematical Physics. 13 (5): 1541–1599. arXiv:0901.2038. doi:10.4310/ATMP.2009.v13.n5.a7. S2CID 15493763.
Bär, Christian; Fredenhagen, Klaus, eds. (2009). Quantum Field Theory on Curved Spacetimes: Concepts and Mathematical Foundations. Lecture Notes in Physics. Vol. 786. Springer. doi:10.1007/978-3-642-02780-2. ISBN 978-3-642-02780-2.
Brunetti, Romeo; Dappiaggi, Claudio; Fredenhagen, Klaus; Yngvason, Jakob, eds. (2015). Advances in Algebraic Quantum Field Theory. Mathematical Physics Studies. Springer. doi:10.1007/978-3-319-21353-8. ISBN 978-3-319-21353-8.
Rejzner, Kasia (2016). Perturbative Algebraic Quantum Field Theory: An Introduction for Mathematicians. Mathematical Physics Studies. Springer. arXiv:1208.1428. Bibcode:2016paqf.book.....R. doi:10.1007/978-3-319-25901-7. ISBN 978-3-319-25901-7.
Hack, Thomas-Paul (2016). Cosmological Applications of Algebraic Quantum Field Theory in Curved Spacetimes. SpringerBriefs in Mathematical Physics. Vol. 6. Springer. arXiv:1506.01869. Bibcode:2016caaq.book.....H. doi:10.1007/978-3-319-21894-6. ISBN 978-3-319-21894-6. S2CID 119657309.
Dütsch, Michael (2019). From Classical Field Theory to Perturbative Quantum Field Theory. Progress in Mathematical Physics. Vol. 74. Birkhäuser. doi:10.1007/978-3-030-04738-2. ISBN 978-3-030-04738-2. S2CID 126907045.
Yau, Donald (2019). Homotopical Quantum Field Theory. World Scientific. arXiv:1802.08101. doi:10.1142/11626. ISBN 978-981-121-287-1. S2CID 119168109.
Dedushenko, Mykola (2023). "Snowmass white paper: The quest to define QFT". International Journal of Modern Physics A. 38 (4n05). arXiv:2203.08053. doi:10.1142/S0217751X23300028. S2CID 247450696.
== External links ==
Local Quantum Physics Crossroads 2.0 – A network of scientists working on local quantum physics
Papers – A database of preprints on algebraic QFT
Algebraic Quantum Field Theory – AQFT resources at the University of Hamburg | Wikipedia/Local_quantum_physics |
In functional analysis, a branch of mathematics, the Borel functional calculus is a functional calculus (that is, an assignment of operators from commutative algebras to functions defined on their spectra), which has particularly broad scope. Thus for instance if T is an operator, applying the squaring function s → s2 to T yields the operator T2. Using the functional calculus for larger classes of functions, we can for example define rigorously the "square root" of the (negative) Laplacian operator −Δ or the exponential
e
i
t
Δ
.
{\displaystyle e^{it\Delta }.}
The 'scope' here means the kind of function of an operator which is allowed. The Borel functional calculus is more general than the continuous functional calculus, and its focus is different than the holomorphic functional calculus.
More precisely, the Borel functional calculus allows for applying an arbitrary Borel function to a self-adjoint operator, in a way that generalizes applying a polynomial function.
== Motivation ==
If T is a self-adjoint operator on a finite-dimensional inner product space H, then H has an orthonormal basis {e1, ..., eℓ} consisting of eigenvectors of T, that is
T
e
k
=
λ
k
e
k
,
1
≤
k
≤
ℓ
.
{\displaystyle Te_{k}=\lambda _{k}e_{k},\qquad 1\leq k\leq \ell .}
Thus, for any positive integer n,
T
n
e
k
=
λ
k
n
e
k
.
{\displaystyle T^{n}e_{k}=\lambda _{k}^{n}e_{k}.}
If only polynomials in T are considered, then one gets the holomorphic functional calculus. The relation also holds for more general functions of T. Given a Borel function h, one can define an operator h(T) by specifying its behavior on the basis:
h
(
T
)
e
k
=
h
(
λ
k
)
e
k
.
{\displaystyle h(T)e_{k}=h(\lambda _{k})e_{k}.}
Generally, any self-adjoint operator T is unitarily equivalent to a multiplication operator; this means that for many purposes, T can be considered as an operator
[
T
ψ
]
(
x
)
=
f
(
x
)
ψ
(
x
)
{\displaystyle [T\psi ](x)=f(x)\psi (x)}
acting on L2 of some measure space. The domain of T consists of those functions whose above expression is in L2. In such a case, one can define analogously
[
h
(
T
)
ψ
]
(
x
)
=
[
h
∘
f
]
(
x
)
ψ
(
x
)
.
{\displaystyle [h(T)\psi ](x)=[h\circ f](x)\psi (x).}
For many technical purposes, the previous formulation is good enough. However, it is desirable to formulate the functional calculus in a way that does not depend on the particular representation of T as a multiplication operator. That's what we do in the next section.
== The bounded functional calculus ==
Formally, the bounded Borel functional calculus of a self adjoint operator T on Hilbert space H is a mapping defined on the space of bounded complex-valued Borel functions f on the real line,
{
π
T
:
L
∞
(
R
,
C
)
→
B
(
H
)
f
↦
f
(
T
)
{\displaystyle {\begin{cases}\pi _{T}:L^{\infty }(\mathbb {R} ,\mathbb {C} )\to {\mathcal {B}}({\mathcal {H}})\\f\mapsto f(T)\end{cases}}}
such that the following conditions hold
πT is an involution-preserving and unit-preserving homomorphism from the ring of complex-valued bounded measurable functions on R.
If ξ is an element of H, then
ν
ξ
:
E
↦
⟨
π
T
(
1
E
)
ξ
,
ξ
⟩
{\displaystyle \nu _{\xi }:E\mapsto \langle \pi _{T}(\mathbf {1} _{E})\xi ,\xi \rangle }
is a countably additive measure on the Borel sets E of R. In the above formula 1E denotes the indicator function of E. These measures νξ are called the spectral measures of T.
If η denotes the mapping z → z on C, then:
π
T
(
[
η
+
i
]
−
1
)
=
[
T
+
i
]
−
1
.
{\displaystyle \pi _{T}\left([\eta +i]^{-1}\right)=[T+i]^{-1}.}
This defines the functional calculus for bounded functions applied to possibly unbounded self-adjoint operators. Using the bounded functional calculus, one can prove part of the Stone's theorem on one-parameter unitary groups:
As an application, we consider the Schrödinger equation, or equivalently, the dynamics of a quantum mechanical system. In non-relativistic quantum mechanics, the Hamiltonian operator H models the total energy observable of a quantum mechanical system S. The unitary group generated by iH corresponds to the time evolution of S.
We can also use the Borel functional calculus to abstractly solve some linear initial value problems such as the heat equation, or Maxwell's equations.
=== Existence of a functional calculus ===
The existence of a mapping with the properties of a functional calculus requires proof. For the case of a bounded self-adjoint operator T, the existence of a Borel functional calculus can be shown in an elementary way as follows:
First pass from polynomial to continuous functional calculus by using the Stone–Weierstrass theorem. The crucial fact here is that, for a bounded self adjoint operator T and a polynomial p,
‖
p
(
T
)
‖
=
sup
λ
∈
σ
(
T
)
|
p
(
λ
)
|
.
{\displaystyle \|p(T)\|=\sup _{\lambda \in \sigma (T)}|p(\lambda )|.}
Consequently, the mapping
p
↦
p
(
T
)
{\displaystyle p\mapsto p(T)}
is an isometry and a densely defined homomorphism on the ring of polynomial functions. Extending by continuity defines f(T) for a continuous function f on the spectrum of T. The Riesz-Markov theorem then allows us to pass from integration on continuous functions to spectral measures, and this is the Borel functional calculus.
Alternatively, the continuous calculus can be obtained via the Gelfand transform, in the context of commutative Banach algebras. Extending to measurable functions is achieved by applying Riesz-Markov, as above. In this formulation, T can be a normal operator.
Given an operator T, the range of the continuous functional calculus h → h(T) is the (abelian) C*-algebra C(T) generated by T. The Borel functional calculus has a larger range, that is the closure of C(T) in the weak operator topology, a (still abelian) von Neumann algebra.
== The general functional calculus ==
We can also define the functional calculus for not necessarily bounded Borel functions h; the result is an operator which in general fails to be bounded. Using the multiplication by a function f model of a self-adjoint operator given by the spectral theorem, this is multiplication by the composition of h with f.
The operator S of the previous theorem is denoted h(T).
More generally, a Borel functional calculus also exists for (bounded) normal operators.
== Resolution of the identity ==
Let
T
{\displaystyle T}
be a self-adjoint operator. If
E
{\displaystyle E}
is a Borel subset of R, and
1
E
{\displaystyle \mathbf {1} _{E}}
is the indicator function of E, then
1
E
(
T
)
{\displaystyle \mathbf {1} _{E}(T)}
is a self-adjoint projection on H. Then mapping
Ω
T
:
E
↦
1
E
(
T
)
{\displaystyle \Omega _{T}:E\mapsto \mathbf {1} _{E}(T)}
is a projection-valued measure. The measure of R with respect to
Ω
T
{\textstyle \Omega _{T}}
is the identity operator on H. In other words, the identity operator can be expressed as the spectral integral
I
=
Ω
T
(
[
−
∞
,
∞
]
)
=
∫
−
∞
∞
d
Ω
T
{\displaystyle I=\Omega _{T}([-\infty ,\infty ])=\int _{-\infty }^{\infty }d\Omega _{T}}
.
Stone's formula expresses the spectral measure
Ω
T
{\displaystyle \Omega _{T}}
in terms of the resolvent
R
T
(
λ
)
≡
(
T
−
λ
I
)
−
1
{\displaystyle R_{T}(\lambda )\equiv \left(T-\lambda I\right)^{-1}}
:
1
2
π
i
lim
ϵ
→
0
+
∫
a
b
[
R
T
(
λ
+
i
ϵ
)
)
−
R
T
(
λ
−
i
ϵ
)
]
d
λ
=
Ω
T
(
(
a
,
b
)
)
+
1
2
[
Ω
T
(
{
a
}
)
+
Ω
T
(
{
b
}
)
]
.
{\displaystyle {\frac {1}{2\pi i}}\lim _{\epsilon \to 0^{+}}\int _{a}^{b}\left[R_{T}(\lambda +i\epsilon ))-R_{T}(\lambda -i\epsilon )\right]\,d\lambda =\Omega _{T}((a,b))+{\frac {1}{2}}\left[\Omega _{T}(\{a\})+\Omega _{T}(\{b\})\right].}
Depending on the source, the resolution of the identity is defined, either as a projection-valued measure
Ω
T
{\displaystyle \Omega _{T}}
, or as a one-parameter family of projection-valued measures
{
Σ
λ
}
{\displaystyle \{\Sigma _{\lambda }\}}
with
−
∞
<
λ
<
∞
{\displaystyle -\infty <\lambda <\infty }
.
In the case of a discrete measure (in particular, when H is finite-dimensional),
I
=
∫
1
d
Ω
T
{\textstyle I=\int 1\,d\Omega _{T}}
can be written as
I
=
∑
i
|
i
⟩
⟨
i
|
{\displaystyle I=\sum _{i}\left|i\right\rangle \left\langle i\right|}
in the Dirac notation, where each
|
i
⟩
{\displaystyle |i\rangle }
is a normalized eigenvector of T. The set
{
|
i
⟩
}
{\displaystyle \{|i\rangle \}}
is an orthonormal basis of H.
In physics literature, using the above as heuristic, one passes to the case when the spectral measure is no longer discrete and write the resolution of identity as
I
=
∫
d
i
|
i
⟩
⟨
i
|
{\displaystyle I=\int \!\!di~|i\rangle \langle i|}
and speak of a "continuous basis", or "continuum of basis states",
{
|
i
⟩
}
{\displaystyle \{|i\rangle \}}
Mathematically, unless rigorous justifications are given, this expression is purely formal.
== References == | Wikipedia/Resolution_of_the_identity |
In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a prediction for the system represented by the state. Knowledge of the quantum state, and the rules for the system's evolution in time, exhausts all that can be known about a quantum system.
Quantum states may be defined differently for different kinds of systems or problems. Two broad categories are
wave functions describing quantum systems using position or momentum variables and
the more abstract vector quantum states.
Historical, educational, and application-focused problems typically feature wave functions; modern professional physics uses the abstract vector states. In both categories, quantum states divide into pure versus mixed states, or into coherent states and incoherent states. Categories with special properties include stationary states for time independence and quantum vacuum states in quantum field theory.
== From the states of classical mechanics ==
As a tool for physics, quantum states grew out of states in classical mechanics. A classical dynamical state consists of a set of dynamical variables with well-defined real values at each instant of time.: 3 For example, the state of a cannon ball would consist of its position and velocity. The state values evolve under equations of motion and thus remain strictly determined. If we know the position of a cannon and the exit velocity of its projectiles, then we can use equations containing the force of gravity to predict the trajectory of a cannon ball precisely.
Similarly, quantum states consist of sets of dynamical variables that evolve under equations of motion. However, the values derived from quantum states are complex numbers, quantized, limited by uncertainty relations,: 159 and only provide a probability distribution for the outcomes for a system. These constraints alter the nature of quantum dynamic variables. For example, the quantum state of an electron in a double-slit experiment would consist of complex values over the detection region and, when squared, only predict the probability distribution of electron counts across the detector.
== Role in quantum mechanics ==
The process of describing a quantum system with quantum mechanics begins with identifying a set of variables defining the quantum state of the system.: 204 The set will contain compatible and incompatible variables. Simultaneous measurement of a complete set of compatible variables prepares the system in a unique state. The state then evolves deterministically according to the equations of motion. Subsequent measurement of the state produces a sample from a probability distribution predicted by the quantum mechanical operator corresponding to the measurement.
The fundamentally statistical or probabilisitic nature of quantum measurements changes the role of quantum states in quantum mechanics compared to classical states in classical mechanics. In classical mechanics, the initial state of one or more bodies is measured; the state evolves according to the equations of motion; measurements of the final state are compared to predictions. In quantum mechanics, ensembles of identically prepared quantum states evolve according to the equations of motion and many repeated measurements are compared to predicted probability distributions.: 204
== Measurements ==
Measurements, macroscopic operations on quantum states, filter the state.: 196 Whatever the input quantum state might be, repeated identical measurements give consistent values. For this reason, measurements 'prepare' quantum states for experiments, placing the system in a partially defined state. Subsequent measurements may either further prepare the system – these are compatible measurements – or it may alter the state, redefining it – these are called incompatible or complementary measurements. For example, we may measure the momentum of a state along the
x
{\displaystyle x}
axis any number of times and get the same result, but if we measure the position after once measuring the momentum, subsequent measurements of momentum are changed. The quantum state appears unavoidably altered by incompatible measurements. This is known as the uncertainty principle.
== Eigenstates and pure states ==
The quantum state after a measurement is in an eigenstate corresponding to that measurement and the value measured.: 202 Other aspects of the state may be unknown. Repeating the measurement will not alter the state. In some cases, compatible measurements can further refine the state, causing it to be an eigenstate corresponding to all these measurements. A full set of compatible measurements produces a pure state. Any state that is not pure is called a mixed state as discussed in more depth below.: 204 : 73
The eigenstate solutions to the Schrödinger equation can be formed into pure states. Experiments rarely produce pure states. Therefore statistical mixtures of solutions must be compared to experiments.: 204
== Representations ==
The same physical quantum state can be expressed mathematically in different ways called representations. The position wave function is one representation often seen first in introductions to quantum mechanics. The equivalent momentum wave function is another wave function based representation. Representations are analogous to coordinate systems: 244 or similar mathematical devices like parametric equations. Selecting a representation will make some aspects of a problem easier at the cost of making other things difficult.
In formal quantum mechanics (see § Formalism in quantum physics below) the theory develops in terms of abstract 'vector space', avoiding any particular representation. This allows many elegant concepts of quantum mechanics to be expressed and to be applied even in cases where no classical analog exists.: 244
== Wave function representations ==
Wave functions represent quantum states, particularly when they are functions of position or of momentum. Historically, definitions of quantum states used wavefunctions before the more formal methods were developed.: 268 The wave function is a complex-valued function of any complete set of commuting or compatible degrees of freedom. For example, one set could be the
x
,
y
,
z
{\displaystyle x,y,z}
spatial coordinates of an electron.
Preparing a system by measuring the complete set of compatible observables produces a pure quantum state. More common, incomplete preparation produces a mixed quantum state. Wave function solutions of Schrödinger's equations of motion for operators corresponding to measurements can readily be expressed as pure states; they must be combined with statistical weights matching experimental preparation to compute the expected probability distribution.: 205
=== Pure states of wave functions ===
Numerical or analytic solutions in quantum mechanics can be expressed as pure states. These solution states, called eigenstates, are labeled with quantized values, typically quantum numbers.
For example, when dealing with the energy spectrum of the electron in a hydrogen atom, the relevant pure states are identified by the principal quantum number n, the angular momentum quantum number ℓ, the magnetic quantum number m, and the spin z-component sz. For another example, if the spin of an electron is measured in any direction, e.g. with a Stern–Gerlach experiment, there are two possible results: up or down. A pure state here is represented by a two-dimensional complex vector
(
α
,
β
)
{\displaystyle (\alpha ,\beta )}
, with a length of one; that is, with
|
α
|
2
+
|
β
|
2
=
1
,
{\displaystyle |\alpha |^{2}+|\beta |^{2}=1,}
where
|
α
|
{\displaystyle |\alpha |}
and
|
β
|
{\displaystyle |\beta |}
are the absolute values of
α
{\displaystyle \alpha }
and
β
{\displaystyle \beta }
.
The postulates of quantum mechanics state that pure states, at a given time t, correspond to vectors in a separable complex Hilbert space, while each measurable physical quantity (such as the energy or momentum of a particle) is associated with a mathematical operator called the observable. The operator serves as a linear function that acts on the states of the system. The eigenvalues of the operator correspond to the possible values of the observable. For example, it is possible to observe a particle with a momentum of 1 kg⋅m/s if and only if one of the eigenvalues of the momentum operator is 1 kg⋅m/s. The corresponding eigenvector (which physicists call an eigenstate) with eigenvalue 1 kg⋅m/s would be a quantum state with a definite, well-defined value of momentum of 1 kg⋅m/s, with no quantum uncertainty. If its momentum were measured, the result is guaranteed to be 1 kg⋅m/s.
On the other hand, a pure state described as a superposition of multiple different eigenstates does in general have quantum uncertainty for the given observable. Using bra–ket notation, this linear combination of eigenstates can be represented as:: 22, 171, 172
|
Ψ
(
t
)
⟩
=
∑
n
C
n
(
t
)
|
Φ
n
⟩
.
{\displaystyle |\Psi (t)\rangle =\sum _{n}C_{n}(t)|\Phi _{n}\rangle .}
The coefficient that corresponds to a particular state in the linear combination is a complex number, thus allowing interference effects between states. The coefficients are time dependent. How a quantum state changes in time is governed by the time evolution operator.
=== Mixed states of wave functions ===
A mixed quantum state corresponds to a probabilistic mixture of pure states; however, different distributions of pure states can generate equivalent (i.e., physically indistinguishable) mixed states. A mixture of quantum states is again a quantum state.
A mixed state for electron spins, in the density-matrix formulation, has the structure of a
2
×
2
{\displaystyle 2\times 2}
matrix that is Hermitian and positive semi-definite, and has trace 1. A more complicated case is given (in bra–ket notation) by the singlet state, which exemplifies quantum entanglement:
|
ψ
⟩
=
1
2
(
|
↑↓
⟩
−
|
↓↑
⟩
)
,
{\displaystyle \left|\psi \right\rangle ={\frac {1}{\sqrt {2}}}{\bigl (}\left|\uparrow \downarrow \right\rangle -\left|\downarrow \uparrow \right\rangle {\bigr )},}
which involves superposition of joint spin states for two particles with spin 1/2. The singlet state satisfies the property that if the particles' spins are measured along the same direction then either the spin of the first particle is observed up and the spin of the second particle is observed down, or the first one is observed down and the second one is observed up, both possibilities occurring with equal probability.
A pure quantum state can be represented by a ray in a projective Hilbert space over the complex numbers, while mixed states are represented by density matrices, which are positive semidefinite operators that act on Hilbert spaces.
The Schrödinger–HJW theorem classifies the multitude of ways to write a given mixed state as a convex combination of pure states.
Before a particular measurement is performed on a quantum system, the theory gives only a probability distribution for the outcome, and the form that this distribution takes is completely determined by the quantum state and the linear operators describing the measurement. Probability distributions for different measurements exhibit tradeoffs exemplified by the uncertainty principle: a state that implies a narrow spread of possible outcomes for one experiment necessarily implies a wide spread of possible outcomes for another.
Statistical mixtures of states are a different type of linear combination. A statistical mixture of states is a statistical ensemble of independent systems. Statistical mixtures represent the degree of knowledge whilst the uncertainty within quantum mechanics is fundamental. Mathematically, a statistical mixture is not a combination using complex coefficients, but rather a combination using real-valued, positive probabilities of different states
Φ
n
{\displaystyle \Phi _{n}}
. A number
P
n
{\displaystyle P_{n}}
represents the probability of a randomly selected system being in the state
Φ
n
{\displaystyle \Phi _{n}}
. Unlike the linear combination case each system is in a definite eigenstate.
The expectation value
⟨
A
⟩
σ
{\displaystyle {\langle A\rangle }_{\sigma }}
of an observable A is a statistical mean of measured values of the observable. It is this mean, and the distribution of probabilities, that is predicted by physical theories.
There is no state that is simultaneously an eigenstate for all observables. For example, we cannot prepare a state such that both the position measurement Q(t) and the momentum measurement P(t) (at the same time t) are known exactly; at least one of them will have a range of possible values. This is the content of the Heisenberg uncertainty relation.
Moreover, in contrast to classical mechanics, it is unavoidable that performing a measurement on the system generally changes its state.: 4 More precisely: After measuring an observable A, the system will be in an eigenstate of A; thus the state has changed, unless the system was already in that eigenstate. This expresses a kind of logical consistency: If we measure A twice in the same run of the experiment, the measurements being directly consecutive in time, then they will produce the same results. This has some strange consequences, however, as follows.
Consider two incompatible observables, A and B, where A corresponds to a measurement earlier in time than B.
Suppose that the system is in an eigenstate of B at the experiment's beginning. If we measure only B, all runs of the experiment will yield the same result.
If we measure first A and then B in the same run of the experiment, the system will transfer to an eigenstate of A after the first measurement, and we will generally notice that the results of B are statistical. Thus: Quantum mechanical measurements influence one another, and the order in which they are performed is important.
Another feature of quantum states becomes relevant if we consider a physical system that consists of multiple subsystems; for example, an experiment with two particles rather than one. Quantum physics allows for certain states, called entangled states, that show certain statistical correlations between measurements on the two particles which cannot be explained by classical theory. For details, see Quantum entanglement. These entangled states lead to experimentally testable properties (Bell's theorem)
that allow us to distinguish between quantum theory and alternative classical (non-quantum) models.
=== Schrödinger picture vs. Heisenberg picture ===
One can take the observables to be dependent on time, while the state σ was fixed once at the beginning of the experiment. This approach is called the Heisenberg picture. (This approach was taken in the later part of the discussion above, with time-varying observables P(t), Q(t).) One can, equivalently, treat the observables as fixed, while the state of the system depends on time; that is known as the Schrödinger picture. (This approach was taken in the earlier part of the discussion above, with a time-varying state
|
Ψ
(
t
)
⟩
=
∑
n
C
n
(
t
)
|
Φ
n
⟩
{\textstyle |\Psi (t)\rangle =\sum _{n}C_{n}(t)|\Phi _{n}\rangle }
.) Conceptually (and mathematically), the two approaches are equivalent; choosing one of them is a matter of convention.
Both viewpoints are used in quantum theory. While non-relativistic quantum mechanics is usually formulated in terms of the Schrödinger picture, the Heisenberg picture is often preferred in a relativistic context, that is, for quantum field theory. Compare with Dirac picture.: 65
== Formalism in quantum physics ==
=== Pure states as rays in a complex Hilbert space ===
Quantum physics is most commonly formulated in terms of linear algebra, as follows. Any given system is identified with some finite- or infinite-dimensional Hilbert space. The pure states correspond to vectors of norm 1. Thus the set of all pure states corresponds to the unit sphere in the Hilbert space, because the unit sphere is defined as the set of all vectors with norm 1.
Multiplying a pure state by a scalar is physically inconsequential (as long as the state is considered by itself). If a vector in a complex Hilbert space
H
{\displaystyle H}
can be obtained from another vector by multiplying by some non-zero complex number, the two vectors in
H
{\displaystyle H}
are said to correspond to the same ray in the projective Hilbert space
P
(
H
)
{\displaystyle \mathbf {P} (H)}
of
H
{\displaystyle H}
. Note that although the word ray is used, properly speaking, a point in the projective Hilbert space corresponds to a line passing through the origin of the Hilbert space, rather than a half-line, or ray in the geometrical sense.
=== Spin ===
The angular momentum has the same dimension (M·L2·T−1) as the Planck constant and, at quantum scale, behaves as a discrete degree of freedom of a quantum system. Most particles possess a kind of intrinsic angular momentum that does not appear at all in classical mechanics and arises from Dirac's relativistic generalization of the theory. Mathematically it is described with spinors. In non-relativistic quantum mechanics the group representations of the Lie group SU(2) are used to describe this additional freedom. For a given particle, the choice of representation (and hence the range of possible values of the spin observable) is specified by a non-negative number S that, in units of the reduced Planck constant ħ, is either an integer (0, 1, 2, ...) or a half-integer (1/2, 3/2, 5/2, ...). For a massive particle with spin S, its spin quantum number m always assumes one of the 2S + 1 possible values in the set
{
−
S
,
−
S
+
1
,
…
,
S
−
1
,
S
}
{\displaystyle \{-S,-S+1,\ldots ,S-1,S\}}
As a consequence, the quantum state of a particle with spin is described by a vector-valued wave function with values in C2S+1. Equivalently, it is represented by a complex-valued function of four variables: one discrete quantum number variable (for the spin) is added to the usual three continuous variables (for the position in space).
=== Many-body states and particle statistics ===
The quantum state of a system of N particles, each potentially with spin, is described by a complex-valued function with four variables per particle, corresponding to 3 spatial coordinates and spin, e.g.
|
ψ
(
r
1
,
m
1
;
…
;
r
N
,
m
N
)
⟩
.
{\displaystyle |\psi (\mathbf {r} _{1},\,m_{1};\;\dots ;\;\mathbf {r} _{N},\,m_{N})\rangle .}
Here, the spin variables mν assume values from the set
{
−
S
ν
,
−
S
ν
+
1
,
…
,
S
ν
−
1
,
S
ν
}
{\displaystyle \{-S_{\nu },\,-S_{\nu }+1,\,\ldots ,\,S_{\nu }-1,\,S_{\nu }\}}
where
S
ν
{\displaystyle S_{\nu }}
is the spin of νth particle.
S
ν
=
0
{\displaystyle S_{\nu }=0}
for a particle that does not exhibit spin.
The treatment of identical particles is very different for bosons (particles with integer spin) versus fermions (particles with half-integer spin). The above N-particle function must either be symmetrized (in the bosonic case) or anti-symmetrized (in the fermionic case) with respect to the particle numbers. If not all N particles are identical, but some of them are, then the function must be (anti)symmetrized separately over the variables corresponding to each group of identical variables, according to its statistics (bosonic or fermionic).
Electrons are fermions with S = 1/2, photons (quanta of light) are bosons with S = 1 (although in the vacuum they are massless and can't be described with Schrödinger mechanics).
When symmetrization or anti-symmetrization is unnecessary, N-particle spaces of states can be obtained simply by tensor products of one-particle spaces, to which we will return later.
=== Basis states of one-particle systems ===
A state
|
ψ
⟩
{\displaystyle |\psi \rangle }
belonging to a separable complex Hilbert space
H
{\displaystyle H}
can always be expressed uniquely as a linear combination of elements of an orthonormal basis of
H
{\displaystyle H}
.
Using bra–ket notation, this means any state
|
ψ
⟩
{\displaystyle |\psi \rangle }
can be written as
|
ψ
⟩
=
∑
i
c
i
|
k
i
⟩
,
=
∑
i
|
k
i
⟩
⟨
k
i
|
ψ
⟩
,
{\displaystyle {\begin{aligned}|\psi \rangle &=\sum _{i}c_{i}|{k_{i}}\rangle ,\\&=\sum _{i}|{k_{i}}\rangle \langle k_{i}|\psi \rangle ,\end{aligned}}}
with complex coefficients
c
i
=
⟨
k
i
|
ψ
⟩
{\displaystyle c_{i}=\langle {k_{i}}|\psi \rangle }
and basis elements
|
k
i
⟩
{\displaystyle |k_{i}\rangle }
. In this case, the normalization condition translates to
⟨
ψ
|
ψ
⟩
=
∑
i
⟨
ψ
|
k
i
⟩
⟨
k
i
|
ψ
⟩
=
∑
i
|
c
i
|
2
=
1.
{\displaystyle \langle \psi |\psi \rangle =\sum _{i}\langle \psi |{k_{i}}\rangle \langle k_{i}|\psi \rangle =\sum _{i}\left|c_{i}\right|^{2}=1.}
In physical terms,
|
ψ
⟩
{\displaystyle |\psi \rangle }
has been expressed as a quantum superposition of the "basis states"
|
k
i
⟩
{\displaystyle |{k_{i}}\rangle }
, i.e., the eigenstates of an observable. In particular, if said observable is measured on the normalized state
|
ψ
⟩
{\displaystyle |\psi \rangle }
, then
|
c
i
|
2
=
|
⟨
k
i
|
ψ
⟩
|
2
,
{\displaystyle |c_{i}|^{2}=|\langle {k_{i}}|\psi \rangle |^{2},}
is the probability that the result of the measurement is
k
i
{\displaystyle k_{i}}
.: 22
In general, the expression for probability always consist of a relation between the quantum state and a portion of the spectrum of the dynamical variable (i.e. random variable) being observed.: 98 : 53 For example, the situation above describes the discrete case as eigenvalues
k
i
{\displaystyle k_{i}}
belong to the point spectrum. Likewise, the wave function is just the eigenfunction of the Hamiltonian operator with corresponding eigenvalue(s)
E
{\displaystyle E}
; the energy of the system.
An example of the continuous case is given by the position operator. The probability measure for a system in state
ψ
{\displaystyle \psi }
is given by:
P
r
(
x
∈
B
|
ψ
)
=
∫
B
⊂
R
|
ψ
(
x
)
|
2
d
x
,
{\displaystyle \mathrm {Pr} (x\in B|\psi )=\int _{B\subset \mathbb {R} }|\psi (x)|^{2}dx,}
where
|
ψ
(
x
)
|
2
{\displaystyle |\psi (x)|^{2}}
is the probability density function for finding a particle at a given position. These examples emphasize the distinction in charactertistics between the state and the observable. That is, whereas
ψ
{\displaystyle \psi }
is a pure state belonging to
H
{\displaystyle H}
, the (generalized) eigenvectors of the position operator do not.
=== Pure states vs. bound states ===
Though closely related, pure states are not the same as bound states belonging to the pure point spectrum of an observable with no quantum uncertainty. A particle is said to be in a bound state if it remains localized in a bounded region of space for all times. A pure state
|
ϕ
⟩
{\displaystyle |\phi \rangle }
is called a bound state if and only if for every
ε
>
0
{\displaystyle \varepsilon >0}
there is a compact set
K
⊂
R
3
{\displaystyle K\subset \mathbb {R} ^{3}}
such that
∫
K
|
ϕ
(
r
,
t
)
|
2
d
3
r
≥
1
−
ε
{\displaystyle \int _{K}|\phi (\mathbf {r} ,t)|^{2}\,\mathrm {d} ^{3}\mathbf {r} \geq 1-\varepsilon }
for all
t
∈
R
{\displaystyle t\in \mathbb {R} }
. The integral represents the probability that a particle is found in a bounded region
K
{\displaystyle K}
at any time
t
{\displaystyle t}
. If the probability remains arbitrarily close to
1
{\displaystyle 1}
then the particle is said to remain in
K
{\displaystyle K}
.
For example, non-normalizable solutions of the free Schrödinger equation can be expressed as functions that are normalizable, using wave packets. These wave packets belong to the pure point spectrum of a corresponding projection operator which, mathematically speaking, constitutes an observable.: 48 However, they are not bound states.
=== Superposition of pure states ===
As mentioned above, quantum states may be superposed. If
|
α
⟩
{\displaystyle |\alpha \rangle }
and
|
β
⟩
{\displaystyle |\beta \rangle }
are two kets corresponding to quantum states, the ket
c
α
|
α
⟩
+
c
β
|
β
⟩
{\displaystyle c_{\alpha }|\alpha \rangle +c_{\beta }|\beta \rangle }
is also a quantum state of the same system. Both
c
α
{\displaystyle c_{\alpha }}
and
c
β
{\displaystyle c_{\beta }}
can be complex numbers; their relative amplitude and relative phase will influence the resulting quantum state.
Writing the superposed state using
c
α
=
A
α
e
i
θ
α
c
β
=
A
β
e
i
θ
β
{\displaystyle c_{\alpha }=A_{\alpha }e^{i\theta _{\alpha }}\ \ c_{\beta }=A_{\beta }e^{i\theta _{\beta }}}
and defining the norm of the state as:
|
c
α
|
2
+
|
c
β
|
2
=
A
α
2
+
A
β
2
=
1
{\displaystyle |c_{\alpha }|^{2}+|c_{\beta }|^{2}=A_{\alpha }^{2}+A_{\beta }^{2}=1}
and extracting the common factors gives:
e
i
θ
α
(
A
α
|
α
⟩
+
1
−
A
α
2
e
i
θ
β
−
i
θ
α
|
β
⟩
)
{\displaystyle e^{i\theta _{\alpha }}\left(A_{\alpha }|\alpha \rangle +{\sqrt {1-A_{\alpha }^{2}}}e^{i\theta _{\beta }-i\theta _{\alpha }}|\beta \rangle \right)}
The overall phase factor in front has no physical effect.: 108 Only the relative phase affects the physical nature of the superposition.
One example of superposition is the double-slit experiment, in which superposition leads to quantum interference. Another example of the importance of relative phase is Rabi oscillations, where the relative phase of two states varies in time due to the Schrödinger equation. The resulting superposition ends up oscillating back and forth between two different states.
=== Mixed states ===
A pure quantum state is a state which can be described by a single ket vector, as described above. A mixed quantum state is a statistical ensemble of pure states (see Quantum statistical mechanics).: 73
Mixed states arise in quantum mechanics in two different situations: first, when the preparation of the system is not fully known, and thus one must deal with a statistical ensemble of possible preparations; and second, when one wants to describe a physical system which is entangled with another, as its state cannot be described by a pure state. In the first case, there could theoretically be another person who knows the full history of the system, and therefore describe the same system as a pure state; in this case, the density matrix is simply used to represent the limited knowledge of a quantum state. In the second case, however, the existence of quantum entanglement theoretically prevents the existence of complete knowledge about the subsystem, and it's impossible for any person to describe the subsystem of an entangled pair as a pure state.
Mixed states inevitably arise from pure states when, for a composite quantum system
H
1
⊗
H
2
{\displaystyle H_{1}\otimes H_{2}}
with an entangled state on it, the part
H
2
{\displaystyle H_{2}}
is inaccessible to the observer.: 121–122 The state of the part
H
1
{\displaystyle H_{1}}
is expressed then as the partial trace over
H
2
{\displaystyle H_{2}}
.
A mixed state cannot be described with a single ket vector.: 691–692 Instead, it is described by its associated density matrix (or density operator), usually denoted ρ. Density matrices can describe both mixed and pure states, treating them on the same footing. Moreover, a mixed quantum state on a given quantum system described by a Hilbert space
H
{\displaystyle H}
can be always represented as the partial trace of a pure quantum state (called a purification) on a larger bipartite system
H
⊗
K
{\displaystyle H\otimes K}
for a sufficiently large Hilbert space
K
{\displaystyle K}
.
The density matrix describing a mixed state is defined to be an operator of the form
ρ
=
∑
s
p
s
|
ψ
s
⟩
⟨
ψ
s
|
{\displaystyle \rho =\sum _{s}p_{s}|\psi _{s}\rangle \langle \psi _{s}|}
where ps is the fraction of the ensemble in each pure state
|
ψ
s
⟩
.
{\displaystyle |\psi _{s}\rangle .}
The density matrix can be thought of as a way of using the one-particle formalism to describe the behavior of many similar particles by giving a probability distribution (or ensemble) of states that these particles can be found in.
A simple criterion for checking whether a density matrix is describing a pure or mixed state is that the trace of ρ2 is equal to 1 if the state is pure, and less than 1 if the state is mixed. Another, equivalent, criterion is that the von Neumann entropy is 0 for a pure state, and strictly positive for a mixed state.
The rules for measurement in quantum mechanics are particularly simple to state in terms of density matrices. For example, the ensemble average (expectation value) of a measurement corresponding to an observable A is given by
⟨
A
⟩
=
∑
s
p
s
⟨
ψ
s
|
A
|
ψ
s
⟩
=
∑
s
∑
i
p
s
a
i
|
⟨
α
i
|
ψ
s
⟩
|
2
=
tr
(
ρ
A
)
{\displaystyle \langle A\rangle =\sum _{s}p_{s}\langle \psi _{s}|A|\psi _{s}\rangle =\sum _{s}\sum _{i}p_{s}a_{i}|\langle \alpha _{i}|\psi _{s}\rangle |^{2}=\operatorname {tr} (\rho A)}
where
|
α
i
⟩
{\displaystyle |\alpha _{i}\rangle }
and
a
i
{\displaystyle a_{i}}
are eigenkets and eigenvalues, respectively, for the operator A, and "tr" denotes trace.: 73 It is important to note that two types of averaging are occurring, one (over
i
{\displaystyle i}
) being the usual expected value of the observable when the quantum is in state
|
ψ
s
⟩
{\displaystyle |\psi _{s}\rangle }
, and the other (over
s
{\displaystyle s}
) being a statistical (said incoherent) average with the probabilities ps that the quantum is in those states.
== Mathematical generalizations ==
States can be formulated in terms of observables, rather than as vectors in a vector space. These are positive normalized linear functionals on a C*-algebra, or sometimes other classes of algebras of observables.
See State on a C*-algebra and Gelfand–Naimark–Segal construction for more details.
== See also ==
== Notes ==
== References ==
== Further reading ==
The concept of quantum states, in particular the content of the section Formalism in quantum physics above, is covered in most standard textbooks on quantum mechanics.
For a discussion of conceptual aspects and a comparison with classical states, see:
Isham, Chris J (1995). Lectures on Quantum Theory: Mathematical and Structural Foundations. Imperial College Press. ISBN 978-1-86094-001-9.
For a more detailed coverage of mathematical aspects, see:
Bratteli, Ola; Robinson, Derek W (1987). Operator Algebras and Quantum Statistical Mechanics 1. Springer. ISBN 978-3-540-17093-8. 2nd edition. In particular, see Sec. 2.3.
For a discussion of purifications of mixed quantum states, see Chapter 2 of John Preskill's lecture notes for Physics 219 at Caltech.
For a discussion of geometric aspects see:
Bengtsson I; Życzkowski K (2006). Geometry of Quantum States. Cambridge: Cambridge University Press., second, revised edition (2017) | Wikipedia/Mixed_state_(physics) |
In quantum mechanics, the Schrödinger equation describes how a system changes with time. It does this by relating changes in the state of the system to the energy in the system (given by an operator called the Hamiltonian). Therefore, once the Hamiltonian is known, the time dynamics are in principle known. All that remains is to plug the Hamiltonian into the Schrödinger equation and solve for the system state as a function of time.
Often, however, the Schrödinger equation is difficult to solve (even with a computer). Therefore, physicists have developed mathematical techniques to simplify these problems and clarify what is happening physically. One such technique is to apply a unitary transformation to the Hamiltonian. Doing so can result in a simplified version of the Schrödinger equation which nonetheless has the same solution as the original.
== Transformation ==
A unitary transformation (or frame change) can be expressed in terms of a time-dependent Hamiltonian
H
(
t
)
{\displaystyle H(t)}
and unitary operator
U
(
t
)
{\displaystyle U(t)}
. Under this change, the Hamiltonian transforms as:
H
→
U
H
U
†
+
i
ℏ
U
˙
U
†
=:
H
˘
(
0
)
{\displaystyle H\to UH{U^{\dagger }}+i\hbar \,{{\dot {U}}U^{\dagger }}=:{\breve {H}}\quad \quad (0)}
.
The Schrödinger equation applies to the new Hamiltonian. Solutions to the untransformed and transformed equations are also related by
U
{\displaystyle U}
. Specifically, if the wave function
ψ
(
t
)
{\displaystyle \psi (t)}
satisfies the original equation, then
U
ψ
(
t
)
{\displaystyle U\psi (t)}
will satisfy the new equation.
=== Derivation ===
Recall that by the definition of a unitary matrix,
U
†
U
=
1
{\displaystyle U^{\dagger }U=1}
. Beginning with the Schrödinger equation,
ψ
˙
=
−
i
ℏ
H
ψ
{\displaystyle {\dot {\psi }}=-{\frac {i}{\hbar }}H\psi }
,
we can therefore insert the identity
U
†
U
=
I
{\displaystyle U^{\dagger }U=I}
at will. In particular, inserting it after
H
/
ℏ
{\displaystyle H/\hbar }
and also premultiplying both sides by
U
{\displaystyle U}
, we get
U
ψ
˙
=
−
i
ℏ
(
U
H
U
†
)
U
ψ
(
1
)
{\displaystyle U{\dot {\psi }}=-{\frac {i}{\hbar }}\left(UHU^{\dagger }\right)U\psi \quad \quad (1)}
.
Next, note that by the product rule,
d
d
t
(
U
ψ
)
=
U
˙
ψ
+
U
ψ
˙
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left(U\psi \right)={\dot {U}}\psi +U{\dot {\psi }}}
.
Inserting another
U
†
U
{\displaystyle U^{\dagger }U}
and rearranging, we get
U
ψ
˙
=
d
d
t
(
U
ψ
)
−
U
˙
U
†
U
ψ
(
2
)
{\displaystyle U{\dot {\psi }}={\frac {\mathrm {d} }{\mathrm {d} t}}{\Big (}U\psi {\Big )}-{\dot {U}}U^{\dagger }U\psi \quad \quad (2)}
.
Finally, combining (1) and (2) above results in the desired transformation:
d
d
t
(
U
ψ
)
=
−
i
ℏ
(
U
H
U
†
+
i
ℏ
U
˙
U
†
)
(
U
ψ
)
(
3
)
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\Big (}U\psi {\Big )}=-{\frac {i}{\hbar }}{\Big (}UH{U^{\dagger }}+i\hbar \,{\dot {U}}{U^{\dagger }}{\Big )}{\Big (}U\psi {\Big )}\quad \quad \left(3\right)}
.
If we adopt the notation
ψ
˘
:=
U
ψ
{\displaystyle {\breve {\psi }}:=U\psi }
to describe the transformed wave function, the equations can be written in a clearer form. For instance,
(
3
)
{\displaystyle (3)}
can be rewritten as
d
d
t
ψ
˘
=
−
i
ℏ
H
˘
ψ
˘
(
4
)
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\breve {\psi }}=-{\frac {i}{\hbar }}{\breve {H}}{\breve {\psi }}\quad \quad \left(4\right)}
,
which can be rewritten in the form of the original Schrödinger equation,
H
˘
ψ
˘
=
i
ℏ
d
ψ
˘
d
t
.
{\displaystyle {\breve {H}}{\breve {\psi }}=i\hbar {\operatorname {d} \!{\breve {\psi }} \over \operatorname {d} \!t}.}
The original wave function can be recovered as
ψ
=
U
†
ψ
˘
{\displaystyle \psi =U^{\dagger }{\breve {\psi }}}
.
=== Relation to the interaction picture ===
Unitary transformations can be seen as a generalization of the interaction (Dirac) picture. In the latter approach, a Hamiltonian is broken into a time-independent part and a time-dependent part,
H
(
t
)
=
H
0
+
V
(
t
)
(
a
)
{\displaystyle H(t)=H_{0}+V(t)\quad \quad (a)}
.
In this case, the Schrödinger equation becomes
ψ
I
˙
=
−
i
ℏ
(
e
i
H
0
t
/
ℏ
V
e
−
i
H
0
t
/
ℏ
)
ψ
I
{\displaystyle {\dot {\psi _{I}}}=-{\frac {i}{\hbar }}\left(e^{iH_{0}t/\hbar }Ve^{-iH_{0}t/\hbar }\right)\psi _{I}}
, with
ψ
I
=
e
i
H
0
t
/
ℏ
ψ
{\displaystyle \psi _{I}=e^{iH_{0}t/\hbar }\psi }
.
The correspondence to a unitary transformation can be shown by choosing
U
(
t
)
=
exp
[
+
i
H
0
t
/
ℏ
]
{\textstyle U(t)=\exp \left[{+iH_{0}t/\hbar }\right]}
. As a result,
U
†
(
t
)
=
exp
[
−
i
H
0
t
/
ℏ
]
.
{\displaystyle {U^{\dagger }}(t)=\exp \left[{-iH_{0}t}/\hbar \right].}
Using the notation from
(
a
)
{\displaystyle (a)}
above, our transformed Hamiltonian becomes
H
˘
=
U
[
H
0
+
V
(
t
)
]
U
†
+
i
ℏ
U
˙
U
†
(
b
)
{\displaystyle {\breve {H}}=U\left[H_{0}+V(t)\right]U^{\dagger }+i\hbar {\dot {U}}U^{\dagger }\quad \quad (b)}
First note that since
U
{\displaystyle U}
is a function of
H
0
{\displaystyle H_{0}}
, the two must commute. Then
U
H
0
U
†
=
H
0
{\displaystyle UH_{0}U^{\dagger }=H_{0}}
,
which takes care of the first term in the transformation in
(
b
)
{\displaystyle (b)}
, i.e.
H
˘
=
H
0
+
U
V
(
t
)
U
†
+
i
ℏ
U
˙
U
†
{\displaystyle {\breve {H}}=H_{0}+UV(t)U^{\dagger }+i\hbar {\dot {U}}U^{\dagger }}
. Next use the chain rule to calculate
i
ℏ
U
˙
U
†
=
i
ℏ
(
d
U
d
t
)
e
−
i
H
0
t
/
ℏ
=
i
ℏ
(
i
H
0
/
ℏ
)
e
+
i
H
0
t
/
ℏ
e
−
i
H
0
t
/
ℏ
=
i
ℏ
(
i
H
0
/
ℏ
)
=
−
H
0
,
{\displaystyle {\begin{aligned}i\hbar {\dot {U}}U^{\dagger }&=i\hbar \left({\operatorname {d} \!U \over \operatorname {d} \!t}\right)e^{-iH_{0}t/\hbar }\\&=i\hbar {\Big (}iH_{0}/\hbar {\Big )}e^{+iH_{0}t/\hbar }e^{-iH_{0}t/\hbar }\\&=i\hbar \left({iH_{0}}/\hbar \right)\\&=-H_{0},\\\end{aligned}}}
which cancels with the other
H
0
{\displaystyle H_{0}}
. Evidently we are left with
H
˘
=
U
V
U
†
{\displaystyle {\breve {H}}=UVU^{\dagger }}
, yielding
ψ
I
˙
=
−
i
ℏ
U
V
U
†
ψ
I
{\displaystyle {\dot {\psi _{I}}}=-{\frac {i}{\hbar }}UVU^{\dagger }\psi _{I}}
as shown above.
When applying a general unitary transformation, however, it is not necessary that
H
(
t
)
{\displaystyle H(t)}
be broken into parts, or even that
U
(
t
)
{\displaystyle U(t)}
be a function of any part of the Hamiltonian.
== Examples ==
=== Rotating frame ===
Consider an atom with two states, ground
|
g
⟩
{\displaystyle |g\rangle }
and excited
|
e
⟩
{\displaystyle |e\rangle }
. The atom has a Hamiltonian
H
=
ℏ
ω
|
e
⟩
⟨
e
|
{\displaystyle H=\hbar \omega {|{e}\rangle \langle {e}|}}
, where
ω
{\displaystyle \omega }
is the frequency of light associated with the ground-to-excited transition. Now suppose we illuminate the atom with a drive at frequency
ω
d
{\displaystyle \omega _{d}}
which couples the two states, and that the time-dependent driven Hamiltonian is
H
/
ℏ
=
ω
|
e
⟩
⟨
e
|
+
Ω
e
i
ω
d
t
|
g
⟩
⟨
e
|
+
Ω
∗
e
−
i
ω
d
t
|
e
⟩
⟨
g
|
{\displaystyle H/\hbar =\omega |e\rangle \langle e|+\Omega \ e^{i\omega _{d}t}|g\rangle \langle e|+\Omega ^{*}\ e^{-i\omega _{d}t}|e\rangle \langle g|}
for some complex drive strength
Ω
{\displaystyle \Omega }
. Because of the competing frequency scales (
ω
{\displaystyle \omega }
,
ω
d
{\displaystyle \omega _{d}}
, and
Ω
{\displaystyle \Omega }
), it is difficult to anticipate the effect of the drive (see driven harmonic motion).
Without a drive, the phase of
|
e
⟩
{\displaystyle |e\rangle }
would oscillate relative to
|
g
⟩
{\displaystyle |g\rangle }
. In the Bloch sphere representation of a two-state system, this corresponds to rotation around the z-axis. Conceptually, we can remove this component of the dynamics by entering a rotating frame of reference defined by the unitary transformation
U
=
e
i
ω
t
|
e
⟩
⟨
e
|
{\displaystyle U=e^{i\omega t|e\rangle \langle e|}}
. Under this transformation, the Hamiltonian becomes
H
/
ℏ
→
Ω
e
i
(
ω
d
−
ω
)
t
|
g
⟩
⟨
e
|
+
Ω
∗
e
i
(
ω
−
ω
d
)
t
|
e
⟩
⟨
g
|
{\displaystyle H/\hbar \to \Omega \,e^{i(\omega _{d}-\omega )t}|g\rangle \langle e|+\Omega ^{*}\,e^{i(\omega -\omega _{d})t}|e\rangle \langle g|}
.
If the driving frequency is equal to the g-e transition's frequency,
ω
d
=
ω
{\displaystyle \omega _{d}=\omega }
, resonance will occur and then the equation above reduces to
H
˘
/
ℏ
=
Ω
|
g
⟩
⟨
e
|
+
Ω
∗
|
e
⟩
⟨
g
|
{\displaystyle {\breve {H}}/\hbar =\Omega \ |g\rangle \langle e|+\Omega ^{*}\ |e\rangle \langle g|}
.
From this it is apparent, even without getting into details, that the dynamics will involve an oscillation between the ground and excited states at frequency
Ω
{\displaystyle \Omega }
.
As another limiting case, suppose the drive is far off-resonant,
|
ω
d
−
ω
|
≫
0
{\displaystyle |\omega _{d}-\omega |\gg 0}
. We can figure out the dynamics in that case without solving the Schrödinger equation directly. Suppose the system starts in the ground state
|
g
⟩
{\displaystyle |g\rangle }
. Initially, the Hamiltonian will populate some component of
|
e
⟩
{\displaystyle |e\rangle }
. A small time later, however, it will populate roughly the same amount of
|
e
⟩
{\displaystyle |e\rangle }
but with completely different phase. Thus the effect of an off-resonant drive will tend to cancel itself out. This can also be expressed by saying that an off-resonant drive is rapidly rotating in the frame of the atom.
These concepts are illustrated in the table below, where the sphere represents the Bloch sphere, the arrow represents the state of the atom, and the hand represents the drive.
=== Displaced frame ===
The example above could also have been analyzed in the interaction picture. The following example, however, is more difficult to analyze without the general formulation of unitary transformations. Consider two harmonic oscillators, between which we would like to engineer a beam splitter interaction,
g
a
b
†
+
g
∗
a
†
b
{\displaystyle g\,ab^{\dagger }+g^{*}\,a^{\dagger }b}
.
This was achieved experimentally with two microwave cavity resonators serving as
a
{\displaystyle a}
and
b
{\displaystyle b}
. Below, we sketch the analysis of a simplified version of this experiment.
In addition to the microwave cavities, the experiment also involved a transmon qubit,
c
{\displaystyle c}
, coupled to both modes. The qubit is driven simultaneously at two frequencies,
ω
1
{\displaystyle \omega _{1}}
and
ω
2
{\displaystyle \omega _{2}}
, for which
ω
1
−
ω
2
=
ω
a
−
ω
b
{\displaystyle \omega _{1}-\omega _{2}=\omega _{a}-\omega _{b}}
.
H
d
r
i
v
e
/
ℏ
=
ℜ
[
ϵ
1
e
i
ω
1
t
+
ϵ
2
e
i
ω
2
t
]
(
c
+
c
†
)
.
{\displaystyle H_{\mathrm {drive} }/\hbar =\Re \left[\epsilon _{1}e^{i\omega _{1}t}+\epsilon _{2}e^{i\omega _{2}t}\right](c+c^{\dagger }).}
In addition, there are many fourth-order terms coupling the modes, but most of them can be neglected. In this experiment, two such terms which will become important are
H
4
/
ℏ
=
g
4
(
e
i
(
ω
b
−
ω
a
)
t
a
b
†
+
h.c.
)
c
†
c
{\displaystyle H_{4}/\hbar =g_{4}{\Big (}e^{i(\omega _{b}-\omega _{a})t}ab^{\dagger }+{\text{h.c.}}{\Big )}c^{\dagger }c}
.
(H.c. is shorthand for the Hermitian conjugate.) We can apply a displacement transformation,
U
=
D
(
−
ξ
1
e
−
i
ω
1
t
−
ξ
2
e
−
i
ω
2
t
)
{\displaystyle U=D(-\xi _{1}e^{-i\omega _{1}t}-\xi _{2}e^{-i\omega _{2}t})}
, to mode
c
{\displaystyle c}
. For carefully chosen amplitudes, this transformation will cancel
H
drive
{\displaystyle H_{\textrm {drive}}}
while also displacing the ladder operator,
c
→
c
+
ξ
1
e
−
i
ω
1
t
+
ξ
2
e
−
i
ω
2
t
{\displaystyle c\to c+\xi _{1}e^{-i\omega _{1}t}+\xi _{2}e^{-i\omega _{2}t}}
. This leaves us with
H
/
ℏ
=
g
4
(
e
i
(
ω
b
−
ω
a
)
t
a
b
†
+
e
i
(
ω
a
−
ω
b
)
t
a
†
b
)
(
c
†
+
ξ
1
∗
e
i
ω
1
t
+
ξ
2
∗
e
i
ω
2
t
)
(
c
+
ξ
1
e
−
i
ω
1
t
+
ξ
2
e
−
i
ω
2
t
)
{\displaystyle H/\hbar =g_{4}{\Big (}e^{i(\omega _{b}-\omega _{a})t}ab^{\dagger }+e^{i(\omega _{a}-\omega _{b})t}a^{\dagger }b{\big )}(c^{\dagger }+\xi _{1}^{*}e^{i\omega _{1}t}+\xi _{2}^{*}e^{i\omega _{2}t})(c+\xi _{1}e^{-i\omega _{1}t}+\xi _{2}e^{-i\omega _{2}t})}
.
Expanding this expression and dropping the rapidly rotating terms, we are left with the desired Hamiltonian,
H
/
ℏ
=
g
4
ξ
1
∗
ξ
2
e
i
(
ω
b
−
ω
a
+
ω
1
−
ω
2
)
t
a
b
†
+
h.c.
=
g
a
b
†
+
g
∗
a
†
b
{\displaystyle H/\hbar =g_{4}\xi _{1}^{*}\xi _{2}e^{i(\omega _{b}-\omega _{a}+\omega _{1}-\omega _{2})t}\ ab^{\dagger }+{\text{h.c.}}=g\,ab^{\dagger }+g^{*}\,a^{\dagger }b}
.
=== Relation to the Baker–Campbell–Hausdorff formula ===
It is common for the operators involved in unitary transformations to be written as exponentials of operators,
U
=
e
X
{\displaystyle U=e^{X}}
, as seen above. Further, the operators in the exponentials commonly obey the relation
X
†
=
−
X
{\displaystyle X^{\dagger }=-X}
, so that the transform of an operator
Y
{\displaystyle Y}
is,
U
Y
U
†
=
e
X
Y
e
−
X
{\displaystyle UYU^{\dagger }=e^{X}Ye^{-X}}
. By now introducing the iterator commutator,
[
(
X
)
n
,
Y
]
≡
[
X
,
⋯
[
X
,
[
X
⏟
n
times
,
Y
]
]
⋯
]
,
[
(
X
)
0
,
Y
]
≡
Y
,
{\displaystyle [(X)^{n},Y]\equiv \underbrace {[X,\dotsb [X,[X} _{n{\text{ times }}},Y]]\dotsb ],\quad [(X)^{0},Y]\equiv Y,}
we can use a special result of the Baker-Campbell-Hausdorff formula to write this transformation compactly as,
e
X
Y
e
−
X
=
∑
n
=
0
∞
[
(
X
)
n
,
Y
]
n
!
,
{\displaystyle e^{X}Ye^{-X}=\sum _{n=0}^{\infty }{\frac {[(X)^{n},Y]}{n!}},}
or, in long form for completeness,
e
X
Y
e
−
X
=
Y
+
[
X
,
Y
]
+
1
2
!
[
X
,
[
X
,
Y
]
]
+
1
3
!
[
X
,
[
X
,
[
X
,
Y
]
]
]
+
⋯
.
{\displaystyle e^{X}Ye^{-X}=Y+\left[X,Y\right]+{\frac {1}{2!}}[X,[X,Y]]+{\frac {1}{3!}}[X,[X,[X,Y]]]+\cdots .}
== References == | Wikipedia/Unitary_transformation_(quantum_mechanics) |
The theory of relativity usually encompasses two interrelated physics theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena in the absence of gravity. General relativity explains the law of gravitation and its relation to the forces of nature. It applies to the cosmological and astrophysical realm, including astronomy.
The theory transformed theoretical physics and astronomy during the 20th century, superseding a 200-year-old theory of mechanics created primarily by Isaac Newton. It introduced concepts including 4-dimensional spacetime as a unified entity of space and time, relativity of simultaneity, kinematic and gravitational time dilation, and length contraction. In the field of physics, relativity improved the science of elementary particles and their fundamental interactions, along with ushering in the nuclear age. With relativity, cosmology and astrophysics predicted extraordinary astronomical phenomena such as neutron stars, black holes, and gravitational waves.
== Development and acceptance ==
Albert Einstein published the theory of special relativity in 1905, building on many theoretical results and empirical findings obtained by Albert A. Michelson, Hendrik Lorentz, Henri Poincaré and others. Max Planck, Hermann Minkowski and others did subsequent work.
Einstein developed general relativity between 1907 and 1915, with contributions by many others after 1915. The final form of general relativity was published in 1916.
The term "theory of relativity" was based on the expression "relative theory" (German: Relativtheorie) used in 1906 by Planck, who emphasized how the theory uses the principle of relativity. In the discussion section of the same paper, Alfred Bucherer used for the first time the expression "theory of relativity" (German: Relativitätstheorie).
By the 1920s, the physics community understood and accepted special relativity. It rapidly became a significant and necessary tool for theorists and experimentalists in the new fields of atomic physics, nuclear physics, and quantum mechanics.
By comparison, general relativity did not appear to be as useful, beyond making minor corrections to predictions of Newtonian gravitation theory. It seemed to offer little potential for experimental test, as most of its assertions were on an astronomical scale. Its mathematics seemed difficult and fully understandable only by a small number of people. Around 1960, general relativity became central to physics and astronomy. New mathematical techniques to apply to general relativity streamlined calculations and made its concepts more easily visualized. As astronomical phenomena were discovered, such as quasars (1963), the 3-kelvin microwave background radiation (1965), pulsars (1967), and the first black hole candidates (1981), the theory explained their attributes, and measurement of them further confirmed the theory.
== Special relativity ==
Special relativity is a theory of the structure of spacetime. It was introduced in Einstein's 1905 paper "On the Electrodynamics of Moving Bodies" (for the contributions of many other physicists and mathematicians, see History of special relativity). Special relativity is based on two postulates which are contradictory in classical mechanics:
The laws of physics are the same for all observers in any inertial frame of reference relative to one another (principle of relativity).
The speed of light in vacuum is the same for all observers, regardless of their relative motion or of the motion of the light source.
The resultant theory copes with experiment better than classical mechanics. For instance, postulate 2 explains the results of the Michelson–Morley experiment. Moreover, the theory has many surprising and counterintuitive consequences. Some of these are:
Relativity of simultaneity: Two events, simultaneous for one observer, may not be simultaneous for another observer if the observers are in relative motion.
Time dilation: Moving clocks are measured to tick more slowly than an observer's "stationary" clock.
Length contraction: Objects are measured to be shortened in the direction that they are moving with respect to the observer.
Maximum speed is finite: No physical object, message or field line can travel faster than the speed of light in vacuum.
The effect of gravity can only travel through space at the speed of light, not faster or instantaneously.
Mass–energy equivalence: E = mc2, energy and mass are equivalent and transmutable.
Relativistic mass, idea used by some researchers.
The defining feature of special relativity is the replacement of the Galilean transformations of classical mechanics by the Lorentz transformations. (See Maxwell's equations of electromagnetism.)
== General relativity ==
General relativity is a theory of gravitation developed by Einstein in the years 1907–1915. The development of general relativity began with the equivalence principle, under which the states of accelerated motion and being at rest in a gravitational field (for example, when standing on the surface of the Earth) are physically identical. The upshot of this is that free fall is inertial motion: an object in free fall is falling because that is how objects move when there is no force being exerted on them, instead of this being due to the force of gravity as is the case in classical mechanics. This is incompatible with classical mechanics and special relativity because in those theories inertially moving objects cannot accelerate with respect to each other, but objects in free fall do so. To resolve this difficulty Einstein first proposed that spacetime is curved. Einstein discussed his idea with mathematician Marcel Grossmann and they concluded that general relativity could be formulated in the context of Riemannian geometry which had been developed in the 1800s.
In 1915, he devised the Einstein field equations which relate the curvature of spacetime with the mass, energy, and any momentum within it.
Some of the consequences of general relativity are:
Gravitational time dilation: Clocks run slower in deeper gravitational wells.
Precession: Orbits precess in a way unexpected in Newton's theory of gravity. (This has been observed in the orbit of Mercury and in binary pulsars).
Light deflection: Rays of light bend in the presence of a gravitational field.
Frame-dragging: Rotating masses "drag along" the spacetime around them.
Expansion of the universe: The universe is expanding, and certain components within the universe can accelerate the expansion.
Technically, general relativity is a theory of gravitation whose defining feature is its use of the Einstein field equations. The solutions of the field equations are metric tensors which define the topology of the spacetime and how objects move inertially.
== Experimental evidence ==
Einstein stated that the theory of relativity belongs to a class of "principle-theories". As such, it employs an analytic method, which means that the elements of this theory are not based on hypothesis but on empirical discovery. By observing natural processes, we understand their general characteristics, devise mathematical models to describe what we observed, and by analytical means we deduce the necessary conditions that have to be satisfied. Measurement of separate events must satisfy these conditions and match the theory's conclusions.
=== Tests of special relativity ===
Relativity is a falsifiable theory: It makes predictions that can be tested by experiment. In the case of special relativity, these include the principle of relativity, the constancy of the speed of light, and time dilation. The predictions of special relativity have been confirmed in numerous tests since Einstein published his paper in 1905, but three experiments conducted between 1881 and 1938 were critical to its validation. These are the Michelson–Morley experiment, the Kennedy–Thorndike experiment, and the Ives–Stilwell experiment. Einstein derived the Lorentz transformations from first principles in 1905, but these three experiments allow the transformations to be induced from experimental evidence.
Maxwell's equations—the foundation of classical electromagnetism—describe light as a wave that moves with a characteristic velocity. The modern view is that light needs no medium of transmission, but Maxwell and his contemporaries were convinced that light waves were propagated in a medium, analogous to sound propagating in air, and ripples propagating on the surface of a pond. This hypothetical medium was called the luminiferous aether, at rest relative to the "fixed stars" and through which the Earth moves. Fresnel's partial ether dragging hypothesis ruled out the measurement of first-order (v/c) effects, and although observations of second-order effects (v2/c2) were possible in principle, Maxwell thought they were too small to be detected with then-current technology.
The Michelson–Morley experiment was designed to detect second-order effects of the "aether wind"—the motion of the aether relative to the Earth. Michelson designed an instrument called the Michelson interferometer to accomplish this. The apparatus was sufficiently accurate to detect the expected effects, but he obtained a null result when the first experiment was conducted in 1881, and again in 1887. Although the failure to detect an aether wind was a disappointment, the results were accepted by the scientific community. In an attempt to salvage the aether paradigm, FitzGerald and Lorentz independently created an ad hoc hypothesis in which the length of material bodies changes according to their motion through the aether. This was the origin of FitzGerald–Lorentz contraction, and their hypothesis had no theoretical basis. The interpretation of the null result of the Michelson–Morley experiment is that the round-trip travel time for light is isotropic (independent of direction), but the result alone is not enough to discount the theory of the aether or validate the predictions of special relativity.
While the Michelson–Morley experiment showed that the velocity of light is isotropic, it said nothing about how the magnitude of the velocity changed (if at all) in different inertial frames. The Kennedy–Thorndike experiment was designed to do that, and was first performed in 1932 by Roy Kennedy and Edward Thorndike. They obtained a null result, and concluded that "there is no effect ... unless the velocity of the solar system in space is no more than about half that of the earth in its orbit". That possibility was thought to be too coincidental to provide an acceptable explanation, so from the null result of their experiment it was concluded that the round-trip time for light is the same in all inertial reference frames.
The Ives–Stilwell experiment was carried out by Herbert Ives and G.R. Stilwell first in 1938 and with better accuracy in 1941. It was designed to test the transverse Doppler effect – the redshift of light from a moving source in a direction perpendicular to its velocity—which had been predicted by Einstein in 1905. The strategy was to compare observed Doppler shifts with what was predicted by classical theory, and look for a Lorentz factor correction. Such a correction was observed, from which was concluded that the frequency of a moving atomic clock is altered according to special relativity.
Those classic experiments have been repeated many times with increased precision. Other experiments include, for instance, relativistic energy and momentum increase at high velocities, experimental testing of time dilation, and modern searches for Lorentz violations.
=== Tests of general relativity ===
General relativity has also been confirmed many times, the classic experiments being the perihelion precession of Mercury's orbit, the deflection of light by the Sun, and the gravitational redshift of light. Other tests confirmed the equivalence principle and frame dragging.
== Modern applications ==
Far from being simply of theoretical interest, relativistic effects are important practical engineering concerns. Satellite-based measurement needs to take into account relativistic effects, as each satellite is in motion relative to an Earth-bound user, and is thus in a different frame of reference under the theory of relativity. Global positioning systems such as GPS, GLONASS, and Galileo, must account for all of the relativistic effects in order to work with precision, such as the consequences of the Earth's gravitational field. This is also the case in the high-precision measurement of time. Instruments ranging from electron microscopes to particle accelerators would not work if relativistic considerations were omitted.
== See also ==
Doubly special relativity
Galilean invariance
List of textbooks on relativity
== References ==
== Further reading ==
== External links ==
The dictionary definition of theory of relativity at Wiktionary
Media related to Theory of relativity at Wikimedia Commons | Wikipedia/Relativity_physics |
The Diósi–Penrose model was introduced as a possible solution to the measurement problem, where the wave function collapse is related to gravity. The model was first suggested by Lajos Diósi when studying how possible gravitational fluctuations may affect the dynamics of quantum systems. Later, following a different line of reasoning, Roger Penrose arrived at an estimation for the collapse time of a superposition due to gravitational effects, which is the same (within an unimportant numerical factor) as that found by Diósi, hence the name Diósi–Penrose model. However, it should be pointed out that while Diósi gave a precise dynamical equation for the collapse, Penrose took a more conservative approach, estimating only the collapse time of a superposition.
== The Diósi model ==
In the Diósi model, the wave-function collapse is induced by the interaction of the system with a classical noise field, where the spatial correlation function of this noise is related to the Newtonian potential. The evolution of the state vector
|
ψ
t
⟩
{\displaystyle |\psi _{t}\rangle }
deviates from the Schrödinger equation and has the typical structure of the collapse models equations:
where
is the mass density function, with
m
j
{\displaystyle m_{j}}
,
x
^
j
{\displaystyle {\hat {\mathbf {x} }}_{j}}
and
μ
R
0
(
x
)
{\displaystyle \mu _{R_{0}}(\mathbf {x} )}
respectively the mass, the position operator and the mass density function of the
j
{\displaystyle j}
-th particle of the system.
R
0
{\displaystyle R_{0}}
is a parameter introduced to smear the mass density function, required since taking a point-like mass distribution
M
point
(
x
)
=
∑
j
=
1
N
m
j
δ
(
x
−
x
^
j
)
{\displaystyle {\mathcal {M}}_{\text{point}}(\mathbf {x} )=\sum _{j=1}^{N}m_{j}\delta (\mathbf {x} -{\hat {\mathbf {x} }}_{j})}
would lead to divergences in the predictions of the model, e.g. an infinite collapse rate or increase of energy. Typically, two different distributions for the mass density
μ
R
0
(
x
−
x
^
j
)
{\displaystyle \mu _{R_{0}}(\mathbf {x} -{\hat {\mathbf {x} }}_{j})}
have been considered in the literature: a spherical or a Gaussian mass density profile, given respectively by
μ
R
0
s
(
x
−
x
^
j
)
=
3
4
π
R
0
3
θ
(
|
x
−
x
^
j
|
−
R
0
)
{\displaystyle \mu _{R_{0}}^{\text{s}}(\mathbf {x} -{\hat {\mathbf {x} }}_{j})={\frac {3}{4\pi R_{0}^{3}}}\theta {\big (}|\mathbf {x} -{\hat {\mathbf {x} }}_{j}|-R_{0}{\big )}}
and
μ
R
0
g
(
x
−
x
^
j
)
=
1
(
2
π
R
0
2
)
3
/
2
exp
(
−
(
x
−
x
^
j
)
2
2
R
0
2
)
.
{\displaystyle \mu _{R_{0}}^{\text{g}}(\mathbf {x} -{\hat {\mathbf {x} }}_{j})={\frac {1}{(2\pi R_{0}^{2})^{3/2}}}\,\exp \left(-{\frac {(\mathbf {x} -{\hat {\mathbf {x} }}_{j})^{2}}{2R_{0}^{2}}}\right).}
Choosing one or another distribution
μ
R
0
(
x
−
x
^
j
)
{\displaystyle \mu _{R_{0}}(\mathbf {x} -{\hat {\mathbf {x} }}_{j})}
does not affect significantly the predictions of the model, as long as the same value for
R
0
{\displaystyle R_{0}}
is considered. The noise field
w
(
x
,
t
)
:=
d
W
(
x
,
t
)
d
t
{\displaystyle w(\mathbf {x} ,t):={\frac {dW(\mathbf {x} ,t)}{dt}}}
in Eq. (1) has zero average and correlation given by
where “
E
{\displaystyle \mathbb {E} }
” denotes the average over the noise. One can then understand from Eq. (1) and (3) in which sense the model is gravity-related: the coupling constant between the system and the noise is proportional to the gravitational constant
G
{\displaystyle G}
, and the spatial correlation of the noise field
w
(
x
,
t
)
{\displaystyle w(\mathbf {x} ,t)}
has the typical form of a Newtonian potential. Similarly to other collapse models, the Diósi–Penrose model shares the following two features:
The model describes a collapse in position.
There is an amplification mechanism, which guarantees that more massive objects localize more effectively.
In order to show these features, it is convenient to write the master equation for the statistical operator
ρ
(
t
)
=
E
[
|
ψ
t
⟩
⟨
ψ
t
|
]
{\displaystyle \rho (t)=\mathbb {E} {\big [}|\psi _{t}\rangle \langle \psi _{t}|{\big ]}}
corresponding to Eq. (1):
It is interesting to point out that this master equation has more recently been re-derived by L. Diósi using a hybrid approach where quantized massive particles interact with classical gravitational fields.
If one considers the master equation in the position basis, introducing
ρ
(
a
→
,
b
→
,
t
)
:=
⟨
a
→
|
ρ
(
t
)
|
b
→
⟩
{\displaystyle \rho ({\vec {\boldsymbol {a}}},{\vec {\boldsymbol {b}}},t):=\langle {\vec {\boldsymbol {a}}}|\rho (t)|{\vec {\boldsymbol {b}}}\rangle }
with
|
a
→
⟩
:=
|
a
1
⟩
⊗
⋯
⊗
|
a
N
⟩
{\displaystyle |{\vec {\boldsymbol {a}}}\rangle :=|{\boldsymbol {a}}_{1}\rangle \otimes \dots \otimes |{\boldsymbol {a}}_{N}\rangle }
, where
|
a
j
⟩
{\displaystyle |{\boldsymbol {a}}_{j}\rangle }
is a position eigenstate of the
j
{\displaystyle j}
-th particle, neglecting the free evolution, one finds
with
where
M
(
x
,
a
→
)
:=
∑
j
m
j
μ
R
0
(
x
−
a
j
)
{\displaystyle {\mathcal {M}}(\mathbf {x} ,{\vec {\boldsymbol {a}}}):=\sum _{j}m_{j}\mu _{R_{0}}(\mathbf {x} -{\boldsymbol {a}}_{j})}
is the mass density when the particles of the system are centered at the points
a
1
{\displaystyle {\boldsymbol {a}}_{1}}
, ...,
a
N
{\displaystyle {\boldsymbol {a}}_{N}}
. Eq. (5) can be solved exactly, and one gets
where
As expected, for the diagonal terms of the density matrix, when
a
→
=
b
→
{\displaystyle {\vec {\boldsymbol {a}}}={\vec {\boldsymbol {b}}}}
, one has
Λ
(
a
→
,
a
→
)
=
0
{\displaystyle \Lambda ({\vec {\boldsymbol {a}}},{\vec {\boldsymbol {a}}})=0}
, i.e. the time of decay goes to infinity, implying that states with well-localized position are not affected by the collapse. On the contrary, the off-diagonal terms
a
→
≠
b
→
{\displaystyle {\vec {\boldsymbol {a}}}\neq {\vec {\boldsymbol {b}}}}
, which are different from zero when a spatial superposition is involved, will decay with a time of decay given by Eq. (8).
To get an idea of the scale at which the gravitationally induced collapse becomes relevant, one can compute the time of decay in Eq. (8) for the case of a sphere with radius
R
0
{\displaystyle R_{0}}
and mass
m
{\displaystyle m}
in a spatial superposition at a distance
d
:=
|
a
−
b
|
{\displaystyle d:=|{\boldsymbol {a}}-{\boldsymbol {b}}|}
. Then the time of decay can be computed) using Eq. (8) with
where
λ
=
d
/
(
2
R
0
)
{\displaystyle \lambda =d/(2R_{0})}
. To give some examples, if one considers a proton, for which
m
≃
1.67
×
10
−
27
{\displaystyle m\simeq 1.67\times 10^{-27}}
kg and
R
0
≃
10
−
15
{\displaystyle R_{0}\simeq 10^{-15}}
m, in a superposition with
d
≫
R
0
{\displaystyle d\gg R_{0}}
, one gets
τ
DP
≃
10
6
{\displaystyle \tau _{\text{DP}}\simeq 10^{6}}
years. On the contrary, for a dust grain with
m
≃
6
×
10
−
12
{\displaystyle m\simeq 6\times 10^{-12}}
kg and
R
0
≃
10
−
5
{\displaystyle R_{0}\simeq 10^{-5}}
m, one gets one gets
τ
DP
≃
10
−
8
{\displaystyle \tau _{\text{DP}}\simeq 10^{-8}}
s. Therefore, contrary to what might be expected considering the weaknesses of gravitational force, the effects of the gravity-related collapse become relevant already at the mesoscopic scale.
Recently, the model have been generalized by including dissipative and non-Markovian effects.
== Penrose's proposal ==
It is well known that general relativity and quantum mechanics, our most fundamental theories for describing the universe, are not compatible, and the unification of the two is still missing. The standard approach to overcome this situation is to try to modify general relativity by quantizing gravity. Penrose suggests an opposite approach, what he calls “gravitization of quantum mechanics”, where quantum mechanics gets modified when gravitational effects become relevant. The reasoning underlying this approach is the following one: take a massive system of well-localized states in space. In this case, the state being well-localized, the induced space–time curvature is well defined. According to quantum mechanics, because of the superposition principle, the system can be placed (at least in principle) in a superposition of two well-localized states, which would lead to a superposition of two different space–times. The key idea is that since space–time metric should be well defined, nature “dislikes” these space–time superpositions and suppresses them by collapsing the wave function to one of the two localized states.
To set these ideas on a more quantitative ground, Penrose suggested that a way for measuring the difference between two space–times, in the Newtonian limit, is
where
g
i
(
r
)
{\displaystyle g_{i}({\boldsymbol {r}})}
is the Newtonian gravitational acceleration at the point where the system is localized around
i
{\displaystyle i}
. The acceleration
g
i
(
r
)
{\displaystyle g_{i}({\boldsymbol {r}})}
can be written in terms of the corresponding gravitational potentials
Φ
i
(
r
)
{\displaystyle \Phi _{i}({\boldsymbol {r}})}
, i.e.
g
i
(
r
)
=
−
∇
Φ
i
(
r
)
{\displaystyle g_{i}({\boldsymbol {r}})=-\nabla \Phi _{i}({\boldsymbol {r}})}
. Using this relation in Eq. (9), together with the Poisson equation
∇
2
Φ
i
(
r
)
=
4
π
G
μ
i
(
r
)
{\displaystyle \nabla ^{2}\Phi _{i}({\boldsymbol {r}})=4\pi G\mu _{i}({\boldsymbol {r}})}
, with
μ
i
(
r
)
{\displaystyle \mu _{i}({\boldsymbol {r}})}
giving the mass density when the state is localized around
i
{\displaystyle i}
, and its solution, one arrives at
The corresponding decay time can be obtained by the Heisenberg time–energy uncertainty:
which, apart for a factor
8
π
{\displaystyle 8\pi }
simply due to the use of different conventions, is exactly the same as the time decay
τ
DP
{\displaystyle \tau _{\text{DP}}}
derived by Diósi's model. This is the reason why the two proposals are named together as the Diósi–Penrose model.
More recently, Penrose suggested a new and quite elegant way to justify the need for a gravity-induced collapse, based on avoiding tensions between the superposition principle and the equivalence principle, the cornerstones of quantum mechanics and general relativity. In order to explain it, let us start by comparing the evolution of a generic state in the presence of uniform gravitational acceleration
g
{\displaystyle {\boldsymbol {g}}}
. One way to perform the calculation, what Penrose calls “Newtonian perspective”, consists in working in an inertial frame, with space–time coordinates
(
r
,
t
)
{\displaystyle ({\boldsymbol {r}},t)}
and solve the Schrödinger equation in presence of the potential
V
(
x
)
=
m
g
⋅
x
{\displaystyle V({\boldsymbol {x}})=m{\boldsymbol {g}}\cdot {\boldsymbol {x}}}
(typically, one chooses the coordinates in such a way that the acceleration
g
{\displaystyle {\boldsymbol {g}}}
is directed along the
z
{\displaystyle z}
axis, in which case
V
(
z
)
=
m
g
z
{\displaystyle V(z)=mgz}
). Alternatively, because of the equivalence principle, one can choose to go in the free-fall reference frame, with coordinates
(
R
,
T
)
{\displaystyle ({\boldsymbol {R}},T)}
related to
(
r
,
t
)
{\displaystyle ({\boldsymbol {r}},t)}
by
R
=
r
+
1
2
g
t
2
{\displaystyle {\boldsymbol {R}}={\boldsymbol {r}}+{\frac {1}{2}}{\boldsymbol {g}}t^{2}}
and
T
=
t
{\displaystyle T=t}
, solve the free Schrödinger equation in that reference frame, and then write the results in terms of the inertial coordinates
(
r
,
t
)
{\displaystyle ({\boldsymbol {r}},t)}
. This is what Penrose calls “Einsteinian perspective”. The solution
Ψ
(
r
,
t
)
{\displaystyle \Psi ({\boldsymbol {r}},t)}
obtained in the Einsteinian perspective and the one
ψ
(
r
,
t
)
{\displaystyle \psi ({\boldsymbol {r}},t)}
obtained in the Newtonian perspective are related to each other by
Since the two wave functions are equivalent apart from an overall phase, they lead to the same physical predictions, which implies that there are no problems in this situation where the gravitational field always has a well-defined value. However, if the space–time metric is not well defined, then we will be in a situation where there is a superposition of a gravitational field corresponding to the acceleration
g
a
{\displaystyle {\boldsymbol {g}}_{a}}
and one corresponding to the acceleration
g
b
{\displaystyle {\boldsymbol {g}}_{b}}
. This does not create problems as long as one sticks to the Newtonian perspective. However, when using the Einstenian perspective, it will imply a phase difference between the two branches of the superposition given by
e
i
m
ℏ
(
1
6
(
g
a
−
g
b
)
2
t
3
+
(
g
a
−
g
b
)
⋅
r
t
)
{\displaystyle e^{i{\frac {m}{\hbar }}\left({\frac {1}{6}}({\boldsymbol {g}}_{a}-{\boldsymbol {g}}_{b})^{2}t^{3}+({\boldsymbol {g}}_{a}-{\boldsymbol {g}}_{b})\cdot {\boldsymbol {r}}\,t\right)}}
. While the term in the exponent linear in the time
t
{\displaystyle t}
does not lead to any conceptual difficulty, the first term, proportional to
t
3
{\displaystyle t^{3}}
, is problematic, since it is a non-relativistic residue of the so-called Unruh effect: in other words, the two terms in the superposition belong to different Hilbert spaces and, strictly speaking, cannot be superposed. Here is where the gravity-induced collapse plays a role, collapsing the superposition when the first term of the phase
1
6
(
g
a
−
g
b
)
2
t
3
{\displaystyle {\frac {1}{6}}(g_{a}-g_{b})^{2}t^{3}}
becomes too large.
Further information on Penrose's idea for the gravity-induced collapse can be also found in the Penrose interpretation.
== Experimental tests and theoretical bounds ==
Since the Diósi–Penrose model predicts deviations from standard quantum mechanics, the model can be tested. The only free parameter of the model is the size of the mass density distribution, given by
R
0
{\displaystyle R_{0}}
. All bounds present in the literature are based on an indirect effect of the gravitational-related collapse: a Brownian-like diffusion induced by the collapse on the motion of the particles. This Brownian-like diffusion is a common feature of all objective-collapse theories and, typically, allows to set the strongest bounds on the parameters of these models. The first bound on
R
0
{\displaystyle R_{0}}
was set by Ghirardi et al., where it was shown that
R
0
>
10
−
15
{\displaystyle R_{0}>10^{-15}}
m to avoid unrealistic heating due to this Brownian-like induced diffusion. Then the bound has been further restricted to
R
0
>
4
×
10
−
14
{\displaystyle R_{0}>4\times 10^{-14}}
m by the analysis of the data from gravitational wave detectors. and later to
R
0
≳
10
−
13
{\displaystyle R_{0}\gtrsim 10^{-13}}
m by studying the heating of neutron stars.
Regarding direct interferometric tests of the model, where a system is prepared in a spatial superposition, there are two proposals currently considered: an optomechanical setup with a mesoscopic mirror to be placed in a superposition by a laser, and experiments involving superpositions of Bose–Einstein condensates.
== See also ==
== References == | Wikipedia/Diósi–Penrose_model |
In theoretical physics, the Dirac–Kähler equation, also known as the Ivanenko–Landau–Kähler equation, is the geometric analogue of the Dirac equation that can be defined on any pseudo-Riemannian manifold using the Laplace–de Rham operator. In four-dimensional flat spacetime, it is equivalent to four copies of the Dirac equation that transform into each other under Lorentz transformations, although this is no longer true in curved spacetime. The geometric structure gives the equation a natural discretization that is equivalent to the staggered fermion formalism in lattice field theory, making Dirac–Kähler fermions the formal continuum limit of staggered fermions. The equation was discovered by Dmitri Ivanenko and Lev Landau in 1928 and later rediscovered by Erich Kähler in 1962.
== Mathematical overview ==
In four dimensional Euclidean spacetime a generic fields of differential forms
Φ
=
∑
H
Φ
H
(
x
)
d
x
H
,
{\displaystyle \Phi =\sum _{H}\Phi _{H}(x)dx_{H},}
is written as a linear combination of sixteen basis forms indexed by
H
{\displaystyle H}
, which runs over the sixteen ordered combinations of indices
{
μ
1
,
…
,
μ
h
}
{\displaystyle \{\mu _{1},\dots ,\mu _{h}\}}
with
μ
1
<
⋯
<
μ
h
{\displaystyle \mu _{1}<\cdots <\mu _{h}}
. Each index runs from one to four. Here
Φ
H
(
x
)
=
Φ
μ
1
…
μ
h
(
x
)
{\displaystyle \Phi _{H}(x)=\Phi _{\mu _{1}\dots \mu _{h}}(x)}
are antisymmetric tensor fields while
d
x
H
{\displaystyle dx_{H}}
are the corresponding differential form basis elements
d
x
H
=
d
x
μ
1
∧
⋯
∧
d
x
μ
h
.
{\displaystyle dx_{H}=dx^{\mu _{1}}\wedge \cdots \wedge dx^{\mu _{h}}.}
Using the Hodge star operator
⋆
{\displaystyle \star }
, the exterior derivative
d
{\displaystyle d}
is related to the codifferential through
δ
=
−
⋆
d
⋆
{\displaystyle \delta =-\star d\star }
. These form the Laplace–de Rham operator
d
−
δ
{\displaystyle d-\delta }
which can be viewed as the square root of the Laplacian operator since
(
d
−
δ
)
2
=
◻
{\displaystyle (d-\delta )^{2}=\square }
. The Dirac–Kähler equation is motivated by noting that this is also the property of the Dirac operator, yielding
This equation is closely related to the usual Dirac equation, a connection which emerges from the close relation between the exterior algebra of differential forms and the Clifford algebra of which Dirac spinors are irreducible representations. For the basis elements to satisfy the Clifford algebra
{
d
x
μ
,
d
x
ν
}
=
2
δ
μ
ν
{\displaystyle \{dx^{\mu },dx^{\nu }\}=2\delta ^{\mu \nu }}
, it is required to introduce a new Clifford product
∨
{\displaystyle \vee }
acting on basis elements as
d
x
μ
∨
d
x
ν
=
d
x
μ
∧
d
x
ν
+
δ
μ
ν
.
{\displaystyle dx_{\mu }\vee dx_{\nu }=dx_{\mu }\wedge dx_{\nu }+\delta _{\mu \nu }.}
Using this product, the action of the Laplace–de Rham operator on differential form basis elements is written as
(
d
−
δ
)
Φ
(
x
)
=
d
x
μ
∨
∂
μ
Φ
(
x
)
.
{\displaystyle (d-\delta )\Phi (x)=dx^{\mu }\vee \partial _{\mu }\Phi (x).}
To acquire the Dirac equation, a change of basis must be performed, where the new basis can be packaged into a matrix
Z
a
b
{\displaystyle Z_{ab}}
defined using the Dirac matrices
Z
a
b
=
∑
H
(
−
1
)
h
(
h
−
1
)
/
2
(
γ
H
)
a
b
T
d
x
H
.
{\displaystyle Z_{ab}=\sum _{H}(-1)^{h(h-1)/2}(\gamma _{H})_{ab}^{T}dx_{H}.}
The matrix
Z
{\displaystyle Z}
is designed to satisfy
d
x
μ
∨
Z
=
γ
μ
T
Z
{\displaystyle dx_{\mu }\vee Z=\gamma _{\mu }^{T}Z}
, decomposing the Clifford algebra into four irreducible copies of the Dirac algebra. This is because in this basis the Clifford product only mixes the column elements indexed by
a
{\displaystyle a}
. Writing the differential form in this basis
Φ
=
∑
a
b
Ψ
(
x
)
a
b
Z
a
b
,
{\displaystyle \Phi =\sum _{ab}\Psi (x)_{ab}Z_{ab},}
transforms the Dirac–Kähler equation into four sets of the Dirac equation indexed by
b
{\displaystyle b}
(
γ
μ
∂
μ
+
m
)
Ψ
(
x
)
b
=
0.
{\displaystyle (\gamma ^{\mu }\partial _{\mu }+m)\Psi (x)_{b}=0.}
The minimally coupled Dirac–Kähler equation is found by replacing the derivative with the covariant derivative
d
x
μ
∨
∂
μ
→
d
x
μ
∨
D
μ
{\displaystyle dx^{\mu }\vee \partial _{\mu }\rightarrow dx^{\mu }\vee D_{\mu }}
leading to
(
d
−
δ
+
m
)
Φ
=
i
A
∨
Φ
.
{\displaystyle (d-\delta +m)\Phi =iA\vee \Phi .}
As before, this is also equivalent to four copies of the Dirac equation. In the abelian case
A
=
e
A
μ
d
x
μ
{\displaystyle A=eA_{\mu }dx^{\mu }}
, while in the non-abelian case there are additional color indices. The Dirac–Kähler fermion
Φ
{\displaystyle \Phi }
also picks up color indices, with it formally corresponding to cross-sections of the Whitney product of the Atiyah–Kähler bundle of differential forms with the vector bundle of local color spaces.
=== Discretization ===
There is a natural way in which to discretize the Dirac–Kähler equation using the correspondence between exterior algebra and simplicial complexes. In four dimensional space a lattice can be considered as a simplicial complex, whose simplexes are constructed using a basis of
h
{\displaystyle h}
-dimensional hypercubes
C
x
,
H
(
h
)
{\displaystyle C_{x,H}^{(h)}}
with a base point
x
{\displaystyle x}
and an orientation determined by
H
{\displaystyle H}
. Then a h-chain is a formal linear combination
C
(
h
)
=
∑
x
,
H
α
x
,
H
C
x
,
H
(
h
)
.
{\displaystyle C^{(h)}=\sum _{x,H}\alpha _{x,H}C_{x,H}^{(h)}.}
The h-chains admit a boundary operator
Δ
C
x
,
H
(
h
)
{\displaystyle \Delta C_{x,H}^{(h)}}
defined as the (h-1)-simplex forming the boundary of the h-chain. A coboundary operator
∇
C
x
,
H
(
h
)
{\displaystyle \nabla C_{x,H}^{(h)}}
can be similarly defined to yield a (h+1)-chain. The dual space of chains consists of
h
{\displaystyle h}
-cochains
Φ
(
h
)
(
C
(
h
)
)
{\displaystyle \Phi ^{(h)}(C^{(h)})}
, which are linear functions acting on the h-chains mapping them to real numbers. The boundary and coboundary operators admit similar structures in dual space called the dual boundary
Δ
^
{\displaystyle {\hat {\Delta }}}
and dual coboundary
∇
^
{\displaystyle {\hat {\nabla }}}
defined to satisfy
(
Δ
^
Φ
)
(
C
)
=
Φ
(
Δ
C
)
,
(
∇
^
Φ
)
(
C
)
=
Φ
(
∇
C
)
.
{\displaystyle ({\hat {\Delta }}\Phi )(C)=\Phi (\Delta C),\ \ \ \ \ \ \ ({\hat {\nabla }}\Phi )(C)=\Phi (\nabla C).}
Under the correspondence between the exterior algebra and simplicial complexes, differential forms are equivalent to cochains, while the exterior derivative and codifferential correspond to the dual boundary and dual coboundary, respectively. Therefore, the Dirac–Kähler equation is written on simplicial complexes as
(
Δ
^
−
∇
^
+
m
)
Φ
(
C
)
=
0.
{\displaystyle ({\hat {\Delta }}-{\hat {\nabla }}+m)\Phi (C)=0.}
The resulting discretized Dirac–Kähler fermion
Φ
(
C
)
{\displaystyle \Phi (C)}
is equivalent to the staggered fermion found in lattice field theory, which can be seen explicitly by an explicit change of basis. This equivalence shows that the continuum Dirac–Kähler fermion is the formal continuum limit of fermion staggered fermions.
== Relation to the Dirac equation ==
As described previously, the Dirac–Kähler equation in flat spacetime is equivalent to four copies of the Dirac equation, despite being a set of equations for antisymmetric tensor fields. The ability of integer spin tensor fields to describe half integer spinor fields is explained by the fact that Lorentz transformations do not commute with the internal Dirac–Kähler
SO
(
2
,
4
)
{\displaystyle {\text{SO}}(2,4)}
symmetry, with the parameters of this symmetry being tensors rather than scalars. This means that the Lorentz transformations mix different spins together and the Dirac fermions are not strictly speaking half-integer spin representations of the Clifford algebra. They instead correspond to a coherent superposition of differential forms. In higher dimensions, particularly on
2
2
n
{\displaystyle 2^{2^{n}}}
dimensional surfaces, the Dirac–Kähler equation is equivalent to
2
2
n
−
1
{\displaystyle 2^{2^{n-1}}}
Dirac equations.
In curved spacetime, the Dirac–Kähler equation no longer decomposes into four Dirac equations. Rather it is a modified Dirac equation acquired if the Dirac operator remained the square root of the Laplace operator, a property not shared by the Dirac equation in curved spacetime. This comes at the expense of Lorentz invariance, although these effects are suppressed by powers of the Planck mass. The equation also differs in that its zero modes on a compact manifold are always guaranteed to exist whenever some of the Betti numbers vanish, being given by the harmonic forms, unlike for the Dirac equation which never has zero modes on a manifold with positive curvature.
== See also ==
Fermion doubling
Lattice QCD
== References == | Wikipedia/Dirac–Kähler_equation |
In mathematics, a positive-definite function is, depending on the context, either of two types of function.
== Definition 1 ==
Let
R
{\displaystyle \mathbb {R} }
be the set of real numbers and
C
{\displaystyle \mathbb {C} }
be the set of complex numbers.
A function
f
:
R
→
C
{\displaystyle f:\mathbb {R} \to \mathbb {C} }
is called positive semi-definite if for all real numbers x1, …, xn the n × n matrix
A
=
(
a
i
j
)
i
,
j
=
1
n
,
a
i
j
=
f
(
x
i
−
x
j
)
{\displaystyle A=\left(a_{ij}\right)_{i,j=1}^{n}~,\quad a_{ij}=f(x_{i}-x_{j})}
is a positive semi-definite matrix.
By definition, a positive semi-definite matrix, such as
A
{\displaystyle A}
, is Hermitian; therefore f(−x) is the complex conjugate of f(x)).
In particular, it is necessary (but not sufficient) that
f
(
0
)
≥
0
,
|
f
(
x
)
|
≤
f
(
0
)
{\displaystyle f(0)\geq 0~,\quad |f(x)|\leq f(0)}
(these inequalities follow from the condition for n = 1, 2.)
A function is negative semi-definite if the inequality is reversed. A function is definite if the weak inequality is replaced with a strong (<, > 0).
=== Examples ===
If
(
X
,
⟨
⋅
,
⋅
⟩
)
{\displaystyle (X,\langle \cdot ,\cdot \rangle )}
is a real inner product space, then
g
y
:
X
→
C
{\displaystyle g_{y}\colon X\to \mathbb {C} }
,
x
↦
exp
(
i
⟨
y
,
x
⟩
)
{\displaystyle x\mapsto \exp(i\langle y,x\rangle )}
is positive definite for every
y
∈
X
{\displaystyle y\in X}
: for all
u
∈
C
n
{\displaystyle u\in \mathbb {C} ^{n}}
and all
x
1
,
…
,
x
n
{\displaystyle x_{1},\ldots ,x_{n}}
we have
u
∗
A
(
g
y
)
u
=
∑
j
,
k
=
1
n
u
k
¯
u
j
e
i
⟨
y
,
x
k
−
x
j
⟩
=
∑
k
=
1
n
u
k
¯
e
i
⟨
y
,
x
k
⟩
∑
j
=
1
n
u
j
e
−
i
⟨
y
,
x
j
⟩
=
|
∑
j
=
1
n
u
j
¯
e
i
⟨
y
,
x
j
⟩
|
2
≥
0.
{\displaystyle u^{*}A^{(g_{y})}u=\sum _{j,k=1}^{n}{\overline {u_{k}}}u_{j}e^{i\langle y,x_{k}-x_{j}\rangle }=\sum _{k=1}^{n}{\overline {u_{k}}}e^{i\langle y,x_{k}\rangle }\sum _{j=1}^{n}u_{j}e^{-i\langle y,x_{j}\rangle }=\left|\sum _{j=1}^{n}{\overline {u_{j}}}e^{i\langle y,x_{j}\rangle }\right|^{2}\geq 0.}
As nonnegative linear combinations of positive definite functions are again positive definite, the cosine function is positive definite as a nonnegative linear combination of the above functions:
cos
(
x
)
=
1
2
(
e
i
x
+
e
−
i
x
)
=
1
2
(
g
1
+
g
−
1
)
.
{\displaystyle \cos(x)={\frac {1}{2}}(e^{ix}+e^{-ix})={\frac {1}{2}}(g_{1}+g_{-1}).}
One can create a positive definite function
f
:
X
→
C
{\displaystyle f\colon X\to \mathbb {C} }
easily from positive definite function
f
:
R
→
C
{\displaystyle f\colon \mathbb {R} \to \mathbb {C} }
for any vector space
X
{\displaystyle X}
: choose a linear function
ϕ
:
X
→
R
{\displaystyle \phi \colon X\to \mathbb {R} }
and define
f
∗
:=
f
∘
ϕ
{\displaystyle f^{*}:=f\circ \phi }
.
Then
u
∗
A
(
f
∗
)
u
=
∑
j
,
k
=
1
n
u
k
¯
u
j
f
∗
(
x
k
−
x
j
)
=
∑
j
,
k
=
1
n
u
k
¯
u
j
f
(
ϕ
(
x
k
)
−
ϕ
(
x
j
)
)
=
u
∗
A
~
(
f
)
u
≥
0
,
{\displaystyle u^{*}A^{(f^{*})}u=\sum _{j,k=1}^{n}{\overline {u_{k}}}u_{j}f^{*}(x_{k}-x_{j})=\sum _{j,k=1}^{n}{\overline {u_{k}}}u_{j}f(\phi (x_{k})-\phi (x_{j}))=u^{*}{\tilde {A}}^{(f)}u\geq 0,}
where
A
~
(
f
)
=
(
f
(
ϕ
(
x
i
)
−
ϕ
(
x
j
)
)
=
f
(
x
~
i
−
x
~
j
)
)
i
,
j
{\displaystyle {\tilde {A}}^{(f)}={\big (}f(\phi (x_{i})-\phi (x_{j}))=f({\tilde {x}}_{i}-{\tilde {x}}_{j}){\big )}_{i,j}}
where
x
~
k
:=
ϕ
(
x
k
)
{\displaystyle {\tilde {x}}_{k}:=\phi (x_{k})}
are distinct as
ϕ
{\displaystyle \phi }
is linear.
=== Bochner's theorem ===
Positive-definiteness arises naturally in the theory of the Fourier transform; it can be seen directly that to be positive-definite it is sufficient for f to be the Fourier transform of a function g on the real line with g(y) ≥ 0.
The converse result is Bochner's theorem, stating that any continuous positive-definite function on the real line is the Fourier transform of a (positive) measure.
==== Applications ====
In statistics, and especially Bayesian statistics, the theorem is usually applied to real functions. Typically, n scalar measurements of some scalar value at points in
R
d
{\displaystyle R^{d}}
are taken and points that are mutually close are required to have measurements that are highly correlated. In practice, one must be careful to ensure that the resulting covariance matrix (an n × n matrix) is always positive-definite. One strategy is to define a correlation matrix A which is then multiplied by a scalar to give a covariance matrix: this must be positive-definite. Bochner's theorem states that if the correlation between two points is dependent only upon the distance between them (via function f), then function f must be positive-definite to ensure the covariance matrix A is positive-definite. See Kriging.
In this context, Fourier terminology is not normally used and instead it is stated that f(x) is the characteristic function of a symmetric probability density function (PDF).
=== Generalization ===
One can define positive-definite functions on any locally compact abelian topological group; Bochner's theorem extends to this context. Positive-definite functions on groups occur naturally in the representation theory of groups on Hilbert spaces (i.e. the theory of unitary representations).
== Definition 2 ==
Alternatively, a function
f
:
R
n
→
R
{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }
is called positive-definite on a neighborhood D of the origin if
f
(
0
)
=
0
{\displaystyle f(0)=0}
and
f
(
x
)
>
0
{\displaystyle f(x)>0}
for every non-zero
x
∈
D
{\displaystyle x\in D}
.
Note that this definition conflicts with definition 1, given above.
In physics, the requirement that
f
(
0
)
=
0
{\displaystyle f(0)=0}
is sometimes dropped (see, e.g., Corney and Olsen).
== See also ==
Positive definiteness
Positive-definite kernel
== References ==
Christian Berg, Christensen, Paul Ressel. Harmonic Analysis on Semigroups, GTM, Springer Verlag.
Z. Sasvári, Positive Definite and Definitizable Functions, Akademie Verlag, 1994
Wells, J. H.; Williams, L. R. Embeddings and extensions in analysis. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 84. Springer-Verlag, New York-Heidelberg, 1975. vii+108 pp.
== Notes ==
== External links ==
"Positive-definite function", Encyclopedia of Mathematics, EMS Press, 2001 [1994] | Wikipedia/Positive-definite_function |
In quantum mechanics, the Lévy-Leblond equation describes the dynamics of a spin-1/2 particle. It is a linearized version of the Schrödinger equation and of the Pauli equation. It was derived by French physicist Jean-Marc Lévy-Leblond in 1967.
Lévy-Leblond equation was obtained under similar heuristic derivations as the Dirac equation, but contrary to the latter, Lévy-Leblond equation is not relativistic. As both equations recover the electron gyromagnetic ratio, it is suggested that spin is not necessarily a relativistic phenomenon.
== Equation ==
For a nonrelativistic spin-1/2 particle of mass m, a representation of the time-independent Lévy-Leblond equation reads:
{
E
ψ
+
(
σ
⋅
p
c
)
χ
=
0
(
σ
⋅
p
c
)
ψ
+
2
m
c
2
χ
=
0
{\displaystyle \left\{{\begin{matrix}E\psi +({\boldsymbol {\sigma }}\cdot \mathbf {p} c)\chi =0\\({\boldsymbol {\sigma }}\cdot \mathbf {p} c)\psi +2mc^{2}\chi =0\end{matrix}}\right.}
where c is the speed of light, E is the nonrelativistic particle energy,
p
=
−
i
ℏ
∇
{\displaystyle \mathbf {p} =-i\hbar \nabla }
is the momentum operator, and
σ
=
(
σ
x
,
σ
y
,
σ
z
)
{\displaystyle {\boldsymbol {\sigma }}=(\sigma _{x},\sigma _{y},\sigma _{z})}
is the vector of Pauli matrices, which is proportional to the spin operator
S
=
1
2
ℏ
σ
{\displaystyle \mathbf {S} ={\tfrac {1}{2}}\hbar {\boldsymbol {\sigma }}}
. Here
ψ
,
χ
{\displaystyle \psi ,\chi }
are two components functions (spinors) describing the wave function of the particle.
By minimal coupling, the equation can be modified to account for the presence of an electromagnetic field,
{
(
E
−
q
V
)
ψ
+
[
σ
⋅
(
p
−
q
A
)
c
]
χ
=
0
[
σ
⋅
(
p
−
q
A
)
c
]
ψ
+
2
m
c
2
χ
=
0
{\displaystyle \left\{{\begin{matrix}(E-qV)\psi +[{\boldsymbol {\sigma }}\cdot (\mathbf {p} -q\mathbf {A} )c]\chi =0\\{[{\boldsymbol {\sigma }}\cdot (\mathbf {p} -q\mathbf {A} )c]}\psi +2mc^{2}\chi =0\end{matrix}}\right.}
where q is the electric charge of the particle. V is the electric potential, and A is the magnetic vector potential. This equation is linear in its spatial derivatives.
== Relation to spin ==
In 1928, Paul Dirac linearized the relativistic dispersion relation and obtained Dirac equation, described by a bispinor. This equation can be decoupled into two spinors in the non-relativistic limit, leading to predict the electron magnetic moment with a gyromagnetic ratio
g
=
2
{\textstyle g=2}
. The success of Dirac theory has led to some textbooks to erroneously claim that spin is necessarily a relativistic phenomena.
Jean-Marc Lévy-Leblond applied the same technique to the non-relativistic energy relation showing that the same prediction of
g
=
2
{\textstyle g=2}
can be obtained. Actually to derive the Pauli equation from Dirac equation one has to pass by Lévy-Leblond equation. Spin is then a result of quantum mechanics and linearization of the equations but not necessarily a relativistic effect.
Lévy-Leblond equation is Galilean invariant. This equation demonstrates that one does not need the full Poincaré group to explain the spin 1/2. In the classical limit where
c
→
∞
{\textstyle c\to \infty }
, quantum mechanics under the Galilean transformation group are enough. Similarly, one can construct a non-relativistic linear equation for any arbitrary spin. Under the same idea one can construct equations for Galilean electromagnetism.
== Relation to other equations ==
=== Schrödinger's and Pauli's equation ===
Taking the second line of Lévy-Leblond equation and inserting it back into the first line, one obtains through the algebra of the Pauli matrices, that
1
2
m
(
σ
⋅
p
)
2
ψ
−
E
ψ
=
[
1
2
m
p
2
−
E
]
ψ
=
0
{\displaystyle {\frac {1}{2m}}({\boldsymbol {\sigma }}\cdot \mathbf {p} )^{2}\psi -E\psi =\left[{\frac {1}{2m}}\mathbf {p} ^{2}-E\right]\psi =0}
,
which is the Schrödinger equation for a two-valued spinor. Note that solving for
χ
{\displaystyle \chi }
also returns another Schrödinger's equation. Pauli's expression for spin-1⁄2 particle in an electromagnetic field can be recovered by minimal coupling:
{
1
2
m
[
σ
⋅
(
p
−
q
A
)
]
2
+
q
V
}
ψ
=
E
ψ
{\displaystyle \left\{{\frac {1}{2m}}[{\boldsymbol {\sigma }}\cdot (\mathbf {p} -q\mathbf {A} )]^{2}+qV\right\}\psi =E\psi }
.
While Lévy-Leblond is linear in its derivatives, Pauli's and Schrödinger's equations are quadratic in the spatial derivatives.
=== Dirac equation ===
Dirac equation can be written as:
{
(
E
−
m
c
2
)
ψ
+
(
σ
⋅
p
c
)
χ
=
0
(
σ
⋅
p
c
)
ψ
+
(
E
+
m
c
2
)
χ
=
0
{\displaystyle \left\{{\begin{matrix}({\mathcal {E}}-mc^{2})\psi +({\boldsymbol {\sigma }}\cdot \mathbf {p} c)\chi =0\\({\boldsymbol {\sigma }}\cdot \mathbf {p} c)\psi +({\mathcal {E}}+mc^{2})\chi =0\end{matrix}}\right.}
where
E
{\textstyle {\mathcal {E}}}
is the total relativistic energy. In the non-relativistic limit,
E
≪
m
c
2
{\textstyle E\ll mc^{2}}
and
E
≈
m
c
2
+
E
+
⋯
{\textstyle {\mathcal {E}}\approx mc^{2}+E+\cdots }
one recovers, Lévy-Leblond equations.
== Heuristic derivation ==
Similar to the historical derivation of Dirac equation by Paul Dirac, one can try to linearize the non-relativistic dispersion relation
E
=
p
2
2
m
{\textstyle E={\frac {\mathbf {p} ^{2}}{2m}}}
. We want two operators Θ and Θ' linear in
p
{\textstyle \mathbf {p} }
(spatial derivatives) and E, like
{
Θ
Ψ
=
[
A
E
+
B
⋅
p
c
+
2
m
c
2
C
]
Ψ
=
0
Θ
′
Ψ
=
[
A
′
E
+
B
′
⋅
p
c
+
2
m
c
2
C
′
]
Ψ
=
0
{\displaystyle \left\{{\begin{matrix}\Theta \Psi =[AE+\mathbf {B} \cdot \mathbf {p} c+2mc^{2}C]\Psi =0\\\Theta '\Psi =[A'E+\mathbf {B} '\cdot \mathbf {p} c+2mc^{2}C']\Psi =0\end{matrix}}\right.}
for some
A
,
A
′
,
B
=
(
B
x
,
B
y
,
B
z
)
,
B
′
=
(
B
x
′
,
B
y
′
,
B
z
′
)
,
C
,
C
′
{\textstyle A,A',\mathbf {B} =(B_{x},B_{y},B_{z}),\mathbf {B} '=(B_{x}',B_{y}',B_{z}'),C,C'}
, such that their product recovers the classical dispersion relation, that is
1
2
m
c
2
Θ
′
Θ
=
E
−
p
2
2
m
{\displaystyle {\frac {1}{2mc^{2}}}\Theta '\Theta =E-{\frac {\mathbf {p} ^{2}}{2m}}}
,
where the factor 2mc2 is arbitrary an it is just there for normalization. By carrying out the product, one find that there is no solution if
A
,
A
′
,
B
i
,
B
i
′
,
C
,
C
′
{\textstyle A,A',B_{i},B_{i}',C,C'}
are one dimensional constants. The lowest dimension where there is a solution is 4. Then
A
,
A
′
,
B
,
B
′
,
C
,
C
′
{\textstyle A,A',\mathbf {B} ,\mathbf {B} ',C,C'}
are matrices that must satisfy the following relations:
{
A
′
A
=
0
C
′
C
=
0
A
′
B
i
+
B
i
′
A
=
0
C
′
B
i
+
B
i
′
C
=
0
A
′
C
+
C
′
A
=
I
4
B
i
′
B
j
+
B
j
′
B
i
=
−
2
δ
i
j
{\displaystyle \left\{{\begin{matrix}A'A=0\\C'C=0\\A'B_{i}+B_{i}'A=0\\C'B_{i}+B_{i}'C=0\\A'C+C'A=I_{4}\\B_{i}'B_{j}+B_{j}'B_{i}=-2\delta _{ij}\end{matrix}}\right.}
these relations can be rearranged to involve the gamma matrices from Clifford algebra.
I
N
{\textstyle I_{N}}
is the Identity matrix of dimension N. One possible representation is
A
=
A
′
=
(
0
0
I
2
0
)
,
B
i
=
−
B
i
′
=
(
σ
i
0
0
σ
i
)
,
C
=
C
′
=
(
0
I
2
0
0
)
{\displaystyle A=A'={\begin{pmatrix}0&0\\I_{2}&0\end{pmatrix}},B_{i}=-B_{i}'={\begin{pmatrix}\sigma _{i}&0\\0&\sigma _{i}\end{pmatrix}},C=C'={\begin{pmatrix}0&I_{2}\\0&0\end{pmatrix}}}
,
such that
Θ
Ψ
=
0
{\textstyle \Theta \Psi =0}
, with
Ψ
=
(
ψ
,
χ
)
{\textstyle \Psi =(\psi ,\chi )}
, returns Lévy-Leblond equation. Other representations can be chosen leading to equivalent equations with different signs or phases.
== References == | Wikipedia/Lévy-Leblond_equation |
In physics, the algebra of physical space (APS) is the use of the Clifford or geometric algebra Cl3,0(R) of the three-dimensional Euclidean space as a model for (3+1)-dimensional spacetime, representing a point in spacetime via a paravector (3-dimensional vector plus a 1-dimensional scalar).
The Clifford algebra Cl3,0(R) has a faithful representation, generated by Pauli matrices, on the spin representation C2; further, Cl3,0(R) is isomorphic to the even subalgebra Cl[0]3,1(R) of the Clifford algebra Cl3,1(R).
APS can be used to construct a compact, unified and geometrical formalism for both classical and quantum mechanics.
APS should not be confused with spacetime algebra (STA), which concerns the Clifford algebra Cl1,3(R) of the four-dimensional Minkowski spacetime.
== Special relativity ==
=== Spacetime position paravector ===
In APS, the spacetime position is represented as the paravector
x
=
x
0
+
x
1
e
1
+
x
2
e
2
+
x
3
e
3
,
{\displaystyle x=x^{0}+x^{1}\mathbf {e} _{1}+x^{2}\mathbf {e} _{2}+x^{3}\mathbf {e} _{3},}
where the time is given by the scalar part x0 = t, and e1, e2, e3 is a basis for position space. Throughout, units such that c = 1 are used, called natural units. In the Pauli matrix representation, the unit basis vectors are replaced by the Pauli matrices and the scalar part by the identity matrix. This means that the Pauli matrix representation of the space-time position is
x
→
(
x
0
+
x
3
x
1
−
i
x
2
x
1
+
i
x
2
x
0
−
x
3
)
{\displaystyle x\rightarrow {\begin{pmatrix}x^{0}+x^{3}&&x^{1}-ix^{2}\\x^{1}+ix^{2}&&x^{0}-x^{3}\end{pmatrix}}}
=== Lorentz transformations and rotors ===
The restricted Lorentz transformations that preserve the direction of time and include rotations and boosts can be performed by an exponentiation of the spacetime rotation biparavector W
L
=
e
W
/
2
.
{\displaystyle L=e^{W/2}.}
In the matrix representation, the Lorentz rotor is seen to form an instance of the SL(2, C) group (special linear group of degree 2 over the complex numbers), which is the double cover of the Lorentz group. The unimodularity of the Lorentz rotor is translated in the following condition in terms of the product of the Lorentz rotor with its Clifford conjugation
L
L
¯
=
L
¯
L
=
1.
{\displaystyle L{\bar {L}}={\bar {L}}L=1.}
This Lorentz rotor can be always decomposed in two factors, one Hermitian B = B†, and the other unitary R† = R−1, such that
L
=
B
R
.
{\displaystyle L=BR.}
The unitary element R is called a rotor because this encodes rotations, and the Hermitian element B encodes boosts.
=== Four-velocity paravector ===
The four-velocity, also called proper velocity, is defined as the derivative of the spacetime position paravector with respect to proper time τ:
u
=
d
x
d
τ
=
d
x
0
d
τ
+
d
d
τ
(
x
1
e
1
+
x
2
e
2
+
x
3
e
3
)
=
d
x
0
d
τ
[
1
+
d
d
x
0
(
x
1
e
1
+
x
2
e
2
+
x
3
e
3
)
]
.
{\displaystyle u={\frac {dx}{d\tau }}={\frac {dx^{0}}{d\tau }}+{\frac {d}{d\tau }}(x^{1}\mathbf {e} _{1}+x^{2}\mathbf {e} _{2}+x^{3}\mathbf {e} _{3})={\frac {dx^{0}}{d\tau }}\left[1+{\frac {d}{dx^{0}}}(x^{1}\mathbf {e} _{1}+x^{2}\mathbf {e} _{2}+x^{3}\mathbf {e} _{3})\right].}
This expression can be brought to a more compact form by defining the ordinary velocity as
v
=
d
d
x
0
(
x
1
e
1
+
x
2
e
2
+
x
3
e
3
)
,
{\displaystyle \mathbf {v} ={\frac {d}{dx^{0}}}(x^{1}\mathbf {e} _{1}+x^{2}\mathbf {e} _{2}+x^{3}\mathbf {e} _{3}),}
and recalling the definition of the gamma factor:
γ
(
v
)
=
1
1
−
|
v
|
2
c
2
,
{\displaystyle \gamma (\mathbf {v} )={\frac {1}{\sqrt {1-{\frac {|\mathbf {v} |^{2}}{c^{2}}}}}},}
so that the proper velocity is more compactly:
u
=
γ
(
v
)
(
1
+
v
)
.
{\displaystyle u=\gamma (\mathbf {v} )(1+\mathbf {v} ).}
The proper velocity is a positive unimodular paravector, which implies the following condition in terms of the Clifford conjugation
u
u
¯
=
1.
{\displaystyle u{\bar {u}}=1.}
The proper velocity transforms under the action of the Lorentz rotor L as
u
→
u
′
=
L
u
L
†
.
{\displaystyle u\rightarrow u^{\prime }=LuL^{\dagger }.}
=== Four-momentum paravector ===
The four-momentum in APS can be obtained by multiplying the proper velocity with the mass as
p
=
m
u
,
{\displaystyle p=mu,}
with the mass shell condition translated into
p
¯
p
=
m
2
.
{\displaystyle {\bar {p}}p=m^{2}.}
== Classical electrodynamics ==
=== Electromagnetic field, potential, and current ===
The electromagnetic field is represented as a bi-paravector F:
F
=
E
+
i
B
,
{\displaystyle F=\mathbf {E} +i\mathbf {B} ,}
with the Hermitian part representing the electric field E and the anti-Hermitian part representing the magnetic field B. In the standard Pauli matrix representation, the electromagnetic field is:
F
→
(
E
3
E
1
−
i
E
2
E
1
+
i
E
2
−
E
3
)
+
i
(
B
3
B
1
−
i
B
2
B
1
+
i
B
2
−
B
3
)
.
{\displaystyle F\rightarrow {\begin{pmatrix}E_{3}&E_{1}-iE_{2}\\E_{1}+iE_{2}&-E_{3}\end{pmatrix}}+i{\begin{pmatrix}B_{3}&B_{1}-iB_{2}\\B_{1}+iB_{2}&-B_{3}\end{pmatrix}}\,.}
The source of the field F is the electromagnetic four-current:
j
=
ρ
+
j
,
{\displaystyle j=\rho +\mathbf {j} \,,}
where the scalar part equals the electric charge density ρ, and the vector part the electric current density j. Introducing the electromagnetic potential paravector defined as:
A
=
ϕ
+
A
,
{\displaystyle A=\phi +\mathbf {A} \,,}
in which the scalar part equals the electric potential ϕ, and the vector part the magnetic potential A. The electromagnetic field is then also:
F
=
∂
A
¯
.
{\displaystyle F=\partial {\bar {A}}.}
The field can be split into electric
E
=
⟨
∂
A
¯
⟩
V
{\displaystyle E=\langle \partial {\bar {A}}\rangle _{V}}
and magnetic
B
=
i
⟨
∂
A
¯
⟩
B
V
{\displaystyle B=i\langle \partial {\bar {A}}\rangle _{BV}}
components. Here,
∂
=
∂
t
+
e
1
∂
x
+
e
2
∂
y
+
e
3
∂
z
{\displaystyle \partial =\partial _{t}+\mathbf {e} _{1}\,\partial _{x}+\mathbf {e} _{2}\,\partial _{y}+\mathbf {e} _{3}\,\partial _{z}}
and F is invariant under a gauge transformation of the form
A
→
A
+
∂
χ
,
{\displaystyle A\rightarrow A+\partial \chi \,,}
where
χ
{\displaystyle \chi }
is a scalar field.
The electromagnetic field is covariant under Lorentz transformations according to the law
F
→
F
′
=
L
F
L
¯
.
{\displaystyle F\rightarrow F^{\prime }=LF{\bar {L}}\,.}
=== Maxwell's equations and the Lorentz force ===
The Maxwell equations can be expressed in a single equation:
∂
¯
F
=
1
ϵ
j
¯
,
{\displaystyle {\bar {\partial }}F={\frac {1}{\epsilon }}{\bar {j}}\,,}
where the overbar represents the Clifford conjugation.
The Lorentz force equation takes the form
d
p
d
τ
=
e
⟨
F
u
⟩
R
.
{\displaystyle {\frac {dp}{d\tau }}=e\langle Fu\rangle _{R}\,.}
=== Electromagnetic Lagrangian ===
The electromagnetic Lagrangian is
L
=
1
2
⟨
F
F
⟩
S
−
⟨
A
j
¯
⟩
S
,
{\displaystyle L={\frac {1}{2}}\langle FF\rangle _{S}-\langle A{\bar {j}}\rangle _{S}\,,}
which is a real scalar invariant.
== Relativistic quantum mechanics ==
The Dirac equation, for an electrically charged particle of mass m and charge e, takes the form:
i
∂
¯
Ψ
e
3
+
e
A
¯
Ψ
=
m
Ψ
¯
†
,
{\displaystyle i{\bar {\partial }}\Psi \mathbf {e} _{3}+e{\bar {A}}\Psi =m{\bar {\Psi }}^{\dagger },}
where e3 is an arbitrary unitary vector, and A is the electromagnetic paravector potential as above. The electromagnetic interaction has been included via minimal coupling in terms of the potential A.
== Classical spinor ==
The differential equation of the Lorentz rotor that is consistent with the Lorentz force is
d
Λ
d
τ
=
e
2
m
c
F
Λ
,
{\displaystyle {\frac {d\Lambda }{d\tau }}={\frac {e}{2mc}}F\Lambda ,}
such that the proper velocity is calculated as the Lorentz transformation of the proper velocity at rest
u
=
Λ
Λ
†
,
{\displaystyle u=\Lambda \Lambda ^{\dagger },}
which can be integrated to find the space-time trajectory
x
(
τ
)
{\displaystyle x(\tau )}
with the additional use of
d
x
d
τ
=
u
.
{\displaystyle {\frac {dx}{d\tau }}=u.}
== See also ==
Paravector
Multivector
wikibooks:Physics Using Geometric Algebra
Dirac equation in the algebra of physical space
Algebra
== References ==
=== Textbooks ===
Baylis, William (2002). Electrodynamics: A Modern Geometric Approach (2nd ed.). Springer. ISBN 0-8176-4025-8.
Baylis, William, ed. (1999) [1996]. Clifford (Geometric) Algebras: with applications to physics, mathematics, and engineering. Springer. ISBN 978-0-8176-3868-9.
Doran, Chris; Lasenby, Anthony (2007) [2003]. Geometric Algebra for Physicists. Cambridge University Press. ISBN 978-1-139-64314-6.
Hestenes, David (1999). New Foundations for Classical Mechanics (2nd ed.). Kluwer. ISBN 0-7923-5514-8.
=== Articles ===
Baylis, W E (2004). "Relativity in introductory physics". Canadian Journal of Physics. 82 (11): 853–873. arXiv:physics/0406158. Bibcode:2004CaJPh..82..853B. doi:10.1139/p04-058. S2CID 35027499.
Baylis, W E; Jones, G (7 January 1989). "The Pauli algebra approach to special relativity". Journal of Physics A: Mathematical and General. 22 (1): 1–15. Bibcode:1989JPhA...22....1B. doi:10.1088/0305-4470/22/1/008.
Baylis, W. E. (1 March 1992). "Classical eigenspinors and the Dirac equation". Physical Review A. 45 (7): 4293–4302. Bibcode:1992PhRvA..45.4293B. doi:10.1103/physreva.45.4293. PMID 9907503.
Baylis, W. E.; Yao, Y. (1 July 1999). "Relativistic dynamics of charges in electromagnetic fields: An eigenspinor approach". Physical Review A. 60 (2): 785–795. Bibcode:1999PhRvA..60..785B. doi:10.1103/physreva.60.785. | Wikipedia/Dirac_equation_in_the_algebra_of_physical_space |
In mathematics, a square-integrable function, also called a quadratically integrable function or
L
2
{\displaystyle L^{2}}
function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite. Thus, square-integrability on the real line
(
−
∞
,
+
∞
)
{\displaystyle (-\infty ,+\infty )}
is defined as follows.
One may also speak of quadratic integrability over bounded intervals such as
[
a
,
b
]
{\displaystyle [a,b]}
for
a
≤
b
{\displaystyle a\leq b}
.
An equivalent definition is to say that the square of the function itself (rather than of its absolute value) is Lebesgue integrable. For this to be true, the integrals of the positive and negative portions of the real part must both be finite, as well as those for the imaginary part.
The vector space of (equivalence classes of) square integrable functions (with respect to Lebesgue measure) forms the
L
p
{\displaystyle L^{p}}
space with
p
=
2.
{\displaystyle p=2.}
Among the
L
p
{\displaystyle L^{p}}
spaces, the class of square integrable functions is unique in being compatible with an inner product, which allows notions like angle and orthogonality to be defined. Along with this inner product, the square integrable functions form a Hilbert space, since all of the
L
p
{\displaystyle L^{p}}
spaces are complete under their respective
p
{\displaystyle p}
-norms.
Often the term is used not to refer to a specific function, but to equivalence classes of functions that are equal almost everywhere.
== Properties ==
The square integrable functions (in the sense mentioned in which a "function" actually means an equivalence class of functions that are equal almost everywhere) form an inner product space with inner product given by
⟨
f
,
g
⟩
=
∫
A
f
(
x
)
g
(
x
)
¯
d
x
,
{\displaystyle \langle f,g\rangle =\int _{A}f(x){\overline {g(x)}}\,\mathrm {d} x,}
where
f
{\displaystyle f}
and
g
{\displaystyle g}
are square integrable functions,
f
(
x
)
¯
{\displaystyle {\overline {f(x)}}}
is the complex conjugate of
f
(
x
)
,
{\displaystyle f(x),}
A
{\displaystyle A}
is the set over which one integrates—in the first definition (given in the introduction above),
A
{\displaystyle A}
is
(
−
∞
,
+
∞
)
{\displaystyle (-\infty ,+\infty )}
, in the second,
A
{\displaystyle A}
is
[
a
,
b
]
{\displaystyle [a,b]}
.
Since
|
a
|
2
=
a
⋅
a
¯
{\displaystyle |a|^{2}=a\cdot {\overline {a}}}
, square integrability is the same as saying
⟨
f
,
f
⟩
<
∞
.
{\displaystyle \langle f,f\rangle <\infty .\,}
It can be shown that square integrable functions form a complete metric space under the metric induced by the inner product defined above.
A complete metric space is also called a Cauchy space, because sequences in such metric spaces converge if and only if they are Cauchy.
A space that is complete under the metric induced by a norm is a Banach space.
Therefore, the space of square integrable functions is a Banach space, under the metric induced by the norm, which in turn is induced by the inner product.
As we have the additional property of the inner product, this is specifically a Hilbert space, because the space is complete under the metric induced by the inner product.
This inner product space is conventionally denoted by
(
L
2
,
⟨
⋅
,
⋅
⟩
2
)
{\displaystyle \left(L_{2},\langle \cdot ,\cdot \rangle _{2}\right)}
and many times abbreviated as
L
2
.
{\displaystyle L_{2}.}
Note that
L
2
{\displaystyle L_{2}}
denotes the set of square integrable functions, but no selection of metric, norm or inner product are specified by this notation.
The set, together with the specific inner product
⟨
⋅
,
⋅
⟩
2
{\displaystyle \langle \cdot ,\cdot \rangle _{2}}
specify the inner product space.
The space of square integrable functions is the
L
p
{\displaystyle L^{p}}
space in which
p
=
2.
{\displaystyle p=2.}
== Examples ==
The function
1
x
n
,
{\displaystyle {\tfrac {1}{x^{n}}},}
defined on
(
0
,
1
)
,
{\displaystyle (0,1),}
is in
L
2
{\displaystyle L^{2}}
for
n
<
1
2
{\displaystyle n<{\tfrac {1}{2}}}
but not for
n
=
1
2
.
{\displaystyle n={\tfrac {1}{2}}.}
The function
1
x
,
{\displaystyle {\tfrac {1}{x}},}
defined on
[
1
,
∞
)
,
{\displaystyle [1,\infty ),}
is square-integrable.
Bounded functions, defined on
[
0
,
1
]
,
{\displaystyle [0,1],}
are square-integrable. These functions are also in
L
p
,
{\displaystyle L^{p},}
for any value of
p
.
{\displaystyle p.}
=== Non-examples ===
The function
1
x
,
{\displaystyle {\tfrac {1}{x}},}
defined on
[
0
,
1
]
,
{\displaystyle [0,1],}
where the value at
0
{\displaystyle 0}
is arbitrary. Furthermore, this function is not in
L
p
{\displaystyle L^{p}}
for any value of
p
{\displaystyle p}
in
[
1
,
∞
)
.
{\displaystyle [1,\infty ).}
== See also ==
Inner product space
L
p
{\displaystyle L^{p}}
space – Function spaces generalizing finite-dimensional p norm spaces
== References == | Wikipedia/Square-integrable_functions |
The Breit equation, or Dirac–Coulomb–Breit equation, is a relativistic wave equation derived by Gregory Breit in 1929 based on the Dirac equation, which formally describes two or more massive spin-1/2 particles (electrons, for example) interacting electromagnetically to the first order in perturbation theory. It accounts for magnetic interactions and retardation effects to the order of 1/c2. When other quantum electrodynamic effects are negligible, this equation has been shown to give results in good agreement with experiment. It was originally derived from the Darwin Lagrangian but later vindicated by the Wheeler–Feynman absorber theory and eventually quantum electrodynamics.
== Introduction ==
The Breit equation is not only an approximation in terms of quantum mechanics, but also in terms of relativity theory as it is not completely invariant with respect to the Lorentz transformation. Just as does the Dirac equation, it treats nuclei as point sources of an external field for the particles it describes. For N particles, the Breit equation has the form (rij is the distance between particle i and j):
where
H
^
D
(
i
)
=
[
q
i
ϕ
(
r
i
)
+
c
∑
s
=
x
,
y
,
z
α
s
(
i
)
π
s
(
I
)
+
α
0
(
I
)
m
0
c
2
]
{\displaystyle {\hat {H}}_{\text{D}}(i)=\left[q_{i}\phi (\mathbf {r} _{i})+c\sum _{s=x,y,z}\alpha _{s}(i)\pi _{s}(I)+\alpha _{0}(I)m_{0}c^{2}\right]}
is the Dirac Hamiltonian (see Dirac equation) for particle i at position
r
i
{\displaystyle \mathbf {r} _{i}}
and
ϕ
(
r
i
)
{\displaystyle \phi (\mathbf {r} _{i})}
is the scalar potential at that position; qi is the charge of the particle, thus for electrons qi = −e.
The one-electron Dirac Hamiltonians of the particles, along with their instantaneous Coulomb interactions 1/rij, form the Dirac–Coulomb operator. To this, Breit added the operator (now known as the (frequency-independent) Breit operator):
B
^
i
j
=
−
1
2
r
i
j
[
α
→
(
i
)
⋅
α
→
(
j
)
+
(
α
→
(
i
)
⋅
r
i
j
)
(
α
→
(
j
)
⋅
r
i
j
)
r
i
j
2
]
,
{\displaystyle {\hat {B}}_{ij}=-{\frac {1}{2r_{ij}}}\left[{\vec {\alpha }}(i)\cdot {\vec {\alpha }}(j)+{\frac {\left({\vec {\alpha }}(i)\cdot \mathbf {r} _{ij}\right)\left({\vec {\alpha }}(j)\cdot \mathbf {r} _{ij}\right)}{r_{ij}^{2}}}\right],}
where the Dirac matrices for electron i: α(i) = [αx(i), αy(i), αz(i)]. The two terms in the Breit operator account for retardation effects to the first order.
The wave function Ψ in the Breit equation is a spinor with 4N elements, since each electron is described by a Dirac bispinor with 4 elements as in the Dirac equation, and the total wave function is the tensor product of these.
== Breit Hamiltonians ==
The total Hamiltonian of the Breit equation, sometimes called the Dirac–Coulomb–Breit Hamiltonian (HDCB) can be decomposed into the following practical energy operators for electrons in electric and magnetic fields (also called the Breit–Pauli Hamiltonian), which have well-defined meanings in the interaction of molecules with magnetic fields (for instance for nuclear magnetic resonance):
B
^
i
j
=
H
^
0
+
H
^
1
+
⋯
+
H
^
6
,
{\displaystyle {\hat {B}}_{ij}={\hat {H}}_{0}+{\hat {H}}_{1}+\dots +{\hat {H}}_{6},}
in which the consecutive partial operators are:
H
^
0
=
∑
i
p
^
i
2
2
m
i
+
V
{\displaystyle {\hat {H}}_{0}=\sum _{i}{\frac {{\hat {p}}_{i}^{2}}{2m_{i}}}+V}
is the nonrelativistic Hamiltonian (
m
i
{\displaystyle m_{i}}
is the stationary mass of particle i).
H
^
1
=
−
1
8
c
2
∑
i
p
^
i
4
m
i
3
{\displaystyle {\hat {H}}_{1}=-{\frac {1}{8c^{2}}}\sum _{i}{\frac {{\hat {p}}_{i}^{4}}{m_{i}^{3}}}}
is connected to the dependence of mass on velocity:
E
k
i
n
2
−
(
m
0
c
2
)
2
=
m
2
v
2
c
2
{\displaystyle E_{\rm {kin}}^{2}-\left(m_{0}c^{2}\right)^{2}=m^{2}v^{2}c^{2}}
.
H
^
2
=
−
∑
i
>
j
q
i
q
j
2
r
i
j
m
i
m
j
c
2
[
p
^
i
⋅
p
^
j
+
(
r
i
j
⋅
p
^
i
)
(
r
i
j
⋅
p
^
j
)
r
i
j
2
]
{\displaystyle {\hat {H}}_{2}=-\sum _{i>j}{\frac {q_{i}q_{j}}{2r_{ij}m_{i}m_{j}c^{2}}}\left[\mathbf {\hat {p}} _{i}\cdot \mathbf {\hat {p}} _{j}+{\frac {(\mathbf {r_{ij}} \cdot \mathbf {\hat {p}} _{i})(\mathbf {r_{ij}} \cdot \mathbf {\hat {p}} _{j})}{r_{ij}^{2}}}\right]}
is a correction that partly accounts for retardation and can be described as the interaction between the magnetic dipole moments of the particles, which arise from the orbital motion of charges (also called orbit–orbit interaction).
H
^
3
=
μ
B
c
∑
i
1
m
i
s
i
⋅
[
F
(
r
i
)
×
p
^
i
+
∑
j
>
i
2
q
i
r
i
j
3
r
i
j
×
p
^
j
]
{\displaystyle {\hat {H}}_{3}={\frac {\mu _{\rm {B}}}{c}}\sum _{i}{\frac {1}{m_{i}}}\mathbf {s} _{i}\cdot \left[\mathbf {F} (\mathbf {r} _{i})\times \mathbf {\hat {p}} _{i}+\sum _{j>i}{\frac {2q_{i}}{r_{ij}^{3}}}\mathbf {r} _{ij}\times \mathbf {\hat {p}} _{j}\right]}
is the classical interaction between the orbital magnetic moments (from the orbital motion of charge) and spin magnetic moments (also called spin–orbit interaction). The first term describes the interaction of a particle's spin with its own orbital moment (F(ri) is the electric field at the particle's position), and the second term between two different particles.
H
^
4
=
i
h
8
π
c
2
∑
i
q
i
m
i
2
p
^
i
⋅
F
(
r
i
)
{\displaystyle {\hat {H}}_{4}={\frac {ih}{8\pi c^{2}}}\sum _{i}{\frac {q_{i}}{m_{i}^{2}}}\mathbf {\hat {p}} _{i}\cdot \mathbf {F} (\mathbf {r} _{i})}
is a nonclassical term characteristic for Dirac theory, sometimes called the Darwin term.
H
^
5
=
4
μ
B
2
∑
i
>
j
{
−
8
π
3
(
s
i
⋅
s
j
)
δ
(
r
i
j
)
+
1
r
i
j
3
[
s
i
⋅
s
j
−
3
(
s
i
⋅
r
i
j
)
(
s
j
⋅
r
i
j
)
r
i
j
2
]
}
{\displaystyle {\hat {H}}_{5}=4\mu _{\rm {B}}^{2}\sum _{i>j}\left\lbrace -{\frac {8\pi }{3}}(\mathbf {s} _{i}\cdot \mathbf {s} _{j})\delta (\mathbf {r} _{ij})+{\frac {1}{r_{ij}^{3}}}\left[\mathbf {s} _{i}\cdot \mathbf {s} _{j}-{\frac {3(\mathbf {s} _{i}\cdot \mathbf {r} _{ij})(\mathbf {s} _{j}\cdot \mathbf {r} _{ij})}{r_{ij}^{2}}}\right]\right\rbrace }
is the magnetic moment spin-spin interaction. The first term is called the contact interaction, because it is nonzero only when the particles are at the same position; the second term is the interaction of the classical dipole-dipole type.
H
^
6
=
2
μ
B
∑
i
[
H
(
r
i
)
⋅
s
i
+
q
i
m
i
c
A
(
r
i
)
⋅
p
^
i
]
{\displaystyle {\hat {H}}_{6}=2\mu _{\rm {B}}\sum _{i}\left[\mathbf {H} (\mathbf {r} _{i})\cdot \mathbf {s} _{i}+{\frac {q_{i}}{m_{i}c}}\mathbf {A} (\mathbf {r} _{i})\cdot \mathbf {\hat {p}} _{i}\right]}
is the interaction between spin and orbital magnetic moments with an external magnetic field H.
where:
V
=
∑
i
>
j
q
i
q
j
r
i
j
{\textstyle V=\sum _{i>j}{\frac {q_{i}q_{j}}{r_{ij}}}}
and
μ
B
=
e
ℏ
2
m
c
{\textstyle \mu _{\rm {B}}={\frac {e\hbar }{2mc}}}
is the Bohr magneton.
== See also ==
Bethe–Salpeter equation
Darwin Lagrangian
Two-body Dirac equations
Positronium
Wheeler–Feynman absorber theory
== References ==
== External links ==
Tensor form of the Breit equation, Institute of Theoretical Physics, Warsaw University.
Solving Nonperturbatively the Breit equation for Parapositronium, Institute of Theoretical Physics, Warsaw University. | Wikipedia/Breit_equation |
The Foldy–Wouthuysen transformation was historically significant and was formulated by Leslie Lawrance Foldy and Siegfried Adolf Wouthuysen in 1949 to understand the nonrelativistic limit of the Dirac equation, the equation for spin-1/2 particles. A detailed general discussion of the Foldy–Wouthuysen-type transformations in particle interpretation of relativistic wave equations is in Acharya and Sudarshan (1960). Its utility in high energy physics is now limited due to the primary applications being in the ultra-relativistic domain where the Dirac field is treated as a quantised field.
== A canonical transform ==
The FW transformation is a unitary transformation of the orthonormal basis in which both the Hamiltonian and the state are represented. The eigenvalues do not change under such a unitary transformation, that is, the physics does not change under such a unitary basis transformation. Therefore, such a unitary transformation can always be applied: in particular a unitary basis transformation may be picked which will put the Hamiltonian in a more pleasant form, at the expense of a change in the state function, which then represents something else. See for example the Bogoliubov transformation, which is an orthogonal basis transform for the same purpose. The suggestion that the FW transform is applicable to the state or the Hamiltonian is thus not correct.
Foldy and Wouthuysen made use of a canonical transform that has now come to be known as the Foldy–Wouthuysen transformation. A brief account of the history of the transformation is to be found in the obituaries of Foldy and Wouthuysen and the biographical memoir of Foldy. Before their work, there was some difficulty in understanding and gathering all the interaction terms of a given order, such as those for a Dirac particle immersed in an external field. With their procedure the physical interpretation of the terms was clear, and it became possible to apply their work in a systematic way to a number of problems that had previously defied solution. The Foldy–Wouthuysen transform was extended to the physically important cases of spin-0 and spin-1 particles, and even generalized to the case of arbitrary spins.
== Description ==
The Foldy–Wouthuysen (FW) transformation is a unitary transformation on a fermion wave function of the form:
where the unitary operator is the 4 × 4 matrix:
Above,
p
^
i
≡
p
i
|
p
|
{\displaystyle {\hat {p}}^{i}\equiv {\frac {p^{i}}{|\mathbf {p} |}}}
is the unit vector oriented in the direction of the fermion momentum. The above are related to the Dirac matrices by β = γ0 and αi = γ0γi, with i = 1, 2, 3. A straightforward series expansion applying the commutativity properties of the Dirac matrices demonstrates that 2 above is true. The inverse
U
−
1
=
e
−
β
α
⋅
p
^
θ
=
I
4
cos
θ
−
β
α
⋅
p
^
sin
θ
{\displaystyle U^{-1}=e^{-\beta {\boldsymbol {\alpha }}\cdot {\hat {\mathbf {p} }}\theta }=\mathbb {I} _{4}\cos \theta -\beta {\boldsymbol {\alpha }}\cdot {\hat {\mathbf {p} }}\sin \theta }
so it is clear that U−1U = I, where I is a 4 × 4 identity matrix.
== Transforming the Dirac Hamiltonian for a free fermion ==
This transformation is of particular interest when applied to the free-fermion Dirac Hamiltonian operator
H
^
0
≡
α
⋅
p
+
β
m
{\displaystyle {\hat {H}}_{0}\equiv {\boldsymbol {\alpha }}\cdot \mathbf {p} +\beta m}
in biunitary fashion, in the form:
Using the commutativity properties of the Dirac matrices, this can be massaged over into the double-angle expression:
This factors out into:
=== Choosing a particular representation: Newton–Wigner ===
Clearly, the FW transformation is a continuous transformation, that is, one may employ any value for θ which one chooses. Choosing a particular value for θ amounts to choosing a particular transformed representation.
One particularly important representation is that in which the transformed Hamiltonian operator Ĥ′0 is diagonalized. A completely diagonal representation can be obtained by choosing θ such that the α · p term in 5 vanishes. This is arranged by choosing:
In the Dirac-Pauli representation where β is a diagonal matrix, 5 is then reduced to a diagonal matrix:
By elementary trigonometry, 6 also implies that:
so that using 8 in 7 and then simplifying now leads to:
Prior to Foldy and Wouthuysen publishing their transformation, it was already known that 9 is the Hamiltonian in the Newton–Wigner (NW) representation (named after Theodore Duddell Newton and Eugene Wigner) of the Dirac equation. What 9 therefore tells us, is that by applying a FW transformation to the Dirac–Pauli representation of Dirac's equation, and then selecting the continuous transformation parameter θ so as to diagonalize the Hamiltonian, one arrives at the NW representation of Dirac's equation, because NW itself already contains the Hamiltonian specified in (9). See this link.
If one considers an on-shell mass—fermion or otherwise—given by m2 = pσpσ, and employs a Minkowski metric tensor for which diag(η) = (+1, −1, −1, −1), it should be apparent that the expression
p
0
=
m
2
+
|
p
|
2
{\displaystyle p^{0}={\sqrt {m^{2}+|\mathbf {p} |^{2}}}}
is equivalent to the E ≡ p0 component of the energy-momentum vector pμ, so that 9 is alternatively specified rather simply by Ĥ′0 = βE.
=== Correspondence between the Dirac–Pauli and Newton–Wigner representations, for a fermion at rest ===
Now consider a fermion at rest, which we may define in this context as a fermion for which |p| = 0. From 6 or 8, this means that cos 2θ = 1, so that θ = 0, ±π, ±2π and, from 2, that the unitary operator U = ±I. Therefore, any operator O in the Dirac–Pauli representation upon which we perform a biunitary transformation, will be given, for an at-rest fermion, by:
Contrasting the original Dirac–Pauli Hamiltonian operator
H
^
0
≡
α
⋅
p
+
β
m
{\displaystyle {\hat {H}}_{0}\equiv {\boldsymbol {\alpha }}\cdot \mathbf {p} +\beta m}
with the NW Hamiltonian 9, we do indeed find the |p| = 0 "at rest" correspondence:
== Transforming the velocity operator ==
=== In the Dirac–Pauli representation ===
Now, consider the velocity operator. To obtain this operator, we must commute the Hamiltonian operator Ĥ′0 with the canonical position operators xi, i.e., we must calculate
v
i
^
≡
i
[
H
^
0
,
x
i
]
{\displaystyle {\hat {v_{i}}}\equiv i\left[{\hat {H}}_{0},x_{i}\right]}
One good way to approach this calculation, is to start by writing the scalar rest mass m as
m
=
γ
0
H
^
0
+
γ
j
p
j
{\displaystyle m=\gamma ^{0}{\hat {H}}_{0}+\gamma ^{j}p_{j}}
and then to mandate that the scalar rest mass commute with the xi. Thus, we may write:
where we have made use of the Heisenberg canonical commutation relationship [xi,pj] = −iηij to reduce terms. Then, multiplying from the left by γ0 and rearranging terms, we arrive at:
Because the canonical relationship
i
[
H
^
0
,
v
^
i
]
≠
0
{\displaystyle i\left[{\hat {H}}_{0},{\hat {v}}_{i}\right]\neq 0}
the above provides the basis for computing an inherent, non-zero acceleration operator, which specifies the oscillatory motion known as zitterbewegung.
=== In the Newton–Wigner representation ===
In the Newton–Wigner representation, we now wish to calculate
v
^
i
′
≡
i
[
H
^
0
′
,
x
i
]
{\displaystyle {\hat {v}}_{i}'\equiv i\left[{\hat {H}}'_{0},x_{i}\right]}
If we use the result at the very end of section 2 above, Ĥ′0 = βp0, then this can be written instead as:
Using the above, we need simply to calculate [p0,xi], then multiply by iβ.
The canonical calculation proceeds similarly to the calculation in section 4 above, but because of the square root expression in p0 = √m2 + |p|2, one additional step is required.
First, to accommodate the square root, we will wish to require that the scalar square mass m2 commute with the canonical coordinates xi, which we write as:
where we again use the Heisenberg canonical relationship [xi,pj] = −iηij. Then, we need an expression for [p0,xi] which will satisfy 15. It is straightforward to verify that:
will satisfy 15 when again employing [xi,pj] = −iηij. Now, we simply return the iβ factor via 14, to arrive at:
This is understood to be the velocity operator in the Newton–Wigner representation. Because:
it is commonly thought that the zitterbewegung motion arising out of 12 vanishes when a fermion is transformed into the Newton–Wigner representation.
== Other applications ==
The powerful machinery of the Foldy–Wouthuysen transform originally developed for the Dirac equation has found applications in many situations such as acoustics, and optics.
It has found applications in very diverse areas such as atomic systems synchrotron radiation and derivation of the Bloch equation for polarized beams.
The application of the Foldy–Wouthuysen transformation in acoustics is very natural; comprehensive and mathematically rigorous accounts.
In the traditional scheme the purpose of expanding the optical Hamiltonian
H
^
=
−
(
n
2
(
r
)
−
p
^
⊥
2
)
1
2
{\displaystyle {\hat {H}}=-\left(n^{2}(r)-{\hat {p}}_{\perp }^{2}\right)^{\frac {1}{2}}}
in a series using
p
^
⊥
2
n
0
2
{\displaystyle {\frac {{\hat {p}}_{\perp }^{2}}{n_{0}^{2}}}}
as the expansion parameter is to understand the propagation of the quasi-paraxial beam in terms of a series of approximations (paraxial plus nonparaxial). Similar is the situation in the case of charged-particle optics. Let us recall that in relativistic quantum mechanics too one has a similar problem of understanding the relativistic wave equations as the nonrelativistic approximation plus the relativistic correction terms in the quasi-relativistic regime. For the Dirac equation (which is first-order in time) this is done most conveniently using the Foldy–Wouthuysen transformation leading to an iterative diagonalization technique. The main framework of the newly developed formalisms of optics (both light optics and charged-particle optics) is based on the transformation technique of Foldy–Wouthuysen theory which casts the Dirac equation in a form displaying the different interaction terms between the Dirac particle and an applied electromagnetic field in a nonrelativistic and easily interpretable form.
In the Foldy–Wouthuysen theory the Dirac equation is decoupled through a canonical transformation into two two-component equations: one reduces to the Pauli equation in the nonrelativistic limit and the other describes the negative-energy states. It is possible to write a Dirac-like matrix representation of Maxwell's equations. In such a matrix form the Foldy–Wouthuysen can be applied.
There is a close algebraic analogy between the Helmholtz equation (governing scalar optics) and the Klein–Gordon equation; and between the matrix form of the Maxwell's equations (governing vector optics) and the Dirac equation. So it is natural to use the powerful machinery of standard quantum mechanics (particularly, the Foldy–Wouthuysen transform) in analyzing these systems.
The suggestion to employ the Foldy–Wouthuysen Transformation technique in the case of the Helmholtz equation was mentioned in the literature as a remark.
It was only in the recent works, that this idea was exploited to analyze the quasiparaxial approximations for specific beam optical system. The Foldy–Wouthuysen technique is ideally suited for the Lie algebraic approach to optics. With all these plus points, the powerful and ambiguity-free expansion, the Foldy–Wouthuysen Transformation is still little used in optics. The technique of the Foldy–Wouthuysen Transformation results in what is known as nontraditional prescriptions of Helmholtz optics and Maxwell optics respectively. The nontraditional approaches give rise to very interesting wavelength-dependent modifications of the paraxial and aberration behaviour. The nontraditional formalism of Maxwell optics provides a unified framework of light beam optics and polarization. The nontraditional prescriptions of light optics are closely analogous with the quantum theory of
charged-particle beam optics. In optics, it has enabled the deeper connections in the wavelength-dependent regime between light optics and charged-particle optics to be seen (see Electron optics).
== See also ==
Relativistic quantum mechanics
== Notes == | Wikipedia/Foldy–Wouthuysen_transformation |
See Ricci calculus and Van der Waerden notation for the notation.
In quantum field theory, the nonlinear Dirac equation is a model of self-interacting Dirac fermions. This model is widely considered in quantum physics as a toy model of self-interacting electrons.
The nonlinear Dirac equation appears in the Einstein–Cartan–Sciama–Kibble theory of gravity, which extends general relativity to matter with intrinsic angular momentum (spin). This theory removes a constraint of the symmetry of the affine connection and treats its antisymmetric part, the torsion tensor, as a variable in varying the action. In the resulting field equations, the torsion tensor is a homogeneous, linear function of the spin tensor. The minimal coupling between torsion and Dirac spinors thus generates an axial-axial, spin–spin interaction in fermionic matter, which becomes significant only at extremely high densities. Consequently, the Dirac equation becomes nonlinear (cubic) in the spinor field, which causes fermions to be spatially extended and may remove the ultraviolet divergence in quantum field theory.
== Models ==
Two common examples are the massive Thirring model and the Soler model.
=== Thirring model ===
The Thirring model was originally formulated as a model in (1 + 1) space-time dimensions and is characterized by the Lagrangian density
L
=
ψ
¯
(
i
∂
/
−
m
)
ψ
−
g
2
(
ψ
¯
γ
μ
ψ
)
(
ψ
¯
γ
μ
ψ
)
,
{\displaystyle {\mathcal {L}}={\overline {\psi }}(i\partial \!\!\!/-m)\psi -{\frac {g}{2}}\left({\overline {\psi }}\gamma ^{\mu }\psi \right)\left({\overline {\psi }}\gamma _{\mu }\psi \right),}
where ψ ∈ C2 is the spinor field, ψ = ψ*γ0 is the Dirac adjoint spinor,
∂
/
=
∑
μ
=
0
,
1
γ
μ
∂
∂
x
μ
,
{\displaystyle \partial \!\!\!/=\sum _{\mu =0,1}\gamma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}\,,}
(Feynman slash notation is used), g is the coupling constant, m is the mass, and γμ are the two-dimensional gamma matrices, finally μ = 0, 1 is an index.
=== Soler model ===
The Soler model was originally formulated in (3 + 1) space-time dimensions. It is characterized by the Lagrangian density
L
=
ψ
¯
(
i
∂
/
−
m
)
ψ
+
g
2
(
ψ
¯
ψ
)
2
,
{\displaystyle {\mathcal {L}}={\overline {\psi }}\left(i\partial \!\!\!/-m\right)\psi +{\frac {g}{2}}\left({\overline {\psi }}\psi \right)^{2},}
using the same notations above, except
∂
/
=
∑
μ
=
0
3
γ
μ
∂
∂
x
μ
,
{\displaystyle \partial \!\!\!/=\sum _{\mu =0}^{3}\gamma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}\,,}
is now the four-gradient operator contracted with the four-dimensional Dirac gamma matrices γμ, so therein μ = 0, 1, 2, 3.
== Einstein–Cartan theory ==
In Einstein–Cartan theory the Lagrangian density for a Dirac spinor field is given by (
c
=
ℏ
=
1
{\displaystyle c=\hbar =1}
)
L
=
−
g
(
ψ
¯
(
i
γ
μ
D
μ
−
m
)
ψ
)
,
{\displaystyle {\mathcal {L}}={\sqrt {-g}}\left({\overline {\psi }}\left(i\gamma ^{\mu }D_{\mu }-m\right)\psi \right),}
where
D
μ
=
∂
μ
+
1
4
ω
ν
ρ
μ
γ
ν
γ
ρ
{\displaystyle D_{\mu }=\partial _{\mu }+{\frac {1}{4}}\omega _{\nu \rho \mu }\gamma ^{\nu }\gamma ^{\rho }}
is the Fock–Ivanenko covariant derivative of a spinor with respect to the affine connection,
ω
μ
ν
ρ
{\displaystyle \omega _{\mu \nu \rho }}
is the spin connection,
g
{\displaystyle g}
is the determinant of the metric tensor
g
μ
ν
{\displaystyle g_{\mu \nu }}
, and the Dirac matrices satisfy
γ
μ
γ
ν
+
γ
ν
γ
μ
=
2
g
μ
ν
I
.
{\displaystyle \gamma ^{\mu }\gamma ^{\nu }+\gamma ^{\nu }\gamma ^{\mu }=2g^{\mu \nu }I.}
The Einstein–Cartan field equations for the spin connection yield an algebraic constraint between the spin connection and the spinor field rather than a partial differential equation, which allows the spin connection to be explicitly eliminated from the theory. The final result is a nonlinear Dirac equation containing an effective "spin-spin" self-interaction,
i
γ
μ
D
μ
ψ
−
m
ψ
=
i
γ
μ
∇
μ
ψ
+
3
κ
8
(
ψ
¯
γ
μ
γ
5
ψ
)
γ
μ
γ
5
ψ
−
m
ψ
=
0
,
{\displaystyle i\gamma ^{\mu }D_{\mu }\psi -m\psi =i\gamma ^{\mu }\nabla _{\mu }\psi +{\frac {3\kappa }{8}}\left({\overline {\psi }}\gamma _{\mu }\gamma ^{5}\psi \right)\gamma ^{\mu }\gamma ^{5}\psi -m\psi =0,}
where
∇
μ
{\displaystyle \nabla _{\mu }}
is the general-relativistic covariant derivative of a spinor, and
κ
{\displaystyle \kappa }
is the Einstein gravitational constant,
8
π
G
c
4
{\textstyle {\frac {8\pi G}{c^{4}}}}
. The cubic term in this equation becomes significant at densities on the order of
m
2
κ
{\textstyle {\frac {m^{2}}{\kappa }}}
.
== See also ==
== References == | Wikipedia/Nonlinear_Dirac_equation |
In mathematical physics, the Dirac algebra is the Clifford algebra
Cl
1
,
3
(
C
)
{\displaystyle {\text{Cl}}_{1,3}(\mathbb {C} )}
. This was introduced by the mathematical physicist P. A. M. Dirac in 1928 in developing the Dirac equation for spin-1/2 particles with a matrix representation of the gamma matrices, which represent the generators of the algebra.
The gamma matrices are a set of four
4
×
4
{\displaystyle 4\times 4}
matrices
{
γ
μ
}
=
{
γ
0
,
γ
1
,
γ
2
,
γ
3
}
{\displaystyle \{\gamma ^{\mu }\}=\{\gamma ^{0},\gamma ^{1},\gamma ^{2},\gamma ^{3}\}}
with entries in
C
{\displaystyle \mathbb {C} }
, that is, elements of
Mat
4
×
4
(
C
)
{\displaystyle {\text{Mat}}_{4\times 4}(\mathbb {C} )}
that satisfy
{
γ
μ
,
γ
ν
}
=
γ
μ
γ
ν
+
γ
ν
γ
μ
=
2
η
μ
ν
,
{\displaystyle \displaystyle \{\gamma ^{\mu },\gamma ^{\nu }\}=\gamma ^{\mu }\gamma ^{\nu }+\gamma ^{\nu }\gamma ^{\mu }=2\eta ^{\mu \nu },}
where by convention, an identity matrix has been suppressed on the right-hand side. The numbers
η
μ
ν
{\displaystyle \eta ^{\mu \nu }\,}
are the components of the Minkowski metric.
For this article we fix the signature to be mostly minus, that is,
(
+
,
−
,
−
,
−
)
{\displaystyle (+,-,-,-)}
.
The Dirac algebra is then the linear span of the identity, the gamma matrices
γ
μ
{\displaystyle \gamma ^{\mu }}
as well as any linearly independent products of the gamma matrices. This forms a finite-dimensional algebra over the field
R
{\displaystyle \mathbb {R} }
or
C
{\displaystyle \mathbb {C} }
, with dimension
16
=
2
4
{\displaystyle 16=2^{4}}
.
== Basis for the algebra ==
The algebra has a basis
I
4
,
{\displaystyle I_{4},}
γ
μ
,
{\displaystyle \gamma ^{\mu },}
γ
μ
γ
ν
,
{\displaystyle \gamma ^{\mu }\gamma ^{\nu },}
γ
μ
γ
ν
γ
ρ
,
{\displaystyle \gamma ^{\mu }\gamma ^{\nu }\gamma ^{\rho },}
γ
μ
γ
ν
γ
ρ
γ
σ
=
γ
0
γ
1
γ
2
γ
3
{\displaystyle \gamma ^{\mu }\gamma ^{\nu }\gamma ^{\rho }\gamma ^{\sigma }=\gamma ^{0}\gamma ^{1}\gamma ^{2}\gamma ^{3}}
where in each expression, each greek index is increasing as we move to the right. In particular, there is no repeated index in the expressions. By dimension counting, the dimension of the algebra is 16.
The algebra can be generated by taking products of the
γ
μ
{\displaystyle \gamma ^{\mu }}
alone: the identity arises as
I
4
=
(
γ
0
)
2
{\displaystyle I_{4}=(\gamma ^{0})^{2}}
while the others are explicitly products of the
γ
μ
{\displaystyle \gamma ^{\mu }}
.
These elements span the space generated by
γ
μ
{\displaystyle \gamma ^{\mu }}
. We conclude that we really do have a basis of the Clifford algebra generated by the
γ
μ
.
{\displaystyle \gamma ^{\mu }.}
== Quadratic powers and Lorentz algebra ==
For the theory in this section, there are many choices of conventions found in the literature, often corresponding to factors of
±
i
{\displaystyle \pm i}
. For clarity, here we will choose conventions to minimise the number of numerical factors needed, but may lead to generators being anti-Hermitian rather than Hermitian.
There is another common way to write the quadratic subspace of the Clifford algebra:
S
μ
ν
=
1
4
[
γ
μ
,
γ
ν
]
{\displaystyle S^{\mu \nu }={\frac {1}{4}}[\gamma ^{\mu },\gamma ^{\nu }]}
with
μ
≠
ν
{\displaystyle \mu \neq \nu }
. Note
S
μ
ν
=
−
S
ν
μ
{\displaystyle S^{\mu \nu }=-S^{\nu \mu }}
.
There is another way to write this which holds even when
μ
=
ν
{\displaystyle \mu =\nu }
:
S
μ
ν
=
1
2
(
γ
μ
γ
ν
−
η
μ
ν
)
.
{\displaystyle S^{\mu \nu }={\frac {1}{2}}(\gamma ^{\mu }\gamma ^{\nu }-\eta ^{\mu \nu }).}
This form can be used to show that the
S
μ
ν
{\displaystyle S^{\mu \nu }}
form a representation of the Lorentz algebra (with real conventions)
[
S
μ
ν
,
S
ρ
σ
]
=
S
μ
σ
η
ν
ρ
−
S
ν
σ
η
μ
ρ
+
S
ν
ρ
η
μ
σ
−
S
μ
ρ
η
ν
σ
.
{\displaystyle [S^{\mu \nu },S^{\rho \sigma }]=S^{\mu \sigma }\eta ^{\nu \rho }-S^{\nu \sigma }\eta ^{\mu \rho }+S^{\nu \rho }\eta ^{\mu \sigma }-S^{\mu \rho }\eta ^{\nu \sigma }.}
=== Physics conventions ===
It is common convention in physics to include a factor of
±
i
{\displaystyle \pm i}
, so that Hermitian conjugation (where transposing is done with respect to the spacetime greek indices) gives a 'Hermitian matrix' of sigma generators
only 6 of which are non-zero due to antisymmetry of the bracket, span the six-dimensional representation space of the tensor (1, 0) ⊕ (0, 1)-representation of the Lorentz algebra inside
C
l
1
,
3
(
R
)
{\displaystyle {\mathcal {Cl}}_{1,3}(\mathbb {R} )}
. Moreover, they have the commutation relations of the Lie algebra,
and hence constitute a representation of the Lorentz algebra (in addition to spanning a representation space) sitting inside
C
l
1
,
3
(
R
)
,
{\displaystyle {\mathcal {Cl}}_{1,3}(\mathbb {R} ),}
the
(
1
2
,
0
)
⊕
(
0
,
1
2
)
{\displaystyle \left({\frac {1}{2}},0\right)\oplus \left(0,{\frac {1}{2}}\right)}
spin representation.
=== Spin(1, 3) ===
The exponential map for matrices is well defined. The
S
μ
ν
{\displaystyle S^{\mu \nu }}
satisfy the Lorentz algebra, and turn out to exponentiate to a representation of the spin group
Spin
(
1
,
3
)
{\displaystyle {\text{Spin}}(1,3)}
of the Lorentz group
SO
(
1
,
3
)
{\displaystyle {\text{SO}}(1,3)}
(strictly, the future-directed part
SO
(
1
,
3
)
+
{\displaystyle {\text{SO}}(1,3)^{+}}
connected to the identity). The
S
μ
ν
{\displaystyle S^{\mu \nu }}
are then the spin generators of this representation.
We emphasize that
S
μ
ν
{\displaystyle S^{\mu \nu }}
is itself a matrix, not the components of a matrix. Its components as a
4
×
4
{\displaystyle 4\times 4}
complex matrix are labelled by convention using greek letters from the start of the alphabet
α
,
β
,
⋯
{\displaystyle \alpha ,\beta ,\cdots }
.
The action of
S
μ
ν
{\displaystyle S^{\mu \nu }}
on a spinor
ψ
{\displaystyle \psi }
, which in this setting is an element of the vector space
C
4
{\displaystyle \mathbb {C} ^{4}}
, is
ψ
↦
S
μ
ν
ψ
{\displaystyle \psi \mapsto S^{\mu \nu }\psi }
, or in components,
ψ
α
↦
(
S
μ
ν
)
α
β
ψ
β
.
{\displaystyle \psi ^{\alpha }\mapsto (S^{\mu \nu })^{\alpha }{}_{\beta }\psi ^{\beta }.}
This corresponds to an infinitesimal Lorentz transformation on a spinor. Then a finite Lorentz transformation, parametrized by the components
ω
μ
ν
{\displaystyle \omega _{\mu \nu }}
(antisymmetric in
μ
,
ν
{\displaystyle \mu ,\nu }
) can be expressed as
S
:=
exp
(
i
2
ω
μ
ν
S
μ
ν
)
.
{\displaystyle S:=\exp \left({\frac {i}{2}}\omega _{\mu \nu }S^{\mu \nu }\right).}
From the property that
(
γ
μ
)
†
=
γ
0
γ
μ
γ
0
,
{\displaystyle (\gamma ^{\mu })^{\dagger }=\gamma ^{0}\gamma ^{\mu }\gamma ^{0},}
it follows that
(
S
μ
ν
)
†
=
−
γ
0
S
μ
ν
γ
0
.
{\displaystyle (S^{\mu \nu })^{\dagger }=-\gamma ^{0}S^{\mu \nu }\gamma ^{0}.}
And
S
{\displaystyle S}
as defined above satisfies
S
†
=
γ
0
S
−
1
γ
0
{\displaystyle S^{\dagger }=\gamma ^{0}S^{-1}\gamma ^{0}}
This motivates the definition of Dirac adjoint for spinors
ψ
{\displaystyle \psi }
, of
ψ
¯
:=
ψ
†
γ
0
{\displaystyle {\bar {\psi }}:=\psi ^{\dagger }\gamma ^{0}}
.
The corresponding transformation for
S
{\displaystyle S}
is
S
¯
:=
γ
0
S
†
γ
0
=
S
−
1
{\displaystyle {\bar {S}}:=\gamma ^{0}S^{\dagger }\gamma ^{0}=S^{-1}}
.
With this, it becomes simple to construct Lorentz invariant quantities for construction of Lagrangians such as the Dirac Lagrangian.
== Quartic power ==
The quartic subspace contains a single basis element,
γ
0
γ
1
γ
2
γ
3
=
1
4
!
ϵ
μ
ν
ρ
σ
γ
μ
γ
ν
γ
ρ
γ
σ
,
{\displaystyle \gamma ^{0}\gamma ^{1}\gamma ^{2}\gamma ^{3}={\frac {1}{4!}}\epsilon _{\mu \nu \rho \sigma }\gamma ^{\mu }\gamma ^{\nu }\gamma ^{\rho }\gamma ^{\sigma },}
where
ϵ
μ
ν
ρ
σ
{\displaystyle \epsilon _{\mu \nu \rho \sigma }}
is the totally antisymmetric tensor such that
ϵ
0123
=
+
1
{\displaystyle \epsilon _{0123}=+1}
by convention.
This is antisymmetric under exchange of any two adjacent gamma matrices.
=== γ5 ===
When considering the complex span, this basis element can alternatively be taken to be
γ
5
:=
i
γ
0
γ
1
γ
2
γ
3
.
{\displaystyle \gamma ^{5}:=i\gamma ^{0}\gamma ^{1}\gamma ^{2}\gamma ^{3}.}
More details can be found here.
=== As a volume form ===
By total antisymmetry of the quartic element, it can be considered to be a volume form. In fact, this observation extends to a discussion of Clifford algebras as a generalization of the exterior algebra: both arise as quotients of the tensor algebra, but the exterior algebra gives a more restrictive quotient, where the anti-commutators all vanish.
== Derivation starting from the Dirac and Klein–Gordon equation ==
The defining form of the gamma elements can be derived if one assumes the covariant form of the Dirac equation:
−
i
ℏ
γ
μ
∂
μ
ψ
+
m
c
ψ
=
0
.
{\displaystyle -i\hbar \gamma ^{\mu }\partial _{\mu }\psi +mc\psi =0\,.}
and the Klein–Gordon equation:
−
∂
t
2
ψ
+
∇
2
ψ
=
m
2
ψ
{\displaystyle -\partial _{t}^{2}\psi +\nabla ^{2}\psi =m^{2}\psi }
to be given, and requires that these equations lead to consistent results.
Derivation from consistency requirement (proof). Multiplying the Dirac equation by its conjugate equation yields:
ψ
†
(
i
ℏ
γ
μ
∂
μ
+
m
c
)
(
−
i
ℏ
γ
ν
∂
ν
+
m
c
)
ψ
=
0
.
{\displaystyle \psi ^{\dagger }(i\hbar \gamma ^{\mu }\partial _{\mu }+mc)(-i\hbar \gamma ^{\nu }\partial _{\nu }+mc)\psi =0\,.}
The demand of consistency with the Klein–Gordon equation leads immediately to:
{
γ
μ
,
γ
ν
}
=
γ
μ
γ
ν
+
γ
ν
γ
μ
=
2
η
μ
ν
I
4
{\displaystyle \displaystyle \{\gamma ^{\mu },\gamma ^{\nu }\}=\gamma ^{\mu }\gamma ^{\nu }+\gamma ^{\nu }\gamma ^{\mu }=2\eta ^{\mu \nu }I_{4}}
where
{
,
}
{\displaystyle \{,\}}
is the anticommutator,
η
μ
ν
{\displaystyle \eta ^{\mu \nu }\,}
is the Minkowski metric with signature (+ − − −) and
I
4
{\displaystyle \ I_{4}\,}
is the 4x4 unit matrix.
== Cl1,3(C) and Cl1,3(R) ==
The Dirac algebra can be regarded as a complexification of the real spacetime algebra Cl1,3(
R
{\displaystyle \mathbb {R} }
):
C
l
1
,
3
(
C
)
=
C
l
1
,
3
(
R
)
⊗
C
.
{\displaystyle \mathrm {Cl} _{1,3}(\mathbb {C} )=\mathrm {Cl} _{1,3}(\mathbb {R} )\otimes \mathbb {C} .}
Cl1,3(
R
{\displaystyle \mathbb {R} }
) differs from Cl1,3(
C
{\displaystyle \mathbb {C} }
): in Cl1,3(
R
{\displaystyle \mathbb {R} }
) only real linear combinations of the gamma matrices and their products are allowed.
Proponents of geometric algebra strive to work with real algebras wherever that is possible. They argue that it is generally possible (and usually enlightening) to identify the presence of an imaginary unit in a physical equation. Such units arise from one of the many quantities in a real Clifford algebra that square to −1, and these have geometric significance because of the properties of the algebra and the interaction of its various subspaces. Some of these proponents also question whether it is necessary or even useful to introduce an additional imaginary unit in the context of the Dirac equation.
In the mathematics of Riemannian geometry, it is conventional to define the Clifford algebra Clp,q(
R
{\displaystyle \mathbb {R} }
) for arbitrary dimensions p,q; the anti-commutation of the Weyl spinors emerges naturally from the Clifford algebra. The Weyl spinors transform under the action of the spin group
S
p
i
n
(
n
)
{\displaystyle \mathrm {Spin} (n)}
. The complexification of the spin group, called the spinc group
S
p
i
n
C
(
n
)
{\displaystyle \mathrm {Spin} ^{\mathbb {C} }(n)}
, is a product
S
p
i
n
(
n
)
×
Z
2
S
1
{\displaystyle \mathrm {Spin} (n)\times _{\mathbb {Z} _{2}}S^{1}}
of the spin group with the circle
S
1
≅
U
(
1
)
{\displaystyle S^{1}\cong U(1)}
with the product
×
Z
2
{\displaystyle \times _{\mathbb {Z} _{2}}}
just a notational device to identify
(
a
,
u
)
∈
S
p
i
n
(
n
)
×
S
1
{\displaystyle (a,u)\in \mathrm {Spin} (n)\times S^{1}}
with
(
−
a
,
−
u
)
.
{\displaystyle (-a,-u).}
The geometric point of this is that it disentangles the real spinor, which is covariant under Lorentz transformations, from the
U
(
1
)
{\displaystyle U(1)}
component, which can be identified with the
U
(
1
)
{\displaystyle U(1)}
fiber of the electromagnetic interaction. The
×
Z
2
{\displaystyle \times _{\mathbb {Z} _{2}}}
is entangling parity and charge conjugation in a manner suitable for relating the Dirac particle/anti-particle states (equivalently, the chiral states in the Weyl basis). The bispinor, insofar as it has linearly independent left and right components, can interact with the electromagnetic field. This is in contrast to the Majorana spinor and the ELKO spinor, which cannot (i.e. they are electrically neutral), as they explicitly constrain the spinor so as to not interact with the
S
1
{\displaystyle S^{1}}
part coming from the complexification. The ELKO spinor (Eigenspinoren des Ladungskonjugationsoperators) is a class 5 Lounesto spinor.: 84
Insofar as the presentation of charge and parity can be a confusing topic in conventional quantum field theory textbooks, the more careful dissection of these topics in a general geometric setting can be elucidating. Standard expositions of the Clifford algebra construct the Weyl spinors from first principles; that they "automatically" anti-commute is an elegant geometric by-product of the construction, completely by-passing any arguments that appeal to the Pauli exclusion principle (or the sometimes common sensation that Grassmann variables have been introduced via ad hoc argumentation.)
In contemporary physics practice, the Dirac algebra continues to be the standard environment the spinors of the Dirac equation "live" in, rather than the spacetime algebra.
== See also ==
Higher-dimensional gamma matrices
Fierz identity
== References ==
Rodrigues, Waldyr A.; Oliveira, Edmundo C. de (2007). The Many Faces of Maxwell, Dirac and Einstein Equations: A Clifford Bundle Approach. Springer Science & Business Media. ISBN 978-3-540-71292-3.
Weinberg, Steven (2005) [2000]. "5 Quantum Fields and Antiparticles §5.4 The Dirac Formulation". The Quantum Theory of Fields: Volume 1, Foundations. Vol. 1. Cambridge University Press. ISBN 978-0-521-67053-1. | Wikipedia/Dirac_algebra |
In quantum field theory, and in the significant subfields of quantum electrodynamics (QED) and quantum chromodynamics (QCD), the two-body Dirac equations (TBDE) of constraint dynamics provide a three-dimensional yet manifestly covariant reformulation of the Bethe–Salpeter equation for two spin-1/2 particles. Such a reformulation is necessary since without it, as shown by Nakanishi, the Bethe–Salpeter equation possesses negative-norm solutions arising from the presence of an essentially relativistic degree of freedom, the relative time. These "ghost" states have spoiled the naive interpretation of the Bethe–Salpeter equation as a quantum mechanical wave equation. The two-body Dirac equations of constraint dynamics rectify this flaw. The forms of these equations can not only be derived from quantum field theory they can also be derived purely in the context of Dirac's constraint dynamics and relativistic mechanics and quantum mechanics. Their structures, unlike the more familiar two-body Dirac equation of Breit, which is a single equation, are that of two simultaneous quantum relativistic wave equations. A single two-body Dirac equation similar to the Breit equation can be derived from the TBDE. Unlike the Breit equation, it is manifestly covariant and free from the types of singularities that prevent a strictly nonperturbative treatment of the Breit equation.
In applications of the TBDE to QED, the two particles interact by way of four-vector potentials derived from the field theoretic electromagnetic interactions between the two particles. In applications to QCD, the two particles interact by way of four-vector potentials and Lorentz invariant scalar interactions, derived in part from the field theoretic chromomagnetic interactions between the quarks and in part by phenomenological considerations. As with the Breit equation a sixteen-component spinor Ψ is used.
== Equations ==
For QED, each equation has the same structure as the ordinary one-body Dirac equation in the presence of an external electromagnetic field, given by the 4-potential
A
μ
{\displaystyle A_{\mu }}
. For QCD, each equation has the same structure as the ordinary one-body Dirac equation in the presence of an external field similar to the electromagnetic field and an additional external field given by in terms of a Lorentz invariant scalar
S
{\displaystyle S}
. In natural units: those two-body equations have the form.
[
(
γ
1
)
μ
(
p
1
−
A
~
1
)
μ
+
m
1
+
S
~
1
]
Ψ
=
0
,
[
(
γ
2
)
μ
(
p
2
−
A
~
2
)
μ
+
m
2
+
S
~
2
]
Ψ
=
0.
{\displaystyle {\begin{aligned}\left[(\gamma _{1})_{\mu }(p_{1}-{\tilde {A}}_{1})^{\mu }+m_{1}+{\tilde {S}}_{1}\right]\Psi &=0,\\[1ex]\left[(\gamma _{2})_{\mu }(p_{2}-{\tilde {A}}_{2})^{\mu }+m_{2}+{\tilde {S}}_{2}\right]\Psi &=0.\end{aligned}}}
where, in coordinate space, pμ is the 4-momentum, related to the 4-gradient by (the metric used here is
η
μ
ν
=
(
−
1
,
1
,
1
,
1
)
{\displaystyle \eta _{\mu \nu }=(-1,1,1,1)}
)
p
μ
=
−
i
∂
∂
x
μ
{\displaystyle p^{\mu }=-i{\frac {\partial }{\partial x_{\mu }}}}
and γμ are the gamma matrices. The two-body Dirac equations (TBDE) have the property that if one of the masses becomes very large, say
m
2
→
∞
{\displaystyle m_{2}\rightarrow \infty }
then the 16-component Dirac equation reduces to the 4-component one-body Dirac equation for particle one in an external potential.
In SI units:
[
(
γ
1
)
μ
(
p
1
−
A
~
1
)
μ
+
m
1
c
+
S
~
1
]
Ψ
=
0
,
[
(
γ
2
)
μ
(
p
2
−
A
~
2
)
μ
+
m
2
c
+
S
~
2
]
Ψ
=
0.
{\displaystyle {\begin{aligned}\left[(\gamma _{1})_{\mu }(p_{1}-{\tilde {A}}_{1})^{\mu }+m_{1}c+{\tilde {S}}_{1}\right]\Psi &=0,\\[1ex]\left[(\gamma _{2})_{\mu }(p_{2}-{\tilde {A}}_{2})^{\mu }+m_{2}c+{\tilde {S}}_{2}\right]\Psi &=0.\end{aligned}}}
where c is the speed of light and
p
μ
=
−
i
ℏ
∂
∂
x
μ
{\displaystyle p^{\mu }=-i\hbar {\frac {\partial }{\partial x_{\mu }}}}
Natural units will be used below. A tilde symbol is used over the two sets of potentials to indicate that they may have additional gamma matrix dependencies not present in the one-body Dirac equation. Any coupling constants such as the electron charge are embodied in the vector potentials.
== Constraint dynamics and the TBDE ==
Constraint dynamics applied to the TBDE requires a particular form of mathematical consistency: the two Dirac operators must commute with each other. This is plausible if one views the two equations as two compatible constraints on the wave function. (See the discussion below on constraint dynamics.) If the two operators did not commute, (as, e.g., with the coordinate and momentum operators
x
,
p
{\displaystyle x,p}
) then the constraints would not be compatible (one could not e.g., have a wave function that satisfied both
x
Ψ
=
0
{\displaystyle x\Psi =0}
and
p
Ψ
=
0
{\displaystyle p\Psi =0}
). This mathematical consistency or compatibility leads to three important properties of the TBDE. The first is a condition that eliminates the dependence on the relative time in the center of momentum (c.m.) frame defined by
P
=
p
1
+
p
2
=
(
w
,
0
→
)
{\displaystyle P=p_{1}+p_{2}=(w,{\vec {0}})}
. (The variable
w
{\displaystyle w}
is the total energy in the c.m. frame.) Stated another way, the relative time is eliminated in a covariant way. In particular, for the two operators to commute, the scalar and four-vector potentials can depend on the relative coordinate
x
=
x
1
−
x
2
{\displaystyle x=x_{1}-x_{2}}
only through its component
x
⊥
{\displaystyle x_{\perp }}
orthogonal to
P
{\displaystyle P}
in which
x
⊥
μ
=
(
η
μ
ν
−
P
μ
P
ν
/
P
2
)
x
ν
,
{\displaystyle x_{\perp }^{\mu }=(\eta ^{\mu \nu }-P^{\mu }P^{\nu }/P^{2})x_{\nu },\,}
P
μ
x
⊥
μ
=
0.
{\displaystyle P_{\mu }x_{\perp }^{\mu }=0.\,}
This implies that in the c.m. frame
x
⊥
=
(
0
,
x
→
=
x
→
1
−
x
→
2
)
{\displaystyle x_{\perp }=(0,{\vec {x}}={\vec {x}}_{1}-{\vec {x}}_{2})}
, which has zero time component.
Secondly, the mathematical consistency condition also eliminates the relative energy in the c.m. frame. It does this by imposing on each Dirac operator a structure such that in a particular combination they lead to this interaction independent form, eliminating in a covariant way the relative energy.
P
⋅
p
Ψ
=
(
−
P
0
p
0
+
P
→
⋅
p
)
Ψ
=
0.
{\displaystyle P\cdot p\Psi =(-P^{0}p^{0}+{\vec {P}}\cdot p)\Psi =0.\,}
In this expression
p
{\displaystyle p}
is the relative momentum having the form
(
p
1
−
p
2
)
/
2
{\displaystyle (p_{1}-p_{2})/2}
for equal masses. In the c.m. frame (
P
0
=
w
,
P
→
=
0
→
{\displaystyle P^{0}=w,{\vec {P}}={\vec {0}}}
), the time component
p
0
{\displaystyle p^{0}}
of the relative momentum, that is the relative energy, is thus eliminated. in the sense that
p
0
Ψ
=
0
{\displaystyle p^{0}\Psi =0}
.
A third consequence of the mathematical consistency is that each of the world scalar
S
~
i
{\displaystyle {\tilde {S}}_{i}}
and four vector
A
~
i
μ
{\displaystyle {\tilde {A}}_{i}^{\mu }}
potentials has a term with a fixed dependence on
γ
1
{\displaystyle \gamma _{1}}
and
γ
2
{\displaystyle \gamma _{2}}
in addition to the gamma matrix independent forms of
S
i
{\displaystyle S_{i}}
and
A
i
μ
{\displaystyle A_{i}^{\mu }}
which appear in the ordinary one-body Dirac equation for scalar and vector potentials.
These extra terms correspond to additional recoil spin-dependence not present in the one-body Dirac equation and vanish when one of the particles becomes very heavy (the so-called static limit).
== More on constraint dynamics: generalized mass shell constraints ==
Constraint dynamics arose from the work of Dirac and Bergmann. This section shows how the elimination of relative time and energy takes place in the c.m. system for the simple system of two relativistic spinless particles. Constraint dynamics was first applied to the classical relativistic two particle system by Todorov, Kalb and Van Alstine, Komar, and Droz–Vincent. With constraint dynamics, these authors found a consistent and covariant approach to relativistic canonical Hamiltonian mechanics that also evades the Currie–Jordan–Sudarshan "No Interaction" theorem. That theorem states that without fields, one cannot have a relativistic Hamiltonian dynamics. Thus, the same covariant three-dimensional approach which allows the quantized version of constraint dynamics to remove quantum ghosts simultaneously circumvents at the classical level the C.J.S. theorem. Consider a constraint on the otherwise independent coordinate and momentum four vectors, written in the form
ϕ
i
(
p
,
x
)
≈
0
{\displaystyle \phi _{i}(p,x)\approx 0}
. The symbol
≈
0
{\displaystyle \approx 0}
is called a weak equality and implies that the constraint is to be imposed only after any needed Poisson brackets are performed. In the presence of such constraints, the total
Hamiltonian
H
{\displaystyle {\mathcal {H}}}
is obtained from the Lagrangian
L
{\displaystyle {\mathcal {L}}}
by adding to the Legendre Hamiltonian
(
p
x
˙
−
L
)
{\displaystyle (p{\dot {x}}-{\mathcal {L}})}
the sum of the constraints times an appropriate set of Lagrange multipliers
(
λ
i
)
{\displaystyle (\lambda _{i})}
.
H
=
p
x
˙
−
L
+
λ
i
ϕ
i
,
{\displaystyle {\mathcal {H}}=p{\dot {x}}-{\mathcal {L}}+\lambda _{i}\phi _{i},}
This total Hamiltonian is traditionally called the Dirac Hamiltonian. Constraints arise naturally from parameter invariant actions of the form
I
=
∫
d
τ
L
(
τ
)
=
∫
d
τ
′
d
τ
d
τ
′
L
(
τ
)
=
∫
d
τ
′
L
(
τ
′
)
.
{\displaystyle I=\int d\tau {\mathcal {L}}(\tau )=\int d\tau '{\frac {d\tau }{d\tau '}}{\mathcal {L}}(\tau )=\int d\tau '{\mathcal {L}}(\tau ').}
In the case of four vector and Lorentz scalar interactions for a single particle the Lagrangian is
L
(
τ
)
=
−
(
m
+
S
(
x
)
)
−
x
˙
2
+
x
˙
⋅
A
(
x
)
{\displaystyle {\mathcal {L}}(\tau )=-(m+S(x)){\sqrt {-{\dot {x}}^{2}}}+{\dot {x}}\cdot A(x)\,}
The canonical momentum is
p
=
∂
L
∂
x
˙
=
(
m
+
S
(
x
)
)
x
˙
−
x
˙
2
+
A
(
x
)
{\displaystyle p={\frac {\partial {\mathcal {L}}}{\partial {\dot {x}}}}={\frac {(m+S(x)){\dot {x}}}{\sqrt {-{\dot {x}}^{2}}}}+A(x)}
and by squaring leads to the generalized mass shell condition or generalized mass shell constraint
(
p
−
A
)
2
+
(
m
+
S
)
2
=
0.
{\displaystyle (p-A)^{2}+(m+S)^{2}=0.\,}
Since, in this case, the Legendre Hamiltonian vanishes
p
⋅
x
˙
−
L
=
0
,
{\displaystyle p\cdot {\dot {x}}-{\mathcal {L}}=0,\,}
the Dirac Hamiltonian is simply the generalized mass constraint (with no interactions it would simply be the ordinary mass shell constraint)
H
=
λ
[
(
p
−
A
)
2
+
(
m
+
S
)
2
]
≡
λ
(
p
2
+
m
2
+
Φ
(
x
,
p
)
)
.
{\displaystyle {\mathcal {H}}=\lambda \left[\left(p-A\right)^{2}+(m+S)^{2}\right]\equiv \lambda (p^{2}+m^{2}+\Phi (x,p)).}
One then postulates that for two bodies the Dirac Hamiltonian is the sum of two such mass shell constraints,
H
i
=
p
i
2
+
m
i
2
+
Φ
i
(
x
1
,
x
2
,
p
1
,
p
2
)
≈
0
,
{\displaystyle {\mathcal {H}}_{i}=p_{i}^{2}+m_{i}^{2}+\Phi _{i}(x_{1},x_{2},p_{1},p_{2})\approx 0,\,}
that is
H
=
λ
1
[
p
1
2
+
m
1
2
+
Φ
1
(
x
1
,
x
2
,
p
1
,
p
2
)
]
+
λ
2
[
p
2
2
+
m
2
2
+
Φ
2
(
x
1
,
x
2
,
p
1
,
p
2
)
]
=
λ
1
H
1
+
λ
2
H
2
,
{\displaystyle {\begin{aligned}{\mathcal {H}}&=\lambda _{1}[p_{1}^{2}+m_{1}^{2}+\Phi _{1}(x_{1},x_{2},p_{1},p_{2})]+\lambda _{2}[p_{2}^{2}+m_{2}^{2}+\Phi _{2}(x_{1},x_{2},p_{1},p_{2})]\\[1ex]&=\lambda _{1}{\mathcal {H}}_{1}+\lambda _{2}{\mathcal {H}}_{2},\end{aligned}}}
and that each constraint
H
i
{\displaystyle {\mathcal {H}}_{i}}
be constant in the proper time associated with
H
{\displaystyle {\mathcal {H}}}
H
˙
i
=
{
H
i
,
H
}
≈
0
{\displaystyle {\dot {\mathcal {H}}}_{i}=\{{\mathcal {H}}_{i},{\mathcal {H}}\}\approx 0\,}
Here the weak equality means that the Poisson bracket could result in terms proportional one of the constraints, the classical Poisson brackets for the relativistic two-body system being defined by
{
O
1
,
O
2
}
=
∂
O
1
∂
x
1
μ
∂
O
2
∂
p
1
μ
−
∂
O
1
∂
p
1
μ
∂
O
2
∂
x
1
μ
+
∂
O
1
∂
x
2
μ
∂
O
2
∂
p
2
μ
−
∂
O
1
∂
p
2
μ
∂
O
2
∂
x
2
μ
.
{\displaystyle \left\{O_{1},O_{2}\right\}={\frac {\partial O_{1}}{\partial x_{1}^{\mu }}}{\frac {\partial O_{2}}{\partial p_{1\mu }}}-{\frac {\partial O_{1}}{\partial p_{1}^{\mu }}}{\frac {\partial O_{2}}{\partial x_{1\mu }}}+{\frac {\partial O_{1}}{\partial x_{2}^{\mu }}}{\frac {\partial O_{2}}{\partial p_{2\mu }}}-{\frac {\partial O_{1}}{\partial p_{2}^{\mu }}}{\frac {\partial O_{2}}{\partial x_{2\mu }}}.}
To see the consequences of having each constraint be a constant of the motion, take, for example
H
˙
1
=
{
H
1
,
H
}
=
λ
1
{
H
1
,
H
1
}
+
{
H
1
,
λ
1
}
H
2
+
λ
2
{
H
2
,
H
1
}
+
{
λ
2
,
H
1
}
H
2
.
{\displaystyle {\dot {\mathcal {H}}}_{1}=\{{\mathcal {H}}_{1},{\mathcal {H}}\}=\lambda _{1}\{{\mathcal {H}}_{1},{\mathcal {H}}_{1}\}+\{{\mathcal {H}}_{1},\lambda _{1}\}{\mathcal {H}}_{2}+\lambda _{2}\{{\mathcal {H}}_{2},{\mathcal {H}}_{1}\}+\{\lambda _{2},{\mathcal {H}}_{1}\}{\mathcal {H}}_{2}.}
Since
{
H
1
,
H
1
}
=
0
{\displaystyle \{{\mathcal {H}}_{1},{\mathcal {H}}_{1}\}=0}
and
H
1
≈
0
{\displaystyle {\mathcal {H}}_{1}\approx 0}
and
H
2
≈
0
{\displaystyle {\mathcal {H}}_{2}\approx 0}
one has
H
˙
1
≈
λ
2
{
H
2
,
H
1
}
≈
0.
{\displaystyle {\dot {\mathcal {H}}}_{1}\approx \lambda _{2}\{{\mathcal {H}}_{2},{\mathcal {H}}_{1}\}\approx 0.}
The simplest solution to this is
Φ
1
=
Φ
2
≡
Φ
(
x
⊥
)
{\displaystyle \Phi _{1}=\Phi _{2}\equiv \Phi (x_{\perp })}
which leads to (note the equality in this case is not a weak one in that no constraint need be imposed after the Poisson bracket is worked out)
{
H
2
,
H
1
}
=
0
{\displaystyle \{{\mathcal {H}}_{2},{\mathcal {H}}_{1}\}=0\,}
(see Todorov, and Wong and Crater ) with the same
x
⊥
{\displaystyle x_{\perp }}
defined above.
== Quantization ==
In addition to replacing classical dynamical variables by their quantum counterparts, quantization of the constraint mechanics takes place by replacing the constraint on the dynamical variables with a restriction on the wave function
H
i
≈
0
→
H
i
Ψ
=
0
,
{\displaystyle {\mathcal {H}}_{i}\approx 0\rightarrow {\mathcal {H}}_{i}\Psi =0,}
H
≈
0
→
H
Ψ
=
0.
{\displaystyle {\mathcal {H}}\approx 0\rightarrow {\mathcal {H}}\Psi =0.}
The first set of equations for i = 1, 2 play the role for spinless particles that the two Dirac equations play for spin-one-half particles. The classical Poisson brackets are replaced by commutators
{
O
1
,
O
2
}
→
1
i
[
O
1
,
O
2
]
.
{\displaystyle \{O_{1},O_{2}\}\rightarrow {\frac {1}{i}}[O_{1},O_{2}].\,}
Thus
[
H
2
,
H
1
]
=
0
,
{\displaystyle [{\mathcal {H}}_{2},{\mathcal {H}}_{1}]=0,\,}
and we see in this case that the constraint formalism leads to the vanishing commutator of the wave operators for the two particles. This is the analogue of the claim stated earlier that the two Dirac operators commute with one another.
== Covariant elimination of the relative energy ==
The vanishing of the above commutator ensures that the dynamics is independent of the relative time in the c.m. frame. In order to covariantly eliminate the relative energy, introduce the relative momentum
p
{\displaystyle p}
defined by
The above definition of the relative momentum forces the orthogonality of the total momentum and the relative momentum,
P
⋅
p
=
0
,
{\displaystyle P\cdot p=0,}
which follows from taking the scalar product of either equation with
P
{\displaystyle P}
.
From Eqs.(1) and (2), this relative momentum can be written in terms of
p
1
{\displaystyle p_{1}}
and
p
2
{\displaystyle p_{2}}
as
p
=
ε
2
−
P
2
p
1
−
ε
1
−
P
2
p
2
{\displaystyle p={\frac {\varepsilon _{2}}{\sqrt {-P^{2}}}}p_{1}-{\frac {\varepsilon _{1}}{\sqrt {-P^{2}}}}p_{2}}
where
ε
1
=
−
p
1
⋅
P
−
P
2
=
−
P
2
+
p
1
2
−
p
2
2
2
−
P
2
{\displaystyle \varepsilon _{1}=-{\frac {p_{1}\cdot P}{\sqrt {-P^{2}}}}=-{\frac {P^{2}+p_{1}^{2}-p_{2}^{2}}{2{\sqrt {-P^{2}}}}}}
ε
2
=
−
p
2
⋅
P
−
P
2
=
−
P
2
+
p
2
2
−
p
1
2
2
−
P
2
{\displaystyle \varepsilon _{2}=-{\frac {p_{2}\cdot P}{\sqrt {-P^{2}}}}=-{\frac {P^{2}+p_{2}^{2}-p_{1}^{2}}{2{\sqrt {-P^{2}}}}}}
are the projections of the momenta
p
1
{\displaystyle p_{1}}
and
p
2
{\displaystyle p_{2}}
along the direction of the total momentum
P
{\displaystyle P}
. Subtracting the two constraints
H
1
Ψ
=
0
{\displaystyle {\mathcal {H}}_{1}\Psi =0}
and
H
2
Ψ
=
0
{\displaystyle {\mathcal {H}}_{2}\Psi =0}
, gives
Thus on these states
Ψ
{\displaystyle \Psi }
ε
1
Ψ
=
−
P
2
+
m
1
2
−
m
2
2
2
−
P
2
Ψ
{\displaystyle \varepsilon _{1}\Psi ={\frac {-P^{2}+m_{1}^{2}-m_{2}^{2}}{2{\sqrt {-P^{2}}}}}\Psi }
ε
2
Ψ
=
−
P
2
+
m
2
2
−
m
1
2
2
−
P
2
Ψ
.
{\displaystyle \varepsilon _{2}\Psi ={\frac {-P^{2}+m_{2}^{2}-m_{1}^{2}}{2{\sqrt {-P^{2}}}}}\Psi .}
The equation
H
Ψ
=
0
{\displaystyle {\mathcal {H}}\Psi =0}
describes both the c.m. motion and the internal relative motion. To characterize the former motion, observe that since the potential
Φ
{\displaystyle \Phi }
depends only on the difference of the two coordinates
[
P
,
H
]
Ψ
=
0.
{\displaystyle [P,{\mathcal {H}}]\Psi =0.}
(This does not require that
[
P
,
λ
i
]
=
0
{\displaystyle [P,\lambda _{i}]=0}
since the
H
i
Ψ
=
0
{\displaystyle {\mathcal {H}}_{i}\Psi =0}
.) Thus, the total momentum
P
{\displaystyle P}
is a constant of motion and
Ψ
{\displaystyle \Psi }
is an eigenstate state characterized by a total momentum
P
′
{\displaystyle P'}
. In the c.m. system
P
′
=
(
w
,
0
→
)
,
{\displaystyle P'=(w,{\vec {0}}),}
with
w
{\displaystyle w}
the invariant center of momentum (c.m.) energy. Thus
and so
Ψ
{\displaystyle \Psi }
is also an eigenstate of c.m. energy operators for each of the two particles,
ε
1
Ψ
=
w
2
+
m
1
2
−
m
2
2
2
w
Ψ
{\displaystyle \varepsilon _{1}\Psi ={\frac {w^{2}+m_{1}^{2}-m_{2}^{2}}{2w}}\Psi }
ε
2
Ψ
=
w
2
+
m
2
2
−
m
1
2
2
w
Ψ
.
{\displaystyle \varepsilon _{2}\Psi ={\frac {w^{2}+m_{2}^{2}-m_{1}^{2}}{2w}}\Psi .}
The relative momentum then satisfies
p
Ψ
=
ε
2
p
1
−
ε
1
p
2
w
Ψ
,
{\displaystyle p\Psi ={\frac {\varepsilon _{2}p_{1}-\varepsilon _{1}p_{2}}{w}}\Psi ,}
so that
p
1
Ψ
=
(
ε
1
w
P
+
p
)
Ψ
,
{\displaystyle p_{1}\Psi =\left({\frac {\varepsilon _{1}}{w}}P+p\right)\Psi ,}
p
2
Ψ
=
(
ε
2
w
P
−
p
)
Ψ
,
{\displaystyle p_{2}\Psi =\left({\frac {\varepsilon _{2}}{w}}P-p\right)\Psi ,}
The above set of equations follow from the constraints
H
i
Ψ
=
0
{\displaystyle {\mathcal {H}}_{i}\Psi =0}
and the definition of the relative momenta given in Eqs.(1) and (2). If instead one chooses to define (for a more general choice see Horwitz),
ε
1
=
w
2
+
m
1
2
−
m
2
2
2
w
,
{\displaystyle \varepsilon _{1}={\frac {w^{2}+m_{1}^{2}-m_{2}^{2}}{2w}},}
ε
2
=
w
2
+
m
2
2
−
m
1
2
2
w
,
{\displaystyle \varepsilon _{2}={\frac {w^{2}+m_{2}^{2}-m_{1}^{2}}{2w}},}
p
=
ε
2
p
1
−
ε
1
p
2
w
,
{\displaystyle p={\frac {\varepsilon _{2}p_{1}-\varepsilon _{1}p_{2}}{w}},}
independent of the wave function, then
and it is straight forward to show that the constraint Eq.(3) leads directly to:
in place of
P
⋅
p
=
0
{\displaystyle P\cdot p=0}
. This conforms with the earlier claim on the vanishing of the relative energy in the c.m. frame made in conjunction with the TBDE. In the second choice the c.m. value of the relative energy is not defined as zero but comes from the original generalized mass shell constraints. The above equations for the relative and constituent four-momentum are the relativistic analogues of the non-relativistic equations
p
→
=
m
2
p
→
1
−
m
1
p
→
2
M
,
p
→
1
=
m
1
M
P
→
+
p
→
,
p
→
2
=
m
2
M
P
→
−
p
→
.
{\displaystyle {\begin{aligned}{\vec {p}}&={\frac {m_{2}{\vec {p}}_{1}-m_{1}{\vec {p}}_{2}}{M}},\\[1ex]{\vec {p}}_{1}&={\frac {m_{1}}{M}}{\vec {P}}+{\vec {p}},\\[1ex]{\vec {p}}_{2}&={\frac {m_{2}}{M}}{\vec {P}}-{\vec {p}}.\end{aligned}}}
== Covariant eigenvalue equation for internal motion ==
Using Eqs.(5),(6),(7), one can write
H
{\displaystyle {\mathcal {H}}}
in terms of
P
{\displaystyle P}
and
p
{\displaystyle p}
H
Ψ
=
{
λ
1
[
−
ε
1
2
+
m
1
2
+
p
2
+
Φ
(
x
⊥
)
]
+
λ
2
[
−
ε
2
2
+
m
2
2
+
p
2
+
Φ
(
x
⊥
)
]
}
Ψ
{\displaystyle {\mathcal {H}}\Psi =\{\lambda _{1}[-\varepsilon _{1}^{2}+m_{1}^{2}+p^{2}+\Phi (x_{\perp })]+\lambda _{2}[-\varepsilon _{2}^{2}+m_{2}^{2}+p^{2}+\Phi (x_{\perp })]\}\Psi }
where
b
2
(
−
P
2
,
m
1
2
,
m
2
2
)
=
ε
1
2
−
m
1
2
=
ε
2
2
−
m
2
2
=
−
1
4
P
2
(
P
4
+
2
P
2
(
m
1
2
+
m
2
2
)
+
(
m
1
2
−
m
2
2
)
2
)
.
{\displaystyle b^{2}(-P^{2},m_{1}^{2},m_{2}^{2})=\varepsilon _{1}^{2}-m_{1}^{2}=\varepsilon _{2}^{2}-m_{2}^{2}\ =-{\frac {1}{4P^{2}}}(P^{4}+2P^{2}(m_{1}^{2}+m_{2}^{2})+(m_{1}^{2}-m_{2}^{2})^{2})\,.}
Eq.(8) contains both the total momentum
P
{\displaystyle P}
[through the
b
2
(
−
P
2
,
m
1
2
,
m
2
2
)
{\displaystyle b^{2}(-P^{2},m_{1}^{2},m_{2}^{2})}
] and the relative momentum
p
{\displaystyle p}
. Using Eq. (4), one obtains the eigenvalue equation
so that
b
2
(
w
2
,
m
1
2
,
m
2
2
)
{\displaystyle b^{2}(w^{2},m_{1}^{2},m_{2}^{2})}
becomes the standard triangle function displaying exact relativistic two-body kinematics:
b
2
(
w
2
,
m
1
2
,
m
2
2
)
=
1
4
w
2
{
w
4
−
2
w
2
(
m
1
2
+
m
2
2
)
+
(
m
1
2
−
m
2
2
)
2
}
.
{\displaystyle b^{2}(w^{2},m_{1}^{2},m_{2}^{2})={\frac {1}{4w^{2}}}\left\{w^{4}-2w^{2}(m_{1}^{2}+m_{2}^{2})+(m_{1}^{2}-m_{2}^{2})^{2}\right\}\,.}
With the above constraint Eqs.(7) on
Ψ
{\displaystyle \Psi }
then
p
2
Ψ
=
p
⊥
2
Ψ
{\displaystyle p^{2}\Psi =p_{\perp }^{2}\Psi }
where
p
⊥
=
p
−
p
⋅
P
P
/
P
2
{\displaystyle p_{\perp }=p-p\cdot PP/P^{2}}
. This allows writing Eq. (9) in the form of an eigenvalue equation
{
p
⊥
2
+
Φ
(
x
⊥
)
}
Ψ
=
b
2
(
w
2
,
m
1
2
,
m
2
2
)
Ψ
,
{\displaystyle \{p_{\perp }^{2}+\Phi (x_{\perp })\}\Psi =b^{2}(w^{2},m_{1}^{2},m_{2}^{2})\Psi \,,}
having a structure very similar to that of the ordinary three-dimensional nonrelativistic Schrödinger equation. It is a manifestly covariant equation, but at the same time its three-dimensional structure is evident. The four-vectors
p
⊥
μ
{\displaystyle p_{\perp }^{\mu }}
and
x
⊥
μ
{\displaystyle x_{\perp }^{\mu }}
have only three independent components since
P
⋅
p
⊥
=
P
⋅
x
⊥
=
0
.
{\displaystyle P\cdot p_{\perp }=P\cdot x_{\perp }=0\,.}
The similarity to the three-dimensional structure of the nonrelativistic Schrödinger equation can be made more explicit by writing the equation in the c.m. frame in which
P
=
(
w
,
0
→
)
,
{\displaystyle P=(w,{\vec {0}}),}
p
⊥
=
(
0
,
p
→
)
,
{\displaystyle p_{\perp }=(0,{\vec {p}}),}
x
⊥
=
(
0
,
x
→
)
.
{\displaystyle x_{\perp }=(0,{\vec {x}}).}
Comparison of the resultant form
with the time independent Schrödinger equation
makes this similarity explicit.
== The two-body relativistic Klein–Gordon equations ==
A plausible structure for the quasipotential
Φ
{\displaystyle \Phi }
can be found by observing that the one-body Klein–Gordon equation
(
p
2
+
m
2
)
ψ
=
(
p
→
2
−
ε
2
+
m
2
)
ψ
=
0
{\displaystyle (p^{2}+m^{2})\psi =({\vec {p}}^{2}-\varepsilon ^{2}+m^{2})\psi =0}
takes the form
(
p
→
2
−
ε
2
+
m
2
+
2
m
S
+
S
2
+
2
ε
A
−
A
2
)
ψ
=
0
{\displaystyle ({\vec {p}}^{2}-\varepsilon ^{2}+m^{2}+2mS+S^{2}+2\varepsilon A-A^{2})\psi =0~}
when one introduces a scalar interaction and timelike vector interaction via
m
→
m
+
S
{\displaystyle m\rightarrow m+S~}
and
ε
→
ε
−
A
{\displaystyle \varepsilon \rightarrow \varepsilon -A}
. In the two-body case, separate classical and quantum field theory
arguments show that when one includes world scalar and vector interactions then
Φ
{\displaystyle \Phi }
depends on two underlying invariant functions
S
(
r
)
{\displaystyle S(r)}
and
A
(
r
)
{\displaystyle A(r)}
through the two-body Klein–Gordon-like potential form with the same general structure, that is
Φ
=
2
m
w
S
+
S
2
+
2
ε
w
A
−
A
2
.
{\displaystyle \Phi =2m_{w}S+S^{2}+2\varepsilon _{w}A-A^{2}.}
Those field theories further yield the c.m. energy dependent forms
m
w
=
m
1
m
2
/
w
,
{\displaystyle m_{w}=m_{1}m_{2}/w,}
and
ε
w
=
(
w
2
−
m
1
2
−
m
2
2
)
/
2
w
,
{\displaystyle \varepsilon _{w}=(w^{2}-m_{1}^{2}-m_{2}^{2})/2w,}
ones that Tododov introduced as the relativistic reduced mass and effective particle energy for a two-body system. Similar to what happens in the nonrelativistic two-body problem, in the relativistic case we have the motion of this effective particle taking place as if it were in an external field (here generated by
S
{\displaystyle S}
and
A
{\displaystyle A}
). The two kinematical variables
m
w
{\displaystyle m_{w}}
and
ε
w
{\displaystyle \varepsilon _{w}}
are related to one another by the Einstein condition
ε
w
2
−
m
w
2
=
b
2
(
w
)
,
{\displaystyle \varepsilon _{w}^{2}-m_{w}^{2}=b^{2}(w),}
If one introduces the four-vectors, including a vector interaction
A
μ
{\displaystyle A^{\mu }}
p
=
ε
w
P
^
+
p
,
{\displaystyle {\mathfrak {p}}=\varepsilon _{w}{\hat {P}}+p,}
A
μ
=
P
^
μ
A
(
r
)
{\displaystyle A^{\mu }={\hat {P}}^{\mu }A(r)}
r
=
x
⊥
2
,
{\displaystyle r={\sqrt {x_{\perp }^{2}}}\,,}
and scalar interaction
S
(
r
)
{\displaystyle S(r)}
, then the following classical minimal constraint form
H
=
(
p
−
A
)
2
+
(
m
w
+
S
)
2
≈
0
,
{\displaystyle {\mathcal {H}}=\left({\mathfrak {p-}}A\right)^{2}+(m_{w}+S)^{2}\approx 0\,,}
reproduces
Notice, that the interaction in this "reduced particle" constraint depends on two invariant scalars,
A
(
r
)
{\displaystyle A(r)}
and
S
(
r
)
{\displaystyle S(r)}
, one guiding the time-like vector interaction and one the scalar interaction.
Is there a set of two-body Klein–Gordon equations analogous to the two-body Dirac equations? The classical relativistic constraints analogous to the quantum two-body Dirac equations (discussed in the introduction) and that have the same structure as the above Klein–Gordon one-body form are
H
1
=
(
p
1
−
A
1
)
2
+
(
m
1
+
S
1
)
2
=
p
1
2
+
m
1
2
+
Φ
1
≈
0
{\displaystyle {\mathcal {H}}_{1}=(p_{1}-A_{1})^{2}+(m_{1}+S_{1})^{2}=p_{1}^{2}+m_{1}^{2}+\Phi _{1}\approx 0}
H
2
=
(
p
1
−
A
2
)
2
+
(
m
2
+
S
2
)
2
=
p
2
2
+
m
2
2
+
Φ
2
≈
0
,
{\displaystyle {\mathcal {H}}_{2}=(p_{1}-A_{2})^{2}+(m_{2}+S_{2})^{2}=p_{2}^{2}+m_{2}^{2}+\Phi _{2}\approx 0,}
p
1
=
ε
1
P
^
+
p
;
p
2
=
ε
2
P
^
−
p
.
{\displaystyle p_{1}=\varepsilon _{1}{\hat {P}}+p;~~p_{2}=\varepsilon _{2}{\hat {P}}-p~.}
Defining structures that display time-like vector and scalar interactions
π
1
=
p
1
−
A
1
=
[
P
^
(
ε
1
−
A
1
)
+
p
]
,
{\displaystyle \pi _{1}=p_{1}-A_{1}=[{\hat {P}}(\varepsilon _{1}-{\mathcal {A}}_{1})+p],}
π
2
=
p
2
−
A
2
=
[
P
^
(
ε
2
−
A
1
)
−
p
]
,
{\displaystyle \pi _{2}=p_{2}-A_{2}=[{\hat {P}}(\varepsilon _{2}-{\mathcal {A}}_{1})-p],}
M
1
=
m
1
+
S
1
,
{\displaystyle M_{1}=m_{1}+S_{1},}
M
2
=
m
2
+
S
2
,
{\displaystyle M_{2}=m_{2}+S_{2},}
gives
H
1
=
π
1
2
+
M
1
2
,
{\displaystyle {\mathcal {H}}_{1}=\pi _{1}^{2}+M_{1}^{2},}
H
2
=
π
2
2
+
M
2
2
.
{\displaystyle {\mathcal {H}}_{2}=\pi _{2}^{2}+M_{2}^{2}.}
Imposing
Φ
1
=
Φ
2
≡
Φ
(
x
⊥
)
=
−
2
p
1
⋅
A
1
+
A
1
2
+
2
m
1
S
1
+
S
1
2
=
−
2
p
2
⋅
A
2
+
A
2
2
+
2
m
2
S
2
+
S
2
2
=
2
ε
w
A
−
A
2
+
2
m
w
S
+
S
2
,
{\displaystyle {\begin{aligned}\Phi _{1}&=\Phi _{2}\equiv \Phi (x_{\perp })\\&=-2p_{1}\cdot A_{1}+A_{1}^{2}+2m_{1}S_{1}+S_{1}^{2}\\&=-2p_{2}\cdot A_{2}+A_{2}^{2}+2m_{2}S_{2}+S_{2}^{2}\\&=2\varepsilon _{w}A-A^{2}+2m_{w}S+S^{2},\end{aligned}}}
and using the constraint
P
⋅
p
≈
0
{\displaystyle P\cdot p\approx 0}
, reproduces Eqs.(12) provided
π
1
2
−
p
2
=
−
(
ε
1
−
A
1
)
2
=
−
ε
1
2
+
2
ε
w
A
−
A
2
,
{\displaystyle \pi _{1}^{2}-p^{2}=-\left(\varepsilon _{1}-{\mathcal {A}}_{1}\right)^{2}=-\varepsilon _{1}^{2}+2\varepsilon _{w}A-A^{2},}
π
2
2
−
p
2
=
−
(
ε
2
−
A
2
)
2
=
−
ε
2
2
+
2
ε
w
A
−
A
2
,
{\displaystyle \pi _{2}^{2}-p^{2}=-\left(\varepsilon _{2}-{\mathcal {A}}_{2}\right)^{2}=-\varepsilon _{2}^{2}+2\varepsilon _{w}A-A^{2},}
M
1
2
=
m
1
2
+
2
m
w
S
+
S
2
,
{\displaystyle M_{1}{}^{2}=m_{1}^{2}+2m_{w}S+S^{2},}
M
2
2
=
m
2
2
+
2
m
w
S
+
S
2
.
{\displaystyle M_{2}^{2}=m_{2}^{2}+2m_{w}S+S^{2}.}
The corresponding Klein–Gordon equations are
(
π
1
2
+
M
1
2
)
ψ
=
0
,
{\displaystyle \left(\pi _{1}^{2}+M_{1}^{2}\right)\psi =0,}
(
π
2
2
+
M
2
2
)
ψ
=
0
,
{\displaystyle \left(\pi _{2}^{2}+M_{2}^{2}\right)\psi =0,}
and each, due to the constraint
P
⋅
p
≈
0
,
{\displaystyle P\cdot p\approx 0,}
is equivalent to
H
ψ
=
(
p
⊥
2
+
Φ
−
b
2
)
ψ
=
0.
{\displaystyle {\mathcal {H}}\psi =\left(p_{\perp }^{2}+\Phi -b^{2}\right)\psi =0.}
== Hyperbolic versus external field form of the two-body Dirac equations ==
For the two body system there are numerous covariant forms of interaction. The simplest way of looking at these is from the point of view of the gamma matrix structures of the corresponding interaction vertices of the single particle exchange diagrams. For scalar, pseudoscalar, vector, pseudovector, and tensor exchanges those matrix structures are respectively
1
1
1
2
;
γ
51
γ
52
;
γ
1
μ
γ
2
μ
;
γ
51
γ
1
μ
γ
52
γ
2
μ
;
σ
1
μ
ν
σ
2
μ
ν
,
{\displaystyle 1_{1}1_{2};\gamma _{51}\gamma _{52};\gamma _{1}^{\mu }\gamma _{2\mu };\gamma _{51}\gamma _{1}^{\mu }\gamma _{52}\gamma _{2\mu };\sigma _{1\mu \nu }\sigma _{2}^{\mu \nu },}
in which
σ
i
μ
ν
=
1
2
i
[
γ
i
μ
,
γ
i
ν
]
;
i
=
1
,
2.
{\displaystyle \sigma _{i\mu \nu }={\frac {1}{2i}}[\gamma _{i\mu },\gamma _{i\nu }];i=1,2.}
The form of the Two-Body Dirac equations which most readily incorporates each or any number of these intereractions in concert is the so-called hyperbolic form of the TBDE. For combined scalar and vector interactions those forms ultimately reduce to the ones given in the first set of equations of this article. Those equations are called the external field-like forms because their appearances are individually the same as those for the usual one-body Dirac equation in the presence of external vector and scalar fields.
The most general hyperbolic form for compatible TBDE is
S
1
ψ
=
(
cosh
(
Δ
)
S
1
+
sinh
(
Δ
)
S
2
)
ψ
=
0
,
{\displaystyle {\mathcal {S}}_{1}\psi =(\cosh(\Delta )\mathbf {S} _{1}+\sinh(\Delta )\mathbf {S} _{2})\psi =0,}
where
Δ
{\displaystyle \Delta }
represents any invariant interaction singly or in combination. It has a matrix structure in addition to coordinate dependence. Depending on what that matrix structure is one has either scalar, pseudoscalar, vector, pseudovector, or tensor interactions. The operators
S
1
{\displaystyle \mathbf {S} _{1}}
and
S
2
{\displaystyle \mathbf {S} _{2}}
are auxiliary constraints satisfying
S
1
ψ
≡
(
S
10
cosh
(
Δ
)
+
S
20
sinh
(
Δ
)
)
ψ
=
0
,
{\displaystyle \mathbf {S} _{1}\psi \equiv ({\mathcal {S}}_{10}\cosh(\Delta )+{\mathcal {S}}_{20}\sinh(\Delta )~)\psi =0,}
in which the
S
i
0
{\displaystyle {\mathcal {S}}_{i0}}
are the free Dirac operators
This, in turn leads to the two compatibility conditions
[
S
1
,
S
2
]
ψ
=
0
,
{\displaystyle \lbrack {\mathcal {S}}_{1},{\mathcal {S}}_{2}]\psi =0,}
and
[
S
1
,
S
2
]
ψ
=
0
,
{\displaystyle \lbrack \mathbf {S} _{1},\mathbf {S} _{2}]\psi =0,}
provided that
Δ
=
Δ
(
x
⊥
)
.
{\displaystyle \Delta =\Delta (x_{\perp }).}
These compatibility conditions do not restrict the gamma matrix structure of
Δ
{\displaystyle \Delta }
. That matrix structure is determined by the type of vertex-vertex structure incorporated in the interaction. For the two types of invariant interactions
Δ
{\displaystyle \Delta }
emphasized in this article they are
Δ
L
(
x
⊥
)
=
−
1
1
1
2
L
(
x
⊥
)
2
O
1
,
scalar
,
{\displaystyle \Delta _{\mathcal {L}}(x_{\perp })=-1_{1}1_{2}{\frac {{\mathcal {L}}(x_{\perp })}{2}}{\mathcal {O}}_{1},{\text{scalar}},}
Δ
G
(
x
⊥
)
=
γ
1
⋅
γ
2
G
(
x
⊥
)
2
O
1
,
vector
,
{\displaystyle \Delta _{\mathcal {G}}(x_{\perp })=\gamma _{1}\cdot \gamma _{2}{\frac {{\mathcal {G}}(x_{\perp })}{2}}{\mathcal {O}}_{1},{\text{vector}},}
O
1
=
−
γ
51
γ
52
.
{\displaystyle {\mathcal {O}}_{1}=-\gamma _{51}\gamma _{52}.}
For general independent scalar and vector interactions
Δ
(
x
⊥
)
=
Δ
L
+
Δ
G
.
{\displaystyle \Delta (x_{\perp })=\Delta _{\mathcal {L}}+\Delta _{\mathcal {G}}.}
The vector interaction specified by the above matrix structure for an electromagnetic-like interaction would correspond to the Feynman gauge.
If one inserts Eq.(14) into (13) and brings the free Dirac operator (15) to the right of the matrix hyperbolic functions and uses standard gamma matrix commutators and anticommutators and
cosh
2
Δ
−
sinh
2
Δ
=
1
{\displaystyle \cosh ^{2}\Delta -\sinh ^{2}\Delta =1}
one arrives at
(
∂
μ
=
∂
/
∂
x
μ
)
,
{\displaystyle \left(\partial _{\mu }=\partial /\partial x^{\mu }\right),}
(
G
γ
1
⋅
P
2
−
E
1
β
1
+
M
1
−
G
i
2
Σ
2
⋅
∂
(
L
β
2
−
G
β
1
)
γ
52
)
ψ
=
0
,
{\displaystyle {\big (}G\gamma _{1}\cdot {\mathcal {P}}_{2}-E_{1}\beta _{1}+M_{1}-G{\frac {i}{2}}\Sigma _{2}\cdot \partial ({\mathcal {L}}\beta _{2}-{\mathcal {G}}\beta _{1})\gamma _{52}{\big )}\psi =0,}
in which
G
=
exp
G
,
{\displaystyle G=\exp {\mathcal {G}},}
β
i
=
−
γ
i
⋅
P
^
,
{\displaystyle \beta _{i}=-\gamma _{i}\cdot {\hat {P}},}
γ
i
⊥
μ
=
(
η
μ
ν
+
P
^
μ
P
^
ν
)
γ
ν
i
,
{\displaystyle \gamma _{i\perp }^{\mu }=(\eta ^{\mu \nu }+{\hat {P}}^{\mu }{\hat {P}}^{\nu })\gamma _{\nu i},}
Σ
i
=
γ
5
i
β
i
γ
⊥
i
,
{\displaystyle \Sigma _{i}=\gamma _{5i}\beta _{i}\gamma _{\perp i},}
P
i
≡
p
⊥
−
i
2
Σ
i
⋅
∂
G
Σ
i
,
i
=
1
,
2.
{\displaystyle {\mathcal {P}}_{i}\equiv p_{\perp }-{\frac {i}{2}}\Sigma _{i}\cdot \partial {\mathcal {G}}\Sigma _{i}\,,\quad i=1,2.}
The (covariant) structure of these equations are analogous to those of a Dirac equation for each of the two particles, with
M
i
{\displaystyle M_{i}}
and
E
i
{\displaystyle E_{i}}
playing the roles that
m
+
S
{\displaystyle m+S}
and
ε
−
A
{\displaystyle \varepsilon -A}
do in the single particle Dirac equation
(
γ
⋅
p
−
β
(
ε
−
A
)
+
m
+
S
)
ψ
=
0.
{\displaystyle (\mathbf {\gamma } \cdot \mathbf {p-} \beta (\varepsilon -A)+m+S)\psi =0.}
Over and above the usual kinetic part
γ
1
⋅
p
⊥
{\displaystyle \gamma _{1}\cdot p_{\perp }}
and time-like vector and scalar potential portions, the spin-dependent modifications involving
Σ
i
⋅
∂
G
Σ
i
{\displaystyle \Sigma _{i}\cdot \partial {\mathcal {G}}\Sigma _{i}}
and the last set of derivative terms are two-body recoil effects absent for the one-body Dirac equation but essential for the compatibility (consistency) of the two-body equations. The connections between what are designated as the vertex invariants
L
,
G
{\displaystyle {\mathcal {L}},{\mathcal {G}}}
and the mass and energy potentials
M
i
,
E
i
{\displaystyle M_{i},E_{i}}
are
M
1
=
m
1
cosh
L
+
m
2
sinh
L
,
{\displaystyle M_{1}=m_{1}\cosh {\mathcal {L}}+m_{2}\sinh {\mathcal {L}},}
M
2
=
m
2
cosh
L
+
m
1
sinh
L
,
{\displaystyle M_{2}=m_{2}\cosh {\mathcal {L}}+m_{1}\sinh {\mathcal {L}},}
E
1
=
ε
1
cosh
G
−
ε
2
sinh
G
,
{\displaystyle E_{1}=\varepsilon _{1}\cosh {\mathcal {G}}-\varepsilon _{2}\sinh {\mathcal {G}},}
E
2
=
ε
2
cosh
G
−
ε
1
sinh
G
.
{\displaystyle E_{2}=\varepsilon _{2}\cosh {\mathcal {G}}-\varepsilon _{1}\sinh {\mathcal {G}}.}
Comparing Eq.(16) with the first equation of this article one finds that the spin-dependent vector interactions are
A
~
1
μ
=
(
(
ε
1
−
E
1
)
)
P
^
μ
+
(
1
−
G
)
p
⊥
μ
−
i
2
∂
G
⋅
γ
2
γ
2
μ
,
{\displaystyle {\tilde {A}}_{1}^{\mu }={\big (}(\varepsilon _{1}-E_{1}){\big )}{\hat {P}}^{\mu }+(1-G)p_{\perp }^{\mu }-{\frac {i}{2}}\partial G\cdot \gamma _{2}\gamma _{2}^{\mu },}
A
2
μ
=
(
(
ε
2
−
E
2
)
)
P
^
μ
−
(
1
−
G
)
p
⊥
μ
+
i
2
∂
G
⋅
γ
1
γ
1
μ
,
{\displaystyle A_{2}^{\mu }={\big (}(\varepsilon _{2}-E_{2}){\big )}{\hat {P}}^{\mu }-(1-G)p_{\perp }^{\mu }+{\frac {i}{2}}\partial G\cdot \gamma _{1}\gamma _{1}^{\mu },}
Note that the first portion of the vector potentials is timelike (parallel to
P
^
μ
)
{\displaystyle {\hat {P}}^{\mu })}
while the next portion is spacelike (perpendicular to
P
^
μ
)
{\displaystyle {\hat {P}}^{\mu })}
. The spin-dependent scalar potentials
S
~
i
{\displaystyle {\tilde {S}}_{i}}
are
S
~
1
=
M
1
−
m
1
−
i
2
G
γ
2
⋅
∂
L
,
{\displaystyle {\tilde {S}}_{1}=M_{1}-m_{1}-{\frac {i}{2}}G\gamma _{2}\cdot \partial {\mathcal {L}},}
S
~
2
=
M
2
−
m
2
+
i
2
G
γ
1
⋅
∂
L
.
{\displaystyle {\tilde {S}}_{2}=M_{2}-m_{2}+{\frac {i}{2}}G\gamma _{1}\cdot \partial {\mathcal {L}}.}
The parametrization for
L
{\displaystyle {\mathcal {L}}}
and
G
{\displaystyle {\mathcal {G}}}
takes advantage of the Todorov effective external potential forms (as seen in the above section on the two-body Klein Gordon equations) and at the same time displays the correct static limit form for the Pauli reduction to Schrödinger-like form. The choice for these parameterizations (as with the two-body Klein Gordon equations) is closely tied to classical or quantum field theories for separate scalar and vector interactions. This amounts to working in the Feynman gauge with the simplest relation between space- and timelike parts of the vector interaction. The mass and energy potentials are respectively
M
i
2
=
m
i
2
+
exp
(
2
G
)
(
2
m
w
S
+
S
2
)
,
{\displaystyle M_{i}^{2}=m_{i}^{2}+\exp(2{\mathcal {G}})(2m_{w}S+S^{2}),}
E
i
2
=
exp
(
2
G
(
A
)
)
(
ε
i
−
A
)
2
,
{\displaystyle E_{i}^{2}=\exp(2{\mathcal {G}}(A))\left(\varepsilon _{i}-A\right)^{2},}
so that
exp
L
=
exp
(
L
(
S
,
A
)
)
=
M
1
+
M
2
m
1
+
m
2
,
{\displaystyle \exp {\mathcal {L}}=\exp({\mathcal {L}}(S,A))={\frac {M_{1}+M_{2}}{m_{1}+m_{2}}},}
G
=
exp
G
=
exp
(
G
(
A
)
)
=
1
(
1
−
2
A
/
w
)
.
{\displaystyle G=\exp {\mathcal {G}}=\exp({\mathcal {G}}(A))={\sqrt {\frac {1}{(1-2A/w)}}}.}
== Applications and limitations ==
The TBDE can be readily applied to two body systems such as positronium, muonium, hydrogen-like atoms, quarkonium, and the two-nucleon system. These applications involve two particles only and do not involve creation or annihilation of particles beyond the two. They involve only elastic processes. Because of the connection between the potentials used in the TBDE and the corresponding quantum field theory, any radiative correction to the lowest order interaction can be incorporated into those potentials. To see how this comes about, consider by contrast how one computes scattering amplitudes without quantum field theory. With no quantum field theory one must come upon potentials by classical arguments or phenomenological considerations. Once one has the potential
V
{\displaystyle V}
between two particles, then one can compute the scattering amplitude
T
{\displaystyle T}
from the Lippmann–Schwinger equation
T
+
V
+
V
G
T
=
0
,
{\displaystyle T+V+VGT=0,}
in which
G
{\displaystyle G}
is a Green function determined from the Schrödinger equation. Because of the similarity between the Schrödinger equation Eq. (11) and the relativistic constraint equation (10), one can derive the same type of equation as the above
T
+
Φ
+
Φ
G
T
=
0
,
{\displaystyle {\mathcal {T}}+\Phi +\Phi {\mathcal {G}}{\mathcal {T}}=0,}
called the quasipotential equation with a
G
{\displaystyle {\mathcal {G}}}
very similar to that given in the Lippmann–Schwinger equation. The difference is that with the quasipotential equation, one starts with the scattering amplitudes
T
{\displaystyle {\mathcal {T}}}
of quantum field theory, as determined from Feynman diagrams and deduces the quasipotential Φ perturbatively. Then one can use that Φ in (10), to compute energy levels of two particle systems that are implied by the field theory. Constraint dynamics provides one of many, in fact an infinite number of, different types of quasipotential equations (three-dimensional truncations of the Bethe–Salpeter equation) differing from one another by the choice of
G
{\displaystyle {\mathcal {G}}}
.
The relatively simple solution to the problem of relative time and energy from the generalized mass shell constraint for two particles, has no simple extension, such as presented here with the
x
⊥
{\displaystyle x_{\perp }}
variable, to either two particles in an external field or to 3 or more particles. Sazdjian has presented a recipe for this extension when the particles are confined and cannot split into clusters of a smaller number of particles with no inter-cluster interactions Lusanna has developed an approach, one that does not involve generalized mass shell constraints with no such restrictions, which extends to N bodies with or without fields. It is formulated on spacelike hypersurfaces and when restricted to the family of hyperplanes orthogonal to the total timelike momentum gives rise to a covariant intrinsic 1-time formulation (with no relative time variables) called the "rest-frame instant form" of dynamics,
== See also ==
== References == | Wikipedia/Two-body_Dirac_equations |
Zeeman energy, or the external field energy, is the potential energy of a magnetised body in an external magnetic field. It is named after the Dutch physicist Pieter Zeeman, primarily known for the Zeeman effect. In SI units, it is given by
E
Z
e
e
m
a
n
=
−
μ
0
∫
V
M
⋅
H
E
x
t
d
V
{\displaystyle E_{\rm {Zeeman}}=-\mu _{0}\int _{V}\,{\textbf {M}}\cdot {\textbf {H}}_{\rm {Ext}}\,\mathrm {d} V}
where HExt is the external field, M the local magnetisation, and the integral is done over the volume of the body. This is the statistical average (over a
unit volume macroscopic sample) of a corresponding microscopic Hamiltonial (energy) for each individual magnetic moment m, which is however experiencing a local induction B:
H
=
−
m
⋅
B
{\displaystyle H=-{\textbf {m}}\cdot {\textbf {B}}}
== References ==
F. Barozzi, F. Gasparini, Fondamenti di Elettrotecnica: Elettromagnetismo, UTET Torino, 1989
Hubert, A. and Schäfer, R. Magnetic domains: the analysis of magnetic microstructures, Springer-Verlag, 1998 | Wikipedia/Zeeman_energy |
In the history of physics, "On the quantum-theoretical reinterpretation of kinematical and mechanical relationships"
(German: Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen), also known as the Umdeutung (reinterpretation) paper, was a breakthrough article in quantum mechanics written by Werner Heisenberg, which appeared in Zeitschrift für Physik in September 1925.
In the article, Heisenberg tried to explain the energy levels of a one-dimensional anharmonic oscillator, avoiding the concrete but unobservable representations of electron orbits by using observable parameters such as transition probabilities for quantum jumps, which necessitated using two indexes corresponding to the initial and final states.: 153
Mathematically, Heisenberg showed the need of non-commutative operators. This insight would later become the basis for Heisenberg's uncertainty principle.
This article was followed by the paper by Max Born and Pascual Jordan of the same year, and by the 'three-man paper' (German: drei Männer Arbeit) by Born, Heisenberg and Jordan in 1926. These articles laid the groundwork for matrix mechanics that would come to substitute old quantum theory, leading to the modern quantum mechanics. Heisenberg received the Nobel Prize in Physics in 1932 for his work on developing quantum mechanics.
== Historical context ==
Heisenberg was 23 years old when he worked on the article while recovering from hay fever on the island of Heligoland, corresponding with Wolfgang Pauli on the subject. When asked for his opinion of the manuscript, Pauli responded favorably, but Heisenberg said that he was still "very uncertain about it". In July 1925, he sent the manuscript to Max Born to review and decide whether to submit it for publication.
When Born read the article, he recognized the formulation as one which could be transcribed and extended to the systematic language of matrices. Born, with the help of his assistant and former student Pascual Jordan, began immediately to make the transcription and extension, and they submitted their results for publication; their manuscript was received for publication just 60 days after Heisenberg’s article. A follow-on article by all three authors extending the theory to multiple dimensions was submitted for publication before the end of the year.
Heisenberg determined to base his quantum mechanics "exclusively upon relationships between quantities that in principle are observable". He observed that one could not then use any statements about such things as "the position and period of revolution of the electron". Rather, to make true progress in understanding the radiation of the simplest case, the radiation of excited hydrogen atoms, one had measurements only of the frequencies and the intensities of the hydrogen bright-line spectrum to work with.
In classical physics, the intensity of each frequency of light produced in a radiating system is equal to the square of the amplitude of the radiation at that frequency, so attention next fell on amplitudes. The classical equations that Heisenberg hoped to use to form quantum theoretical equations would first yield the amplitudes, and in classical physics one could compute the intensities simply by squaring the amplitudes. But Heisenberg saw that "the simplest and most natural assumption would be": 275f </ref> to follow the lead provided by recent work in computing light dispersion done by Hans Kramers. The work he had done assisting Kramers in the previous year: paper 3 now gave him an important clue about how to model what happened to excited hydrogen gas when it radiated light and what happened when incoming radiation of one frequency excited atoms in a dispersive medium and then the energy delivered by the incoming light was re-radiated – sometimes at the original frequency but often at two lower frequencies the sum of which equalled the original frequency. According to their model, an electron that had been driven to a higher energy state by accepting the energy of an incoming photon might return in one step to its equilibrium position, re-radiating a photon of the same frequency, or it might return in more than one step, radiating one photon for each step in its return to its equilibrium state. Because of the way factors cancel out in deriving the new equation based on these considerations, the result turns out to be relatively simple.
Also included in the manuscript was the Heisenberg commutator, his law of multiplication needed to describe certain properties of atoms, whereby the product of two physical quantities did not commute. Therefore, PQ would differ from QP where, for example, P was an electron's momentum, and Q its position. Paul Dirac, who had received a proof copy in August 1925, realized that the commutative law had not been fully developed, and he produced an algebraic formulation to express the same results in more logical form.
== Heisenberg's multiplication rule ==
By means of an intense series of mathematical analogies that some physicists have termed "magical", Heisenberg wrote out an equation that is the quantum mechanical analog for the classical computation of intensities:: 266 : 5
C
(
n
,
n
−
b
)
=
∑
a
A
(
n
,
n
−
a
)
B
(
n
−
a
,
n
−
b
)
{\displaystyle C(n,n-b)=\sum _{a}A(n,n-a)B(n-a,n-b)}
This general format indicates that some term C is to be computed by summing up all of the products of some group of terms A by some related group of terms B. There will potentially be an infinite series of A terms and their matching B terms. Each of these multiplications has as its factors two measurements that pertain to sequential downward transitions between energy states of an electron. This type of rule differentiates matrix mechanics from the kind of physics familiar in everyday life because the important values are where (in what energy state or "orbital") the electron begins and in what energy state it ends, not what the electron is doing while in one or another state.
If A and B both refer to lists of frequencies, for instance, the calculation proceeds as follows:
Multiply the frequency for a change of energy from state n to state n − a by the frequency for a change of energy from state n − a to state n − b, and to that add the product found by multiplying the frequency for a change of energy from state n − a to state n − b by the frequency for a change of energy from state n − b to state n − c, and so forth. Symbolically, that is:
f
(
n
,
n
−
a
)
f
(
n
−
a
,
n
−
b
)
+
f
(
n
−
a
,
n
−
b
)
f
(
n
−
b
,
n
−
c
)
+
⋯
{\displaystyle f(n,n-a)f(n-a,n-b)+f(n-a,n-b)f(n-b,n-c)+\cdots }
(According to the convention used, n − a represents a higher energy state than n, so a transition from n to n − a would indicate that an electron has accepted energy from an incoming photon and has risen to a higher orbital, while a transition from n − a to n would represent an electron falling to a lower orbital and emitting a photon.)
It would be easy to perform each individual step of this process for some measured quantity. For instance, the boxed formula at the head of this article gives each needed wavelength in sequence. The values calculated could very easily be filled into a grid as described below. However, since the series is infinite, nobody could do the entire set of calculations.
Heisenberg originally devised this equation to enable himself to multiply two measurements of the same kind (amplitudes), so it happened not to matter in which order they were multiplied. Heisenberg noticed, however that if he tried to use the same schema to multiply two variables, such as momentum, p, and displacement, q, then "a significant difficulty arises".: 266 It turns out that multiplying a matrix of p by a matrix of q gives a different result from multiplying a matrix of q by a matrix of p. It only made a tiny bit of difference, but that difference could never be reduced below a certain limit, and that limit involved the Planck constant, h. More on that later. Below is a very short sample of what the calculations would be, placed into grids that are called matrices. Heisenberg's teacher saw almost immediately that his work should be expressed in a matrix format because mathematicians already were familiar with how to do computations involving matrices in an efficient way. (Since Heisenberg was interested in photon radiation, the illustrations will be given in terms of electrons going from a higher energy level to a lower level, e.g., n ← n − 1, instead of going from a lower level to a higher level, e.g., n → n − 1.)
Y
(
n
,
n
−
b
)
=
∑
a
p
(
n
,
n
−
a
)
q
(
n
−
a
,
n
−
b
)
{\displaystyle Y(n,n-b)=\sum _{a}p(n,n-a)q(n-a,n-b)}
(equation for the conjugate variables momentum and position)
Matrix of p
Matrix of q
The matrix for the product of the above two matrices as specified by the relevant equation in the Umdeutung paper is
where
and so forth.
If the matrices were reversed, the following values would result
and so forth.
== Development of matrix mechanics ==
Werner Heisenberg used the idea that since classical physics is correct when it applies to phenomena in the world of things larger than atoms and molecules, it must stand as a special case of a more inclusive quantum theoretical model. So he hoped that he could modify quantum physics in such a way that when the parameters were on the scale of everyday objects it would look just like classical physics, but when the parameters were pulled down to the atomic scale the discontinuities seen in things like the widely spaced frequencies of the visible hydrogen bright line spectrum would come back into sight.
The one thing that people at that time most wanted to understand about hydrogen radiation was how to predict or account for the intensities of the lines in its spectrum. Although Heisenberg did not know it at the time, the general format he worked out to express his new way of working with quantum theoretical calculations can serve as a recipe for two matrices and how to multiply them.: Ch 12
The Umdeutung paper does not mention matrices. Heisenberg's great advance was the "scheme which was capable in principle of determining uniquely the relevant physical qualities (transition frequencies and amplitudes)": 2 of hydrogen radiation.
After Heisenberg wrote the Umdeutung paper, he turned it over to one of his senior colleagues for any needed corrections and went on vacation. Max Born puzzled over the equations and the non-commuting equations that Heisenberg had found troublesome and disturbing. After several days he realized that these equations amounted to directions for writing out matrices.
By consideration of ... examples. .. [Heisenberg] found this rule ... This was in the summer of 1925. Heisenberg ... took leave of absence ... and handed over his paper to me for publication ... Heisenberg's rule of multiplication left me no peace, and after a week of intensive thought and trial, I suddenly remembered an algebraic theory....Such quadratic arrays are quite familiar to mathematicians and are called matrices, in association with a definite rule of multiplication. I applied this rule to Heisenberg's quantum condition and found that it agreed for the diagonal elements. It was easy to guess what the remaining elements must be, namely, null; and immediately there stood before me the strange formula
Q
P
−
P
Q
=
i
h
2
π
{\displaystyle {QP-PQ={\frac {ih}{2\pi }}}}
The symbol Q is the matrix for displacement, P is the matrix for momentum, i stands for the square root of negative one, and h is the Planck constant.: A Born and a few colleagues took up the task of working everything out in matrix form before Heisenberg returned from his time off, and within a few months the new quantum mechanics in matrix form formed the basis for another paper. This relation is now known as Heisenberg's uncertainty principle.
When quantities such as position and momentum are mentioned in the context of Heisenberg's matrix mechanics, a statement such as pq ≠ qp does not refer to a single value of p and a single value q but to a matrix (grid of values arranged in a defined way) of values of position and a matrix of values of momentum. So multiplying p times q or q times p is really talking about the matrix multiplication of the two matrices. When two matrices are multiplied, the answer is a third matrix.
Paul Dirac decided that the essence of Heisenberg's work lay in the very feature that Heisenberg had originally found problematical – the fact of non-commutativity such as that between multiplication of a momentum matrix by a displacement matrix and multiplication of a displacement matrix by a momentum matrix. That insight led Dirac in new and productive directions.
== See also ==
History of quantum mechanics
Mathematical formulation of quantum mechanics
== References ==
== Further reading ==
Werner Heisenberg (1925). "Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen" (PDF). Zeitschrift für Physik (in German). 33 (1): 879–893. Bibcode:1925ZPhy...33..879H. doi:10.1007/BF01328377. S2CID 186238950.
An English translation "Quantum-theoretical reinterpretation of kinematic and mechanical relations" by B. L. van der Waerden may be found in B. L. van der Waerden, ed. (1968). Sources of Quantum Mechanics. New York: Dover. pp. 261–276. ISBN 0-486-61881-1.
Dirac, P. A. M. (1925). "The Fundamental Equations of Quantum Mechanics". Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character. 109 (752): 642–653. Bibcode:1925RSPSA.109..642D. doi:10.1098/rspa.1925.0150. ISSN 0950-1207. JSTOR 94441. The crucial reinterpretation synthesis of Heisenberg's paper, which introduces the contemporary language employed now. | Wikipedia/Heisenberg's_entryway_to_matrix_mechanics |
In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates (q, p) → (Q, P) that preserves the form of Hamilton's equations. This is sometimes known as form invariance. Although Hamilton's equations are preserved, it need not preserve the explicit form of the Hamiltonian itself. Canonical transformations are useful in their own right, and also form the basis for the Hamilton–Jacobi equations (a useful method for calculating conserved quantities) and Liouville's theorem (itself the basis for classical statistical mechanics).
Since Lagrangian mechanics is based on generalized coordinates, transformations of the coordinates q → Q do not affect the form of Lagrange's equations and, hence, do not affect the form of Hamilton's equations if the momentum is simultaneously changed by a Legendre transformation into
P
i
=
∂
L
∂
Q
˙
i
,
{\displaystyle P_{i}={\frac {\partial L}{\partial {\dot {Q}}_{i}}}\ ,}
where
{
(
P
1
,
Q
1
)
,
(
P
2
,
Q
2
)
,
(
P
3
,
Q
3
)
,
…
}
{\displaystyle \left\{\ (P_{1},Q_{1}),\ (P_{2},Q_{2}),\ (P_{3},Q_{3}),\ \ldots \ \right\}}
are the new co‑ordinates, grouped in canonical conjugate pairs of momenta
P
i
{\displaystyle P_{i}}
and corresponding positions
Q
i
,
{\displaystyle Q_{i},}
for
i
=
1
,
2
,
…
N
,
{\displaystyle i=1,2,\ldots \ N,}
with
N
{\displaystyle N}
being the number of degrees of freedom in both co‑ordinate systems.
Therefore, coordinate transformations (also called point transformations) are a type of canonical transformation. However, the class of canonical transformations is much broader, since the old generalized coordinates, momenta and even time may be combined to form the new generalized coordinates and momenta. Canonical transformations that do not include the time explicitly are called restricted canonical transformations (many textbooks consider only this type).
Modern mathematical descriptions of canonical transformations are considered under the broader topic of symplectomorphism which covers the subject with advanced mathematical prerequisites such as cotangent bundles, exterior derivatives and symplectic manifolds.
== Notation ==
Boldface variables such as q represent a list of N generalized coordinates that need not transform like a vector under rotation and similarly p represents the corresponding generalized momentum, e.g.,
q
≡
(
q
1
,
q
2
,
…
,
q
N
−
1
,
q
N
)
p
≡
(
p
1
,
p
2
,
…
,
p
N
−
1
,
p
N
)
.
{\displaystyle {\begin{aligned}\mathbf {q} &\equiv \left(q_{1},q_{2},\ldots ,q_{N-1},q_{N}\right)\\\mathbf {p} &\equiv \left(p_{1},p_{2},\ldots ,p_{N-1},p_{N}\right).\end{aligned}}}
A dot over a variable or list signifies the time derivative, e.g.,
q
˙
≡
d
q
d
t
{\displaystyle {\dot {\mathbf {q} }}\equiv {\frac {d\mathbf {q} }{dt}}}
and the equalities are read to be satisfied for all coordinates, for example:
p
˙
=
−
∂
f
∂
q
⟺
p
i
˙
=
−
∂
f
∂
q
i
(
i
=
1
,
…
,
N
)
.
{\displaystyle {\dot {\mathbf {p} }}=-{\frac {\partial f}{\partial \mathbf {q} }}\quad \Longleftrightarrow \quad {\dot {p_{i}}}=-{\frac {\partial f}{\partial {q_{i}}}}\quad (i=1,\dots ,N).}
The dot product notation between two lists of the same number of coordinates is a shorthand for the sum of the products of corresponding components, e.g.,
p
⋅
q
≡
∑
k
=
1
N
p
k
q
k
.
{\displaystyle \mathbf {p} \cdot \mathbf {q} \equiv \sum _{k=1}^{N}p_{k}q_{k}.}
The dot product (also known as an "inner product") maps the two coordinate lists into one variable representing a single numerical value. The coordinates after transformation are similarly labelled with Q for transformed generalized coordinates and P for transformed generalized momentum.
== Conditions for restricted canonical transformation ==
Restricted canonical transformations are coordinate transformations where transformed coordinates Q and P do not have explicit time dependence, i.e.,
Q
=
Q
(
q
,
p
)
{\textstyle \mathbf {Q} =\mathbf {Q} (\mathbf {q} ,\mathbf {p} )}
and
P
=
P
(
q
,
p
)
{\textstyle \mathbf {P} =\mathbf {P} (\mathbf {q} ,\mathbf {p} )}
. The functional form of Hamilton's equations is
p
˙
=
−
∂
H
∂
q
,
q
˙
=
∂
H
∂
p
{\displaystyle {\begin{aligned}{\dot {\mathbf {p} }}&=-{\frac {\partial H}{\partial \mathbf {q} }}\,,&{\dot {\mathbf {q} }}&={\frac {\partial H}{\partial \mathbf {p} }}\end{aligned}}}
In general, a transformation (q, p) → (Q, P) does not preserve the form of Hamilton's equations but in the absence of time dependence in transformation, some simplifications are possible. Following the formal definition for a canonical transformation, it can be shown that for this type of transformation, the new Hamiltonian (sometimes called the Kamiltonian) can be expressed as:
K
(
Q
,
P
,
t
)
=
H
(
q
(
Q
,
P
)
,
p
(
Q
,
P
)
,
t
)
+
∂
G
∂
t
(
t
)
{\displaystyle K(\mathbf {Q} ,\mathbf {P} ,t)=H(q(\mathbf {Q} ,\mathbf {P} ),p(\mathbf {Q} ,\mathbf {P} ),t)+{\frac {\partial G}{\partial t}}(t)}
where it differs by a partial time derivative of a function known as a generator, which reduces to being only a function of time for restricted canonical transformations.
In addition to leaving the form of the Hamiltonian unchanged, it is also permits the use of the unchanged Hamiltonian in the Hamilton's equations of motion due to the above form as:
P
˙
=
−
∂
K
∂
Q
=
−
(
∂
H
∂
Q
)
Q
,
P
,
t
Q
˙
=
∂
K
∂
P
=
(
∂
H
∂
P
)
Q
,
P
,
t
{\displaystyle {\begin{alignedat}{3}{\dot {\mathbf {P} }}&=-{\frac {\partial K}{\partial \mathbf {Q} }}&&=-\left({\frac {\partial H}{\partial \mathbf {Q} }}\right)_{\mathbf {Q} ,\mathbf {P} ,t}\\{\dot {\mathbf {Q} }}&=\,\,\,\,{\frac {\partial K}{\partial \mathbf {P} }}&&=\,\,\,\,\,\left({\frac {\partial H}{\partial \mathbf {P} }}\right)_{\mathbf {Q} ,\mathbf {P} ,t}\\\end{alignedat}}}
Although canonical transformations refers to a more general set of transformations of phase space corresponding with less permissive transformations of the Hamiltonian, it provides simpler conditions to obtain results that can be further generalized. All of the following conditions, with the exception of bilinear invariance condition, can be generalized for canonical transformations, including time dependance.
=== Indirect conditions ===
Since restricted transformations have no explicit time dependence (by definition), the time derivative of a new generalized coordinate Qm is
Q
˙
m
=
∂
Q
m
∂
q
⋅
q
˙
+
∂
Q
m
∂
p
⋅
p
˙
=
∂
Q
m
∂
q
⋅
∂
H
∂
p
−
∂
Q
m
∂
p
⋅
∂
H
∂
q
=
{
Q
m
,
H
}
{\displaystyle {\begin{aligned}{\dot {Q}}_{m}&={\frac {\partial Q_{m}}{\partial \mathbf {q} }}\cdot {\dot {\mathbf {q} }}+{\frac {\partial Q_{m}}{\partial \mathbf {p} }}\cdot {\dot {\mathbf {p} }}\\&={\frac {\partial Q_{m}}{\partial \mathbf {q} }}\cdot {\frac {\partial H}{\partial \mathbf {p} }}-{\frac {\partial Q_{m}}{\partial \mathbf {p} }}\cdot {\frac {\partial H}{\partial \mathbf {q} }}\\&=\lbrace Q_{m},H\rbrace \end{aligned}}}
where {⋅, ⋅} is the Poisson bracket.
Similarly for the identity for the conjugate momentum, Pm using the form of the "Kamiltonian" it follows that:
∂
K
(
Q
,
P
,
t
)
∂
P
m
=
∂
K
(
Q
(
q
,
p
)
,
P
(
q
,
p
)
,
t
)
∂
q
⋅
∂
q
∂
P
m
+
∂
K
(
Q
(
q
,
p
)
,
P
(
q
,
p
)
,
t
)
∂
p
⋅
∂
p
∂
P
m
=
∂
H
(
q
,
p
,
t
)
∂
q
⋅
∂
q
∂
P
m
+
∂
H
(
q
,
p
,
t
)
∂
p
⋅
∂
p
∂
P
m
=
∂
H
∂
q
⋅
∂
q
∂
P
m
+
∂
H
∂
p
⋅
∂
p
∂
P
m
{\displaystyle {\begin{aligned}{\frac {\partial K(\mathbf {Q} ,\mathbf {P} ,t)}{\partial P_{m}}}&={\frac {\partial K(\mathbf {Q} (\mathbf {q} ,\mathbf {p} ),\mathbf {P} (\mathbf {q} ,\mathbf {p} ),t)}{\partial \mathbf {q} }}\cdot {\frac {\partial \mathbf {q} }{\partial P_{m}}}+{\frac {\partial K(\mathbf {Q} (\mathbf {q} ,\mathbf {p} ),\mathbf {P} (\mathbf {q} ,\mathbf {p} ),t)}{\partial \mathbf {p} }}\cdot {\frac {\partial \mathbf {p} }{\partial P_{m}}}\\[1ex]&={\frac {\partial H(\mathbf {q} ,\mathbf {p} ,t)}{\partial \mathbf {q} }}\cdot {\frac {\partial \mathbf {q} }{\partial P_{m}}}+{\frac {\partial H(\mathbf {q} ,\mathbf {p} ,t)}{\partial \mathbf {p} }}\cdot {\frac {\partial \mathbf {p} }{\partial P_{m}}}\\[1ex]&={\frac {\partial H}{\partial \mathbf {q} }}\cdot {\frac {\partial \mathbf {q} }{\partial P_{m}}}+{\frac {\partial H}{\partial \mathbf {p} }}\cdot {\frac {\partial \mathbf {p} }{\partial P_{m}}}\end{aligned}}}
Due to the form of the Hamiltonian equations of motion,
P
˙
=
−
∂
K
∂
Q
Q
˙
=
∂
K
∂
P
{\displaystyle {\begin{aligned}{\dot {\mathbf {P} }}&=-{\frac {\partial K}{\partial \mathbf {Q} }}\\{\dot {\mathbf {Q} }}&=\,\,\,\,{\frac {\partial K}{\partial \mathbf {P} }}\end{aligned}}}
if the transformation is canonical, the two derived results must be equal, resulting in the equations:
(
∂
Q
m
∂
p
n
)
q
,
p
=
−
(
∂
q
n
∂
P
m
)
Q
,
P
(
∂
Q
m
∂
q
n
)
q
,
p
=
(
∂
p
n
∂
P
m
)
Q
,
P
{\displaystyle {\begin{aligned}\left({\frac {\partial Q_{m}}{\partial p_{n}}}\right)_{\mathbf {q} ,\mathbf {p} }&=-\left({\frac {\partial q_{n}}{\partial P_{m}}}\right)_{\mathbf {Q} ,\mathbf {P} }\\\left({\frac {\partial Q_{m}}{\partial q_{n}}}\right)_{\mathbf {q} ,\mathbf {p} }&=\left({\frac {\partial p_{n}}{\partial P_{m}}}\right)_{\mathbf {Q} ,\mathbf {P} }\end{aligned}}}
The analogous argument for the generalized momenta Pm leads to two other sets of equations:
(
∂
P
m
∂
p
n
)
q
,
p
=
(
∂
q
n
∂
Q
m
)
Q
,
P
(
∂
P
m
∂
q
n
)
q
,
p
=
−
(
∂
p
n
∂
Q
m
)
Q
,
P
{\displaystyle {\begin{aligned}\left({\frac {\partial P_{m}}{\partial p_{n}}}\right)_{\mathbf {q} ,\mathbf {p} }&=\left({\frac {\partial q_{n}}{\partial Q_{m}}}\right)_{\mathbf {Q} ,\mathbf {P} }\\\left({\frac {\partial P_{m}}{\partial q_{n}}}\right)_{\mathbf {q} ,\mathbf {p} }&=-\left({\frac {\partial p_{n}}{\partial Q_{m}}}\right)_{\mathbf {Q} ,\mathbf {P} }\end{aligned}}}
These are the indirect conditions to check whether a given transformation is canonical.
=== Symplectic condition ===
Sometimes the Hamiltonian relations are represented as:
η
˙
=
J
∇
η
H
{\displaystyle {\dot {\eta }}=J\nabla _{\eta }H}
Where
J
:=
(
0
I
n
−
I
n
0
)
,
{\textstyle J:={\begin{pmatrix}0&I_{n}\\-I_{n}&0\\\end{pmatrix}},}
and
η
=
[
q
1
⋮
q
n
p
1
⋮
p
n
]
{\textstyle \mathbf {\eta } ={\begin{bmatrix}q_{1}\\\vdots \\q_{n}\\p_{1}\\\vdots \\p_{n}\\\end{bmatrix}}}
. Similarly, let
ε
=
[
Q
1
⋮
Q
n
P
1
⋮
P
n
]
{\textstyle \mathbf {\varepsilon } ={\begin{bmatrix}Q_{1}\\\vdots \\Q_{n}\\P_{1}\\\vdots \\P_{n}\\\end{bmatrix}}}
.
From the relation of partial derivatives, converting the
η
˙
=
J
∇
η
H
{\displaystyle {\dot {\eta }}=J\nabla _{\eta }H}
relation in terms of partial derivatives with new variables gives
η
˙
=
J
(
M
T
∇
ε
H
)
{\displaystyle {\dot {\eta }}=J(M^{T}\nabla _{\varepsilon }H)}
where
M
:=
∂
(
Q
,
P
)
∂
(
q
,
p
)
{\textstyle M:={\frac {\partial (\mathbf {Q} ,\mathbf {P} )}{\partial (\mathbf {q} ,\mathbf {p} )}}}
. Similarly for
ε
˙
{\textstyle {\dot {\varepsilon }}}
,
ε
˙
=
M
η
˙
=
M
J
M
T
∇
ε
H
{\displaystyle {\dot {\varepsilon }}=M{\dot {\eta }}=MJM^{T}\nabla _{\varepsilon }H}
Due to form of the Hamiltonian equations for
ε
˙
{\textstyle {\dot {\varepsilon }}}
,
ε
˙
=
J
∇
ε
K
=
J
∇
ε
H
{\displaystyle {\dot {\varepsilon }}=J\nabla _{\varepsilon }K=J\nabla _{\varepsilon }H}
where
∇
ε
K
=
∇
ε
H
{\textstyle \nabla _{\varepsilon }K=\nabla _{\varepsilon }H}
can be used due to the form of Kamiltonian. Equating the two equations gives the symplectic condition as:
M
J
M
T
=
J
{\displaystyle MJM^{T}=J}
The left hand side of the above is called the Poisson matrix of
ε
{\displaystyle \varepsilon }
, denoted as
P
(
ε
)
=
M
J
M
T
{\textstyle {\mathcal {P}}(\varepsilon )=MJM^{T}}
. Similarly, a Lagrange matrix of
η
{\displaystyle \eta }
can be constructed as
L
(
η
)
=
M
T
J
M
{\textstyle {\mathcal {L}}(\eta )=M^{T}JM}
. It can be shown that the symplectic condition is also equivalent to
M
T
J
M
=
J
{\textstyle M^{T}JM=J}
by using the
J
−
1
=
−
J
{\textstyle J^{-1}=-J}
property. The set of all matrices
M
{\textstyle M}
which satisfy symplectic conditions form a symplectic group. The symplectic conditions are equivalent with indirect conditions as they both lead to the equation
ε
˙
=
J
∇
ε
H
{\textstyle {\dot {\varepsilon }}=J\nabla _{\varepsilon }H}
, which is used in both of the derivations.
=== Invariance of the Poisson bracket ===
The Poisson bracket which is defined as:
{
u
,
v
}
η
:=
∑
i
=
1
n
(
∂
u
∂
q
i
∂
v
∂
p
i
−
∂
u
∂
p
i
∂
v
∂
q
i
)
{\displaystyle \{u,v\}_{\eta }:=\sum _{i=1}^{n}\left({\frac {\partial u}{\partial q_{i}}}{\frac {\partial v}{\partial p_{i}}}-{\frac {\partial u}{\partial p_{i}}}{\frac {\partial v}{\partial q_{i}}}\right)}
can be represented in matrix form as:
{
u
,
v
}
η
:=
(
∇
η
u
)
T
J
(
∇
η
v
)
{\displaystyle \{u,v\}_{\eta }:=(\nabla _{\eta }u)^{T}J(\nabla _{\eta }v)}
Hence using partial derivative relations and symplectic condition gives:
{
u
,
v
}
η
=
(
∇
η
u
)
T
J
(
∇
η
v
)
=
(
M
T
∇
ε
u
)
T
J
(
M
T
∇
ε
v
)
=
(
∇
ε
u
)
T
M
J
M
T
(
∇
ε
v
)
=
(
∇
ε
u
)
T
J
(
∇
ε
v
)
=
{
u
,
v
}
ε
{\displaystyle \{u,v\}_{\eta }=(\nabla _{\eta }u)^{T}J(\nabla _{\eta }v)=(M^{T}\nabla _{\varepsilon }u)^{T}J(M^{T}\nabla _{\varepsilon }v)=(\nabla _{\varepsilon }u)^{T}MJM^{T}(\nabla _{\varepsilon }v)=(\nabla _{\varepsilon }u)^{T}J(\nabla _{\varepsilon }v)=\{u,v\}_{\varepsilon }}
The symplectic condition can also be recovered by taking
u
=
ε
i
{\textstyle u=\varepsilon _{i}}
and
v
=
ε
j
{\textstyle v=\varepsilon _{j}}
which shows that
(
M
J
M
T
)
i
j
=
J
i
j
{\textstyle (MJM^{T})_{ij}=J_{ij}}
. Thus these conditions are equivalent to symplectic conditions. Furthermore, it can be seen that
P
i
j
(
ε
)
=
{
ε
i
,
ε
j
}
η
=
(
M
J
M
T
)
i
j
{\textstyle {\mathcal {P}}_{ij}(\varepsilon )=\{\varepsilon _{i},\varepsilon _{j}\}_{\eta }=(MJM^{T})_{ij}}
, which is also the result of explicitly calculating the matrix element by expanding it.
=== Invariance of the Lagrange bracket ===
The Lagrange bracket which is defined as:
[
u
,
v
]
η
:=
∑
i
=
1
n
(
∂
q
i
∂
u
∂
p
i
∂
v
−
∂
p
i
∂
u
∂
q
i
∂
v
)
{\displaystyle [u,v]_{\eta }:=\sum _{i=1}^{n}\left({\frac {\partial q_{i}}{\partial u}}{\frac {\partial p_{i}}{\partial v}}-{\frac {\partial p_{i}}{\partial u}}{\frac {\partial q_{i}}{\partial v}}\right)}
can be represented in matrix form as:
[
u
,
v
]
η
:=
(
∂
η
∂
u
)
T
J
(
∂
η
∂
v
)
{\displaystyle [u,v]_{\eta }:=\left({\frac {\partial \eta }{\partial u}}\right)^{T}J\left({\frac {\partial \eta }{\partial v}}\right)}
Using similar derivation, gives:
[
u
,
v
]
ε
=
(
∂
u
ε
)
T
J
(
∂
v
ε
)
=
(
M
∂
u
η
)
T
J
(
M
∂
v
η
)
=
(
∂
u
η
)
T
M
T
J
M
(
∂
v
η
)
=
(
∂
u
η
)
T
J
(
∂
v
η
)
=
[
u
,
v
]
η
{\displaystyle [u,v]_{\varepsilon }=(\partial _{u}\varepsilon )^{T}\,J\,(\partial _{v}\varepsilon )=(M\,\partial _{u}\eta )^{T}\,J\,(M\,\partial _{v}\eta )=(\partial _{u}\eta )^{T}\,M^{T}JM\,(\partial _{v}\eta )=(\partial _{u}\eta )^{T}\,J\,(\partial _{v}\eta )=[u,v]_{\eta }}
The symplectic condition can also be recovered by taking
u
=
η
i
{\textstyle u=\eta _{i}}
and
v
=
η
j
{\textstyle v=\eta _{j}}
which shows that
(
M
T
J
M
)
i
j
=
J
i
j
{\textstyle (M^{T}JM)_{ij}=J_{ij}}
. Thus these conditions are equivalent to symplectic conditions. Furthermore, it can be seen that
L
i
j
(
η
)
=
[
η
i
,
η
j
]
ε
=
(
M
T
J
M
)
i
j
{\textstyle {\mathcal {L}}_{ij}(\eta )=[\eta _{i},\eta _{j}]_{\varepsilon }=(M^{T}JM)_{ij}}
, which is also the result of explicitly calculating the matrix element by expanding it.
=== Bilinear invariance conditions ===
These set of conditions only apply to restricted canonical transformations or canonical transformations that are independent of time variable.
Consider arbitrary variations of two kinds, in a single pair of generalized coordinate and the corresponding momentum:
d
ε
=
(
d
q
1
,
d
p
1
,
0
,
0
,
…
)
,
δ
ε
=
(
δ
q
1
,
δ
p
1
,
0
,
0
,
…
)
.
{\textstyle d\varepsilon =(dq_{1},dp_{1},0,0,\ldots ),\quad \delta \varepsilon =(\delta q_{1},\delta p_{1},0,0,\ldots ).}
The area of the infinitesimal parallelogram is given by:
δ
a
(
12
)
=
d
q
1
δ
p
1
−
δ
q
1
d
p
1
=
(
δ
ε
)
T
J
d
ε
.
{\textstyle \delta a(12)=dq_{1}\delta p_{1}-\delta q_{1}dp_{1}={(\delta \varepsilon )}^{T}\,J\,d\varepsilon .}
It follows from the
M
T
J
M
=
J
{\textstyle M^{T}JM=J}
symplectic condition that the infinitesimal area is conserved under canonical transformation:
δ
a
(
12
)
=
(
δ
ε
)
T
J
d
ε
=
(
M
δ
η
)
T
J
M
d
η
=
(
δ
η
)
T
M
T
J
M
d
η
=
(
δ
η
)
T
J
d
η
=
δ
A
(
12
)
.
{\textstyle \delta a(12)={(\delta \varepsilon )}^{T}\,J\,d\varepsilon ={(M\delta \eta )}^{T}\,J\,Md\eta ={(\delta \eta )}^{T}\,M^{T}JM\,d\eta ={(\delta \eta )}^{T}\,J\,d\eta =\delta A(12).}
Note that the new coordinates need not be completely oriented in one coordinate momentum plane.
Hence, the condition is more generally stated as an invariance of the form
(
d
ε
)
T
J
δ
ε
{\textstyle {(d\varepsilon )}^{T}\,J\,\delta \varepsilon }
under canonical transformation, expanded as:
∑
δ
q
⋅
d
p
−
δ
p
⋅
d
q
=
∑
δ
Q
⋅
d
P
−
δ
P
⋅
d
Q
{\displaystyle \sum \delta q\cdot dp-\delta p\cdot dq=\sum \delta Q\cdot dP-\delta P\cdot dQ}
If the above is obeyed for any arbitrary variations, it would be only possible if the indirect conditions are met.
The form of the equation,
v
T
J
w
{\textstyle {v}^{T}\,J\,w}
is also known as a symplectic product of the vectors
v
{\textstyle {v}}
and
w
{\textstyle w}
and the bilinear invariance condition can be stated as a local conservation of the symplectic product.
== Liouville's theorem ==
The indirect conditions allow us to prove Liouville's theorem, which states that the volume in phase space is conserved under canonical transformations, i.e.,
∫
d
q
d
p
=
∫
d
Q
d
P
{\displaystyle \int \mathrm {d} \mathbf {q} \,\mathrm {d} \mathbf {p} =\int \mathrm {d} \mathbf {Q} \,\mathrm {d} \mathbf {P} }
By calculus, the latter integral must equal the former times the determinant of Jacobian M
∫
d
Q
d
P
=
∫
det
(
M
)
d
q
d
p
{\displaystyle \int \mathrm {d} \mathbf {Q} \,\mathrm {d} \mathbf {P} =\int \det(M)\,\mathrm {d} \mathbf {q} \,\mathrm {d} \mathbf {p} }
Where
M
:=
∂
(
Q
,
P
)
∂
(
q
,
p
)
{\textstyle M:={\frac {\partial (\mathbf {Q} ,\mathbf {P} )}{\partial (\mathbf {q} ,\mathbf {p} )}}}
Exploiting the "division" property of Jacobians yields
M
≡
∂
(
Q
,
P
)
∂
(
q
,
P
)
/
∂
(
q
,
p
)
∂
(
q
,
P
)
{\displaystyle M\equiv {\frac {\partial (\mathbf {Q} ,\mathbf {P} )}{\partial (\mathbf {q} ,\mathbf {P} )}}\left/{\frac {\partial (\mathbf {q} ,\mathbf {p} )}{\partial (\mathbf {q} ,\mathbf {P} )}}\right.}
Eliminating the repeated variables gives
M
≡
∂
(
Q
)
∂
(
q
)
/
∂
(
p
)
∂
(
P
)
{\displaystyle M\equiv {\frac {\partial (\mathbf {Q} )}{\partial (\mathbf {q} )}}\left/{\frac {\partial (\mathbf {p} )}{\partial (\mathbf {P} )}}\right.}
Application of the indirect conditions above yields
det
(
M
)
=
1
{\displaystyle \operatorname {det} (M)=1}
.
== Generating function approach ==
To guarantee a valid transformation between (q, p, H) and (Q, P, K), we may resort to a direct generating function approach. Both sets of variables must obey Hamilton's principle. That is the action integral over the Lagrangians
L
q
p
=
p
⋅
q
˙
−
H
(
q
,
p
,
t
)
{\displaystyle {\mathcal {L}}_{qp}=\mathbf {p} \cdot {\dot {\mathbf {q} }}-H(\mathbf {q} ,\mathbf {p} ,t)}
and
L
Q
P
=
P
⋅
Q
˙
−
K
(
Q
,
P
,
t
)
{\displaystyle {\mathcal {L}}_{QP}=\mathbf {P} \cdot {\dot {\mathbf {Q} }}-K(\mathbf {Q} ,\mathbf {P} ,t)}
, obtained from the respective Hamiltonian via an "inverse" Legendre transformation, must be stationary in both cases (so that one can use the Euler–Lagrange equations to arrive at Hamiltonian equations of motion of the designated form; as it is shown for example here):
δ
∫
t
1
t
2
[
p
⋅
q
˙
−
H
(
q
,
p
,
t
)
]
d
t
=
0
δ
∫
t
1
t
2
[
P
⋅
Q
˙
−
K
(
Q
,
P
,
t
)
]
d
t
=
0
{\displaystyle {\begin{aligned}\delta \int _{t_{1}}^{t_{2}}\left[\mathbf {p} \cdot {\dot {\mathbf {q} }}-H(\mathbf {q} ,\mathbf {p} ,t)\right]dt&=0\\\delta \int _{t_{1}}^{t_{2}}\left[\mathbf {P} \cdot {\dot {\mathbf {Q} }}-K(\mathbf {Q} ,\mathbf {P} ,t)\right]dt&=0\end{aligned}}}
One way for both variational integral equalities to be satisfied is to have
λ
[
p
⋅
q
˙
−
H
(
q
,
p
,
t
)
]
=
P
⋅
Q
˙
−
K
(
Q
,
P
,
t
)
+
d
G
d
t
{\displaystyle \lambda \left[\mathbf {p} \cdot {\dot {\mathbf {q} }}-H(\mathbf {q} ,\mathbf {p} ,t)\right]=\mathbf {P} \cdot {\dot {\mathbf {Q} }}-K(\mathbf {Q} ,\mathbf {P} ,t)+{\frac {dG}{dt}}}
Lagrangians are not unique: one can always multiply by a constant λ and add a total time derivative dG/dt and yield the same equations of motion (as discussed on Wikibooks). In general, the scaling factor λ is set equal to one; canonical transformations for which λ ≠ 1 are called extended canonical transformations. dG/dt is kept, otherwise the problem would be rendered trivial and there would be not much freedom for the new canonical variables to differ from the old ones.
Here G is a generating function of one old canonical coordinate (q or p), one new canonical coordinate (Q or P) and (possibly) the time t. Thus, there are four basic types of generating functions (although mixtures of these four types can exist), depending on the choice of variables. As will be shown below, the generating function will define a transformation from old to new canonical coordinates, and any such transformation (q, p) → (Q, P) is guaranteed to be canonical.
The various generating functions and its properties tabulated below is discussed in detail:
=== Type 1 generating function ===
The type 1 generating function G1 depends only on the old and new generalized coordinates
G
≡
G
1
(
q
,
Q
,
t
)
{\textstyle G\equiv G_{1}(\mathbf {q} ,\mathbf {Q} ,t)}
. To derive the implicit transformation, we expand the defining equation above
p
⋅
q
˙
−
H
(
q
,
p
,
t
)
=
P
⋅
Q
˙
−
K
(
Q
,
P
,
t
)
+
∂
G
1
∂
t
+
∂
G
1
∂
q
⋅
q
˙
+
∂
G
1
∂
Q
⋅
Q
˙
{\displaystyle \mathbf {p} \cdot {\dot {\mathbf {q} }}-H(\mathbf {q} ,\mathbf {p} ,t)=\mathbf {P} \cdot {\dot {\mathbf {Q} }}-K(\mathbf {Q} ,\mathbf {P} ,t)+{\frac {\partial G_{1}}{\partial t}}+{\frac {\partial G_{1}}{\partial \mathbf {q} }}\cdot {\dot {\mathbf {q} }}+{\frac {\partial G_{1}}{\partial \mathbf {Q} }}\cdot {\dot {\mathbf {Q} }}}
Since the new and old coordinates are each independent, the following 2N + 1 equations must hold
p
=
∂
G
1
∂
q
P
=
−
∂
G
1
∂
Q
K
=
H
+
∂
G
1
∂
t
{\displaystyle {\begin{aligned}\mathbf {p} &={\frac {\partial G_{1}}{\partial \mathbf {q} }}\\\mathbf {P} &=-{\frac {\partial G_{1}}{\partial \mathbf {Q} }}\\K&=H+{\frac {\partial G_{1}}{\partial t}}\end{aligned}}}
These equations define the transformation (q, p) → (Q, P) as follows: The first set of N equations
p
=
∂
G
1
∂
q
{\textstyle \ \mathbf {p} ={\frac {\ \partial G_{1}\ }{\partial \mathbf {q} }}\ }
define relations between the new generalized coordinates Q and the old canonical coordinates (q, p). Ideally, one can invert these relations to obtain formulae for each Qk as a function of the old canonical coordinates. Substitution of these formulae for the Q coordinates into the second set of N equations
P
=
−
∂
G
1
∂
Q
{\textstyle \mathbf {P} =-{\frac {\partial G_{1}}{\partial \mathbf {Q} }}}
yields analogous formulae for the new generalized momenta P in terms of the old canonical coordinates (q, p). We then invert both sets of formulae to obtain the old canonical coordinates (q, p) as functions of the new canonical coordinates (Q, P). Substitution of the inverted formulae into the final equation
K
=
H
+
∂
G
1
∂
t
{\textstyle K=H+{\frac {\partial G_{1}}{\partial t}}}
yields a formula for K as a function of the new canonical coordinates (Q, P).
In practice, this procedure is easier than it sounds, because the generating function is usually simple. For example, let
G
1
≡
q
⋅
Q
{\textstyle G_{1}\equiv \mathbf {q} \cdot \mathbf {Q} }
. This results in swapping the generalized coordinates for the momenta and vice versa
p
=
∂
G
1
∂
q
=
Q
P
=
−
∂
G
1
∂
Q
=
−
q
{\displaystyle {\begin{aligned}\mathbf {p} &={\frac {\partial G_{1}}{\partial \mathbf {q} }}=\mathbf {Q} \\\mathbf {P} &=-{\frac {\partial G_{1}}{\partial \mathbf {Q} }}=-\mathbf {q} \end{aligned}}}
and K = H. This example illustrates how independent the coordinates and momenta are in the Hamiltonian formulation; they are equivalent variables.
=== Type 2 generating function ===
The type 2 generating function
G
2
(
q
,
P
,
t
)
{\displaystyle G_{2}(\mathbf {q} ,\mathbf {P} ,t)}
depends only on the old generalized coordinates and the new generalized momenta
G
≡
G
2
(
q
,
P
,
t
)
−
Q
⋅
P
{\textstyle G\equiv G_{2}(\mathbf {q} ,\mathbf {P} ,t)-\mathbf {Q} \cdot \mathbf {P} }
where the
−
Q
⋅
P
{\displaystyle -\mathbf {Q} \cdot \mathbf {P} }
terms represent a Legendre transformation to change the right-hand side of the equation below. To derive the implicit transformation, we expand the defining equation above
p
⋅
q
˙
−
H
(
q
,
p
,
t
)
=
−
Q
⋅
P
˙
−
K
(
Q
,
P
,
t
)
+
∂
G
2
∂
t
+
∂
G
2
∂
q
⋅
q
˙
+
∂
G
2
∂
P
⋅
P
˙
{\displaystyle \mathbf {p} \cdot {\dot {\mathbf {q} }}-H(\mathbf {q} ,\mathbf {p} ,t)=-\mathbf {Q} \cdot {\dot {\mathbf {P} }}-K(\mathbf {Q} ,\mathbf {P} ,t)+{\frac {\partial G_{2}}{\partial t}}+{\frac {\partial G_{2}}{\partial \mathbf {q} }}\cdot {\dot {\mathbf {q} }}+{\frac {\partial G_{2}}{\partial \mathbf {P} }}\cdot {\dot {\mathbf {P} }}}
Since the old coordinates and new momenta are each independent, the following 2N + 1 equations must hold
p
=
∂
G
2
∂
q
Q
=
∂
G
2
∂
P
K
=
H
+
∂
G
2
∂
t
{\displaystyle {\begin{aligned}\mathbf {p} &={\frac {\partial G_{2}}{\partial \mathbf {q} }}\\\mathbf {Q} &={\frac {\partial G_{2}}{\partial \mathbf {P} }}\\K&=H+{\frac {\partial G_{2}}{\partial t}}\end{aligned}}}
These equations define the transformation (q, p) → (Q, P) as follows: The first set of N equations
p
=
∂
G
2
∂
q
{\textstyle \mathbf {p} ={\frac {\partial G_{2}}{\partial \mathbf {q} }}}
define relations between the new generalized momenta P and the old canonical coordinates (q, p). Ideally, one can invert these relations to obtain formulae for each Pk as a function of the old canonical coordinates. Substitution of these formulae for the P coordinates into the second set of N equations
Q
=
∂
G
2
∂
P
{\textstyle \mathbf {Q} ={\frac {\partial G_{2}}{\partial \mathbf {P} }}}
yields analogous formulae for the new generalized coordinates Q in terms of the old canonical coordinates (q, p). We then invert both sets of formulae to obtain the old canonical coordinates (q, p) as functions of the new canonical coordinates (Q, P). Substitution of the inverted formulae into the final equation
K
=
H
+
∂
G
2
∂
t
{\textstyle K=H+{\frac {\partial G_{2}}{\partial t}}}
yields a formula for K as a function of the new canonical coordinates (Q, P).
In practice, this procedure is easier than it sounds, because the generating function is usually simple. For example, let
G
2
≡
g
(
q
;
t
)
⋅
P
{\textstyle G_{2}\equiv \mathbf {g} (\mathbf {q} ;t)\cdot \mathbf {P} }
where g is a set of N functions. This results in a point transformation of the generalized coordinates
Q
=
∂
G
2
∂
P
=
g
(
q
;
t
)
{\textstyle \mathbf {Q} ={\frac {\partial G_{2}}{\partial \mathbf {P} }}=\mathbf {g} (\mathbf {q} ;t)}
.
=== Type 3 generating function ===
The type 3 generating function
G
3
(
p
,
Q
,
t
)
{\displaystyle G_{3}(\mathbf {p} ,\mathbf {Q} ,t)}
depends only on the old generalized momenta and the new generalized coordinates
G
≡
G
3
(
p
,
Q
,
t
)
+
q
⋅
p
{\textstyle G\equiv G_{3}(\mathbf {p} ,\mathbf {Q} ,t)+\mathbf {q} \cdot \mathbf {p} }
where the
q
⋅
p
{\displaystyle \mathbf {q} \cdot \mathbf {p} }
terms represent a Legendre transformation to change the left-hand side of the equation below. To derive the implicit transformation, we expand the defining equation above
−
q
⋅
p
˙
−
H
(
q
,
p
,
t
)
=
P
⋅
Q
˙
−
K
(
Q
,
P
,
t
)
+
∂
G
3
∂
t
+
∂
G
3
∂
p
⋅
p
˙
+
∂
G
3
∂
Q
⋅
Q
˙
{\displaystyle -\mathbf {q} \cdot {\dot {\mathbf {p} }}-H(\mathbf {q} ,\mathbf {p} ,t)=\mathbf {P} \cdot {\dot {\mathbf {Q} }}-K(\mathbf {Q} ,\mathbf {P} ,t)+{\frac {\partial G_{3}}{\partial t}}+{\frac {\partial G_{3}}{\partial \mathbf {p} }}\cdot {\dot {\mathbf {p} }}+{\frac {\partial G_{3}}{\partial \mathbf {Q} }}\cdot {\dot {\mathbf {Q} }}}
Since the new and old coordinates are each independent, the following 2N + 1 equations must hold
q
=
−
∂
G
3
∂
p
P
=
−
∂
G
3
∂
Q
K
=
H
+
∂
G
3
∂
t
{\displaystyle {\begin{aligned}\mathbf {q} &=-{\frac {\partial G_{3}}{\partial \mathbf {p} }}\\\mathbf {P} &=-{\frac {\partial G_{3}}{\partial \mathbf {Q} }}\\K&=H+{\frac {\partial G_{3}}{\partial t}}\end{aligned}}}
These equations define the transformation (q, p) → (Q, P) as follows: The first set of N equations
q
=
−
∂
G
3
∂
p
{\textstyle \mathbf {q} =-{\frac {\partial G_{3}}{\partial \mathbf {p} }}}
define relations between the new generalized coordinates Q and the old canonical coordinates (q, p). Ideally, one can invert these relations to obtain formulae for each Qk as a function of the old canonical coordinates. Substitution of these formulae for the Q coordinates into the second set of N equations
P
=
−
∂
G
3
∂
Q
{\textstyle \mathbf {P} =-{\frac {\partial G_{3}}{\partial \mathbf {Q} }}}
yields analogous formulae for the new generalized momenta P in terms of the old canonical coordinates (q, p). We then invert both sets of formulae to obtain the old canonical coordinates (q, p) as functions of the new canonical coordinates (Q, P). Substitution of the inverted formulae into the final equation
K
=
H
+
∂
G
3
∂
t
{\textstyle K=H+{\frac {\partial G_{3}}{\partial t}}}
yields a formula for K as a function of the new canonical coordinates (Q, P).
In practice, this procedure is easier than it sounds, because the generating function is usually simple.
=== Type 4 generating function ===
The type 4 generating function
G
4
(
p
,
P
,
t
)
{\displaystyle G_{4}(\mathbf {p} ,\mathbf {P} ,t)}
depends only on the old and new generalized momenta
G
≡
G
4
(
p
,
P
,
t
)
+
q
⋅
p
−
Q
⋅
P
{\textstyle G\equiv G_{4}(\mathbf {p} ,\mathbf {P} ,t)+\mathbf {q} \cdot \mathbf {p} -\mathbf {Q} \cdot \mathbf {P} }
where the
q
⋅
p
−
Q
⋅
P
{\displaystyle \mathbf {q} \cdot \mathbf {p} -\mathbf {Q} \cdot \mathbf {P} }
terms represent a Legendre transformation to change both sides of the equation below. To derive the implicit transformation, we expand the defining equation above
−
q
⋅
p
˙
−
H
(
q
,
p
,
t
)
=
−
Q
⋅
P
˙
−
K
(
Q
,
P
,
t
)
+
∂
G
4
∂
t
+
∂
G
4
∂
p
⋅
p
˙
+
∂
G
4
∂
P
⋅
P
˙
{\displaystyle -\mathbf {q} \cdot {\dot {\mathbf {p} }}-H(\mathbf {q} ,\mathbf {p} ,t)=-\mathbf {Q} \cdot {\dot {\mathbf {P} }}-K(\mathbf {Q} ,\mathbf {P} ,t)+{\frac {\partial G_{4}}{\partial t}}+{\frac {\partial G_{4}}{\partial \mathbf {p} }}\cdot {\dot {\mathbf {p} }}+{\frac {\partial G_{4}}{\partial \mathbf {P} }}\cdot {\dot {\mathbf {P} }}}
Since the new and old coordinates are each independent, the following 2N + 1 equations must hold
q
=
−
∂
G
4
∂
p
Q
=
∂
G
4
∂
P
K
=
H
+
∂
G
4
∂
t
{\displaystyle {\begin{aligned}\mathbf {q} &=-{\frac {\partial G_{4}}{\partial \mathbf {p} }}\\\mathbf {Q} &={\frac {\partial G_{4}}{\partial \mathbf {P} }}\\K&=H+{\frac {\partial G_{4}}{\partial t}}\end{aligned}}}
These equations define the transformation (q, p) → (Q, P) as follows: The first set of N equations
q
=
−
∂
G
4
∂
p
{\textstyle \mathbf {q} =-{\frac {\partial G_{4}}{\partial \mathbf {p} }}}
define relations between the new generalized momenta P and the old canonical coordinates (q, p). Ideally, one can invert these relations to obtain formulae for each Pk as a function of the old canonical coordinates. Substitution of these formulae for the P coordinates into the second set of N equations
Q
=
∂
G
4
∂
P
{\textstyle \mathbf {Q} ={\frac {\partial G_{4}}{\partial \mathbf {P} }}}
yields analogous formulae for the new generalized coordinates Q in terms of the old canonical coordinates (q, p). We then invert both sets of formulae to obtain the old canonical coordinates (q, p) as functions of the new canonical coordinates (Q, P). Substitution of the inverted formulae into the final equation
K
=
H
+
∂
G
4
∂
t
{\textstyle K=H+{\frac {\partial G_{4}}{\partial t}}}
yields a formula for K as a function of the new canonical coordinates (Q, P).
=== Limitations on the four types of generating functions ===
Considering
G
2
(
q
,
P
,
t
)
{\displaystyle G_{2}(\mathbf {q} ,\mathbf {P} ,t)}
as an example, using generating function of second kind:
p
i
=
∂
G
2
∂
q
i
{\textstyle {p}_{i}={\frac {\partial G_{2}}{\partial {q}_{i}}}}
and
Q
i
=
∂
G
2
∂
P
i
{\textstyle {Q}_{i}={\frac {\partial G_{2}}{\partial {P}_{i}}}}
, the first set of equations consisting of variables
p
{\textstyle \mathbf {p} }
,
q
{\textstyle \mathbf {q} }
and
P
{\textstyle \mathbf {P} }
has to be inverted to get
P
(
q
,
p
)
{\textstyle \mathbf {P} (\mathbf {q} ,\mathbf {p} )}
. This process is possible when the matrix defined by
a
i
j
=
∂
p
i
(
q
,
P
)
∂
P
j
{\textstyle a_{ij}={\frac {\partial {p}_{i}(\mathbf {q} ,\mathbf {P} )}{\partial P_{j}}}}
is non-singular using the inverse function theorem, and can be restated as the following relation.
|
∂
2
G
2
∂
P
1
∂
q
1
⋯
∂
2
G
2
∂
P
1
∂
q
n
⋮
⋱
⋮
∂
2
G
2
∂
P
n
∂
q
1
⋯
∂
2
G
2
∂
P
n
∂
q
n
|
≠
0
{\displaystyle \left|{\begin{array}{l l l}{\displaystyle {\frac {\partial ^{2}G_{2}}{\partial P_{1}\partial q_{1}}}}&{\cdots }&{\displaystyle {\frac {\partial ^{2}G_{2}}{\partial P_{1}\partial q_{n}}}}\\{\quad \vdots }&{\ddots }&{\quad \vdots }\\{\displaystyle {\frac {\partial ^{2}G_{2}}{\partial P_{n}\partial q_{1}}}}&{\cdots }&{\displaystyle {\frac {\partial ^{2}G_{2}}{\partial P_{n}\partial q_{n}}}}\end{array}}\right|{\neq 0}}
Hence, restrictions are placed on generating functions to have the matrices:
[
∂
2
G
1
∂
Q
j
∂
q
i
]
{\textstyle \left[{\frac {\partial ^{2}G_{1}}{\partial Q_{j}\partial q_{i}}}\right]}
,
[
∂
2
G
2
∂
P
j
∂
q
i
]
{\textstyle \left[{\frac {\partial ^{2}G_{2}}{\partial P_{j}\partial q_{i}}}\right]}
,
[
∂
2
G
3
∂
p
j
∂
Q
i
]
{\textstyle \left[{\frac {\partial ^{2}G_{3}}{\partial p_{j}\partial Q_{i}}}\right]}
and
[
∂
2
G
4
∂
p
j
∂
P
i
]
{\textstyle \left[{\frac {\partial ^{2}G_{4}}{\partial p_{j}\partial P_{i}}}\right]}
, being non-singular. These conditions also correspond to local invertibility of the coordinates. From these restrictions, it can be stated that type 1 and type 4 generating functions always have a non-singular
[
∂
Q
i
(
q
,
p
)
∂
p
j
]
{\textstyle \left[{\frac {\partial Q_{i}(\mathbf {q} ,\mathbf {p} )}{\partial p_{j}}}\right]}
matrix whereas type 2 and type 3 generating functions always have a non-singular
[
∂
P
i
(
q
,
p
)
∂
p
j
]
{\textstyle \left[{\frac {\partial P_{i}(\mathbf {q} ,\mathbf {p} )}{\partial p_{j}}}\right]}
matrix. Hence, the canonical transformations resulting from these four generating functions alone are not completely general.
=== Generalized use of generating functions ===
In other words, since (Q, P) and (q, p) are each 2N independent functions, it follows that to have generating function of the form
G
1
(
q
,
Q
,
t
)
{\textstyle G_{1}(\mathbf {q} ,\mathbf {Q} ,t)}
and
G
4
(
p
,
P
,
t
)
{\displaystyle G_{4}(\mathbf {p} ,\mathbf {P} ,t)}
or
G
2
(
q
,
P
,
t
)
{\displaystyle G_{2}(\mathbf {q} ,\mathbf {P} ,t)}
and
G
3
(
p
,
Q
,
t
)
{\displaystyle G_{3}(\mathbf {p} ,\mathbf {Q} ,t)}
, the corresponding Jacobian matrices
[
∂
Q
i
∂
p
j
]
{\textstyle \left[{\frac {\partial Q_{i}}{\partial p_{j}}}\right]}
and
[
∂
P
i
∂
p
j
]
{\textstyle \left[{\frac {\partial P_{i}}{\partial p_{j}}}\right]}
are restricted to be non singular, ensuring that the generating function is a function of 2N + 1 independent variables. However, as a feature of canonical transformations, it is always possible to choose 2N such independent functions from sets (q, p) or (Q, P), to form a generating function representation of canonical transformations, including the time variable. Hence, it can be proven that every finite canonical transformation can be given as a closed but implicit form that is a variant of the given four simple forms.
== Canonical transformation conditions ==
=== Canonical transformation relations ===
From:
K
=
H
+
∂
G
∂
t
{\displaystyle K=H+{\frac {\partial G}{\partial t}}}
, calculate
∂
(
K
−
H
)
∂
P
{\textstyle {\frac {\partial (K-H)}{\partial P}}}
:
(
∂
(
K
−
H
)
∂
P
)
Q
,
P
,
t
=
∂
K
∂
P
−
∂
H
∂
p
∂
p
∂
P
−
∂
H
∂
q
∂
q
∂
P
−
∂
H
∂
t
(
∂
t
∂
P
)
Q
,
P
,
t
=
Q
˙
+
p
˙
∂
q
∂
P
−
q
˙
∂
p
∂
P
=
∂
Q
∂
t
+
∂
Q
∂
q
⋅
q
˙
+
∂
Q
∂
p
⋅
p
˙
+
p
˙
∂
q
∂
P
−
q
˙
∂
p
∂
P
=
q
˙
(
∂
Q
∂
q
−
∂
p
∂
P
)
+
p
˙
(
∂
q
∂
P
+
∂
Q
∂
p
)
+
∂
Q
∂
t
{\displaystyle {\begin{aligned}\left({\frac {\partial (K-H)}{\partial P}}\right)_{Q,P,t}&={\frac {\partial K}{\partial P}}-{\frac {\partial H}{\partial p}}{\frac {\partial p}{\partial P}}-{\frac {\partial H}{\partial q}}{\frac {\partial q}{\partial P}}-{\frac {\partial H}{\partial t}}\left({\frac {\partial t}{\partial P}}\right)_{Q,P,t}\\&={\dot {Q}}+{\dot {p}}{\frac {\partial q}{\partial P}}-{\dot {q}}{\frac {\partial p}{\partial P}}\\&={\frac {\partial Q}{\partial t}}+{\frac {\partial Q}{\partial q}}\cdot {\dot {q}}+{\frac {\partial Q}{\partial p}}\cdot {\dot {p}}+{\dot {p}}{\frac {\partial q}{\partial P}}-{\dot {q}}{\frac {\partial p}{\partial P}}\\&={\dot {q}}\left({\frac {\partial Q}{\partial q}}-{\frac {\partial p}{\partial P}}\right)+{\dot {p}}\left({\frac {\partial q}{\partial P}}+{\frac {\partial Q}{\partial p}}\right)+{\frac {\partial Q}{\partial t}}\end{aligned}}}
Since the left hand side is
∂
(
K
−
H
)
∂
P
=
∂
∂
P
(
∂
G
∂
t
)
|
Q
,
P
,
t
{\textstyle {\frac {\partial (K-H)}{\partial P}}={\frac {\partial }{\partial P}}\left({\frac {\partial G}{\partial t}}\right){\bigg |}_{Q,P,t}}
which is independent of dynamics of the particles, equating coefficients of
q
˙
{\textstyle {\dot {q}}}
and
p
˙
{\textstyle {\dot {p}}}
to zero, canonical transformation rules are obtained. This step is equivalent to equating the left hand side as
∂
(
K
−
H
)
∂
P
=
∂
Q
∂
t
{\textstyle {\frac {\partial (K-H)}{\partial P}}={\frac {\partial Q}{\partial t}}}
.
Since the left hand side is
∂
(
K
−
H
)
∂
P
=
∂
∂
P
(
∂
G
∂
t
)
|
Q
,
P
,
t
{\textstyle {\frac {\partial (K-H)}{\partial P}}={\frac {\partial }{\partial P}}\left({\frac {\partial G}{\partial t}}\right){\bigg |}_{Q,P,t}}
which is independent of dynamics of the particles, equating coefficients of
q
˙
{\textstyle {\dot {q}}}
and
p
˙
{\textstyle {\dot {p}}}
to zero, canonical transformation rules are obtained. This step is equivalent to equating the left hand side as
∂
(
K
−
H
)
∂
P
=
∂
Q
∂
t
{\textstyle {\frac {\partial (K-H)}{\partial P}}={\frac {\partial Q}{\partial t}}}
.
Similarly:
(
∂
(
K
−
H
)
∂
Q
)
Q
,
P
,
t
=
∂
K
∂
Q
−
∂
H
∂
p
∂
p
∂
Q
−
∂
H
∂
q
∂
q
∂
Q
−
∂
H
∂
t
(
∂
t
∂
Q
)
Q
,
P
,
t
=
−
P
˙
+
p
˙
∂
q
∂
Q
−
q
˙
∂
p
∂
Q
=
−
∂
P
∂
t
−
∂
P
∂
q
⋅
q
˙
−
∂
P
∂
p
⋅
p
˙
+
p
˙
∂
q
∂
Q
−
q
˙
∂
p
∂
Q
=
−
(
q
˙
(
∂
P
∂
q
+
∂
p
∂
Q
)
+
p
˙
(
∂
P
∂
p
−
∂
q
∂
Q
)
+
∂
P
∂
t
)
{\displaystyle {\begin{aligned}\left({\frac {\partial (K-H)}{\partial Q}}\right)_{Q,P,t}&={\frac {\partial K}{\partial Q}}-{\frac {\partial H}{\partial p}}{\frac {\partial p}{\partial Q}}-{\frac {\partial H}{\partial q}}{\frac {\partial q}{\partial Q}}-{\frac {\partial H}{\partial t}}\left({\frac {\partial t}{\partial Q}}\right)_{Q,P,t}\\&=-{\dot {P}}+{\dot {p}}{\frac {\partial q}{\partial Q}}-{\dot {q}}{\frac {\partial p}{\partial Q}}\\&=-{\frac {\partial P}{\partial t}}-{\frac {\partial P}{\partial q}}\cdot {\dot {q}}-{\frac {\partial P}{\partial p}}\cdot {\dot {p}}+{\dot {p}}{\frac {\partial q}{\partial Q}}-{\dot {q}}{\frac {\partial p}{\partial Q}}\\&=-\left({\dot {q}}\left({\frac {\partial P}{\partial q}}+{\frac {\partial p}{\partial Q}}\right)+{\dot {p}}\left({\frac {\partial P}{\partial p}}-{\frac {\partial q}{\partial Q}}\right)+{\frac {\partial P}{\partial t}}\right)\end{aligned}}}
Similarly the canonical transformation rules are obtained by equating the left hand side as
∂
(
K
−
H
)
∂
Q
=
−
∂
P
∂
t
{\textstyle {\frac {\partial (K-H)}{\partial Q}}=-{\frac {\partial P}{\partial t}}}
.
The above two relations can be combined in matrix form as:
J
(
∇
ε
∂
G
∂
t
)
=
∂
ε
∂
t
{\textstyle J\left(\nabla _{\varepsilon }{\frac {\partial G}{\partial t}}\right)={\frac {\partial \varepsilon }{\partial t}}}
(which will also retain same form for extended canonical transformation) where the result
∂
G
∂
t
=
K
−
H
{\textstyle {\frac {\partial G}{\partial t}}=K-H}
, has been used. The canonical transformation relations are hence said to be equivalent to
J
(
∇
ε
∂
G
∂
t
)
=
∂
ε
∂
t
{\textstyle J\left(\nabla _{\varepsilon }{\frac {\partial G}{\partial t}}\right)={\frac {\partial \varepsilon }{\partial t}}}
in this context.
The canonical transformation relations can now be restated to include time dependance:
(
∂
Q
m
∂
p
n
)
q
,
p
,
t
=
−
(
∂
q
n
∂
P
m
)
Q
,
P
,
t
(
∂
Q
m
∂
q
n
)
q
,
p
,
t
=
(
∂
p
n
∂
P
m
)
Q
,
P
,
t
{\displaystyle {\begin{aligned}\left({\frac {\partial Q_{m}}{\partial p_{n}}}\right)_{\mathbf {q} ,\mathbf {p} ,t}&=-\left({\frac {\partial q_{n}}{\partial P_{m}}}\right)_{\mathbf {Q} ,\mathbf {P} ,t}\\\left({\frac {\partial Q_{m}}{\partial q_{n}}}\right)_{\mathbf {q} ,\mathbf {p} ,t}&=\left({\frac {\partial p_{n}}{\partial P_{m}}}\right)_{\mathbf {Q} ,\mathbf {P} ,t}\end{aligned}}}
(
∂
P
m
∂
p
n
)
q
,
p
,
t
=
(
∂
q
n
∂
Q
m
)
Q
,
P
,
t
(
∂
P
m
∂
q
n
)
q
,
p
,
t
=
−
(
∂
p
n
∂
Q
m
)
Q
,
P
,
t
{\displaystyle {\begin{aligned}\left({\frac {\partial P_{m}}{\partial p_{n}}}\right)_{\mathbf {q} ,\mathbf {p} ,t}&=\left({\frac {\partial q_{n}}{\partial Q_{m}}}\right)_{\mathbf {Q} ,\mathbf {P} ,t}\\\left({\frac {\partial P_{m}}{\partial q_{n}}}\right)_{\mathbf {q} ,\mathbf {p} ,t}&=-\left({\frac {\partial p_{n}}{\partial Q_{m}}}\right)_{\mathbf {Q} ,\mathbf {P} ,t}\end{aligned}}}
Since
∂
(
K
−
H
)
∂
P
=
∂
Q
∂
t
{\textstyle {\frac {\partial (K-H)}{\partial P}}={\frac {\partial Q}{\partial t}}}
and
∂
(
K
−
H
)
∂
Q
=
−
∂
P
∂
t
{\textstyle {\frac {\partial (K-H)}{\partial Q}}=-{\frac {\partial P}{\partial t}}}
, if Q and P do not explicitly depend on time,
K
=
H
+
∂
G
∂
t
(
t
)
{\textstyle K=H+{\frac {\partial G}{\partial t}}(t)}
can be taken. The analysis of restricted canonical transformations is hence consistent with this generalization.
=== Symplectic condition ===
Applying transformation of co-ordinates formula for
∇
η
H
=
M
T
∇
ε
H
{\displaystyle \nabla _{\eta }H=M^{T}\nabla _{\varepsilon }H}
, in Hamiltonian's equations gives:
η
˙
=
J
∇
η
H
=
J
(
M
T
∇
ε
H
)
{\displaystyle {\dot {\eta }}=J\nabla _{\eta }H=J(M^{T}\nabla _{\varepsilon }H)}
Similarly for
ε
˙
{\textstyle {\dot {\varepsilon }}}
:
ε
˙
=
M
η
˙
+
∂
ε
∂
t
=
M
J
M
T
∇
ε
H
+
∂
ε
∂
t
{\displaystyle {\dot {\varepsilon }}=M{\dot {\eta }}+{\frac {\partial \varepsilon }{\partial t}}=MJM^{T}\nabla _{\varepsilon }H+{\frac {\partial \varepsilon }{\partial t}}}
or:
ε
˙
=
J
∇
ε
K
=
J
∇
ε
H
+
J
∇
ε
(
∂
G
∂
t
)
{\displaystyle {\dot {\varepsilon }}=J\nabla _{\varepsilon }K=J\nabla _{\varepsilon }H+J\nabla _{\varepsilon }\left({\frac {\partial G}{\partial t}}\right)}
Where the last terms of each equation cancel due to
J
(
∇
ε
∂
G
∂
t
)
=
∂
ε
∂
t
{\textstyle J\left(\nabla _{\varepsilon }{\frac {\partial G}{\partial t}}\right)={\frac {\partial \varepsilon }{\partial t}}}
condition from canonical transformations. Hence leaving the symplectic relation:
M
J
M
T
=
J
{\textstyle MJM^{T}=J}
which is also equivalent with the condition
M
T
J
M
=
J
{\textstyle M^{T}JM=J}
. It follows from the above two equations that the symplectic condition implies the equation
J
(
∇
ε
∂
G
∂
t
)
=
∂
ε
∂
t
{\textstyle J\left(\nabla _{\varepsilon }{\frac {\partial G}{\partial t}}\right)={\frac {\partial \varepsilon }{\partial t}}}
, from which the indirect conditions can be recovered. Thus, symplectic conditions and indirect conditions can be said to be equivalent in the context of using generating functions.
=== Invariance of the Poisson and Lagrange brackets ===
Since
P
i
j
(
ε
)
=
{
ε
i
,
ε
j
}
η
=
(
M
J
M
T
)
i
j
=
J
i
j
{\textstyle {\mathcal {P}}_{ij}(\varepsilon )=\{\varepsilon _{i},\varepsilon _{j}\}_{\eta }=(MJM^{T})_{ij}=J_{ij}}
and
L
i
j
(
η
)
=
[
η
i
,
η
j
]
ε
=
(
M
T
J
M
)
i
j
=
J
i
j
{\textstyle {\mathcal {L}}_{ij}(\eta )=[\eta _{i},\eta _{j}]_{\varepsilon }=(M^{T}JM)_{ij}=J_{ij}}
where the symplectic condition is used in the last equalities. Using
{
ε
i
,
ε
j
}
ε
=
[
η
i
,
η
j
]
η
=
J
i
j
{\textstyle \{\varepsilon _{i},\varepsilon _{j}\}_{\varepsilon }=[\eta _{i},\eta _{j}]_{\eta }=J_{ij}}
, the equalities
{
ε
i
,
ε
j
}
η
=
{
ε
i
,
ε
j
}
ε
{\textstyle \{\varepsilon _{i},\varepsilon _{j}\}_{\eta }=\{\varepsilon _{i},\varepsilon _{j}\}_{\varepsilon }}
and
[
η
i
,
η
j
]
ε
=
[
η
i
,
η
j
]
η
{\textstyle [\eta _{i},\eta _{j}]_{\varepsilon }=[\eta _{i},\eta _{j}]_{\eta }}
are obtained which imply the invariance of Poisson and Lagrange brackets.
== Extended canonical transformation ==
=== Canonical transformation relations ===
By solving for:
λ
[
p
⋅
q
˙
−
H
(
q
,
p
,
t
)
]
=
P
⋅
Q
˙
−
K
(
Q
,
P
,
t
)
+
d
G
d
t
{\displaystyle \lambda \left[\mathbf {p} \cdot {\dot {\mathbf {q} }}-H(\mathbf {q} ,\mathbf {p} ,t)\right]=\mathbf {P} \cdot {\dot {\mathbf {Q} }}-K(\mathbf {Q} ,\mathbf {P} ,t)+{\frac {dG}{dt}}}
with various forms of generating function, the relation between K and H goes as
∂
G
∂
t
=
K
−
λ
H
{\textstyle {\frac {\partial G}{\partial t}}=K-\lambda H}
instead, which also applies for
λ
=
1
{\textstyle \lambda =1}
case.
All results presented below can also be obtained by replacing
q
→
λ
q
{\textstyle q\rightarrow {\sqrt {\lambda }}q}
,
p
→
λ
p
{\textstyle p\rightarrow {\sqrt {\lambda }}p}
and
H
→
λ
H
{\textstyle H\rightarrow {\lambda }H}
from known solutions, since it retains the form of Hamilton's equations. The extended canonical transformations are hence said to be result of a canonical transformation (
λ
=
1
{\textstyle \lambda =1}
) and a trivial canonical transformation (
λ
≠
1
{\textstyle \lambda \neq 1}
) which has
M
J
M
T
=
λ
J
{\textstyle MJM^{T}=\lambda J}
(for the given example,
M
=
λ
I
{\textstyle M={\sqrt {\lambda }}I}
which satisfies the condition).
Using same steps previously used in previous generalization, with
∂
G
∂
t
=
K
−
λ
H
{\textstyle {\frac {\partial G}{\partial t}}=K-\lambda H}
in the general case, and retaining the equation
J
(
∇
ε
∂
g
∂
t
)
=
∂
ε
∂
t
{\textstyle J\left(\nabla _{\varepsilon }{\frac {\partial g}{\partial t}}\right)={\frac {\partial \varepsilon }{\partial t}}}
, extended canonical transformation partial differential relations are obtained as:
(
∂
Q
m
∂
p
n
)
q
,
p
,
t
=
−
λ
(
∂
q
n
∂
P
m
)
Q
,
P
,
t
(
∂
Q
m
∂
q
n
)
q
,
p
,
t
=
λ
(
∂
p
n
∂
P
m
)
Q
,
P
,
t
{\displaystyle {\begin{aligned}\left({\frac {\partial Q_{m}}{\partial p_{n}}}\right)_{\mathbf {q} ,\mathbf {p} ,t}&=-\lambda \left({\frac {\partial q_{n}}{\partial P_{m}}}\right)_{\mathbf {Q} ,\mathbf {P} ,t}\\\left({\frac {\partial Q_{m}}{\partial q_{n}}}\right)_{\mathbf {q} ,\mathbf {p} ,t}&=\lambda \left({\frac {\partial p_{n}}{\partial P_{m}}}\right)_{\mathbf {Q} ,\mathbf {P} ,t}\end{aligned}}}
(
∂
P
m
∂
p
n
)
q
,
p
,
t
=
λ
(
∂
q
n
∂
Q
m
)
Q
,
P
,
t
(
∂
P
m
∂
q
n
)
q
,
p
,
t
=
−
λ
(
∂
p
n
∂
Q
m
)
Q
,
P
,
t
{\displaystyle {\begin{aligned}\left({\frac {\partial P_{m}}{\partial p_{n}}}\right)_{\mathbf {q} ,\mathbf {p} ,t}&=\lambda \left({\frac {\partial q_{n}}{\partial Q_{m}}}\right)_{\mathbf {Q} ,\mathbf {P} ,t}\\\left({\frac {\partial P_{m}}{\partial q_{n}}}\right)_{\mathbf {q} ,\mathbf {p} ,t}&=-\lambda \left({\frac {\partial p_{n}}{\partial Q_{m}}}\right)_{\mathbf {Q} ,\mathbf {P} ,t}\end{aligned}}}
=== Symplectic condition ===
Following the same steps to derive the symplectic conditions, as:
η
˙
=
J
∇
η
H
=
J
(
M
T
∇
ε
H
)
{\displaystyle {\dot {\eta }}=J\nabla _{\eta }H=J(M^{T}\nabla _{\varepsilon }H)}
and
ε
˙
=
M
η
˙
+
∂
ε
∂
t
=
M
J
M
T
∇
ε
H
+
∂
ε
∂
t
{\displaystyle {\dot {\varepsilon }}=M{\dot {\eta }}+{\frac {\partial \varepsilon }{\partial t}}=MJM^{T}\nabla _{\varepsilon }H+{\frac {\partial \varepsilon }{\partial t}}}
where using
∂
G
∂
t
=
K
−
λ
H
{\textstyle {\frac {\partial G}{\partial t}}=K-\lambda H}
instead gives:
ε
˙
=
J
∇
ε
K
=
λ
J
∇
ε
H
+
J
∇
ε
(
∂
G
∂
t
)
{\displaystyle {\dot {\varepsilon }}=J\nabla _{\varepsilon }K=\lambda J\nabla _{\varepsilon }H+J\nabla _{\varepsilon }\left({\frac {\partial G}{\partial t}}\right)}
The second part of each equation cancel. Hence the condition for extended canonical transformation instead becomes:
M
J
M
T
=
λ
J
{\textstyle MJM^{T}=\lambda J}
.
=== Poisson and Lagrange brackets ===
The Poisson brackets are changed as follows:
{
u
,
v
}
η
=
(
∇
η
u
)
T
J
(
∇
η
v
)
=
(
M
T
∇
ε
u
)
T
J
(
M
T
∇
ε
v
)
=
(
∇
ε
u
)
T
M
J
M
T
(
∇
ε
v
)
=
λ
(
∇
ε
u
)
T
J
(
∇
ε
v
)
=
λ
{
u
,
v
}
ε
{\displaystyle \{u,v\}_{\eta }=(\nabla _{\eta }u)^{T}J(\nabla _{\eta }v)=(M^{T}\nabla _{\varepsilon }u)^{T}J(M^{T}\nabla _{\varepsilon }v)=(\nabla _{\varepsilon }u)^{T}MJM^{T}(\nabla _{\varepsilon }v)=\lambda (\nabla _{\varepsilon }u)^{T}J(\nabla _{\varepsilon }v)=\lambda \{u,v\}_{\varepsilon }}
whereas, the Lagrange brackets are changed as:
[
u
,
v
]
ε
=
(
∂
u
ε
)
T
J
(
∂
v
ε
)
=
(
M
∂
u
η
)
T
J
(
M
∂
v
η
)
=
(
∂
u
η
)
T
M
T
J
M
(
∂
v
η
)
=
λ
(
∂
u
η
)
T
J
(
∂
v
η
)
=
λ
[
u
,
v
]
η
{\displaystyle [u,v]_{\varepsilon }=(\partial _{u}\varepsilon )^{T}\,J\,(\partial _{v}\varepsilon )=(M\,\partial _{u}\eta )^{T}\,J\,(M\,\partial _{v}\eta )=(\partial _{u}\eta )^{T}\,M^{T}JM\,(\partial _{v}\eta )=\lambda (\partial _{u}\eta )^{T}\,J\,(\partial _{v}\eta )=\lambda [u,v]_{\eta }}
Hence, the Poisson bracket scales by the inverse of
λ
{\textstyle \lambda }
whereas the Lagrange bracket scales by a factor of
λ
{\textstyle \lambda }
.
== Infinitesimal canonical transformation ==
Consider the canonical transformation that depends on a continuous parameter
α
{\displaystyle \alpha }
, as follows:
Q
(
q
,
p
,
t
;
α
)
Q
(
q
,
p
,
t
;
0
)
=
q
P
(
q
,
p
,
t
;
α
)
with
P
(
q
,
p
,
t
;
0
)
=
p
{\displaystyle {\begin{aligned}&Q(q,p,t;\alpha )\quad \quad \quad &Q(q,p,t;0)=q\\&P(q,p,t;\alpha )\quad \quad {\text{with}}\quad &P(q,p,t;0)=p\\\end{aligned}}}
For infinitesimal values of
α
{\displaystyle \alpha }
, the corresponding transformations are called as infinitesimal canonical transformations which are also known as differential canonical transformations.
=== Explicit construction ===
Consider the following generating function:
G
2
(
q
,
P
,
t
)
=
q
P
+
α
G
(
q
,
P
,
t
)
{\displaystyle G_{2}(q,P,t)=qP+\alpha G(q,P,t)}
Since for
α
=
0
{\displaystyle \alpha =0}
,
G
2
=
q
P
{\displaystyle G_{2}=qP}
has the resulting canonical transformation,
Q
=
q
{\displaystyle Q=q}
and
P
=
p
{\displaystyle P=p}
, this type of generating function can be used for infinitesimal canonical transformation by restricting
α
{\displaystyle \alpha }
to an infinitesimal value.
From the conditions of generators of second type:
p
=
∂
G
2
∂
q
=
P
+
α
∂
G
∂
q
(
q
,
P
,
t
)
Q
=
∂
G
2
∂
P
=
q
+
α
∂
G
∂
P
(
q
,
P
,
t
)
{\displaystyle {\begin{aligned}{p}&={\frac {\partial G_{2}}{\partial {q}}}=P+\alpha {\frac {\partial G}{\partial {q}}}(q,P,t)\\{Q}&={\frac {\partial G_{2}}{\partial {P}}}=q+\alpha {\frac {\partial G}{\partial {P}}}(q,P,t)\\\end{aligned}}}
Since
P
=
P
(
q
,
p
,
t
;
α
)
{\displaystyle P=P(q,p,t;\alpha )}
, changing the variables of the function
G
{\displaystyle G}
to
G
(
q
,
p
,
t
)
{\displaystyle G(q,p,t)}
and neglecting terms of higher order of
α
{\displaystyle \alpha }
, gives:
p
=
P
+
α
∂
G
∂
q
(
q
,
p
,
t
)
Q
=
q
+
α
∂
G
∂
p
(
q
,
p
,
t
)
{\displaystyle {\begin{aligned}{p}&=P+\alpha {\frac {\partial G}{\partial {q}}}(q,p,t)\\{Q}&=q+\alpha {\frac {\partial G}{\partial p}}(q,p,t)\\\end{aligned}}}
Infinitesimal canonical transformations can also be derived using the matrix form of the symplectic condition. The function
G
(
q
,
p
,
t
)
{\displaystyle G(q,p,t)}
is very significant in infinitesimal canonical transformations and is referred to as the generator of infinitesimal canonical transformation.
=== Active and passive transformations ===
In the active view of transformations, the coordinate system is changed without the physical system changing, whereas in the passive view of transformation, the coordinate system is retained and the physical system is said to undergo transformations.
==== Active view of transformation ====
Thus, using the relations from infinitesimal canonical transformations, the change in the system states under active view of the canonical transformation is said to be:
δ
q
=
α
∂
G
∂
p
(
q
,
p
,
t
)
and
δ
p
=
−
α
∂
G
∂
q
(
q
,
p
,
t
)
,
{\displaystyle {\begin{aligned}&\delta q=\alpha {\frac {\partial G}{\partial p}}(q,p,t)\quad {\text{and}}\quad \delta p=-\alpha {\frac {\partial G}{\partial q}}(q,p,t),\\\end{aligned}}}
or as
δ
η
=
α
J
∇
η
G
{\displaystyle \delta \eta =\alpha J\nabla _{\eta }G}
in matrix form.
For any function
u
(
η
)
{\displaystyle u(\eta )}
, it changes under active view of the transformation according to:
δ
u
=
u
(
η
+
δ
η
)
−
u
(
η
)
=
(
∇
η
u
)
T
δ
η
=
α
(
∇
η
u
)
T
J
(
∇
η
G
)
=
α
{
u
,
G
}
.
{\displaystyle \delta u=u(\eta +\delta \eta )-u(\eta )=(\nabla _{\eta }u)^{T}\delta \eta =\alpha (\nabla _{\eta }u)^{T}J(\nabla _{\eta }G)=\alpha \{u,G\}.}
==== Passive view of transformation ====
Considering the change of Hamiltonians in the passive view, i.e., for a fixed point,
K
(
Q
=
q
0
,
P
=
p
0
,
t
)
−
H
(
q
=
q
0
,
p
=
p
0
,
t
)
=
(
H
(
q
0
′
,
p
0
′
,
t
)
+
∂
G
2
∂
t
)
−
H
(
q
0
,
p
0
,
t
)
=
−
δ
H
+
α
∂
G
∂
t
=
α
(
{
G
,
H
}
+
∂
G
∂
t
)
=
α
d
G
d
t
{\displaystyle K(Q=q_{0},P=p_{0},t)-H(q=q_{0},p=p_{0},t)=\left(H(q_{0}',p_{0}',t)+{\frac {\partial G_{2}}{\partial t}}\right)-H(q_{0},p_{0},t)=-\delta H+\alpha {\frac {\partial G}{\partial t}}=\alpha \left(\{G,H\}+{\frac {\partial G}{\partial t}}\right)=\alpha {\frac {dG}{dt}}}
where
(
q
=
q
0
′
,
p
=
p
0
′
)
{\textstyle (q=q_{0}',p=p_{0}')}
are mapped to the point,
(
Q
=
q
0
,
P
=
p
0
)
{\textstyle (Q=q_{0},P=p_{0})}
by the infinitesimal canonical transformation, and similar change of variables for
G
(
q
,
P
,
t
)
{\displaystyle G(q,P,t)}
to
G
(
q
,
p
,
t
)
{\displaystyle G(q,p,t)}
is considered up-to first order of
α
{\displaystyle \alpha }
. Hence, if the Hamiltonian is invariant for infinitesimal canonical transformations, its generator is a constant of motion.
=== Generators of dynamical symmetry transformations ===
Consider the transformation where the change of coordinates also depends on the generalized velocities.
q
r
→
q
r
+
δ
q
r
δ
q
r
=
ϵ
ϕ
r
(
q
,
q
˙
,
t
)
{\displaystyle {\begin{aligned}q^{r}\to q^{r}+\delta q^{r}\\\delta q^{r}=\epsilon \phi ^{r}(q,{\dot {q}},t)\\\end{aligned}}}
If the above is a dynamical symmetry, then the lagrangian changes by:
δ
L
=
ϵ
d
d
t
F
(
q
,
q
˙
,
t
)
{\displaystyle \delta L=\epsilon {\frac {d}{dt}}F(q,{\dot {q}},t)}
and the new Lagrangian is said to be dynamically equivalent to the old Lagrangian as it ensures the resultant equations of motion being the same. The change in generalized velocity and momentum term can be derived as:
p
=
∂
L
∂
q
˙
,
q
˙
=
d
q
d
t
δ
p
r
=
∂
2
L
∂
q
s
∂
q
˙
r
δ
q
s
+
∂
2
L
∂
q
˙
s
∂
q
˙
r
δ
q
˙
s
,
δ
q
˙
r
=
ϵ
∂
ϕ
r
∂
q
s
q
˙
s
+
ϵ
∂
ϕ
r
∂
q
˙
s
q
¨
s
+
ϵ
∂
ϕ
r
∂
t
{\displaystyle {\begin{aligned}p={\frac {\partial L}{\partial {\dot {q}}}},\quad &{\dot {q}}={\frac {dq}{dt}}\\\delta p_{r}={\frac {\partial ^{2}L}{\partial q^{s}\partial {\dot {q}}^{r}}}\delta q^{s}+{\frac {\partial ^{2}L}{\partial {\dot {q}}^{s}\partial {\dot {q}}^{r}}}\delta {\dot {q}}^{s},\quad &\delta {\dot {q}}^{r}=\epsilon {\frac {\partial \phi ^{r}}{\partial q^{s}}}{\dot {q}}^{s}+\epsilon {\frac {\partial \phi ^{r}}{\partial {\dot {q}}^{s}}}{\ddot {q}}^{s}+\epsilon {\frac {\partial \phi ^{r}}{\partial t}}\\\end{aligned}}}
==== Generator of transformation ====
Using the change in Lagrangian property of a dynamical symmetry:
d
d
t
F
=
∂
F
∂
q
r
q
˙
r
+
∂
F
∂
q
˙
r
q
¨
r
+
∂
F
∂
t
=
δ
L
ϵ
=
(
∂
L
∂
q
r
ϕ
r
+
∂
L
∂
q
˙
r
∂
ϕ
r
∂
t
)
+
p
s
∂
ϕ
s
∂
q
r
q
˙
r
+
p
s
∂
ϕ
s
∂
q
˙
r
q
¨
r
{\displaystyle {\frac {d}{dt}}F={\frac {\partial F}{\partial q^{r}}}{\dot {q}}^{r}+{\frac {\partial F}{\partial {\dot {q}}^{r}}}{\ddot {q}}^{r}+{\frac {\partial F}{\partial t}}={\frac {\delta L}{\epsilon }}=\left({\frac {\partial L}{\partial q^{r}}}\phi ^{r}+{\frac {\partial L}{\partial {\dot {q}}^{r}}}{\frac {\partial \phi ^{r}}{\partial t}}\right)+p_{s}{\frac {\partial \phi ^{s}}{\partial q^{r}}}{\dot {q}}^{r}+p_{s}{\frac {\partial \phi ^{s}}{\partial {\dot {q}}^{r}}}{\ddot {q}}^{r}}
Since the
q
¨
{\displaystyle {\ddot {q}}}
terms appear only once in either side, it's coefficients must be equal for this to be true, giving the relation:
p
s
∂
ϕ
s
∂
q
˙
r
=
∂
F
∂
q
˙
r
{\textstyle p_{s}{\frac {\partial \phi ^{s}}{\partial {\dot {q}}^{r}}}={\frac {\partial F}{\partial {\dot {q}}^{r}}}}
using which, it can be shown that
{
q
r
,
ϵ
(
p
s
ϕ
s
−
F
)
}
=
δ
q
r
,
{
p
r
,
ϵ
(
p
s
ϕ
s
−
F
)
}
=
δ
p
r
+
ϵ
(
∂
L
∂
q
s
−
d
d
t
∂
L
∂
q
˙
s
)
∂
ϕ
s
∂
q
˙
r
{\displaystyle \{q^{r},\epsilon (p_{s}\phi ^{s}-F)\}=\delta q^{r},\quad \{p_{r},\epsilon (p_{s}\phi ^{s}-F)\}=\delta p_{r}+\epsilon \left({\frac {\partial L}{\partial q^{s}}}-{\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q}}^{s}}}\right){\frac {\partial \phi ^{s}}{\partial {\dot {q}}^{r}}}}
Hence, the term
p
ϕ
−
F
{\displaystyle p\phi -F}
generates the canonical dynamical symmetry transformation if either the Euler Lagrange relation gives zero, or if
∂
ϕ
s
∂
q
˙
r
=
0
∀
s
,
r
{\displaystyle {\frac {\partial \phi _{s}}{\partial {\dot {q}}^{r}}}=0\,\forall s,r}
which is a infinitesimal point transformation. Note that in the point transformation condition, the quantity generates the transformation regardless of if the Euler Lagrange equations are satisfied and since they do not depend on the dynamics of the problem are said to be a purely kinematic relation.
==== Noether Invariant ====
Using Euler Lagrange relation for the provided Lagrangian, the invariants of motion can be derived as:
δ
L
−
ϵ
d
d
t
F
(
q
,
q
˙
,
t
)
=
ϵ
ϕ
(
∂
∂
q
−
d
d
t
∂
∂
q
˙
)
L
=
0
+
ϵ
d
d
t
(
ϕ
∂
∂
q
˙
L
−
F
)
=
ϵ
d
d
t
(
ϕ
∂
∂
q
˙
L
−
F
)
=
0
{\displaystyle \delta L-\epsilon {\frac {d}{dt}}F(q,{\dot {q}},t)=\epsilon \phi {\cancelto {=0}{\left({\frac {\partial }{\partial q}}-{\frac {d}{dt}}{\frac {\partial }{\partial {\dot {q}}}}\right)L}}+\epsilon {\frac {d}{dt}}\left(\phi {\frac {\partial }{\partial {\dot {q}}}}L-F\right)=\epsilon {\frac {d}{dt}}\left(\phi {\frac {\partial }{\partial {\dot {q}}}}L-F\right)=0}
Hence
(
ϕ
∂
∂
q
˙
L
−
F
)
=
p
ϕ
−
F
{\displaystyle \left(\phi {\frac {\partial }{\partial {\dot {q}}}}L-F\right)=p\phi -F}
is a constant of motion. Hence, the derived Noether invariant also generates the same symmetry transformation as shown previously.
=== Examples of ICT ===
==== Time evolution ====
Taking
G
(
q
,
p
,
t
)
=
H
(
q
,
p
,
t
)
{\displaystyle G(q,p,t)=H(q,p,t)}
and
α
=
d
t
{\displaystyle \alpha =dt}
, then
δ
η
=
(
J
∇
η
H
)
d
t
=
η
˙
d
t
=
d
η
{\displaystyle \delta \eta =(J\nabla _{\eta }H)dt={\dot {\eta }}dt=d\eta }
. Thus the continuous application of such a transformation maps the coordinates
η
(
τ
)
{\displaystyle \eta (\tau )}
to
η
(
τ
+
t
)
{\displaystyle \eta (\tau +t)}
. Hence if the Hamiltonian is time translation invariant, i.e. does not have explicit time dependence, its value is conserved for the motion.
==== Translation ====
Taking
G
(
q
,
p
,
t
)
=
p
k
{\displaystyle G(q,p,t)=p_{k}}
,
δ
p
i
=
0
{\displaystyle \delta p_{i}=0}
and
δ
q
i
=
α
δ
i
k
{\displaystyle \delta q_{i}=\alpha \delta _{ik}}
. Hence, the canonical momentum generates a shift in the corresponding generalized coordinate and if the Hamiltonian is invariant of translation, the momentum is a constant of motion.
==== Rotation ====
Consider an orthogonal system for an N-particle system:
q
=
(
x
1
,
y
1
,
z
1
,
…
,
x
n
,
y
n
,
z
n
)
,
p
=
(
p
1
x
,
p
1
y
,
p
1
z
,
…
,
p
n
x
,
p
n
y
,
p
n
z
)
.
{\displaystyle {\begin{array}{l}{\mathbf {q} =\left(x_{1},y_{1},z_{1},\ldots ,x_{n},y_{n},z_{n}\right),}\\{\mathbf {p} =\left(p_{1x},p_{1y},p_{1z},\ldots ,p_{nx},p_{ny},p_{nz}\right).}\end{array}}}
Choosing the generator to be:
G
=
L
z
=
∑
i
=
1
n
(
x
i
p
i
y
−
y
i
p
i
x
)
{\displaystyle G=L_{z}=\sum _{i=1}^{n}\left(x_{i}p_{iy}-y_{i}p_{ix}\right)}
and the infinitesimal value of
α
=
δ
ϕ
{\displaystyle \alpha =\delta \phi }
, then the change in the coordinates is given for x by:
δ
x
i
=
{
x
i
,
G
}
δ
ϕ
=
∑
j
{
x
i
,
x
j
p
j
y
−
y
j
p
j
x
}
δ
ϕ
=
∑
j
(
{
x
i
,
x
j
p
j
y
}
⏟
=
0
−
{
x
i
,
y
j
p
j
x
}
)
δ
ϕ
=
−
∑
j
y
j
{
x
i
,
p
j
x
}
⏟
=
δ
i
j
δ
ϕ
=
−
y
i
δ
ϕ
{\displaystyle {\begin{array}{c}{\delta x_{i}=\{x_{i},G\}\delta \phi =\displaystyle \sum _{j}\{x_{i},x_{j}p_{jy}-y_{j}p_{jx}\}\delta \phi =\displaystyle \sum _{j}(\underbrace {\{x_{i},x_{j}p_{jy}\}} _{=0}-{\{x_{i},y_{j}p_{jx}\}}})\delta \phi \\{=\displaystyle -\sum _{j}y_{j}\underbrace {\{x_{i},p_{jx}\}} _{=\delta _{ij}}\delta \phi =-y_{i}\delta \phi }\end{array}}}
and similarly for y:
δ
y
i
=
{
y
i
,
G
}
δ
ϕ
=
∑
j
{
y
i
,
x
j
p
j
y
−
y
j
p
j
x
}
δ
ϕ
=
∑
j
(
{
y
i
,
x
j
p
j
y
}
−
{
y
i
,
y
j
p
j
x
}
⏟
=
0
)
δ
ϕ
=
∑
j
x
j
{
y
i
,
p
j
y
}
⏟
=
δ
i
j
δ
ϕ
=
x
i
δ
ϕ
,
{\displaystyle {\begin{array}{c}\delta y_{i}=\{y_{i},G\}\delta \phi =\displaystyle \sum _{j}\{y_{i},x_{j}p_{jy}-y_{j}p_{jx}\}\delta \phi =\displaystyle \sum _{j}(\{y_{i},x_{j}p_{jy}\}-\underbrace {\{y_{i},y_{j}p_{jx}\}} _{=0})\delta \phi \\{=\displaystyle \sum _{j}x_{j}\underbrace {\{y_{i},p_{jy}\}} _{=\delta _{ij}}\delta \phi =x_{i}\delta \phi \,,}\end{array}}}
whereas the z component of all particles is unchanged:
δ
z
i
=
{
z
i
,
G
}
δ
ϕ
=
∑
j
{
z
i
,
x
j
p
j
y
−
y
j
p
j
x
}
δ
ϕ
=
0
{\textstyle \delta z_{i}=\left\{z_{i},G\right\}\delta \phi =\sum _{j}\left\{z_{i},x_{j}p_{jy}-y_{j}p_{jx}\right\}\delta \phi =0}
.
These transformations correspond to rotation about the z axis by angle
δ
ϕ
{\displaystyle \delta \phi }
in its first order approximation. Hence, repeated application of the infinitesimal canonical transformation generates a rotation of system of particles about the z axis. If the Hamiltonian is invariant under rotation about the z axis, the generator, the component of angular momentum along the axis of rotation, is an invariant of motion.
== One parameter subgroup of Canonical transformations ==
Allowing the values of
α
{\displaystyle \alpha }
to take continuous range of values in:
Q
(
q
,
p
,
t
;
α
)
Q
(
q
,
p
,
t
;
0
)
=
q
P
(
q
,
p
,
t
;
α
)
with
P
(
q
,
p
,
t
;
0
)
=
p
{\displaystyle {\begin{aligned}&Q(q,p,t;\alpha )\quad \quad \quad &Q(q,p,t;0)=q\\&P(q,p,t;\alpha )\quad \quad {\text{with}}\quad &P(q,p,t;0)=p\\\end{aligned}}}
which can be expressed as
ϵ
μ
(
η
,
t
;
α
)
{\displaystyle \epsilon ^{\mu }(\eta ,t;\alpha )}
where
ϵ
μ
(
η
,
t
;
0
)
=
η
μ
{\displaystyle \epsilon ^{\mu }(\eta ,t;0)=\eta ^{\mu }}
.
One parameter subgroup of Canonical transformations are those where the generator of the transformation can be used to generate coordinates where
ϵ
μ
(
ϵ
(
η
,
t
;
α
1
)
;
α
2
)
=
ϵ
μ
(
η
,
t
;
α
1
+
α
2
)
{\displaystyle \epsilon ^{\mu }(\epsilon (\eta ,t;\alpha _{1});\alpha _{2})=\epsilon ^{\mu }(\eta ,t;\alpha _{1}+\alpha _{2})}
is satisfied, i.e. composition of two canonical transformations of parameter
α
1
{\displaystyle \alpha _{1}}
and
α
2
{\displaystyle \alpha _{2}}
are the same as that of a single canonical transformation of parameter
α
1
+
α
2
{\displaystyle \alpha _{1}+\alpha _{2}}
.
The condition on the transformations of the one parameter subgroup kind can be expressed equivalently as a differential equation:
δ
ϵ
μ
(
η
,
t
;
α
)
=
δ
α
{
ϵ
ν
,
G
}
=
δ
α
J
μ
ν
∂
G
∂
ϵ
ν
(
ϵ
(
η
,
t
;
α
)
,
t
)
⟹
d
ϵ
μ
(
η
,
t
;
α
)
d
α
=
J
μ
ν
∂
G
∂
ϵ
ν
(
ϵ
(
η
,
t
;
α
)
,
t
)
{\displaystyle \delta \epsilon ^{\mu }(\eta ,t;\alpha )=\delta \alpha \{\epsilon ^{\nu },G\}=\delta \alpha J^{\mu \nu }{\frac {\partial G}{\partial \epsilon ^{\nu }}}(\epsilon (\eta ,t;\alpha ),t)\implies {\frac {d\epsilon ^{\mu }(\eta ,t;\alpha )}{d\alpha }}=J^{\mu \nu }{\frac {\partial G}{\partial \epsilon ^{\nu }}}(\epsilon (\eta ,t;\alpha ),t)}
for all
η
{\displaystyle \eta }
given that the generator has no explicit dependance on
α
{\displaystyle \alpha }
. The conditions
ϵ
μ
(
ϵ
(
η
,
t
;
α
1
)
;
α
2
)
=
ϵ
μ
(
η
,
t
;
α
1
+
α
2
)
{\displaystyle \epsilon ^{\mu }(\epsilon (\eta ,t;\alpha _{1});\alpha _{2})=\epsilon ^{\mu }(\eta ,t;\alpha _{1}+\alpha _{2})}
can be recovered since this equation is trivially satisfied when
α
2
=
0
{\displaystyle \alpha _{2}=0}
which is considered initial values and the differential equations of both sides are of the same form implying the relation due to uniqueness of solutions with given initial values. Hence one parameter subgroups of canonical transformations are extension of infinitesimal canonical transformations to finite values of
α
{\displaystyle \alpha }
by using the same functional form of its generator independent of parameter
α
{\displaystyle \alpha }
.
As a consequence of the generator having no explicit dependance on
α
{\displaystyle \alpha }
, the generator is also implicitly independent of
α
{\displaystyle \alpha }
.
d
G
(
ϵ
(
η
;
α
)
,
t
)
d
α
=
{
G
,
G
}
=
0
,
∀
α
⟹
G
(
ϵ
(
η
;
α
)
,
t
)
=
G
(
η
,
t
)
{\displaystyle {\frac {dG(\epsilon (\eta ;\alpha ),t)}{d\alpha }}=\{G,G\}=0,\,\forall \alpha \implies G(\epsilon (\eta ;\alpha ),t)=G(\eta ,t)}
This can be used to express the differential equation as:
d
ϵ
μ
(
η
,
t
;
α
)
d
α
=
{
ϵ
μ
(
η
,
t
;
α
)
,
G
(
η
,
t
)
}
η
=:
−
G
~
ϵ
μ
{\displaystyle {\frac {d\epsilon ^{\mu }(\eta ,t;\alpha )}{d\alpha }}=\{\epsilon ^{\mu }(\eta ,t;\alpha ),G(\eta ,t)\}_{\eta }=:-{\tilde {G}}\epsilon ^{\mu }}
where the linear differential operator is defined as
G
~
:=
(
∇
η
G
)
T
J
∇
η
{\displaystyle {\tilde {G}}:=(\nabla _{\eta }G)^{T}J\nabla _{\eta }}
.
=== Active view of transformation ===
Upon iteratively solving the differential equation, the solution of the differential equation follows as:
ϵ
(
η
,
t
;
α
)
=
η
+
α
{
η
,
G
(
η
,
t
)
}
+
1
2
!
α
2
{
{
η
,
G
(
η
,
t
)
}
,
G
(
η
,
t
)
}
+
⋯
=
e
−
α
G
~
η
{\displaystyle \epsilon (\eta ,t;\alpha )=\eta +\alpha \{\eta ,G(\eta ,t)\}+{\frac {1}{2!}}\alpha ^{2}\{\{\eta ,G(\eta ,t)\},G(\eta ,t)\}+\cdots =e^{-\alpha {\tilde {G}}}\eta }
Change in function values
d
f
(
ϵ
(
η
;
α
)
,
t
)
d
α
=
{
f
(
ϵ
(
η
;
α
)
,
t
)
,
G
(
η
,
t
)
}
η
=:
−
G
~
f
(
ϵ
(
η
;
α
)
,
t
)
{\displaystyle {\frac {df(\epsilon (\eta ;\alpha ),t)}{d\alpha }}=\{f(\epsilon (\eta ;\alpha ),t),G(\eta ,t)\}_{\eta }=:-{\tilde {G}}f(\epsilon (\eta ;\alpha ),t)}
by taking repeatedly in steps and using
ϵ
(
η
,
t
;
0
)
=
η
{\displaystyle \epsilon (\eta ,t;0)=\eta }
we get similarly
f
(
e
−
α
G
~
η
,
t
)
=
f
(
ϵ
(
η
;
α
)
,
t
)
=
f
(
η
,
t
)
+
α
{
f
(
η
,
t
)
,
G
(
η
,
t
)
}
+
1
2
!
α
2
{
{
f
(
η
,
t
)
,
G
(
η
,
t
)
}
,
G
(
η
,
t
)
}
+
⋯
=
e
−
α
G
~
f
(
η
,
t
)
{\displaystyle f(e^{-\alpha {\tilde {G}}}\eta ,t)=f(\epsilon (\eta ;\alpha ),t)=f(\eta ,t)+\alpha \{f(\eta ,t),G(\eta ,t)\}+{\frac {1}{2!}}\alpha ^{2}\{\{f(\eta ,t),G(\eta ,t)\},G(\eta ,t)\}+\cdots =e^{-\alpha {\tilde {G}}}f(\eta ,t)}
=== Passive view of transformation ===
Change in a function can be invoked by preserving its values on the same physical states in phase space as
f
(
ϵ
,
t
)
=
f
(
ϵ
(
η
;
α
)
,
t
)
=
f
′
(
ϵ
(
η
;
α
+
δ
α
)
,
t
)
=
f
′
(
ϵ
′
,
t
)
{\displaystyle f(\epsilon ,t)=f(\epsilon (\eta ;\alpha ),t)=f'(\epsilon (\eta ;\alpha +\delta \alpha ),t)=f'(\epsilon ',t)}
can be expressed as upto first order as:
δ
′
f
=
f
′
(
ϵ
)
−
f
(
ϵ
)
=
f
′
(
ϵ
)
−
f
′
(
ϵ
′
)
≈
f
(
ϵ
(
η
;
α
−
δ
α
)
)
−
f
(
ϵ
(
η
;
α
)
)
=
−
δ
α
{
f
,
G
}
{\displaystyle \delta 'f=f'(\epsilon )-f(\epsilon )=f'(\epsilon )-f'(\epsilon ')\approx f(\epsilon (\eta ;\alpha -\delta \alpha ))-f(\epsilon (\eta ;\alpha ))=-\delta \alpha \{f,G\}}
Including the change in the function as some explicit dependance on parameter of transformation
α
{\displaystyle \alpha }
, it can be expressed as
f
(
ϵ
,
t
;
α
)
{\displaystyle f(\epsilon ,t;\alpha )}
where it is explicitly dependant on
α
{\displaystyle \alpha }
such that
∂
f
(
ϵ
,
t
;
α
)
∂
α
=
−
{
f
,
G
}
{\displaystyle {\frac {\partial f(\epsilon ,t;\alpha )}{\partial \alpha }}=-\{f,G\}}
which indicates that the function transforms oppositely to that due to the coordinates to preserve well defined mapping from a physical point in phase space to its scalar values. It is also possible that functions transform without needing to preserve its values on the same physical states in phase space. Such as, for example, the Hamiltonian whose explicit dependance on the canonical transformation can be different from the above form, restated from its previous derivation as
∂
H
(
ϵ
,
t
;
α
)
∂
α
=
d
G
d
t
{\displaystyle {\frac {\partial H(\epsilon ,t;\alpha )}{\partial \alpha }}={\frac {dG}{dt}}}
which is similar to previous relation but also accounts for any explicit time dependence of the generator. Hence, if the Hamiltonian is invariant in passive view for infinitesimal canonical transformations, its generator is a constant of motion.
== Motion as canonical transformation ==
Motion itself (or, equivalently, a shift in the time origin) is a canonical transformation. If
Q
(
t
)
≡
q
(
t
+
τ
)
{\displaystyle \mathbf {Q} (t)\equiv \mathbf {q} (t+\tau )}
and
P
(
t
)
≡
p
(
t
+
τ
)
{\displaystyle \mathbf {P} (t)\equiv \mathbf {p} (t+\tau )}
, then Hamilton's principle is automatically satisfied
δ
∫
t
1
t
2
[
P
⋅
Q
˙
−
K
(
Q
,
P
,
t
)
]
d
t
=
δ
∫
t
1
+
τ
t
2
+
τ
[
p
⋅
q
˙
−
H
(
q
,
p
,
t
+
τ
)
]
d
t
=
0
{\displaystyle \delta \int _{t_{1}}^{t_{2}}\left[\mathbf {P} \cdot {\dot {\mathbf {Q} }}-K(\mathbf {Q} ,\mathbf {P} ,t)\right]dt=\delta \int _{t_{1}+\tau }^{t_{2}+\tau }\left[\mathbf {p} \cdot {\dot {\mathbf {q} }}-H(\mathbf {q} ,\mathbf {p} ,t+\tau )\right]dt=0}
since a valid trajectory
(
q
(
t
)
,
p
(
t
)
)
{\displaystyle (\mathbf {q} (t),\mathbf {p} (t))}
should always satisfy Hamilton's principle, regardless of the endpoints.
== Examples ==
The translation
Q
(
q
,
p
)
=
q
+
a
,
P
(
q
,
p
)
=
p
+
b
{\displaystyle \mathbf {Q} (\mathbf {q} ,\mathbf {p} )=\mathbf {q} +\mathbf {a} ,\mathbf {P} (\mathbf {q} ,\mathbf {p} )=\mathbf {p} +\mathbf {b} }
where
a
,
b
{\displaystyle \mathbf {a} ,\mathbf {b} }
are two constant vectors is a canonical transformation. Indeed, the Jacobian matrix is the identity, which is symplectic:
I
T
J
I
=
J
{\displaystyle I^{\text{T}}JI=J}
.
Set
x
=
(
q
,
p
)
{\displaystyle \mathbf {x} =(q,p)}
and
X
=
(
Q
,
P
)
{\displaystyle \mathbf {X} =(Q,P)}
, the transformation
X
(
x
)
=
R
x
{\displaystyle \mathbf {X} (\mathbf {x} )=R\mathbf {x} }
where
R
∈
S
O
(
2
)
{\displaystyle R\in SO(2)}
is a rotation matrix of order 2 is canonical. Keeping in mind that special orthogonal matrices obey
R
T
R
=
I
{\displaystyle R^{\text{T}}R=I}
it's easy to see that the Jacobian is symplectic. However, this example only works in dimension 2:
S
O
(
2
)
{\displaystyle SO(2)}
is the only special orthogonal group in which every matrix is symplectic. Note that the rotation here acts on
(
q
,
p
)
{\displaystyle (q,p)}
and not on
q
{\displaystyle q}
and
p
{\displaystyle p}
independently, so these are not the same as a physical rotation of an orthogonal spatial coordinate system.
The transformation
(
Q
(
q
,
p
)
,
P
(
q
,
p
)
)
=
(
q
+
f
(
p
)
,
p
)
{\displaystyle (Q(q,p),P(q,p))=(q+f(p),p)}
, where
f
(
p
)
{\displaystyle f(p)}
is an arbitrary function of
p
{\displaystyle p}
, is canonical. Jacobian matrix is indeed given by
∂
X
∂
x
=
[
1
f
′
(
p
)
0
1
]
{\displaystyle {\frac {\partial X}{\partial x}}={\begin{bmatrix}1&f'(p)\\0&1\end{bmatrix}}}
which is symplectic.
== Modern mathematical description ==
In mathematical terms, canonical coordinates are any coordinates on the phase space (cotangent bundle) of the system that allow the canonical one-form to be written as
∑
i
p
i
d
q
i
{\displaystyle \sum _{i}p_{i}\,dq^{i}}
up to a total differential (exact form). The change of variable between one set of canonical coordinates and another is a canonical transformation. The index of the generalized coordinates q is written here as a superscript (
q
i
{\displaystyle q^{i}}
), not as a subscript as done above (
q
i
{\displaystyle q_{i}}
). The superscript conveys the contravariant transformation properties of the generalized coordinates, and does not mean that the coordinate is being raised to a power. Further details may be found at the symplectomorphism article.
== History ==
The first major application of the canonical transformation was in 1846, by Charles Delaunay, in the study of the Earth-Moon-Sun system. This work resulted in the publication of a pair of large volumes as Mémoires by the French Academy of Sciences, in 1860 and 1867.
== See also ==
Symplectomorphism
Hamilton–Jacobi equation
Liouville's theorem (Hamiltonian)
Mathieu transformation
Linear canonical transformation
== Notes ==
== References ==
Goldstein, Herbert; Poole, Charles P.; Safko, John L. (2007). Classical mechanics (3rd ed.). Upper Saddle River, N.J: Pearson [u.a.] ISBN 978-0-321-18897-7.
Landau, L. D.; Lifshitz, E. M. (1975) [1939]. Mechanics. Translated by Bell, S. J.; Sykes, J. B. (3rd ed.). Amsterdam: Elsevier. ISBN 978-0-7506-28969.
Giacaglia, Georgio Eugenio Oscare (1972). Perturbation Methods in Non-Linear Systems. New York: Springer-Verlag. ISBN 3-540-90054-3. LCCN 72-87714.
Lanczos, Cornelius (2012-04-24). The Variational Principles of Mechanics. Courier Corporation. ISBN 978-0-486-13470-3.
Lurie, Anatolii I. (2002). Analytical Mechanics (1st ed.). Springer-Verlag Berlin. ISBN 978-3-642-53650-2.
Gupta, Praveen P.; Gupta, Sanjay (2008). Rigid Dynamics (10th ed.). Krishna Prakashan Media.
Johns, Oliver Davis (2005). Analytical Mechanics for Relativity and Quantum Mechanics. Oxford University Press. ISBN 978-0-19-856726-4.
Lemos, Nivaldo A (2018). Analytical mechanics. Cambridge University Press. ISBN 978-1-108-41658-0.
Hand, Louis N.; Finch, Janet D. (1999). Analytical Mechanics (1st ed.). Cambridge University Press. ISBN 978-0521573276.
Sudarshan, E C George; Mukunda, N (2010). Classical Dynamics: A Modern Perspective. Wiley. ISBN 9780471835400. | Wikipedia/Canonical_transformation |
In atomic physics, the Bohr model or Rutherford–Bohr model was a model of the atom that incorporated some early quantum concepts. Developed from 1911 to 1918 by Niels Bohr and building on Ernest Rutherford's nuclear model, it supplanted the plum pudding model of J. J. Thomson only to be replaced by the quantum atomic model in the 1920s. It consists of a small, dense nucleus surrounded by orbiting electrons. It is analogous to the structure of the Solar System, but with attraction provided by electrostatic force rather than gravity, and with the electron energies quantized (assuming only discrete values).
In the history of atomic physics, it followed, and ultimately replaced, several earlier models, including Joseph Larmor's Solar System model (1897), Jean Perrin's model (1901), the cubical model (1902), Hantaro Nagaoka's Saturnian model (1904), the plum pudding model (1904), Arthur Haas's quantum model (1910), the Rutherford model (1911), and John William Nicholson's nuclear quantum model (1912). The improvement over the 1911 Rutherford model mainly concerned the new quantum mechanical interpretation introduced by Haas and Nicholson, but forsaking any attempt to explain radiation according to classical physics.
The model's key success lies in explaining the Rydberg formula for hydrogen's spectral emission lines. While the Rydberg formula had been known experimentally, it did not gain a theoretical basis until the Bohr model was introduced. Not only did the Bohr model explain the reasons for the structure of the Rydberg formula, it also provided a justification for the fundamental physical constants that make up the formula's empirical results.
The Bohr model is a relatively primitive model of the hydrogen atom, compared to the valence shell model. As a theory, it can be derived as a first-order approximation of the hydrogen atom using the broader and much more accurate quantum mechanics and thus may be considered to be an obsolete scientific theory. However, because of its simplicity, and its correct results for selected systems (see below for application), the Bohr model is still commonly taught to introduce students to quantum mechanics or energy level diagrams before moving on to the more accurate, but more complex, valence shell atom. A related quantum model was proposed by Arthur Erich Haas in 1910 but was rejected until the 1911 Solvay Congress where it was thoroughly discussed. The quantum theory of the period between Planck's discovery of the quantum (1900) and the advent of a mature quantum mechanics (1925) is often referred to as the old quantum theory.
== Background ==
Until the second decade of the 20th century, atomic models were generally speculative. Even the concept of atoms, let alone atoms with internal structure, faced opposition from some scientists.: 2
=== Planetary models ===
In the late 1800s speculations on the possible structure of the atom included planetary models with orbiting charged electrons.: 35
These models faced a significant constraint.
In 1897, Joseph Larmor showed that an accelerating charge would radiate power according to classical electrodynamics, a result known as the Larmor formula. Since electrons forced to remain in orbit are continuously accelerating, they would be mechanically unstable. Larmor noted that electromagnetic effect of multiple electrons, suitable arranged, would cancel each other. Thus subsequent atomic models based on classical electrodynamics needed to adopt such special multi-electron arrangements.: 113
=== Thomson's atom model ===
When Bohr began his work on a new atomic theory in the summer of 1912: 237 the atomic model proposed by J. J. Thomson, now known as the plum pudding model, was the best available.: 37 Thomson proposed a model with electrons rotating in coplanar rings within an atomic-sized, positively-charged, spherical volume. Thomson showed that this model was mechanically stable by lengthy calculations and was electrodynamically stable under his original assumption of thousands of electrons per atom. Moreover, he suggested that the particularly stable configurations of electrons in rings was connected to chemical properties of the atoms. He developed a formula for the scattering of beta particles that seemed to match experimental results.: 38
However Thomson himself later showed that the atom had a factor of a thousand fewer electrons, challenging the stability argument and forcing the poorly understood positive sphere to have most of the atom's mass. Thomson was also unable to explain the many lines in atomic spectra.: 18
=== Rutherford nuclear model ===
In 1908, Hans Geiger and Ernest Marsden demonstrated that alpha particle occasionally scatter at large angles, a result inconsistent with Thomson's model.
In 1911 Ernest Rutherford developed a new scattering model, showing that the observed large angle scattering could be explained by a compact, highly charged mass at the center of the atom.
Rutherford scattering did not involve the electrons and thus his model of the atom was incomplete.
Bohr begins his first paper on his atomic model by describing Rutherford's atom as consisting of a small, dense, positively charged nucleus attracting negatively charged electrons.
=== Atomic spectra ===
By the early twentieth century, it was expected that the atom would account for the many atomic spectral lines. These lines were summarized in empirical formula by Johann Balmer and Johannes Rydberg. In 1897, Lord Rayleigh showed that vibrations of electrical systems predicted spectral lines that depend on the square of the vibrational frequency, contradicting the empirical formula which depended directly on the frequency.: 18
In 1907 Arthur W. Conway showed that, rather than the entire atom vibrating, vibrations of only one of the electrons in the system described by Thomson might be sufficient to account for spectral series.: II:106 Although Bohr's model would also rely on just the electron to explain the spectrum, he did not assume an electrodynamical model for the atom.
The other important advance in the understanding of atomic spectra was the Rydberg–Ritz combination principle which related atomic spectral line frequencies to differences between 'terms', special frequencies characteristic of each element.: 173 Bohr would recognize the terms as energy levels of the atom divided by the Planck constant, leading to the modern view that the spectral lines result from energy differences.: 847
=== Haas atomic model ===
In 1910, Arthur Erich Haas proposed a model of the hydrogen atom with an electron circulating on the surface of a sphere of positive charge. The model resembled Thomson's plum pudding model, but Haas added a radical new twist: he constrained the electron's potential energy,
E
pot
{\displaystyle E_{\text{pot}}}
, on a sphere of radius a to equal the frequency, f, of the electron's orbit on the sphere times the Planck constant:: 197
E
pot
=
−
e
2
a
=
h
f
{\displaystyle E_{\text{pot}}={\frac {-e^{2}}{a}}=hf}
where e represents the charge on the electron and the sphere. Haas combined this constraint with the balance-of-forces equation. The attractive force between the electron and the sphere balances the centrifugal force:
e
2
a
2
=
m
a
(
2
π
f
)
2
{\displaystyle {\frac {e^{2}}{a^{2}}}=ma(2\pi f)^{2}}
where m is the mass of the electron. This combination relates the radius of the sphere to the Planck constant:
a
=
h
2
4
π
2
e
2
m
{\displaystyle a={\frac {h^{2}}{4\pi ^{2}e^{2}m}}}
Haas solved for the Planck constant using the then-current value for the radius of the hydrogen atom.
Three years later, Bohr would use similar equations with different interpretation. Bohr took the Planck constant as given value and used the equations to predict, a, the radius of the electron orbiting in the ground state of the hydrogen atom. This value is now called the Bohr radius.: 197
=== Influence of the Solvay Conference ===
The first Solvay Conference, in 1911, was one of the first international physics conferences. Nine Nobel or future Nobel laureates attended, including
Ernest Rutherford, Bohr's mentor.: 271
Bohr did not attend but he read the Solvay reports and discussed them with Rutherford.: 233
The subject of the conference was the theory of radiation and the energy quanta of Max Planck's oscillators.
Planck's lecture at the conference ended with comments about atoms and the discussion that followed it concerned atomic models. Hendrik Lorentz raised the question of the composition of the atom based on Haas's model, a form of Thomson's plum pudding model with a quantum modification. Lorentz explained that the size of atoms could be taken to determine the Planck constant as Haas had done or the Planck constant could be taken as determining the size of atoms.: 273 Bohr would adopt the second path.
The discussions outlined the need for the quantum theory to be included in the atom. Planck explicitly mentions the failings of classical mechanics.: 273 While Bohr had already expressed a similar opinion in his PhD thesis, at Solvay the leading scientists of the day discussed a break with classical theories.: 244 Bohr's first paper on his atomic model cites the Solvay proceedings saying: "Whatever the alteration in the laws of motion of the electrons may be, it seems necessary to introduce in the laws in question a quantity foreign to the classical electrodynamics, i.e. Planck's constant, or as it often is called the elementary quantum of action." Encouraged by the Solvay discussions, Bohr would assume the atom was stable and abandon the efforts to stabilize classical models of the atom: 199
=== Nicholson atom theory ===
In 1911 John William Nicholson published a model of the atom which would influence Bohr's model. Nicholson developed his model based on the analysis of astrophysical spectroscopy. He connected the observed spectral line frequencies with the orbits of electrons in his atoms. The connection he adopted associated the atomic electron orbital angular momentum with the Planck constant.
Whereas Planck focused on a quantum of energy, Nicholson's angular momentum quantum relates to orbital frequency.
This new concept gave Planck constant an atomic meaning for the first time.: 169 In his 1913 paper Bohr cites Nicholson as finding quantized angular momentum important for the atom.
The other critical influence of Nicholson work was his detailed analysis of spectra. Before Nicholson's work Bohr thought the spectral data was not useful for understanding atoms. In comparing his work to Nicholson's, Bohr came to understand the spectral data and their value. When he then learned from a friend about Balmer's compact formula for the spectral line data, Bohr quickly realized his model would match it in detail.: 178
Nicholson's model was based on classical electrodynamics along the lines of J.J. Thomson's plum pudding model but his negative electrons orbiting a positive nucleus rather than circulating in a sphere. To avoid immediate collapse of this system he required that electrons come in pairs so the rotational acceleration of each electron was matched across the orbit.: 163 By 1913 Bohr had already shown, from the analysis of alpha particle energy loss, that hydrogen had only a single electron not a matched pair.: 195 Bohr's atomic model would abandon classical electrodynamics.
Nicholson's model of radiation was quantum but was attached to the orbits of the electrons. Bohr quantization would associate it with differences in energy levels of his model of hydrogen rather than the orbital frequency.
=== Bohr's previous work ===
Bohr completed his PhD in 1911 with a thesis 'Studies on the Electron Theory of Metals', an application of the classical electron theory of Hendrik Lorentz. Bohr noted two deficits of the classical model. The first concerned the specific heat of metals which James Clerk Maxwell noted in 1875: every additional degree of freedom in a theory of metals, like subatomic electrons, cause more disagreement with experiment. The second, the classical theory could not explain magnetism.: 194
After his PhD, Bohr worked briefly in the lab of JJ Thomson before moving to Rutherford's lab in Manchester to study radioactivity. He arrived just after Rutherford completed his proposal of a compact nuclear core for atoms. Charles Galton Darwin, also at Manchester, had just completed an analysis of alpha particle energy loss in metals, concluding the electron collisions where the dominant cause of loss. Bohr showed in a subsequent paper that Darwin's results would improve by accounting for electron binding energy. Importantly this allowed Bohr to conclude that hydrogen atoms have a single electron.: 195
== Development ==
Next, Bohr was told by his friend, Hans Hansen, that the Balmer series is calculated using the Balmer formula, an empirical equation discovered by Johann Balmer in 1885 that described wavelengths of some spectral lines of hydrogen. This was further generalized by Johannes Rydberg in 1888, resulting in what is now known as the Rydberg formula.
After this, Bohr declared, "everything became clear".
In 1913 Niels Bohr put forth three postulates to provide an electron model consistent with Rutherford's nuclear model:
The electron is able to revolve in certain stable orbits around the nucleus without radiating any energy, contrary to what classical electromagnetism suggests. These stable orbits are called stationary orbits and are attained at certain discrete distances from the nucleus. The electron cannot have any other orbit in between the discrete ones.
The stationary orbits are attained at distances for which the angular momentum of the revolving electron is an integer multiple of the reduced Planck constant:
m
e
v
r
=
n
ℏ
{\displaystyle m_{\mathrm {e} }vr=n\hbar }
, where
n
=
1
,
2
,
3
,
.
.
.
{\displaystyle n=1,2,3,...}
is called the principal quantum number, and
ℏ
=
h
/
2
π
{\displaystyle \hbar =h/2\pi }
. The lowest value of
n
{\displaystyle n}
is 1; this gives the smallest possible orbital radius, known as the Bohr radius, of 0.0529 nm for hydrogen. Once an electron is in this lowest orbit, it can get no closer to the nucleus. Starting from the angular momentum quantum rule as Bohr admits is previously given by Nicholson in his 1912 paper, Bohr was able to calculate the energies of the allowed orbits of the hydrogen atom and other hydrogen-like atoms and ions. These orbits are associated with definite energies and are also called energy shells or energy levels. In these orbits, the electron's acceleration does not result in radiation and energy loss. The Bohr model of an atom was based upon Planck's quantum theory of radiation.
Electrons can only gain and lose energy by jumping from one allowed orbit to another, absorbing or emitting electromagnetic radiation with a frequency
ν
{\displaystyle \nu }
determined by the energy difference of the levels according to the Planck relation:
Δ
E
=
E
2
−
E
1
=
h
ν
{\displaystyle \Delta E=E_{2}-E_{1}=h\nu }
, where
h
{\displaystyle h}
is the Planck constant.
Other points are:
Like Einstein's theory of the photoelectric effect, Bohr's formula assumes that during a quantum jump a discrete amount of energy is radiated. However, unlike Einstein, Bohr stuck to the classical Maxwell theory of the electromagnetic field. Quantization of the electromagnetic field was explained by the discreteness of the atomic energy levels; Bohr did not believe in the existence of photons.
According to the Maxwell theory the frequency
ν
{\displaystyle \nu }
of classical radiation is equal to the rotation frequency
ν
{\displaystyle \nu }
rot of the electron in its orbit, with harmonics at integer multiples of this frequency. This result is obtained from the Bohr model for jumps between energy levels
E
n
{\displaystyle E_{n}}
and
E
n
−
k
{\displaystyle E_{n-k}}
when
k
{\displaystyle k}
is much smaller than
n
{\displaystyle n}
. These jumps reproduce the frequency of the
k
{\displaystyle k}
-th harmonic of orbit
n
{\displaystyle n}
. For sufficiently large values of
n
{\displaystyle n}
(so-called Rydberg states), the two orbits involved in the emission process have nearly the same rotation frequency, so that the classical orbital frequency is not ambiguous. But for small
n
{\displaystyle n}
(or large
k
{\displaystyle k}
), the radiation frequency has no unambiguous classical interpretation. This marks the birth of the correspondence principle, requiring quantum theory to agree with the classical theory only in the limit of large quantum numbers.
The Bohr–Kramers–Slater theory (BKS theory) is a failed attempt to extend the Bohr model, which violates the conservation of energy and momentum in quantum jumps, with the conservation laws only holding on average.
Bohr's condition, that the angular momentum be an integer multiple of
ℏ
{\displaystyle \hbar }
, was later reinterpreted in 1924 by de Broglie as a standing wave condition: the electron is described by a wave and a whole number of wavelengths must fit along the circumference of the electron's orbit:
n
λ
=
2
π
r
.
{\displaystyle n\lambda =2\pi r.}
According to de Broglie's hypothesis, matter particles such as the electron behave as waves. The de Broglie wavelength of an electron is
λ
=
h
m
v
,
{\displaystyle \lambda ={\frac {h}{mv}},}
which implies that
n
h
m
v
=
2
π
r
,
{\displaystyle {\frac {nh}{mv}}=2\pi r,}
or
n
h
2
π
=
m
v
r
,
{\displaystyle {\frac {nh}{2\pi }}=mvr,}
where
m
v
r
{\displaystyle mvr}
is the angular momentum of the orbiting electron. Writing
ℓ
{\displaystyle \ell }
for this angular momentum, the previous equation becomes
ℓ
=
n
h
2
π
,
{\displaystyle \ell ={\frac {nh}{2\pi }},}
which is Bohr's second postulate.
Bohr described angular momentum of the electron orbit as
2
/
h
{\displaystyle 2/h}
while de Broglie's wavelength of
λ
=
h
/
p
{\displaystyle \lambda =h/p}
described
h
{\displaystyle h}
divided by the electron momentum. In 1913, however, Bohr justified his rule by appealing to the correspondence principle, without providing any sort of wave interpretation. In 1913, the wave behavior of matter particles such as the electron was not suspected.
In 1925, a new kind of mechanics was proposed, quantum mechanics, in which Bohr's model of electrons traveling in quantized orbits was extended into a more accurate model of electron motion. The new theory was proposed by Werner Heisenberg. Another form of the same theory, wave mechanics, was discovered by the Austrian physicist Erwin Schrödinger independently, and by different reasoning. Schrödinger employed de Broglie's matter waves, but sought wave solutions of a three-dimensional wave equation describing electrons that were constrained to move about the nucleus of a hydrogen-like atom, by being trapped by the potential of the positive nuclear charge.
== Electron energy levels ==
The Bohr model gives almost exact results only for a system where two charged points orbit each other at speeds much less than that of light. This not only involves one-electron systems such as the hydrogen atom, singly ionized helium, and doubly ionized lithium, but it includes positronium and Rydberg states of any atom where one electron is far away from everything else. It can be used for K-line X-ray transition calculations if other assumptions are added (see Moseley's law below). In high energy physics, it can be used to calculate the masses of heavy quark mesons.
Calculation of the orbits requires two assumptions.
Classical mechanics
The electron is held in a circular orbit by electrostatic attraction. The centripetal force is equal to the Coulomb force.
m
e
v
2
r
=
Z
k
e
e
2
r
2
,
{\displaystyle {\frac {m_{\mathrm {e} }v^{2}}{r}}={\frac {Zk_{\mathrm {e} }e^{2}}{r^{2}}},}
where me is the electron's mass, e is the elementary charge, ke is the Coulomb constant and Z is the atom's atomic number. It is assumed here that the mass of the nucleus is much larger than the electron mass (which is a good assumption). This equation determines the electron's speed at any radius:
v
=
Z
k
e
e
2
m
e
r
.
{\displaystyle v={\sqrt {\frac {Zk_{\mathrm {e} }e^{2}}{m_{\mathrm {e} }r}}}.}
It also determines the electron's total energy at any radius:
E
=
−
1
2
m
e
v
2
.
{\displaystyle E=-{\frac {1}{2}}m_{\mathrm {e} }v^{2}.}
The total energy is negative and inversely proportional to r. This means that it takes energy to pull the orbiting electron away from the proton. For infinite values of r, the energy is zero, corresponding to a motionless electron infinitely far from the proton. The total energy is half the potential energy, the difference being the kinetic energy of the electron. This is also true for noncircular orbits by the virial theorem.
A quantum rule
The angular momentum L = mevr is an integer multiple of ħ:
m
e
v
r
=
n
ℏ
.
{\displaystyle m_{\mathrm {e} }vr=n\hbar .}
=== Derivation ===
In classical mechanics, if an electron is orbiting around an atom with period T, and if its coupling to the electromagnetic field is weak, so that the orbit doesn't decay very much in one cycle, it will emit electromagnetic radiation in a pattern repeating at every period, so that the Fourier transform of the pattern will only have frequencies which are multiples of 1/T.
However, in quantum mechanics, the quantization of angular momentum leads to discrete energy levels of the orbits, and the emitted frequencies are quantized according to the energy differences between these levels. This discrete nature of energy levels introduces a fundamental departure from the classical radiation law, giving rise to distinct spectral lines in the emitted radiation.
Bohr assumes that the electron is circling the nucleus in an elliptical orbit obeying the rules of classical mechanics, but with no loss of radiation due to the Larmor formula.
Denoting the total energy as E, the electron charge as −e, the nucleus charge as K = Ze, the electron mass as me, half the major axis of the ellipse as a, he starts with these equations:: 3
E is assumed to be negative, because a positive energy is required to unbind the electron from the nucleus and put it at rest at an infinite distance.
Eq. (1a) is obtained from equating the centripetal force to the Coulombian force acting between the nucleus and the electron, considering that
E
=
T
+
U
{\displaystyle E=T+U}
(where T is the average kinetic energy and U the average electrostatic potential), and that for Kepler's second law, the average separation between the electron and the nucleus is a.
Eq. (1b) is obtained from the same premises of eq. (1a) plus the virial theorem, stating that, for an elliptical orbit,
Then Bohr assumes that
|
E
|
{\displaystyle \vert E\vert }
is an integer multiple of the energy of a quantum of light with half the frequency of the electron's revolution frequency,: 4 i.e.:
From eq. (1a, 1b, 2), it descends:
He further assumes that the orbit is circular, i.e.
a
=
r
{\displaystyle a=r}
, and, denoting the angular momentum of the electron as L, introduces the equation:
Eq. (4) stems from the virial theorem, and from the classical mechanics relationships between the angular momentum, the kinetic energy and the frequency of revolution.
From eq. (1c, 2, 4), it stems:
where:
that is:
This results states that the angular momentum of the electron is an integer multiple of the reduced Planck constant.: 15
Substituting the expression for the velocity gives an equation for r in terms of n:
m
e
k
e
Z
e
2
m
e
r
r
=
n
ℏ
,
{\displaystyle m_{\text{e}}{\sqrt {\dfrac {k_{\text{e}}Ze^{2}}{m_{\text{e}}r}}}r=n\hbar ,}
so that the allowed orbit radius at any n is
r
n
=
n
2
ℏ
2
Z
k
e
e
2
m
e
.
{\displaystyle r_{n}={\frac {n^{2}\hbar ^{2}}{Zk_{\mathrm {e} }e^{2}m_{\mathrm {e} }}}.}
The smallest possible value of r in the hydrogen atom (Z = 1) is called the Bohr radius and is equal to:
r
1
=
ℏ
2
k
e
e
2
m
e
≈
5.29
×
10
−
11
m
=
52.9
p
m
.
{\displaystyle r_{1}={\frac {\hbar ^{2}}{k_{\mathrm {e} }e^{2}m_{\mathrm {e} }}}\approx 5.29\times 10^{-11}~\mathrm {m} =52.9~\mathrm {pm} .}
The energy of the n-th level for any atom is determined by the radius and quantum number:
E
=
−
Z
k
e
e
2
2
r
n
=
−
Z
2
(
k
e
e
2
)
2
m
e
2
ℏ
2
n
2
≈
−
13.6
Z
2
n
2
e
V
.
{\displaystyle E=-{\frac {Zk_{\mathrm {e} }e^{2}}{2r_{n}}}=-{\frac {Z^{2}(k_{\mathrm {e} }e^{2})^{2}m_{\mathrm {e} }}{2\hbar ^{2}n^{2}}}\approx {\frac {-13.6\ Z^{2}}{n^{2}}}~\mathrm {eV} .}
An electron in the lowest energy level of hydrogen (n = 1) therefore has about 13.6 eV less energy than a motionless electron infinitely far from the nucleus. The next energy level (n = 2) is −3.4 eV. The third (n = 3) is −1.51 eV, and so on. For larger values of n, these are also the binding energies of a highly excited atom with one electron in a large circular orbit around the rest of the atom. The hydrogen formula also coincides with the Wallis product.
The combination of natural constants in the energy formula is called the Rydberg energy (RE):
R
E
=
(
k
e
e
2
)
2
m
e
2
ℏ
2
.
{\displaystyle R_{\mathrm {E} }={\frac {(k_{\mathrm {e} }e^{2})^{2}m_{\mathrm {e} }}{2\hbar ^{2}}}.}
This expression is clarified by interpreting it in combinations that form more natural units:
m
e
c
2
{\displaystyle m_{\mathrm {e} }c^{2}}
is the rest mass energy of the electron (511 keV),
k
e
e
2
ℏ
c
=
α
≈
1
137
{\displaystyle {\frac {k_{\mathrm {e} }e^{2}}{\hbar c}}=\alpha \approx {\frac {1}{137}}}
is the fine-structure constant,
R
E
=
1
2
(
m
e
c
2
)
α
2
{\displaystyle R_{\mathrm {E} }={\frac {1}{2}}(m_{\mathrm {e} }c^{2})\alpha ^{2}}
.
Since this derivation is with the assumption that the nucleus is orbited by one electron, we can generalize this result by letting the nucleus have a charge q = Ze, where Z is the atomic number. This will now give us energy levels for hydrogenic (hydrogen-like) atoms, which can serve as a rough order-of-magnitude approximation of the actual energy levels. So for nuclei with Z protons, the energy levels are (to a rough approximation):
E
n
=
−
Z
2
R
E
n
2
.
{\displaystyle E_{n}=-{\frac {Z^{2}R_{\mathrm {E} }}{n^{2}}}.}
The actual energy levels cannot be solved analytically for more than one electron (see n-body problem) because the electrons are not only affected by the nucleus but also interact with each other via the Coulomb force.
When Z = 1/α (Z ≈ 137), the motion becomes highly relativistic, and Z2 cancels the α2 in R; the orbit energy begins to be comparable to rest energy. Sufficiently large nuclei, if they were stable, would reduce their charge by creating a bound electron from the vacuum, ejecting the positron to infinity. This is the theoretical phenomenon of electromagnetic charge screening which predicts a maximum nuclear charge. Emission of such positrons has been observed in the collisions of heavy ions to create temporary super-heavy nuclei.
The Bohr formula properly uses the reduced mass of electron and proton in all situations, instead of the mass of the electron,
m
red
=
m
e
m
p
m
e
+
m
p
=
m
e
1
1
+
m
e
/
m
p
.
{\displaystyle m_{\text{red}}={\frac {m_{\mathrm {e} }m_{\mathrm {p} }}{m_{\mathrm {e} }+m_{\mathrm {p} }}}=m_{\mathrm {e} }{\frac {1}{1+m_{\mathrm {e} }/m_{\mathrm {p} }}}.}
However, these numbers are very nearly the same, due to the much larger mass of the proton, about 1836.1 times the mass of the electron, so that the reduced mass in the system is the mass of the electron multiplied by the constant 1836.1/(1 + 1836.1) = 0.99946. This fact was historically important in convincing Rutherford of the importance of Bohr's model, for it explained the fact that the frequencies of lines in the spectra for singly ionized helium do not differ from those of hydrogen by a factor of exactly 4, but rather by 4 times the ratio of the reduced mass for the hydrogen vs. the helium systems, which was much closer to the experimental ratio than exactly 4.
For positronium, the formula uses the reduced mass also, but in this case, it is exactly the electron mass divided by 2. For any value of the radius, the electron and the positron are each moving at half the speed around their common center of mass, and each has only one fourth the kinetic energy. The total kinetic energy is half what it would be for a single electron moving around a heavy nucleus.
E
n
=
R
E
2
n
2
{\displaystyle E_{n}={\frac {R_{\mathrm {E} }}{2n^{2}}}}
(positronium).
== Rydberg formula ==
Beginning in late 1860s, Johann Balmer and later Johannes Rydberg and Walther Ritz developed increasingly accurate empirical formula matching measured atomic spectral lines.
Critical for Bohr's later work, Rydberg expressed his formula in terms of wave-number, equivalent to frequency. These formula contained a constant,
R
{\displaystyle R}
, now known the Rydberg constant and a pair of integers indexing the lines:: 247
ν
=
R
(
1
m
2
−
1
n
2
)
.
{\displaystyle \nu =R\left({\frac {1}{m^{2}}}-{\frac {1}{n^{2}}}\right).}
Despite many attempts, no theory of the atom could reproduce these relatively simple formula.: 169
In Bohr's theory describing the energies of transitions or quantum jumps between orbital energy levels is able to explain these formula. For the hydrogen atom Bohr starts with his derived formula for the energy released as a free electron moves into a stable circular orbit indexed by
τ
{\displaystyle \tau }
:
W
τ
=
2
π
2
m
e
4
h
2
τ
2
{\displaystyle W_{\tau }={\frac {2\pi ^{2}me^{4}}{h^{2}\tau ^{2}}}}
The energy difference between two such levels is then:
h
ν
=
W
τ
2
−
W
τ
1
=
2
π
2
m
e
4
h
2
(
1
τ
2
2
−
1
τ
1
2
)
{\displaystyle h\nu =W_{\tau _{2}}-W_{\tau _{1}}={\frac {2\pi ^{2}me^{4}}{h^{2}}}\left({\frac {1}{\tau _{2}^{2}}}-{\frac {1}{\tau _{1}^{2}}}\right)}
Therefore, Bohr's theory gives the Rydberg formula and moreover the numerical value the Rydberg constant for hydrogen in terms of more fundamental constants of nature, including the electron's charge, the electron's mass, and the Planck constant:: 31
c
R
H
=
2
π
2
m
e
4
h
3
.
{\displaystyle cR_{\text{H}}={\frac {2\pi ^{2}me^{4}}{h^{3}}}.}
Since the energy of a photon is
E
=
h
c
λ
,
{\displaystyle E={\frac {hc}{\lambda }},}
these results can be expressed in terms of the wavelength of the photon given off:
1
λ
=
R
(
1
n
f
2
−
1
n
i
2
)
.
{\displaystyle {\frac {1}{\lambda }}=R\left({\frac {1}{n_{\text{f}}^{2}}}-{\frac {1}{n_{\text{i}}^{2}}}\right).}
Bohr's derivation of the Rydberg constant, as well as the concomitant agreement of Bohr's formula with experimentally observed spectral lines of the Lyman (nf = 1), Balmer (nf = 2), and Paschen (nf = 3) series, and successful theoretical prediction of other lines not yet observed, was one reason that his model was immediately accepted.: 34
To apply to atoms with more than one electron, the Rydberg formula can be modified by replacing Z with Z − b or n with n − b where b is constant representing a screening effect due to the inner-shell and other electrons (see Electron shell and the later discussion of the "Shell Model of the Atom" below). This was established empirically before Bohr presented his model.
== Shell model (heavier atoms) ==
Bohr's original three papers in 1913 described mainly the electron configuration in lighter elements. Bohr called his electron shells, "rings" in 1913. Atomic orbitals within shells did not exist at the time of his planetary model. Bohr explains in Part 3 of his famous 1913 paper that the maximum electrons in a shell is eight, writing: "We see, further, that a ring of n electrons cannot rotate in a single ring round a nucleus of charge ne unless n < 8." For smaller atoms, the electron shells would be filled as follows: "rings of electrons will only join together if they contain equal numbers of electrons; and that accordingly the numbers of electrons on inner rings will only be 2, 4, 8". However, in larger atoms the innermost shell would contain eight electrons, "on the other hand, the periodic system of the elements strongly suggests that already in neon N = 10 an inner ring of eight electrons will occur". Bohr wrote "From the above we are led to the following possible scheme for the arrangement of the electrons in light atoms:"
In Bohr's third 1913 paper Part III called "Systems Containing Several Nuclei", he says that two atoms form molecules on a symmetrical plane and he reverts to describing hydrogen. The 1913 Bohr model did not discuss higher elements in detail and John William Nicholson was one of the first to prove in 1914 that it couldn't work for lithium, but was an attractive theory for hydrogen and ionized helium.
In 1921, following the work of chemists and others involved in work on the periodic table, Bohr extended the model of hydrogen to give an approximate model for heavier atoms. This gave a physical picture that reproduced many known atomic properties for the first time although these properties were proposed contemporarily with the identical work of chemist Charles Rugeley Bury
Bohr's partner in research during 1914 to 1916 was Walther Kossel who corrected Bohr's work to show that electrons interacted through the outer rings, and Kossel called the rings: "shells". Irving Langmuir is credited with the first viable arrangement of electrons in shells with only two in the first shell and going up to eight in the next according to the octet rule of 1904, although Kossel had already predicted a maximum of eight per shell in 1916. Heavier atoms have more protons in the nucleus, and more electrons to cancel the charge. Bohr took from these chemists the idea that each discrete orbit could only hold a certain number of electrons. Per Kossel, after that the orbit is full, the next level would have to be used. This gives the atom a shell structure designed by Kossel, Langmuir, and Bury, in which each shell corresponds to a Bohr orbit.
This model is even more approximate than the model of hydrogen, because it treats the electrons in each shell as non-interacting. But the repulsions of electrons are taken into account somewhat by the phenomenon of screening. The electrons in outer orbits do not only orbit the nucleus, but they also move around the inner electrons, so the effective charge Z that they feel is reduced by the number of the electrons in the inner orbit.
For example, the lithium atom has two electrons in the lowest 1s orbit, and these orbit at Z = 2. Each one sees the nuclear charge of Z = 3 minus the screening effect of the other, which crudely reduces the nuclear charge by 1 unit. This means that the innermost electrons orbit at approximately 1/2 the Bohr radius. The outermost electron in lithium orbits at roughly the Bohr radius, since the two inner electrons reduce the nuclear charge by 2. This outer electron should be at nearly one Bohr radius from the nucleus. Because the electrons strongly repel each other, the effective charge description is very approximate; the effective charge Z doesn't usually come out to be an integer.
The shell model was able to qualitatively explain many of the mysterious properties of atoms which became codified in the late 19th century in the periodic table of the elements. One property was the size of atoms, which could be determined approximately by measuring the viscosity of gases and density of pure crystalline solids. Atoms tend to get smaller toward the right in the periodic table, and become much larger at the next line of the table. Atoms to the right of the table tend to gain electrons, while atoms to the left tend to lose them. Every element on the last column of the table is chemically inert (noble gas).
In the shell model, this phenomenon is explained by shell-filling. Successive atoms become smaller because they are filling orbits of the same size, until the orbit is full, at which point the next atom in the table has a loosely bound outer electron, causing it to expand. The first Bohr orbit is filled when it has two electrons, which explains why helium is inert. The second orbit allows eight electrons, and when it is full the atom is neon, again inert. The third orbital contains eight again, except that in the more correct Sommerfeld treatment (reproduced in modern quantum mechanics) there are extra "d" electrons. The third orbit may hold an extra 10 d electrons, but these positions are not filled until a few more orbitals from the next level are filled (filling the n = 3 d-orbitals produces the 10 transition elements). The irregular filling pattern is an effect of interactions between electrons, which are not taken into account in either the Bohr or Sommerfeld models and which are difficult to calculate even in the modern treatment.
== Moseley's law and calculation (K-alpha X-ray emission lines) ==
Niels Bohr said in 1962: "You see actually the Rutherford work was not taken seriously. We cannot understand today, but it was not taken seriously at all. There was no mention of it any place. The great change came from Moseley."
In 1913, Henry Moseley found an empirical relationship between the strongest X-ray line emitted by atoms under electron bombardment (then known as the K-alpha line), and their atomic number Z. Moseley's empiric formula was found to be derivable from Rydberg's formula and later Bohr's formula (Moseley actually mentions only Ernest Rutherford and Antonius Van den Broek in terms of models as these had been published before Moseley's work and Moseley's 1913 paper was published the same month as the first Bohr model paper). The two additional assumptions that [1] this X-ray line came from a transition between energy levels with quantum numbers 1 and 2, and [2], that the atomic number Z when used in the formula for atoms heavier than hydrogen, should be diminished by 1, to (Z − 1)2.
Moseley wrote to Bohr, puzzled about his results, but Bohr was not able to help. At that time, he thought that the postulated innermost "K" shell of electrons should have at least four electrons, not the two which would have neatly explained the result. So Moseley published his results without a theoretical explanation.
It was Walther Kossel in 1914 and in 1916 who explained that in the periodic table new elements would be created as electrons were added to the outer shell. In Kossel's paper, he writes: "This leads to the conclusion that the electrons, which are added further, should be put into concentric rings or shells, on each of which ... only a certain number of electrons—namely, eight in our case—should be arranged. As soon as one ring or shell is completed, a new one has to be started for the next element; the number of electrons, which are most easily accessible, and lie at the outermost periphery, increases again from element to element and, therefore, in the formation of each new shell the chemical periodicity is repeated." Later, chemist Langmuir realized that the effect was caused by charge screening, with an inner shell containing only 2 electrons. In his 1919 paper, Irving Langmuir postulated the existence of "cells" which could each only contain two electrons each, and these were arranged in "equidistant layers".
In the Moseley experiment, one of the innermost electrons in the atom is knocked out, leaving a vacancy in the lowest Bohr orbit, which contains a single remaining electron. This vacancy is then filled by an electron from the next orbit, which has n=2. But the n=2 electrons see an effective charge of Z − 1, which is the value appropriate for the charge of the nucleus, when a single electron remains in the lowest Bohr orbit to screen the nuclear charge +Z, and lower it by −1 (due to the electron's negative charge screening the nuclear positive charge). The energy gained by an electron dropping from the second shell to the first gives Moseley's law for K-alpha lines,
E
=
h
ν
=
E
i
−
E
f
=
R
E
(
Z
−
1
)
2
(
1
1
2
−
1
2
2
)
,
{\displaystyle E=h\nu =E_{i}-E_{f}=R_{\mathrm {E} }(Z-1)^{2}\left({\frac {1}{1^{2}}}-{\frac {1}{2^{2}}}\right),}
or
f
=
ν
=
R
v
(
3
4
)
(
Z
−
1
)
2
=
(
2.46
×
10
15
Hz
)
(
Z
−
1
)
2
.
{\displaystyle f=\nu =R_{\mathrm {v} }\left({\frac {3}{4}}\right)(Z-1)^{2}=(2.46\times 10^{15}~{\text{Hz}})(Z-1)^{2}.}
Here, Rv = RE/h is the Rydberg constant, in terms of frequency equal to 3.28×1015 Hz. For values of Z between 11 and 31 this latter relationship had been empirically derived by Moseley, in a simple (linear) plot of the square root of X-ray frequency against atomic number (however, for silver, Z = 47, the experimentally obtained screening term should be replaced by 0.4). Notwithstanding its restricted validity, Moseley's law not only established the objective meaning of atomic number, but as Bohr noted, it also did more than the Rydberg derivation to establish the validity of the Rutherford/Van den Broek/Bohr nuclear model of the atom, with atomic number (place on the periodic table) standing for whole units of nuclear charge. Van den Broek had published his model in January 1913 showing the periodic table was arranged according to charge while Bohr's atomic model was not published until July 1913.
The K-alpha line of Moseley's time is now known to be a pair of close lines, written as (Kα1 and Kα2) in Siegbahn notation.
== Shortcomings ==
The Bohr model gives an incorrect value L=ħ for the ground state orbital angular momentum: The angular momentum in the true ground state is known to be zero from experiment. Although mental pictures fail somewhat at these levels of scale, an electron in the lowest modern "orbital" with no orbital momentum, may be thought of as not to revolve "around" the nucleus at all, but merely to go tightly around it in an ellipse with zero area (this may be pictured as "back and forth", without striking or interacting with the nucleus). This is only reproduced in a more sophisticated semiclassical treatment like Sommerfeld's. Still, even the most sophisticated semiclassical model fails to explain the fact that the lowest energy state is spherically symmetric – it doesn't point in any particular direction.
In modern quantum mechanics, the electron in hydrogen is a spherical cloud of probability that grows denser near the nucleus. The rate-constant of probability-decay in hydrogen is equal to the inverse of the Bohr radius, but since Bohr worked with circular orbits, not zero area ellipses, the fact that these two numbers exactly agree is considered a "coincidence". (However, many such coincidental agreements are found between the semiclassical vs. full quantum mechanical treatment of the atom; these include identical energy levels in the hydrogen atom and the derivation of a fine-structure constant, which arises from the relativistic Bohr–Sommerfeld model (see below) and which happens to be equal to an entirely different concept, in full modern quantum mechanics).
The Bohr model also failed to explain:
Much of the spectra of larger atoms. At best, it can make predictions about the K-alpha and some L-alpha X-ray emission spectra for larger atoms, if two additional ad hoc assumptions are made. Emission spectra for atoms with a single outer-shell electron (atoms in the lithium group) can also be approximately predicted. Also, if the empiric electron–nuclear screening factors for many atoms are known, many other spectral lines can be deduced from the information, in similar atoms of differing elements, via the Ritz–Rydberg combination principles (see Rydberg formula). All these techniques essentially make use of Bohr's Newtonian energy-potential picture of the atom.
The relative intensities of spectral lines; although in some simple cases, Bohr's formula or modifications of it, was able to provide reasonable estimates (for example, calculations by Kramers for the Stark effect).
The existence of fine structure and hyperfine structure in spectral lines, which are known to be due to a variety of relativistic and subtle effects, as well as complications from electron spin.
The Zeeman effect – changes in spectral lines due to external magnetic fields; these are also due to more complicated quantum principles interacting with electron spin and orbital magnetic fields.
Doublets and triplets appear in the spectra of some atoms as very close pairs of lines. Bohr's model cannot say why some energy levels should be very close together.
Multi-electron atoms do not have energy levels predicted by the model. It does not work for (neutral) helium.
== Refinements ==
Several enhancements to the Bohr model were proposed, most notably the Sommerfeld or Bohr–Sommerfeld models, which suggested that electrons travel in elliptical orbits around a nucleus instead of the Bohr model's circular orbits. This model supplemented the quantized angular momentum condition of the Bohr model with an additional radial quantization condition, the Wilson–Sommerfeld quantization condition
∫
0
T
p
r
d
q
r
=
n
h
,
{\displaystyle \int _{0}^{T}p_{\text{r}}\,dq_{\text{r}}=nh,}
where pr is the radial momentum canonically conjugate to the coordinate qr, which is the radial position, and T is one full orbital period. The integral is the action of action-angle coordinates. This condition, suggested by the correspondence principle, is the only one possible, since the quantum numbers are adiabatic invariants.
The Bohr–Sommerfeld model was fundamentally inconsistent and led to many paradoxes. The magnetic quantum number measured the tilt of the orbital plane relative to the xy plane, and it could only take a few discrete values. This contradicted the obvious fact that an atom could have any orientation relative to the coordinates, without restriction. The Sommerfeld quantization can be performed in different canonical coordinates and sometimes gives different answers. The incorporation of radiation corrections was difficult, because it required finding action-angle coordinates for a combined radiation/atom system, which is difficult when the radiation is allowed to escape. The whole theory did not extend to non-integrable motions, which meant that many systems could not be treated even in principle. In the end, the model was replaced by the modern quantum-mechanical treatment of the hydrogen atom, which was first given by Wolfgang Pauli in 1925, using Heisenberg's matrix mechanics. The current picture of the hydrogen atom is based on the atomic orbitals of wave mechanics, which Erwin Schrödinger developed in 1926.
However, this is not to say that the Bohr–Sommerfeld model was without its successes. Calculations based on the Bohr–Sommerfeld model were able to accurately explain a number of more complex atomic spectral effects. For example, up to first-order perturbations, the Bohr model and quantum mechanics make the same predictions for the spectral line splitting in the Stark effect. At higher-order perturbations, however, the Bohr model and quantum mechanics differ, and measurements of the Stark effect under high field strengths helped confirm the correctness of quantum mechanics over the Bohr model. The prevailing theory behind this difference lies in the shapes of the orbitals of the electrons, which vary according to the energy state of the electron.
The Bohr–Sommerfeld quantization conditions lead to questions in modern mathematics. Consistent semiclassical quantization condition requires a certain type of structure on the phase space, which places topological limitations on the types of symplectic manifolds which can be quantized. In particular, the symplectic form should be the curvature form of a connection of a Hermitian line bundle, which is called a prequantization.
Bohr also updated his model in 1922, assuming that certain numbers of electrons (for example, 2, 8, and 18) correspond to stable "closed shells".
== Model of the chemical bond ==
Niels Bohr proposed a model of the atom and a model of the chemical bond. According to his model for a diatomic molecule, the electrons of the atoms of the molecule form a rotating ring whose plane is perpendicular to the axis of the molecule and equidistant from the atomic nuclei. The dynamic equilibrium of the molecular system is achieved through the balance of forces between the forces of attraction of nuclei to the plane of the ring of electrons and the forces of mutual repulsion of the nuclei. The Bohr model of the chemical bond took into account the Coulomb repulsion – the electrons in the ring are at the maximum distance from each other.
== Symbolism of planetary atomic models ==
Although Bohr's atomic model was superseded by quantum models in the 1920s, the visual image of electrons orbiting a nucleus has remained the popular concept of atoms.
The concept of an atom as a tiny planetary system has been widely used as a symbol for atoms and even for "atomic" energy (even though this is more properly considered nuclear energy).: 58 Examples of its use over the past century include but are not limited to:
The logo of the United States Atomic Energy Commission, which was in part responsible for its later usage in relation to nuclear fission technology in particular.
The flag of the International Atomic Energy Agency is a "crest-and-spinning-atom emblem", enclosed in olive branches.
The US minor league baseball Albuquerque Isotopes' logo shows baseballs as electrons orbiting a large letter "A".
A similar symbol, the atomic whirl, was chosen as the symbol for the American Atheists, and has come to be used as a symbol of atheism in general.
The Unicode Miscellaneous Symbols code point U+269B (⚛) for an atom looks like a planetary atom model.
The television show The Big Bang Theory uses a planetary-like image in its print logo.
The JavaScript library React uses planetary-like image as its logo.
On maps, it is generally used to indicate a nuclear power installation.
== See also ==
== References ==
=== Footnotes ===
=== Primary sources ===
Bohr, N. (July 1913). "I. On the constitution of atoms and molecules". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 26 (151): 1–25. Bibcode:1913PMag...26....1B. doi:10.1080/14786441308634955.
Bohr, N. (September 1913). "XXXVII. On the constitution of atoms and molecules". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 26 (153): 476–502. Bibcode:1913PMag...26..476B. doi:10.1080/14786441308634993.
Bohr, N. (1 November 1913). "LXXIII. On the constitution of atoms and molecules". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 26 (155): 857–875. Bibcode:1913PMag...26..857B. doi:10.1080/14786441308635031.
Bohr, N. (October 1913). "The Spectra of Helium and Hydrogen". Nature. 92 (2295): 231–232. Bibcode:1913Natur..92..231B. doi:10.1038/092231d0. S2CID 11988018.
Bohr, N. (March 1921). "Atomic Structure". Nature. 107 (2682): 104–107. Bibcode:1921Natur.107..104B. doi:10.1038/107104a0. S2CID 4035652.
A. Einstein (1917). "Zum Quantensatz von Sommerfeld und Epstein". Verhandlungen der Deutschen Physikalischen Gesellschaft. 19: 82–92. Reprinted in The Collected Papers of Albert Einstein, A. Engel translator, (1997) Princeton University Press, Princeton. 6 p. 434. (provides an elegant reformulation of the Bohr–Sommerfeld quantization conditions, as well as an important insight into the quantization of non-integrable (chaotic) dynamical systems.)
de Broglie, Maurice; Langevin, Paul; Solvay, Ernest; Einstein, Albert (1912). La théorie du rayonnement et les quanta : rapports et discussions de la réunion tenue à Bruxelles, du 30 octobre au 3 novembre 1911, sous les auspices de M.E. Solvay (in French). Gauthier-Villars. OCLC 1048217622.
== Further reading ==
Linus Carl Pauling (1970). "Chapter 5-1". General Chemistry (3rd ed.). San Francisco: W.H. Freeman & Co.
Reprint: Linus Pauling (1988). General Chemistry. New York: Dover Publications. ISBN 0-486-65622-5.
George Gamow (1985). "Chapter 2". Thirty Years That Shook Physics. Dover Publications.
Walter J. Lehmann (1972). "Chapter 18". Atomic and Molecular Structure: the development of our concepts. John Wiley and Sons. ISBN 0-471-52440-9.
Paul Tipler and Ralph Llewellyn (2002). Modern Physics (4th ed.). W. H. Freeman. ISBN 0-7167-4345-0.
Klaus Hentschel: Elektronenbahnen, Quantensprünge und Spektren, in: Charlotte Bigg & Jochen Hennig (eds.) Atombilder. Ikonografien des Atoms in Wissenschaft und Öffentlichkeit des 20. Jahrhunderts, Göttingen: Wallstein-Verlag 2009, pp. 51–61
Steven and Susan Zumdahl (2010). "Chapter 7.4". Chemistry (8th ed.). Brooks/Cole. ISBN 978-0-495-82992-8.
Kragh, Helge (November 2011). "Conceptual objections to the Bohr atomic theory — do electrons have a 'free will'?". The European Physical Journal H. 36 (3): 327–352. Bibcode:2011EPJH...36..327K. doi:10.1140/epjh/e2011-20031-x. S2CID 120859582.
== External links ==
Standing waves in Bohr's atomic model—An interactive simulation to intuitively explain the quantization condition of standing waves in Bohr's atomic mode | Wikipedia/Bohr_theory |
In solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have (called band gaps or forbidden bands).
Band theory derives these bands and band gaps by examining the allowed quantum mechanical wave functions for an electron in a large, periodic lattice of atoms or molecules. Band theory has been successfully used to explain many physical properties of solids, such as electrical resistivity and optical absorption, and forms the foundation of the understanding of all solid-state devices (transistors, solar cells, etc.).
== Why bands and band gaps occur ==
The formation of electronic bands and band gaps can be illustrated with two complementary models for electrons in solids.: 161 The first one is the nearly free electron model, in which the electrons are assumed to move almost freely within the material. In this model, the electronic states resemble free electron plane waves, and are only slightly perturbed by the crystal lattice. This model explains the origin of the electronic dispersion relation, but the explanation for band gaps is subtle in this model.: 121
The second model starts from the opposite limit, in which the electrons are tightly bound to individual atoms. The electrons of a single, isolated atom occupy atomic orbitals with discrete energy levels. If two atoms come close enough so that their atomic orbitals overlap, the electrons can tunnel between the atoms. This tunneling splits (hybridizes) the atomic orbitals into molecular orbitals with different energies.: 117–122
Similarly, if a large number N of identical atoms come together to form a solid, such as a crystal lattice, the atoms' atomic orbitals overlap with the nearby orbitals. Each discrete energy level splits into N levels, each with a different energy. Since the number of atoms in a macroscopic piece of solid is a very large number (N ≈ 1022), the number of orbitals that hybridize with each other is very large. For this reason, the adjacent levels are very closely spaced in energy (of the order of 10−22 eV), and can be considered to form a continuum, an energy band.
This formation of bands is mostly a feature of the outermost electrons (valence electrons) in the atom, which are the ones involved in chemical bonding and electrical conductivity. The inner electron orbitals do not overlap to a significant degree, so their bands are very narrow.
Band gaps are essentially leftover ranges of energy not covered by any band, a result of the finite widths of the energy bands. The bands have different widths, with the widths depending upon the degree of overlap in the atomic orbitals from which they arise. Two adjacent bands may simply not be wide enough to fully cover the range of energy. For example, the bands associated with core orbitals (such as 1s electrons) are extremely narrow due to the small overlap between adjacent atoms. As a result, there tend to be large band gaps between the core bands. Higher bands involve comparatively larger orbitals with more overlap, becoming progressively wider at higher energies so that there are no band gaps at higher energies.
== Basic concepts ==
=== Assumptions and limits of band structure theory ===
Band theory is only an approximation to the quantum state of a solid, which applies to solids consisting of many identical atoms or molecules bonded together. These are the assumptions necessary for band theory to be valid:
Infinite-size system: For the bands to be continuous, the piece of material must consist of a large number of atoms. Since a macroscopic piece of material contains on the order of 1022 atoms, this is not a serious restriction; band theory even applies to microscopic-sized transistors in integrated circuits. With modifications, the concept of band structure can also be extended to systems which are only "large" along some dimensions, such as two-dimensional electron systems.
Homogeneous system: Band structure is an intrinsic property of a material, which assumes that the material is homogeneous. Practically, this means that the chemical makeup of the material must be uniform throughout the piece.
Non-interactivity: The band structure describes "single electron states". The existence of these states assumes that the electrons travel in a static potential without dynamically interacting with lattice vibrations, other electrons, photons, etc.
The above assumptions are broken in a number of important practical situations, and the use of band structure requires one to keep a close check on the limitations of band theory:
Inhomogeneities and interfaces: Near surfaces, junctions, and other inhomogeneities, the bulk band structure is disrupted. Not only are there local small-scale disruptions (e.g., surface states or dopant states inside the band gap), but also local charge imbalances. These charge imbalances have electrostatic effects that extend deeply into semiconductors, insulators, and the vacuum (see doping, band bending).
Along the same lines, most electronic effects (capacitance, electrical conductance, electric-field screening) involve the physics of electrons passing through surfaces and/or near interfaces. The full description of these effects, in a band structure picture, requires at least a rudimentary model of electron-electron interactions (see space charge, band bending).
Small systems: For systems which are small along every dimension (e.g., a small molecule or a quantum dot), there is no continuous band structure. The crossover between small and large dimensions is the realm of mesoscopic physics.
Strongly correlated materials (for example, Mott insulators) simply cannot be understood in terms of single-electron states. The electronic band structures of these materials are poorly defined (or at least, not uniquely defined) and may not provide useful information about their physical state.
=== Crystalline symmetry and wavevectors ===
Band structure calculations take advantage of the periodic nature of a crystal lattice, exploiting its symmetry. The single-electron Schrödinger equation is solved for an electron in a lattice-periodic potential, giving Bloch electrons as solutions
ψ
n
k
(
r
)
=
e
i
k
⋅
r
u
n
k
(
r
)
,
{\displaystyle \psi _{n\mathbf {k} }(\mathbf {r} )=e^{i\mathbf {k} \cdot \mathbf {r} }u_{n\mathbf {k} }(\mathbf {r} ),}
where k is called the wavevector. For each value of k, there are multiple solutions to the Schrödinger equation labelled by n, the band index, which simply numbers the energy bands.
Each of these energy levels evolves smoothly with changes in k, forming a smooth band of states. For each band we can define a function En(k), which is the dispersion relation for electrons in that band.
The wavevector takes on any value inside the Brillouin zone, which is a polyhedron in wavevector (reciprocal lattice) space that is related to the crystal's lattice.
Wavevectors outside the Brillouin zone simply correspond to states that are physically identical to those states within the Brillouin zone.
Special high symmetry points/lines in the Brillouin zone are assigned labels like Γ, Δ, Λ, Σ (see Fig 1).
It is difficult to visualize the shape of a band as a function of wavevector, as it would require a plot in four-dimensional space, E vs. kx, ky, kz. In scientific literature it is common to see band structure plots which show the values of En(k) for values of k along straight lines connecting symmetry points, often labelled Δ, Λ, Σ, or [100], [111], and [110], respectively. Another method for visualizing band structure is to plot a constant-energy isosurface in wavevector space, showing all of the states with energy equal to a particular value. The isosurface of states with energy equal to the Fermi level is known as the Fermi surface.
Energy band gaps can be classified using the wavevectors of the states surrounding the band gap:
Direct band gap: the lowest-energy state above the band gap has the same k as the highest-energy state beneath the band gap.
Indirect band gap: the closest states above and beneath the band gap do not have the same k value.
==== Asymmetry: Band structures in non-crystalline solids ====
Although electronic band structures are usually associated with crystalline materials, quasi-crystalline and amorphous solids may also exhibit band gaps. These are somewhat more difficult to study theoretically since they lack the simple symmetry of a crystal, and it is not usually possible to determine a precise dispersion relation. As a result, virtually all of the existing theoretical work on the electronic band structure of solids has focused on crystalline materials.
=== Density of states ===
The density of states function g(E) is defined as the number of electronic states per unit volume, per unit energy, for electron energies near E.
The density of states function is important for calculations of effects based on band theory.
In Fermi's Golden Rule, a calculation for the rate of optical absorption, it provides both the number of excitable electrons and the number of final states for an electron. It appears in calculations of electrical conductivity where it provides the number of mobile states, and in computing electron scattering rates where it provides the number of final states after scattering.
For energies inside a band gap, g(E) = 0.
=== Filling of bands ===
At thermodynamic equilibrium, the likelihood of a state of energy E being filled with an electron is given by the Fermi–Dirac distribution, a thermodynamic distribution that takes into account the Pauli exclusion principle:
f
(
E
)
=
1
1
+
e
(
E
−
μ
)
/
k
B
T
{\displaystyle f(E)={\frac {1}{1+e^{{(E-\mu )}/{k_{\text{B}}T}}}}}
where:
kBT is the product of the Boltzmann constant and temperature, and
µ is the total chemical potential of electrons, or Fermi level (in semiconductor physics, this quantity is more often denoted EF). The Fermi level of a solid is directly related to the voltage on that solid, as measured with a voltmeter. Conventionally, in band structure plots the Fermi level is taken to be the zero of energy (an arbitrary choice).
The density of electrons in the material is simply the integral of the Fermi–Dirac distribution times the density of states:
N
/
V
=
∫
−
∞
∞
g
(
E
)
f
(
E
)
d
E
{\displaystyle N/V=\int _{-\infty }^{\infty }g(E)f(E)\,dE}
Although there are an infinite number of bands and thus an infinite number of states, there are only a finite number of electrons to place in these bands.
The preferred value for the number of electrons is a consequence of electrostatics: even though the surface of a material can be charged, the internal bulk of a material prefers to be charge neutral.
The condition of charge neutrality means that N/V must match the density of protons in the material. For this to occur, the material electrostatically adjusts itself, shifting its band structure up or down in energy (thereby shifting g(E)), until it is at the correct equilibrium with respect to the Fermi level.
==== Names of bands near the Fermi level (conduction band, valence band) ====
A solid has an infinite number of allowed bands, just as an atom has infinitely many energy levels. However, most of the bands simply have too high energy, and are usually disregarded under ordinary circumstances.
Conversely, there are very low energy bands associated with the core orbitals (such as 1s electrons). These low-energy core bands are also usually disregarded since they remain filled with electrons at all times, and are therefore inert.
Likewise, materials have several band gaps throughout their band structure.
The most important bands and band gaps—those relevant for electronics and optoelectronics—are those with energies near the Fermi level.
The bands and band gaps near the Fermi level are given special names, depending on the material:
In a semiconductor or band insulator, the Fermi level is surrounded by a band gap, referred to as the band gap (to distinguish it from the other band gaps in the band structure). The closest band above the band gap is called the conduction band, and the closest band beneath the band gap is called the valence band. The name "valence band" was coined by analogy to chemistry, since in semiconductors (and insulators) the valence band is built out of the valence orbitals.
In a metal or semimetal, the Fermi level is inside of one or more allowed bands. In semimetals the bands are usually referred to as "conduction band" or "valence band" depending on whether the charge transport is more electron-like or hole-like, by analogy to semiconductors. In many metals, however, the bands are neither electron-like nor hole-like, and often just called "valence band" as they are made of valence orbitals. The band gaps in a metal's band structure are not important for low energy physics, since they are too far from the Fermi level.
== Theory in crystals ==
The ansatz is the special case of electron waves in a periodic crystal lattice using Bloch's theorem as treated generally in the dynamical theory of diffraction. Every crystal is a periodic structure which can be characterized by a Bravais lattice, and for each Bravais lattice we can determine the reciprocal lattice, which encapsulates the periodicity in a set of three reciprocal lattice vectors (b1, b2, b3). Now, any periodic potential V(r) which shares the same periodicity as the direct lattice can be expanded out as a Fourier series whose only non-vanishing components are those associated with the reciprocal lattice vectors. So the expansion can be written as:
V
(
r
)
=
∑
K
V
K
e
i
K
⋅
r
{\displaystyle V(\mathbf {r} )=\sum _{\mathbf {K} }{V_{\mathbf {K} }e^{i\mathbf {K} \cdot \mathbf {r} }}}
where K = m1b1 + m2b2 + m3b3 for any set of integers (m1, m2, m3).
From this theory, an attempt can be made to predict the band structure of a particular material, however most ab initio methods for electronic structure calculations fail to predict the observed band gap.
=== Nearly free electron approximation ===
In the nearly free electron approximation, interactions between electrons are completely ignored. This approximation allows use of Bloch's Theorem which states that electrons in a periodic potential have wavefunctions and energies which are periodic in wavevector up to a constant phase shift between neighboring reciprocal lattice vectors. The consequences of periodicity are described mathematically by the Bloch's theorem, which states that the eigenstate wavefunctions have the form
Ψ
n
,
k
(
r
)
=
e
i
k
⋅
r
u
n
(
r
)
{\displaystyle \Psi _{n,\mathbf {k} }(\mathbf {r} )=e^{i\mathbf {k} \cdot \mathbf {r} }u_{n}(\mathbf {r} )}
where the Bloch function
u
n
(
r
)
{\displaystyle u_{n}(\mathbf {r} )}
is periodic over the crystal lattice, that is,
u
n
(
r
)
=
u
n
(
r
−
R
)
.
{\displaystyle u_{n}(\mathbf {r} )=u_{n}(\mathbf {r} -\mathbf {R} ).}
Here index n refers to the nth energy band, wavevector k is related to the direction of motion of the electron, r is the position in the crystal, and R is the location of an atomic site.: 179
The NFE model works particularly well in materials like metals where distances between neighbouring atoms are small. In such materials the overlap of atomic orbitals and potentials on neighbouring atoms is relatively large. In that case the wave function of the electron can be approximated by a (modified) plane wave. The band structure of a metal like aluminium even gets close to the empty lattice approximation.
=== Tight binding model ===
The opposite extreme to the nearly free electron approximation assumes the electrons in the crystal behave much like an assembly of constituent atoms. This tight binding model assumes the solution to the time-independent single electron Schrödinger equation
Ψ
{\displaystyle \Psi }
is well approximated by a linear combination of atomic orbitals
ψ
n
(
r
)
{\displaystyle \psi _{n}(\mathbf {r} )}
.: 245–248
Ψ
(
r
)
=
∑
n
,
R
b
n
,
R
ψ
n
(
r
−
R
)
,
{\displaystyle \Psi (\mathbf {r} )=\sum _{n,\mathbf {R} }b_{n,\mathbf {R} }\psi _{n}(\mathbf {r} -\mathbf {R} ),}
where the coefficients
b
n
,
R
{\displaystyle b_{n,\mathbf {R} }}
are selected to give the best approximate solution of this form. Index n refers to an atomic energy level and R refers to an atomic site. A more accurate approach using this idea employs Wannier functions, defined by:: Eq. 42 p. 267
a
n
(
r
−
R
)
=
V
C
(
2
π
)
3
∫
BZ
d
k
e
−
i
k
⋅
(
R
−
r
)
u
n
k
;
{\displaystyle a_{n}(\mathbf {r} -\mathbf {R} )={\frac {V_{C}}{(2\pi )^{3}}}\int _{\text{BZ}}d\mathbf {k} e^{-i\mathbf {k} \cdot (\mathbf {R} -\mathbf {r} )}u_{n\mathbf {k} };}
in which
u
n
k
{\displaystyle u_{n\mathbf {k} }}
is the periodic part of the Bloch's theorem and the integral is over the Brillouin zone. Here index n refers to the n-th energy band in the crystal. The Wannier functions are localized near atomic sites, like atomic orbitals, but being defined in terms of Bloch functions they are accurately related to solutions based upon the crystal potential. Wannier functions on different atomic sites R are orthogonal. The Wannier functions can be used to form the Schrödinger solution for the n-th energy band as:
Ψ
n
,
k
(
r
)
=
∑
R
e
−
i
k
⋅
(
R
−
r
)
a
n
(
r
−
R
)
.
{\displaystyle \Psi _{n,\mathbf {k} }(\mathbf {r} )=\sum _{\mathbf {R} }e^{-i\mathbf {k} \cdot (\mathbf {R} -\mathbf {r} )}a_{n}(\mathbf {r} -\mathbf {R} ).}
The TB model works well in materials with limited overlap between atomic orbitals and potentials on neighbouring atoms. Band structures of materials like Si, GaAs, SiO2 and diamond for instance are well described by TB-Hamiltonians on the basis of atomic sp3 orbitals. In transition metals a mixed TB-NFE model is used to describe the broad NFE conduction band and the narrow embedded TB d-bands. The radial functions of the atomic orbital part of the Wannier functions are most easily calculated by the use of pseudopotential methods. NFE, TB or combined NFE-TB band structure calculations,
sometimes extended with wave function approximations based on pseudopotential methods, are often used as an economic starting point for further calculations.
=== KKR model ===
The KKR method, also called "multiple scattering theory" or Green's function method, finds the stationary values of the inverse transition matrix T rather than the Hamiltonian. A variational implementation was suggested by Korringa, Kohn and Rostocker, and is often referred to as the Korringa–Kohn–Rostoker method. The most important features of the KKR or Green's function formulation are (1) it separates the two aspects of the problem: structure (positions of the atoms) from the scattering (chemical identity of the atoms); and (2) Green's functions provide a natural approach to a localized description of electronic properties that can be adapted to alloys and other disordered system. The simplest form of this approximation centers non-overlapping spheres (referred to as muffin tins) on the atomic positions. Within these regions, the potential experienced by an electron is approximated to be spherically symmetric about the given nucleus. In the remaining interstitial region, the screened potential is approximated as a constant. Continuity of the potential between the atom-centered spheres and interstitial region is enforced.
=== Density-functional theory ===
In recent physics literature, a large majority of the electronic structures and band plots are calculated using density-functional theory (DFT), which is not a model but rather a theory, i.e., a microscopic first-principles theory of condensed matter physics that tries to cope with the electron-electron many-body problem via the introduction of an exchange-correlation term in the functional of the electronic density. DFT-calculated bands are in many cases found to be in agreement with experimentally measured bands, for example by angle-resolved photoemission spectroscopy (ARPES). In particular, the band shape is typically well reproduced by DFT. But there are also systematic errors in DFT bands when compared to experiment results. In particular, DFT seems to systematically underestimate by about 30-40% the band gap in insulators and semiconductors.
It is commonly believed that DFT is a theory to predict ground state properties of a system only (e.g. the total energy, the atomic structure, etc.), and that excited state properties cannot be determined by DFT. This is a misconception. In principle, DFT can determine any property (ground state or excited state) of a system given a functional that maps the ground state density to that property. This is the essence of the Hohenberg–Kohn theorem. In practice, however, no known functional exists that maps the ground state density to excitation energies of electrons within a material. Thus, what in the literature is quoted as a DFT band plot is a representation of the DFT Kohn–Sham energies, i.e., the energies of a fictive non-interacting system, the Kohn–Sham system, which has no physical interpretation at all. The Kohn–Sham electronic structure must not be confused with the real, quasiparticle electronic structure of a system, and there is no Koopmans' theorem holding for Kohn–Sham energies, as there is for Hartree–Fock energies, which can be truly considered as an approximation for quasiparticle energies. Hence, in principle, Kohn–Sham based DFT is not a band theory, i.e., not a theory suitable for calculating bands and band-plots. In principle time-dependent DFT can be used to calculate the true band structure although in practice this is often difficult. A popular approach is the use of hybrid functionals, which incorporate a portion of Hartree–Fock exact exchange; this produces a substantial improvement in predicted bandgaps of semiconductors, but is less reliable for metals and wide-bandgap materials.
=== Green's function methods and the ab initio GW approximation ===
To calculate the bands including electron-electron interaction many-body effects, one can resort to so-called Green's function methods. Indeed, knowledge of the Green's function of a system provides both ground (the total energy) and also excited state observables of the system. The poles of the Green's function are the quasiparticle energies, the bands of a solid. The Green's function can be calculated by solving the Dyson equation once the self-energy of the system is known. For real systems like solids, the self-energy is a very complex quantity and usually approximations are needed to solve the problem. One such approximation is the GW approximation, so called from the mathematical form the self-energy takes as the product Σ = GW of the Green's function G and the dynamically screened interaction W. This approach is more pertinent when addressing the calculation of band plots (and also quantities beyond, such as the spectral function) and can also be formulated in a completely ab initio way. The GW approximation seems to provide band gaps of insulators and semiconductors in agreement with experiment, and hence to correct the systematic DFT underestimation.
=== Dynamical mean-field theory ===
Although the nearly free electron approximation is able to describe many properties of electron band structures, one consequence of this theory is that it predicts the same number of electrons in each unit cell. If the number of electrons is odd, we would then expect that there is an unpaired electron in each unit cell, and thus that the valence band is not fully occupied, making the material a conductor. However, materials such as CoO that have an odd number of electrons per unit cell are insulators, in direct conflict with this result. This kind of material is known as a Mott insulator, and requires inclusion of detailed electron-electron interactions (treated only as an averaged effect on the crystal potential in band theory) to explain the discrepancy. The Hubbard model is an approximate theory that can include these interactions. It can be treated non-perturbatively within the so-called dynamical mean-field theory, which attempts to bridge the gap between the nearly free electron approximation and the atomic limit. Formally, however, the states are not non-interacting in this case and the concept of a band structure is not adequate to describe these cases.
=== Others ===
Calculating band structures is an important topic in theoretical solid state physics. In addition to the models mentioned above, other models include the following:
Empty lattice approximation: the "band structure" of a region of free space that has been divided into a lattice.
k·p perturbation theory is a technique that allows a band structure to be approximately described in terms of just a few parameters. The technique is commonly used for semiconductors, and the parameters in the model are often determined by experiment.
The Kronig–Penney model, a one-dimensional rectangular well model useful for illustration of band formation. While simple, it predicts many important phenomena, but is not quantitative.
Hubbard model
The band structure has been generalised to wavevectors that are complex numbers, resulting in what is called a complex band structure, which is of interest at surfaces and interfaces.
Each model describes some types of solids very well, and others poorly. The nearly free electron model works well for metals, but poorly for non-metals. The tight binding model is extremely accurate for ionic insulators, such as metal halide salts (e.g. NaCl).
== Band diagrams ==
To understand how band structure changes relative to the Fermi level in real space, a band structure plot is often first simplified in the form of a band diagram. In a band diagram the vertical axis is energy while the horizontal axis represents real space. Horizontal lines represent energy levels, while blocks represent energy bands. When the horizontal lines in these diagram are slanted then the energy of the level or band changes with distance. Diagrammatically, this depicts the presence of an electric field within the crystal system. Band diagrams are useful in relating the general band structure properties of different materials to one another when placed in contact with each other.
== See also ==
Band-gap engineering – the process of altering a material's band structure
Felix Bloch – pioneer in the theory of band structure
Alan Herries Wilson – pioneer in the theory of band structure
== References ==
== Further reading ==
Ashcroft, Neil and N. David Mermin, Solid State Physics, ISBN 0-03-083993-9
Harrison, Walter A., Elementary Electronic Structure, ISBN 981-238-708-0
Harrison, Walter A.; W. A. Benjamin Pseudopotentials in the theory of metals, (New York) 1966
Marder, Michael P., Condensed Matter Physics, ISBN 0-471-17779-2
Martin, Richard, Electronic Structure: Basic Theory and Practical Methods, ISBN 0-521-78285-6
Millman, Jacob; Arvin Gabriel, Microelectronics, ISBN 0-07-463736-3, Tata McGraw-Hill Edition.
Nemoshkalenko, V. V., and N. V. Antonov, Computational Methods in Solid State Physics, ISBN 90-5699-094-2
Omar, M. Ali, Elementary Solid State Physics: Principles and Applications, ISBN 0-201-60733-6
Singh, Jasprit, Electronic and Optoelectronic Properties of Semiconductor Structures Chapters 2 and 3, ISBN 0-521-82379-X
Vasileska, Dragica, Tutorial on Bandstructure Methods (2008) | Wikipedia/Energy_band |
In quantum mechanics, a doublet is a composite quantum state of a system with an effective spin of 1/2, such that there are two allowed values of the spin component, −1/2 and +1/2. Quantum systems with two possible states are sometimes called two-level systems. Essentially all occurrences of doublets in nature arise from rotational symmetry; spin 1/2 is associated with the fundamental representation of the Lie group SU(2).
== History and applications ==
The term "doublet" dates back to the early 19th century, when it was observed that certain spectral lines of an ionized, excited gas would split into two under the influence of a strong magnetic field, in an effect known as the anomalous Zeeman effect. Such spectral lines were observed not only in the laboratory, but also in astronomical spectroscopy observations, allowing astronomers to deduce the existence of, and measure the strength of magnetic fields around the Sun, stars and galaxies. Conversely, it was the observation of doublets in spectroscopy that allowed physicists to deduce that the electron had a spin, and that furthermore, the magnitude of the spin had to be 1/2. See the history section of the article on Spin (physics) for greater detail.
Doublets continue to play an important role in physics. For example, the healthcare technology of magnetic resonance imaging is based on nuclear magnetic resonance. In this technology, a spectroscopic doublet occurs in a spin-1/2 atomic nucleus, whose doublet splitting is in the radio-frequency range. By applying both a magnetic field and carefully tuning a radio-frequency transmitter, the nuclear spins will flip and re-emit radiation, in an effect known as the Rabi cycle. The strength and frequency of the emitted radio waves allow the concentration of such nuclei to be measured. Another potential application is the use of doublets as the emitting layer in light emitting diodes (LEDs). These materials have the advantage of having 100% theoretical quantum efficiency based on spin statistics whereas singlet systems and triplet systems have significantly lower efficiencies or rely on noble metals such as Pt and Ir to emit light.
== See also ==
Singlet state
Triplet state
Spin multiplicity
== References == | Wikipedia/Doublet_(physics) |
In solid-state physics, the work function (sometimes spelled workfunction) is the minimum thermodynamic work (i.e., energy) needed to remove an electron from a solid to a point in the vacuum immediately outside the solid surface. Here "immediately" means that the final electron position is far from the surface on the atomic scale, but still too close to the solid to be influenced by ambient electric fields in the vacuum.
The work function is not a characteristic of a bulk material, but rather a property of the surface of the material (depending on crystal face and contamination).
== Definition ==
The work function W for a given surface is defined by the difference
W
=
−
e
ϕ
−
E
F
,
{\displaystyle W=-e\phi -E_{\rm {F}},}
where −e is the charge of an electron, ϕ is the electrostatic potential in the vacuum nearby the surface, and EF is the Fermi level (electrochemical potential of electrons) inside the material. The term −eϕ is the energy of an electron at rest in the vacuum nearby the surface.
In practice, one directly controls EF by the voltage applied to the material through electrodes, and the work function is generally a fixed characteristic of the surface material. Consequently, this means that when a voltage is applied to a material, the electrostatic potential ϕ produced in the vacuum will be somewhat lower than the applied voltage, the difference depending on the work function of the material surface. Rearranging the above equation, one has
ϕ
=
V
−
W
e
{\displaystyle \phi =V-{\frac {W}{e}}}
where V = −EF / e is the voltage of the material (as measured by a voltmeter, through an attached electrode), relative to an electrical ground that is defined as having zero Fermi level. The fact that ϕ depends on the material surface means that the space between two dissimilar conductors will have a built-in electric field, when those conductors are in total equilibrium with each other (electrically shorted to each other, and with equal temperatures).
The work function refers to removal of an electron to a position that is far enough from the surface (many nm) that the force between the electron and its image charge in the surface can be neglected. The electron must also be close to the surface compared to the nearest edge of a crystal facet, or to any other change in the surface structure, such as a change in the material composition, surface coating or reconstruction. The built-in electric field that results from these structures, and any other ambient electric field present in the vacuum are excluded in defining the work function.
== Applications ==
Thermionic emission
In thermionic electron guns, the work function and temperature of the hot cathode are critical parameters in determining the amount of current that can be emitted. Tungsten, the common choice for vacuum tube filaments, can survive to high temperatures but its emission is somewhat limited due to its relatively high work function (approximately 4.5 eV). By coating the tungsten with a substance of lower work function (e.g., thorium or barium oxide), the emission can be greatly increased. This prolongs the lifetime of the filament by allowing operation at lower temperatures (for more information, see hot cathode).
Band bending models in solid-state electronics
The behavior of a solid-state device is strongly dependent on the size of various Schottky barriers and band offsets in the junctions of differing materials, such as metals, semiconductors, and insulators. Some commonly used heuristic approaches to predict the band alignment between materials, such as Anderson's rule and the Schottky–Mott rule, are based on the thought experiment of two materials coming together in vacuum, such that the surfaces charge up and adjust their work functions to become equal just before contact. In reality these work function heuristics are inaccurate due to their neglect of numerous microscopic effects. However, they provide a convenient estimate until the true value can be determined by experiment.
Equilibrium electric fields in vacuum chambers
Variation in work function between different surfaces causes a non-uniform electrostatic potential in the vacuum. Even on an ostensibly uniform surface, variations in W known as patch potentials are always present due to microscopic inhomogeneities. Patch potentials have disrupted sensitive apparatus that rely on a perfectly uniform vacuum, such as Casimir force experiments and the Gravity Probe B experiment. Critical apparatus may have surfaces covered with molybdenum, which shows low variations in work function between different crystal faces.
Contact electrification
If two conducting surfaces are moved relative to each other, and there is potential difference in the space between them, then an electric current will be driven. This is because the surface charge on a conductor depends on the magnitude of the electric field, which in turn depends on the distance between the surfaces. The externally observed electrical effects are largest when the conductors are separated by the smallest distance without touching (once brought into contact, the charge will instead flow internally through the junction between the conductors). Since two conductors in equilibrium can have a built-in potential difference due to work function differences, this means that bringing dissimilar conductors into contact, or pulling them apart, will drive electric currents. These contact currents can damage sensitive microelectronic circuitry and occur even when the conductors would be grounded in the absence of motion.
== Measurement ==
Certain physical phenomena are highly sensitive to the value of the work function. The observed data from these effects can be fitted to simplified theoretical models, allowing one to extract a value of the work function. These phenomenologically extracted work functions may be slightly different from the thermodynamic definition given above. For inhomogeneous surfaces, the work function varies from place to place, and different methods will yield different values of the typical "work function" as they average or select differently among the microscopic work functions.
Many techniques have been developed based on different physical effects to measure the electronic work function of a sample. One may distinguish between two groups of experimental methods for work function measurements: absolute and relative.
Absolute methods employ electron emission from the sample induced by photon absorption (photoemission), by high temperature (thermionic emission), due to an electric field (field electron emission), or using electron tunnelling.
Relative methods make use of the contact potential difference between the sample and a reference electrode. Experimentally, either an anode current of a diode is used or the displacement current between the sample and reference, created by an artificial change in the capacitance between the two, is measured (the Kelvin Probe method, Kelvin probe force microscope). However, absolute work function values can be obtained if the tip is first calibrated against a reference sample.
=== Methods based on thermionic emission ===
The work function is important in the theory of thermionic emission, where thermal fluctuations provide enough energy to "evaporate" electrons out of a hot material (called the 'emitter') into the vacuum. If these electrons are absorbed by another, cooler material (called the collector) then a measurable electric current will be observed. Thermionic emission can be used to measure the work function of both the hot emitter and cold collector. Generally, these measurements involve fitting to Richardson's law, and so they must be carried out in a low temperature and low current regime where space charge effects are absent.
In order to move from the hot emitter to the vacuum, an electron's energy must exceed the emitter Fermi level by an amount
E
b
a
r
r
i
e
r
=
W
e
{\displaystyle E_{\rm {barrier}}=W_{\rm {e}}}
determined simply by the thermionic work function of the emitter.
If an electric field is applied towards the surface of the emitter, then all of the escaping electrons will be accelerated away from the emitter and absorbed into whichever material is applying the electric field.
According to Richardson's law the emitted current density (per unit area of emitter), Je (A/m2), is related to the absolute temperature Te of the emitter by the equation:
J
e
=
−
A
e
T
e
2
e
−
E
b
a
r
r
i
e
r
/
k
T
e
{\displaystyle J_{\rm {e}}=-A_{\rm {e}}T_{\rm {e}}^{2}e^{-E_{\rm {barrier}}/kT_{\rm {e}}}}
where k is the Boltzmann constant and the proportionality constant Ae is the Richardson's constant of the emitter.
In this case, the dependence of Je on Te can be fitted to yield We.
==== Work function of cold electron collector ====
The same setup can be used to instead measure the work function in the collector, simply by adjusting the applied voltage.
If an electric field is applied away from the emitter instead, then most of the electrons coming from the emitter will simply be reflected back to the emitter. Only the highest energy electrons will have enough energy to reach the collector, and the height of the potential barrier in this case depends on the collector's work function, rather than the emitter's.
The current is still governed by Richardson's law. However, in this case the barrier height does not depend on We. The barrier height now depends on the work function of the collector, as well as any additional applied voltages:
E
b
a
r
r
i
e
r
=
W
c
−
e
(
Δ
V
c
e
−
Δ
V
S
)
{\displaystyle E_{\rm {barrier}}=W_{\rm {c}}-e(\Delta V_{\rm {ce}}-\Delta V_{\rm {S}})}
where Wc is the collector's thermionic work function, ΔVce is the applied collector–emitter voltage, and ΔVS is the Seebeck voltage in the hot emitter (the influence of ΔVS is often omitted, as it is a small contribution of order 10 mV).
The resulting current density Jc through the collector (per unit of collector area) is again given by Richardson's Law, except now
J
c
=
A
T
e
2
e
−
E
b
a
r
r
i
e
r
/
k
T
e
{\displaystyle J_{\rm {c}}=AT_{\rm {e}}^{2}e^{-E_{\rm {barrier}}/kT_{\rm {e}}}}
where A is a Richardson-type constant that depends on the collector material but may also depend on the emitter material, and the diode geometry.
In this case, the dependence of Jc on Te, or on ΔVce, can be fitted to yield Wc.
This retarding potential method is one of the simplest and oldest methods of measuring work functions, and is advantageous since the measured material (collector) is not required to survive high temperatures.
=== Methods based on photoemission ===
The photoelectric work function is the minimum photon energy required to liberate an electron from a substance, in the photoelectric effect.
If the photon's energy is greater than the substance's work function, photoelectric emission occurs and the electron is liberated from the surface.
Similar to the thermionic case described above, the liberated electrons can be extracted into a collector and produce a detectable current, if an electric field is applied into the surface of the emitter.
Excess photon energy results in a liberated electron with non-zero kinetic energy.
It is expected that the minimum photon energy
ℏ
ω
{\displaystyle \hbar \omega }
required to liberate an electron (and generate a current) is
ℏ
ω
=
W
e
{\displaystyle \hbar \omega =W_{\rm {e}}}
where We is the work function of the emitter.
Photoelectric measurements require a great deal of care, as an incorrectly designed experimental geometry can result in an erroneous measurement of work function. This may be responsible for the large variation in work function values in scientific literature.
Moreover, the minimum energy can be misleading in materials where there are no actual electron states at the Fermi level that are available for excitation. For example, in a semiconductor the minimum photon energy would actually correspond to the valence band edge rather than work function.
Of course, the photoelectric effect may be used in the retarding mode, as with the thermionic apparatus described above. In the retarding case, the dark collector's work function is measured instead.
=== Kelvin probe method ===
The Kelvin probe technique relies on the detection of an electric field (gradient in ϕ) between a sample material and probe material.
The electric field can be varied by the voltage ΔVsp that is applied to the probe relative to the sample.
If the voltage is chosen such that the electric field is eliminated (the flat vacuum condition), then
e
Δ
V
s
p
=
W
s
−
W
p
,
when
ϕ
is flat
.
{\displaystyle e\Delta V_{\rm {sp}}=W_{\rm {s}}-W_{\rm {p}},\quad {\text{when}}~\phi ~{\text{is flat}}.}
Since the experimenter controls and knows ΔVsp, then finding the flat vacuum condition gives directly the work function difference between the two materials.
The only question is, how to detect the flat vacuum condition?
Typically, the electric field is detected by varying the distance between the sample and probe. When the distance is changed but ΔVsp is held constant, a current will flow due to the change in capacitance. This current is proportional to the vacuum electric field, and so when the electric field is neutralized no current will flow.
Although the Kelvin probe technique only measures a work function difference, it is possible to obtain an absolute work function by first calibrating the probe against a reference material (with known work function) and then using the same probe to measure a desired sample.
The Kelvin probe technique can be used to obtain work function maps of a surface with extremely high spatial resolution, by using a sharp tip for the probe (see Kelvin probe force microscope).
== Work functions of elements ==
The work function depends on the configurations of atoms at the surface of the material. For example, on polycrystalline silver the work function is 4.26 eV, but on silver crystals it varies for different crystal faces as (100) face: 4.64 eV, (110) face: 4.52 eV, (111) face: 4.74 eV. Ranges for typical surfaces are shown in the table below.
== Physical factors that determine the work function ==
Due to the complications described in the modelling section below, it is difficult to theoretically predict the work function with accuracy. However, various trends have been identified. The work function tends to be smaller for metals with an open lattice, and larger for metals in which the atoms are closely packed. It is somewhat higher on dense crystal faces than open crystal faces, also depending on surface reconstructions for the given crystal face.
=== Surface dipole ===
The work function is not simply dependent on the "internal vacuum level" inside the material (i.e., its average electrostatic potential), because of the formation of an atomic-scale electric double layer at the surface. This surface electric dipole gives a jump in the electrostatic potential between the material and the vacuum.
A variety of factors are responsible for the surface electric dipole. Even with a completely clean surface, the electrons can spread slightly into the vacuum, leaving behind a slightly positively charged layer of material. This primarily occurs in metals, where the bound electrons do not encounter a hard wall potential at the surface but rather a gradual ramping potential due to image charge attraction. The amount of surface dipole depends on the detailed layout of the atoms at the surface of the material, leading to the variation in work function for different crystal faces.
=== Doping and electric field effect (semiconductors) ===
In a semiconductor, the work function is sensitive to the doping level at the surface of the semiconductor. Since the doping near the surface can also be controlled by electric fields, the work function of a semiconductor is also sensitive to the electric field in the vacuum.
The reason for the dependence is that, typically, the vacuum level and the conduction band edge retain a fixed spacing independent of doping. This spacing is called the electron affinity (note that this has a different meaning than the electron affinity of chemistry); in silicon for example the electron affinity is 4.05 eV. If the electron affinity EEA and the surface's band-referenced Fermi level EF-EC are known, then the work function is given by
W
=
E
E
A
+
E
C
−
E
F
{\displaystyle W=E_{\rm {EA}}+E_{\rm {C}}-E_{\rm {F}}}
where EC is taken at the surface.
From this one might expect that by doping the bulk of the semiconductor, the work function can be tuned. In reality, however, the energies of the bands near the surface are often pinned to the Fermi level, due to the influence of surface states. If there is a large density of surface states, then the work function of the semiconductor will show a very weak dependence on doping or electric field.
=== Theoretical models of metal work functions ===
Theoretical modeling of the work function is difficult, as an accurate model requires a careful treatment of both electronic many body effects and surface chemistry; both of these topics are already complex in their own right.
One of the earliest successful models for metal work function trends was the jellium model, which allowed for oscillations in electronic density nearby the abrupt surface (these are similar to Friedel oscillations) as well as the tail of electron density extending outside the surface. This model showed why the density of conduction electrons (as represented by the Wigner–Seitz radius rs) is an important parameter in determining work function.
The jellium model is only a partial explanation, as its predictions still show significant deviation from real work functions. More recent models have focused on including more accurate forms of electron exchange and correlation effects, as well as including the crystal face dependence (this requires the inclusion of the actual atomic lattice, something that is neglected in the jellium model).
=== Temperature dependence of the electron work function ===
The electron behavior in metals varies with temperature and is largely reflected by the electron work function. A theoretical model for predicting the temperature dependence of the electron work function, developed by Rahemi et al. explains the underlying mechanism and predicts this temperature dependence for various crystal structures via calculable and measurable parameters. In general, as the temperature increases, the EWF decreases via
φ
(
T
)
=
φ
0
−
γ
(
k
B
T
)
2
φ
0
{\textstyle \varphi (T)=\varphi _{0}-\gamma {\frac {(k_{\text{B}}T)^{2}}{\varphi _{0}}}}
and
γ
{\displaystyle \gamma }
is a calculable material property which is dependent on the crystal structure (for example, BCC, FCC).
φ
0
{\displaystyle \varphi _{0}}
is the electron work function at T=0 and
k
B
{\displaystyle k_{\text{B}}}
is constant throughout the change.
== References ==
== Further reading ==
Ashcroft; Mermin (1976). Solid State Physics. Thomson Learning, Inc.
Goldstein, Newbury; et al. (2003). Scanning Electron Microscopy and X-Ray Microanalysis. New York: Springer.
For a quick reference to values of work function of the elements:
Michaelson, Herbert B. (1977). "The work function of the elements and its periodicity". J. Appl. Phys. 48 (11): 4729. Bibcode:1977JAP....48.4729M. doi:10.1063/1.323539. S2CID 122357835.
== External links ==
Work function of polymeric insulators (Table 2.1)
Work function of diamond and doped carbon Archived 2012-06-29 at the Wayback Machine
Work functions of common metals
Work functions of various metals for the photoelectric effect
Physics of free surfaces of semiconductors | Wikipedia/Work_function |
Quantum mechanics is a difficult subject to teach due to its counterintuitive nature. As the subject is now offered by advanced secondary schools, educators have applied scientific methodology to the process of teaching quantum mechanics, in order to identify common misconceptions and ways of improving students' understanding.
== Common learning difficulties ==
Students' misconceptions range from fully classical physics thinking, mixed models, to quasi-quantum ideas. For example, if the concept that quantum mechanics does not describe a path for electrons or photons is misunderstood, students may believe that they follow specific trajectories (classical), or sinusoidal paths (mixes), or are simultaneously wave and particles (quasi-quantum: "in which students understand that quantum objects can behave as both particles and waves, but still have difficulty describing events in a nondeterministic way"). Among the concepts most often misunderstood are:
the postulates of quantum mechanics provide no description for the trajectories for electrons or photons,
amplitude of a wave is not a measure of energy,
most bound states have no corresponding classical orbits,
in practice, quantum mechanics gives probabilisitic rather than deterministic results,
intrinsic uncertainty rather than measurement error.
Issues also arise from misunderstanding classical concepts related to quantum concepts, such as the difference between light energy and light intensity.
== Teaching strategies ==
=== Mathematics ===
Quantum mechanics can be taught with a focus on different interpretations, different models, or via mathematical techniques. Studies have shown that focus on non-mathematical concepts can lead to adequate understanding.
=== Digital and multi-media ===
Despite the fundamental impossibility of directly viewing quantum states, multimedia visualizations are an important tool in education.
Interactive media provides an alternative experience beyond everyday personal experience as a tool for understanding quantum mechanics. Among the multimedia sites that have been studied with positive results are QuVis and Phet.
=== History and philosophy of science as educational guides ===
In introducing history as part of the process of teaching quantum mechanics sets up a potential conflict of goals: accurate history or pedagogical clarity. Studies have shown that teaching through history helps students recognize that the counterintuitive issues are fundamental rather than simply something they don't understand. Specifically discussing the historical debates on quantum concepts drives home the idea the quantum differs from classical. Discussing the philosophy of science introduces the idea that language derived from everyday experience limits our ability to describe quantum phenomena.
=== Directly discussing the meanings of words ===
Mohan analyzes two widely used representative quantum mechanics textbooks against the learning challenges reported by Krijtenburg-Lewerissa and others. Both texts adopt language ('waves' and 'particles') familiar to students in other contexts without directly exploring the significant shifts in meaning required by quantum mechanics. Mohan attributes some of the learning challenges to this unexplored application of inappropriate language.
== Teaching for quantum computing ==
N. David Mermin reports that an unconventional strategy based on abstract but simple math concepts is sufficient to teach quantum mechanics to students interested in quantum computing application rather than physics. Many of the issues that confound students of physics to not apply to this case and the mathematical background of quantum computing resembles the background already taught in computer science. Mermin develops notation and operations with classical bits then introduces quantum bits as superpositions of two classical states. He never needs to discuss even the Planck constant, which he suggests is important for quantum computer hardware but not software.
== Teaching based on quantum optics ==
Philipp Blitzenbauer engages students through simple but intrinsically quantum single-photon experiments. The approach avoids the ambiguous classical vs quantum character of photons in optical interference experiments like the double slit. Students exposed to quantum mechanics in this way avoid developing misconceptions apparent among students in the control group.
== See also ==
Physics education
Physics education research
Introduction to quantum mechanics
Mermin's device
List of textbooks about quantum mechanics
== Notes ==
== References == | Wikipedia/Teaching_quantum_mechanics |
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines field theory and the principle of relativity with ideas behind quantum mechanics.: xi QFT is used in particle physics to construct physical models of subatomic particles and in condensed matter physics to construct models of quasiparticles. The current standard model of particle physics is based on QFT.
== History ==
Quantum field theory emerged from the work of generations of theoretical physicists spanning much of the 20th century. Its development began in the 1920s with the description of interactions between light and electrons, culminating in the first quantum field theory—quantum electrodynamics. A major theoretical obstacle soon followed with the appearance and persistence of various infinities in perturbative calculations, a problem only resolved in the 1950s with the invention of the renormalization procedure. A second major barrier came with QFT's apparent inability to describe the weak and strong interactions, to the point where some theorists called for the abandonment of the field theoretic approach. The development of gauge theory and the completion of the Standard Model in the 1970s led to a renaissance of quantum field theory.
=== Theoretical background ===
Quantum field theory results from the combination of classical field theory, quantum mechanics, and special relativity.: xi A brief overview of these theoretical precursors follows.
The earliest successful classical field theory is one that emerged from Newton's law of universal gravitation, despite the complete absence of the concept of fields from his 1687 treatise Philosophiæ Naturalis Principia Mathematica. The force of gravity as described by Isaac Newton is an "action at a distance"—its effects on faraway objects are instantaneous, no matter the distance. In an exchange of letters with Richard Bentley, however, Newton stated that "it is inconceivable that inanimate brute matter should, without the mediation of something else which is not material, operate upon and affect other matter without mutual contact".: 4 It was not until the 18th century that mathematical physicists discovered a convenient description of gravity based on fields—a numerical quantity (a vector in the case of gravitational field) assigned to every point in space indicating the action of gravity on any particle at that point. However, this was considered merely a mathematical trick.: 18
Fields began to take on an existence of their own with the development of electromagnetism in the 19th century. Michael Faraday coined the English term "field" in 1845. He introduced fields as properties of space (even when it is devoid of matter) having physical effects. He argued against "action at a distance", and proposed that interactions between objects occur via space-filling "lines of force". This description of fields remains to this day.: 301 : 2
The theory of classical electromagnetism was completed in 1864 with Maxwell's equations, which described the relationship between the electric field, the magnetic field, electric current, and electric charge. Maxwell's equations implied the existence of electromagnetic waves, a phenomenon whereby electric and magnetic fields propagate from one spatial point to another at a finite speed, which turns out to be the speed of light. Action-at-a-distance was thus conclusively refuted.: 19
Despite the enormous success of classical electromagnetism, it was unable to account for the discrete lines in atomic spectra, nor for the distribution of blackbody radiation in different wavelengths. Max Planck's study of blackbody radiation marked the beginning of quantum mechanics. He treated atoms, which absorb and emit electromagnetic radiation, as tiny oscillators with the crucial property that their energies can only take on a series of discrete, rather than continuous, values. These are known as quantum harmonic oscillators. This process of restricting energies to discrete values is called quantization.: Ch.2 Building on this idea, Albert Einstein proposed in 1905 an explanation for the photoelectric effect, that light is composed of individual packets of energy called photons (the quanta of light). This implied that the electromagnetic radiation, while being waves in the classical electromagnetic field, also exists in the form of particles.
In 1913, Niels Bohr introduced the Bohr model of atomic structure, wherein electrons within atoms can only take on a series of discrete, rather than continuous, energies. This is another example of quantization. The Bohr model successfully explained the discrete nature of atomic spectral lines. In 1924, Louis de Broglie proposed the hypothesis of wave–particle duality, that microscopic particles exhibit both wave-like and particle-like properties under different circumstances. Uniting these scattered ideas, a coherent discipline, quantum mechanics, was formulated between 1925 and 1926, with important contributions from Max Planck, Louis de Broglie, Werner Heisenberg, Max Born, Erwin Schrödinger, Paul Dirac, and Wolfgang Pauli.: 22–23
In the same year as his paper on the photoelectric effect, Einstein published his theory of special relativity, built on Maxwell's electromagnetism. New rules, called Lorentz transformations, were given for the way time and space coordinates of an event change under changes in the observer's velocity, and the distinction between time and space was blurred.: 19 It was proposed that all physical laws must be the same for observers at different velocities, i.e. that physical laws be invariant under Lorentz transformations.
Two difficulties remained. Observationally, the Schrödinger equation underlying quantum mechanics could explain the stimulated emission of radiation from atoms, where an electron emits a new photon under the action of an external electromagnetic field, but it was unable to explain spontaneous emission, where an electron spontaneously decreases in energy and emits a photon even without the action of an external electromagnetic field. Theoretically, the Schrödinger equation could not describe photons and was inconsistent with the principles of special relativity—it treats time as an ordinary number while promoting spatial coordinates to linear operators.
=== Quantum electrodynamics ===
Quantum field theory naturally began with the study of electromagnetic interactions, as the electromagnetic field was the only known classical field as of the 1920s.: 1
Through the works of Born, Heisenberg, and Pascual Jordan in 1925–1926, a quantum theory of the free electromagnetic field (one with no interactions with matter) was developed via canonical quantization by treating the electromagnetic field as a set of quantum harmonic oscillators.: 1 With the exclusion of interactions, however, such a theory was yet incapable of making quantitative predictions about the real world.: 22
In his seminal 1927 paper The quantum theory of the emission and absorption of radiation, Dirac coined the term quantum electrodynamics (QED), a theory that adds upon the terms describing the free electromagnetic field an additional interaction term between electric current density and the electromagnetic vector potential. Using first-order perturbation theory, he successfully explained the phenomenon of spontaneous emission. According to the uncertainty principle in quantum mechanics, quantum harmonic oscillators cannot remain stationary, but they have a non-zero minimum energy and must always be oscillating, even in the lowest energy state (the ground state). Therefore, even in a perfect vacuum, there remains an oscillating electromagnetic field having zero-point energy. It is this quantum fluctuation of electromagnetic fields in the vacuum that "stimulates" the spontaneous emission of radiation by electrons in atoms. Dirac's theory was hugely successful in explaining both the emission and absorption of radiation by atoms; by applying second-order perturbation theory, it was able to account for the scattering of photons, resonance fluorescence and non-relativistic Compton scattering. Nonetheless, the application of higher-order perturbation theory was plagued with problematic infinities in calculations.: 71
In 1928, Dirac wrote down a wave equation that described relativistic electrons: the Dirac equation. It had the following important consequences: the spin of an electron is 1/2; the electron g-factor is 2; it led to the correct Sommerfeld formula for the fine structure of the hydrogen atom; and it could be used to derive the Klein–Nishina formula for relativistic Compton scattering. Although the results were fruitful, the theory also apparently implied the existence of negative energy states, which would cause atoms to be unstable, since they could always decay to lower energy states by the emission of radiation.: 71–72
The prevailing view at the time was that the world was composed of two very different ingredients: material particles (such as electrons) and quantum fields (such as photons). Material particles were considered to be eternal, with their physical state described by the probabilities of finding each particle in any given region of space or range of velocities. On the other hand, photons were considered merely the excited states of the underlying quantized electromagnetic field, and could be freely created or destroyed. It was between 1928 and 1930 that Jordan, Eugene Wigner, Heisenberg, Pauli, and Enrico Fermi discovered that material particles could also be seen as excited states of quantum fields. Just as photons are excited states of the quantized electromagnetic field, so each type of particle had its corresponding quantum field: an electron field, a proton field, etc. Given enough energy, it would now be possible to create material particles. Building on this idea, Fermi proposed in 1932 an explanation for beta decay known as Fermi's interaction. Atomic nuclei do not contain electrons per se, but in the process of decay, an electron is created out of the surrounding electron field, analogous to the photon created from the surrounding electromagnetic field in the radiative decay of an excited atom.: 22–23
It was realized in 1929 by Dirac and others that negative energy states implied by the Dirac equation could be removed by assuming the existence of particles with the same mass as electrons but opposite electric charge. This not only ensured the stability of atoms, but it was also the first proposal of the existence of antimatter. Indeed, the evidence for positrons was discovered in 1932 by Carl David Anderson in cosmic rays. With enough energy, such as by absorbing a photon, an electron-positron pair could be created, a process called pair production; the reverse process, annihilation, could also occur with the emission of a photon. This showed that particle numbers need not be fixed during an interaction. Historically, however, positrons were at first thought of as "holes" in an infinite electron sea, rather than a new kind of particle, and this theory was referred to as the Dirac hole theory.: 72 : 23 QFT naturally incorporated antiparticles in its formalism.: 24
=== Infinities and renormalization ===
Robert Oppenheimer showed in 1930 that higher-order perturbative calculations in QED always resulted in infinite quantities, such as the electron self-energy and the vacuum zero-point energy of the electron and photon fields, suggesting that the computational methods at the time could not properly deal with interactions involving photons with extremely high momenta.: 25 It was not until 20 years later that a systematic approach to remove such infinities was developed.
A series of papers was published between 1934 and 1938 by Ernst Stueckelberg that established a relativistically invariant formulation of QFT. In 1947, Stueckelberg also independently developed a complete renormalization procedure. Such achievements were not understood and recognized by the theoretical community.
Faced with these infinities, John Archibald Wheeler and Heisenberg proposed, in 1937 and 1943 respectively, to supplant the problematic QFT with the so-called S-matrix theory. Since the specific details of microscopic interactions are inaccessible to observations, the theory should only attempt to describe the relationships between a small number of observables (e.g. the energy of an atom) in an interaction, rather than be concerned with the microscopic minutiae of the interaction. In 1945, Richard Feynman and Wheeler daringly suggested abandoning QFT altogether and proposed action-at-a-distance as the mechanism of particle interactions.: 26
In 1947, Willis Lamb and Robert Retherford measured the minute difference in the 2S1/2 and 2P1/2 energy levels of the hydrogen atom, also called the Lamb shift. By ignoring the contribution of photons whose energy exceeds the electron mass, Hans Bethe successfully estimated the numerical value of the Lamb shift.: 28 Subsequently, Norman Myles Kroll, Lamb, James Bruce French, and Victor Weisskopf again confirmed this value using an approach in which infinities cancelled other infinities to result in finite quantities. However, this method was clumsy and unreliable and could not be generalized to other calculations.
The breakthrough eventually came around 1950 when a more robust method for eliminating infinities was developed by Julian Schwinger, Richard Feynman, Freeman Dyson, and Shinichiro Tomonaga. The main idea is to replace the calculated values of mass and charge, infinite though they may be, by their finite measured values. This systematic computational procedure is known as renormalization and can be applied to arbitrary order in perturbation theory. As Tomonaga said in his Nobel lecture:Since those parts of the modified mass and charge due to field reactions [become infinite], it is impossible to calculate them by the theory. However, the mass and charge observed in experiments are not the original mass and charge but the mass and charge as modified by field reactions, and they are finite. On the other hand, the mass and charge appearing in the theory are… the values modified by field reactions. Since this is so, and particularly since the theory is unable to calculate the modified mass and charge, we may adopt the procedure of substituting experimental values for them phenomenologically... This procedure is called the renormalization of mass and charge… After long, laborious calculations, less skillful than Schwinger's, we obtained a result... which was in agreement with [the] Americans'.
By applying the renormalization procedure, calculations were finally made to explain the electron's anomalous magnetic moment (the deviation of the electron g-factor from 2) and vacuum polarization. These results agreed with experimental measurements to a remarkable degree, thus marking the end of a "war against infinities".
At the same time, Feynman introduced the path integral formulation of quantum mechanics and Feynman diagrams.: 2 The latter can be used to visually and intuitively organize and to help compute terms in the perturbative expansion. Each diagram can be interpreted as paths of particles in an interaction, with each vertex and line having a corresponding mathematical expression, and the product of these expressions gives the scattering amplitude of the interaction represented by the diagram.: 5
It was with the invention of the renormalization procedure and Feynman diagrams that QFT finally arose as a complete theoretical framework.: 2
=== Non-renormalizability ===
Given the tremendous success of QED, many theorists believed, in the few years after 1949, that QFT could soon provide an understanding of all microscopic phenomena, not only the interactions between photons, electrons, and positrons. Contrary to this optimism, QFT entered yet another period of depression that lasted for almost two decades.: 30
The first obstacle was the limited applicability of the renormalization procedure. In perturbative calculations in QED, all infinite quantities could be eliminated by redefining a small (finite) number of physical quantities (namely the mass and charge of the electron). Dyson proved in 1949 that this is only possible for a small class of theories called "renormalizable theories", of which QED is an example. However, most theories, including the Fermi theory of the weak interaction, are "non-renormalizable". Any perturbative calculation in these theories beyond the first order would result in infinities that could not be removed by redefining a finite number of physical quantities.: 30
The second major problem stemmed from the limited validity of the Feynman diagram method, which is based on a series expansion in perturbation theory. In order for the series to converge and low-order calculations to be a good approximation, the coupling constant, in which the series is expanded, must be a sufficiently small number. The coupling constant in QED is the fine-structure constant α ≈ 1/137, which is small enough that only the simplest, lowest order, Feynman diagrams need to be considered in realistic calculations. In contrast, the coupling constant in the strong interaction is roughly of the order of one, making complicated, higher order, Feynman diagrams just as important as simple ones. There was thus no way of deriving reliable quantitative predictions for the strong interaction using perturbative QFT methods.: 31
With these difficulties looming, many theorists began to turn away from QFT. Some focused on symmetry principles and conservation laws, while others picked up the old S-matrix theory of Wheeler and Heisenberg. QFT was used heuristically as guiding principles, but not as a basis for quantitative calculations.: 31
=== Source theory ===
Schwinger, however, took a different route. For more than a decade he and his students had been nearly the only exponents of field theory,: 454 but in 1951 he found a way around the problem of the infinities with a new method using external sources as currents coupled to gauge fields. Motivated by the former findings, Schwinger kept pursuing this approach in order to "quantumly" generalize the classical process of coupling external forces to the configuration space parameters known as Lagrange multipliers. He summarized his source theory in 1966 then expanded the theory's applications to quantum electrodynamics in his three volume-set titled: Particles, Sources, and Fields. Developments in pion physics, in which the new viewpoint was most successfully applied, convinced him of the great advantages of mathematical simplicity and conceptual clarity that its use bestowed.
In source theory there are no divergences, and no renormalization. It may be regarded as the calculational tool of field theory, but it is more general. Using source theory, Schwinger was able to calculate the anomalous magnetic moment of the electron, which he had done in 1947, but this time with no ‘distracting remarks’ about infinite quantities.: 467
Schwinger also applied source theory to his QFT theory of gravity, and was able to reproduce all four of Einstein's classic results: gravitational red shift, deflection and slowing of light by gravity, and the perihelion precession of Mercury. The neglect of source theory by the physics community was a major disappointment for Schwinger:The lack of appreciation of these facts by others was depressing, but understandable. -J. SchwingerSee "the shoes incident" between J. Schwinger and S. Weinberg.
=== Standard model ===
In 1954, Yang Chen-Ning and Robert Mills generalized the local symmetry of QED, leading to non-Abelian gauge theories (also known as Yang–Mills theories), which are based on more complicated local symmetry groups.: 5 In QED, (electrically) charged particles interact via the exchange of photons, while in non-Abelian gauge theory, particles carrying a new type of "charge" interact via the exchange of massless gauge bosons. Unlike photons, these gauge bosons themselves carry charge.: 32
Sheldon Glashow developed a non-Abelian gauge theory that unified the electromagnetic and weak interactions in 1960. In 1964, Abdus Salam and John Clive Ward arrived at the same theory through a different path. This theory, nevertheless, was non-renormalizable.
Peter Higgs, Robert Brout, François Englert, Gerald Guralnik, Carl Hagen, and Tom Kibble proposed in their famous Physical Review Letters papers that the gauge symmetry in Yang–Mills theories could be broken by a mechanism called spontaneous symmetry breaking, through which originally massless gauge bosons could acquire mass.: 5–6
By combining the earlier theory of Glashow, Salam, and Ward with the idea of spontaneous symmetry breaking, Steven Weinberg wrote down in 1967 a theory describing electroweak interactions between all leptons and the effects of the Higgs boson. His theory was at first mostly ignored,: 6 until it was brought back to light in 1971 by Gerard 't Hooft's proof that non-Abelian gauge theories are renormalizable. The electroweak theory of Weinberg and Salam was extended from leptons to quarks in 1970 by Glashow, John Iliopoulos, and Luciano Maiani, marking its completion.
Harald Fritzsch, Murray Gell-Mann, and Heinrich Leutwyler discovered in 1971 that certain phenomena involving the strong interaction could also be explained by non-Abelian gauge theory. Quantum chromodynamics (QCD) was born. In 1973, David Gross, Frank Wilczek, and Hugh David Politzer showed that non-Abelian gauge theories are "asymptotically free", meaning that under renormalization, the coupling constant of the strong interaction decreases as the interaction energy increases. (Similar discoveries had been made numerous times previously, but they had been largely ignored.) : 11 Therefore, at least in high-energy interactions, the coupling constant in QCD becomes sufficiently small to warrant a perturbative series expansion, making quantitative predictions for the strong interaction possible.: 32
These theoretical breakthroughs brought about a renaissance in QFT. The full theory, which includes the electroweak theory and chromodynamics, is referred to today as the Standard Model of elementary particles. The Standard Model successfully describes all fundamental interactions except gravity, and its many predictions have been met with remarkable experimental confirmation in subsequent decades.: 3 The Higgs boson, central to the mechanism of spontaneous symmetry breaking, was finally detected in 2012 at CERN, marking the complete verification of the existence of all constituents of the Standard Model.
=== Other developments ===
The 1970s saw the development of non-perturbative methods in non-Abelian gauge theories. The 't Hooft–Polyakov monopole was discovered theoretically by 't Hooft and Alexander Polyakov, flux tubes by Holger Bech Nielsen and Poul Olesen, and instantons by Polyakov and coauthors. These objects are inaccessible through perturbation theory.: 4
Supersymmetry also appeared in the same period. The first supersymmetric QFT in four dimensions was built by Yuri Golfand and Evgeny Likhtman in 1970, but their result failed to garner widespread interest due to the Iron Curtain. Supersymmetry theories only took off in the theoretical community after the work of Julius Wess and Bruno Zumino in 1973,: 7 but to date have not been widely accepted as part of the Standard Model due to lack of experimental evidence.
Among the four fundamental interactions, gravity remains the only one that lacks a consistent QFT description. Various attempts at a theory of quantum gravity led to the development of string theory,: 6 itself a type of two-dimensional QFT with conformal symmetry. Joël Scherk and John Schwarz first proposed in 1974 that string theory could be the quantum theory of gravity.
=== Condensed-matter-physics ===
Although quantum field theory arose from the study of interactions between elementary particles, it has been successfully applied to other physical systems, particularly to many-body systems in condensed matter physics.
Historically, the Higgs mechanism of spontaneous symmetry breaking was a result of Yoichiro Nambu's application of superconductor theory to elementary particles, while the concept of renormalization came out of the study of second-order phase transitions in matter.
Soon after the introduction of photons, Einstein performed the quantization procedure on vibrations in a crystal, leading to the first quasiparticle—phonons. Lev Landau claimed that low-energy excitations in many condensed matter systems could be described in terms of interactions between a set of quasiparticles. The Feynman diagram method of QFT was naturally well suited to the analysis of various phenomena in condensed matter systems.
Gauge theory is used to describe the quantization of magnetic flux in superconductors, the resistivity in the quantum Hall effect, as well as the relation between frequency and voltage in the AC Josephson effect.
== Principles ==
For simplicity, natural units are used in the following sections, in which the reduced Planck constant ħ and the speed of light c are both set to one.
=== Classical fields ===
A classical field is a function of spatial and time coordinates. Examples include the gravitational field in Newtonian gravity g(x, t) and the electric field E(x, t) and magnetic field B(x, t) in classical electromagnetism. A classical field can be thought of as a numerical quantity assigned to every point in space that changes in time. Hence, it has infinitely many degrees of freedom.
Many phenomena exhibiting quantum mechanical properties cannot be explained by classical fields alone. Phenomena such as the photoelectric effect are best explained by discrete particles (photons), rather than a spatially continuous field. The goal of quantum field theory is to describe various quantum mechanical phenomena using a modified concept of fields.
Canonical quantization and path integrals are two common formulations of QFT.: 61 To motivate the fundamentals of QFT, an overview of classical field theory follows.
The simplest classical field is a real scalar field — a real number at every point in space that changes in time. It is denoted as ϕ(x, t), where x is the position vector, and t is the time. Suppose the Lagrangian of the field,
L
{\displaystyle L}
, is
L
=
∫
d
3
x
L
=
∫
d
3
x
[
1
2
ϕ
˙
2
−
1
2
(
∇
ϕ
)
2
−
1
2
m
2
ϕ
2
]
,
{\displaystyle L=\int d^{3}x\,{\mathcal {L}}=\int d^{3}x\,\left[{\frac {1}{2}}{\dot {\phi }}^{2}-{\frac {1}{2}}(\nabla \phi )^{2}-{\frac {1}{2}}m^{2}\phi ^{2}\right],}
where
L
{\displaystyle {\mathcal {L}}}
is the Lagrangian density,
ϕ
˙
{\displaystyle {\dot {\phi }}}
is the time-derivative of the field, ∇ is the gradient operator, and m is a real parameter (the "mass" of the field). Applying the Euler–Lagrange equation on the Lagrangian:: 16
∂
∂
t
[
∂
L
∂
(
∂
ϕ
/
∂
t
)
]
+
∑
i
=
1
3
∂
∂
x
i
[
∂
L
∂
(
∂
ϕ
/
∂
x
i
)
]
−
∂
L
∂
ϕ
=
0
,
{\displaystyle {\frac {\partial }{\partial t}}\left[{\frac {\partial {\mathcal {L}}}{\partial (\partial \phi /\partial t)}}\right]+\sum _{i=1}^{3}{\frac {\partial }{\partial x^{i}}}\left[{\frac {\partial {\mathcal {L}}}{\partial (\partial \phi /\partial x^{i})}}\right]-{\frac {\partial {\mathcal {L}}}{\partial \phi }}=0,}
we obtain the equations of motion for the field, which describe the way it varies in time and space:
(
∂
2
∂
t
2
−
∇
2
+
m
2
)
ϕ
=
0.
{\displaystyle \left({\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2}+m^{2}\right)\phi =0.}
This is known as the Klein–Gordon equation.: 17
The Klein–Gordon equation is a wave equation, so its solutions can be expressed as a sum of normal modes (obtained via Fourier transform) as follows:
ϕ
(
x
,
t
)
=
∫
d
3
p
(
2
π
)
3
1
2
ω
p
(
a
p
e
−
i
ω
p
t
+
i
p
⋅
x
+
a
p
∗
e
i
ω
p
t
−
i
p
⋅
x
)
,
{\displaystyle \phi (\mathbf {x} ,t)=\int {\frac {d^{3}p}{(2\pi )^{3}}}{\frac {1}{\sqrt {2\omega _{\mathbf {p} }}}}\left(a_{\mathbf {p} }e^{-i\omega _{\mathbf {p} }t+i\mathbf {p} \cdot \mathbf {x} }+a_{\mathbf {p} }^{*}e^{i\omega _{\mathbf {p} }t-i\mathbf {p} \cdot \mathbf {x} }\right),}
where a is a complex number (normalized by convention), * denotes complex conjugation, and ωp is the frequency of the normal mode:
ω
p
=
|
p
|
2
+
m
2
.
{\displaystyle \omega _{\mathbf {p} }={\sqrt {|\mathbf {p} |^{2}+m^{2}}}.}
Thus each normal mode corresponding to a single p can be seen as a classical harmonic oscillator with frequency ωp.: 21,26
=== Canonical quantization ===
The quantization procedure for the above classical field to a quantum operator field is analogous to the promotion of a classical harmonic oscillator to a quantum harmonic oscillator.
The displacement of a classical harmonic oscillator is described by
x
(
t
)
=
1
2
ω
a
e
−
i
ω
t
+
1
2
ω
a
∗
e
i
ω
t
,
{\displaystyle x(t)={\frac {1}{\sqrt {2\omega }}}ae^{-i\omega t}+{\frac {1}{\sqrt {2\omega }}}a^{*}e^{i\omega t},}
where a is a complex number (normalized by convention), and ω is the oscillator's frequency. Note that x is the displacement of a particle in simple harmonic motion from the equilibrium position, not to be confused with the spatial label x of a quantum field.
For a quantum harmonic oscillator, x(t) is promoted to a linear operator
x
^
(
t
)
{\displaystyle {\hat {x}}(t)}
:
x
^
(
t
)
=
1
2
ω
a
^
e
−
i
ω
t
+
1
2
ω
a
^
†
e
i
ω
t
.
{\displaystyle {\hat {x}}(t)={\frac {1}{\sqrt {2\omega }}}{\hat {a}}e^{-i\omega t}+{\frac {1}{\sqrt {2\omega }}}{\hat {a}}^{\dagger }e^{i\omega t}.}
Complex numbers a and a* are replaced by the annihilation operator
a
^
{\displaystyle {\hat {a}}}
and the creation operator
a
^
†
{\displaystyle {\hat {a}}^{\dagger }}
, respectively, where † denotes Hermitian conjugation. The commutation relation between the two is
[
a
^
,
a
^
†
]
=
1.
{\displaystyle \left[{\hat {a}},{\hat {a}}^{\dagger }\right]=1.}
The Hamiltonian of the simple harmonic oscillator can be written as
H
^
=
ℏ
ω
a
^
†
a
^
+
1
2
ℏ
ω
.
{\displaystyle {\hat {H}}=\hbar \omega {\hat {a}}^{\dagger }{\hat {a}}+{\frac {1}{2}}\hbar \omega .}
The vacuum state
|
0
⟩
{\displaystyle |0\rangle }
, which is the lowest energy state, is defined by
a
^
|
0
⟩
=
0
{\displaystyle {\hat {a}}|0\rangle =0}
and has energy
1
2
ℏ
ω
.
{\displaystyle {\frac {1}{2}}\hbar \omega .}
One can easily check that
[
H
^
,
a
^
†
]
=
ℏ
ω
a
^
†
,
{\displaystyle [{\hat {H}},{\hat {a}}^{\dagger }]=\hbar \omega {\hat {a}}^{\dagger },}
which implies that
a
^
†
{\displaystyle {\hat {a}}^{\dagger }}
increases the energy of the simple harmonic oscillator by
ℏ
ω
{\displaystyle \hbar \omega }
. For example, the state
a
^
†
|
0
⟩
{\displaystyle {\hat {a}}^{\dagger }|0\rangle }
is an eigenstate of energy
3
ℏ
ω
/
2
{\displaystyle 3\hbar \omega /2}
.
Any energy eigenstate state of a single harmonic oscillator can be obtained from
|
0
⟩
{\displaystyle |0\rangle }
by successively applying the creation operator
a
^
†
{\displaystyle {\hat {a}}^{\dagger }}
:: 20 and any state of the system can be expressed as a linear combination of the states
|
n
⟩
∝
(
a
^
†
)
n
|
0
⟩
.
{\displaystyle |n\rangle \propto \left({\hat {a}}^{\dagger }\right)^{n}|0\rangle .}
A similar procedure can be applied to the real scalar field ϕ, by promoting it to a quantum field operator
ϕ
^
{\displaystyle {\hat {\phi }}}
, while the annihilation operator
a
^
p
{\displaystyle {\hat {a}}_{\mathbf {p} }}
, the creation operator
a
^
p
†
{\displaystyle {\hat {a}}_{\mathbf {p} }^{\dagger }}
and the angular frequency
ω
p
{\displaystyle \omega _{\mathbf {p} }}
are now for a particular p:
ϕ
^
(
x
,
t
)
=
∫
d
3
p
(
2
π
)
3
1
2
ω
p
(
a
^
p
e
−
i
ω
p
t
+
i
p
⋅
x
+
a
^
p
†
e
i
ω
p
t
−
i
p
⋅
x
)
.
{\displaystyle {\hat {\phi }}(\mathbf {x} ,t)=\int {\frac {d^{3}p}{(2\pi )^{3}}}{\frac {1}{\sqrt {2\omega _{\mathbf {p} }}}}\left({\hat {a}}_{\mathbf {p} }e^{-i\omega _{\mathbf {p} }t+i\mathbf {p} \cdot \mathbf {x} }+{\hat {a}}_{\mathbf {p} }^{\dagger }e^{i\omega _{\mathbf {p} }t-i\mathbf {p} \cdot \mathbf {x} }\right).}
Their commutation relations are:: 21
[
a
^
p
,
a
^
q
†
]
=
(
2
π
)
3
δ
(
p
−
q
)
,
[
a
^
p
,
a
^
q
]
=
[
a
^
p
†
,
a
^
q
†
]
=
0
,
{\displaystyle \left[{\hat {a}}_{\mathbf {p} },{\hat {a}}_{\mathbf {q} }^{\dagger }\right]=(2\pi )^{3}\delta (\mathbf {p} -\mathbf {q} ),\quad \left[{\hat {a}}_{\mathbf {p} },{\hat {a}}_{\mathbf {q} }\right]=\left[{\hat {a}}_{\mathbf {p} }^{\dagger },{\hat {a}}_{\mathbf {q} }^{\dagger }\right]=0,}
where δ is the Dirac delta function. The vacuum state
|
0
⟩
{\displaystyle |0\rangle }
is defined by
a
^
p
|
0
⟩
=
0
,
for all
p
.
{\displaystyle {\hat {a}}_{\mathbf {p} }|0\rangle =0,\quad {\text{for all }}\mathbf {p} .}
Any quantum state of the field can be obtained from
|
0
⟩
{\displaystyle |0\rangle }
by successively applying creation operators
a
^
p
†
{\displaystyle {\hat {a}}_{\mathbf {p} }^{\dagger }}
(or by a linear combination of such states), e.g. : 22
(
a
^
p
3
†
)
3
a
^
p
2
†
(
a
^
p
1
†
)
2
|
0
⟩
.
{\displaystyle \left({\hat {a}}_{\mathbf {p} _{3}}^{\dagger }\right)^{3}{\hat {a}}_{\mathbf {p} _{2}}^{\dagger }\left({\hat {a}}_{\mathbf {p} _{1}}^{\dagger }\right)^{2}|0\rangle .}
While the state space of a single quantum harmonic oscillator contains all the discrete energy states of one oscillating particle, the state space of a quantum field contains the discrete energy levels of an arbitrary number of particles. The latter space is known as a Fock space, which can account for the fact that particle numbers are not fixed in relativistic quantum systems. The process of quantizing an arbitrary number of particles instead of a single particle is often also called second quantization.: 19
The foregoing procedure is a direct application of non-relativistic quantum mechanics and can be used to quantize (complex) scalar fields, Dirac fields,: 52 vector fields (e.g. the electromagnetic field), and even strings. However, creation and annihilation operators are only well defined in the simplest theories that contain no interactions (so-called free theory). In the case of the real scalar field, the existence of these operators was a consequence of the decomposition of solutions of the classical equations of motion into a sum of normal modes. To perform calculations on any realistic interacting theory, perturbation theory would be necessary.
The Lagrangian of any quantum field in nature would contain interaction terms in addition to the free theory terms. For example, a quartic interaction term could be introduced to the Lagrangian of the real scalar field:: 77
L
=
1
2
(
∂
μ
ϕ
)
(
∂
μ
ϕ
)
−
1
2
m
2
ϕ
2
−
λ
4
!
ϕ
4
,
{\displaystyle {\mathcal {L}}={\frac {1}{2}}(\partial _{\mu }\phi )\left(\partial ^{\mu }\phi \right)-{\frac {1}{2}}m^{2}\phi ^{2}-{\frac {\lambda }{4!}}\phi ^{4},}
where μ is a spacetime index,
∂
0
=
∂
/
∂
t
,
∂
1
=
∂
/
∂
x
1
{\displaystyle \partial _{0}=\partial /\partial t,\ \partial _{1}=\partial /\partial x^{1}}
, etc. The summation over the index μ has been omitted following the Einstein notation. If the parameter λ is sufficiently small, then the interacting theory described by the above Lagrangian can be considered as a small perturbation from the free theory.
=== Path integrals ===
The path integral formulation of QFT is concerned with the direct computation of the scattering amplitude of a certain interaction process, rather than the establishment of operators and state spaces. To calculate the probability amplitude for a system to evolve from some initial state
|
ϕ
I
⟩
{\displaystyle |\phi _{I}\rangle }
at time t = 0 to some final state
|
ϕ
F
⟩
{\displaystyle |\phi _{F}\rangle }
at t = T, the total time T is divided into N small intervals. The overall amplitude is the product of the amplitude of evolution within each interval, integrated over all intermediate states. Let H be the Hamiltonian (i.e. generator of time evolution), then: 10
⟨
ϕ
F
|
e
−
i
H
T
|
ϕ
I
⟩
=
∫
d
ϕ
1
∫
d
ϕ
2
⋯
∫
d
ϕ
N
−
1
⟨
ϕ
F
|
e
−
i
H
T
/
N
|
ϕ
N
−
1
⟩
⋯
⟨
ϕ
2
|
e
−
i
H
T
/
N
|
ϕ
1
⟩
⟨
ϕ
1
|
e
−
i
H
T
/
N
|
ϕ
I
⟩
.
{\displaystyle \langle \phi _{F}|e^{-iHT}|\phi _{I}\rangle =\int d\phi _{1}\int d\phi _{2}\cdots \int d\phi _{N-1}\,\langle \phi _{F}|e^{-iHT/N}|\phi _{N-1}\rangle \cdots \langle \phi _{2}|e^{-iHT/N}|\phi _{1}\rangle \langle \phi _{1}|e^{-iHT/N}|\phi _{I}\rangle .}
Taking the limit N → ∞, the above product of integrals becomes the Feynman path integral:: 282 : 12
⟨
ϕ
F
|
e
−
i
H
T
|
ϕ
I
⟩
=
∫
D
ϕ
(
t
)
exp
{
i
∫
0
T
d
t
L
}
,
{\displaystyle \langle \phi _{F}|e^{-iHT}|\phi _{I}\rangle =\int {\mathcal {D}}\phi (t)\,\exp \left\{i\int _{0}^{T}dt\,L\right\},}
where L is the Lagrangian involving ϕ and its derivatives with respect to spatial and time coordinates, obtained from the Hamiltonian H via Legendre transformation. The initial and final conditions of the path integral are respectively
ϕ
(
0
)
=
ϕ
I
,
ϕ
(
T
)
=
ϕ
F
.
{\displaystyle \phi (0)=\phi _{I},\quad \phi (T)=\phi _{F}.}
In other words, the overall amplitude is the sum over the amplitude of every possible path between the initial and final states, where the amplitude of a path is given by the exponential in the integrand.
=== Two-point correlation function ===
In calculations, one often encounters expression like
⟨
0
|
T
{
ϕ
(
x
)
ϕ
(
y
)
}
|
0
⟩
or
⟨
Ω
|
T
{
ϕ
(
x
)
ϕ
(
y
)
}
|
Ω
⟩
{\displaystyle \langle 0|T\{\phi (x)\phi (y)\}|0\rangle \quad {\text{or}}\quad \langle \Omega |T\{\phi (x)\phi (y)\}|\Omega \rangle }
in the free or interacting theory, respectively. Here,
x
{\displaystyle x}
and
y
{\displaystyle y}
are position four-vectors,
T
{\displaystyle T}
is the time ordering operator that shuffles its operands so the time-components
x
0
{\displaystyle x^{0}}
and
y
0
{\displaystyle y^{0}}
increase from right to left, and
|
Ω
⟩
{\displaystyle |\Omega \rangle }
is the ground state (vacuum state) of the interacting theory, different from the free ground state
|
0
⟩
{\displaystyle |0\rangle }
. This expression represents the probability amplitude for the field to propagate from y to x, and goes by multiple names, like the two-point propagator, two-point correlation function, two-point Green's function or two-point function for short.: 82
The free two-point function, also known as the Feynman propagator, can be found for the real scalar field by either canonical quantization or path integrals to be: 31,288 : 23
⟨
0
|
T
{
ϕ
(
x
)
ϕ
(
y
)
}
|
0
⟩
≡
D
F
(
x
−
y
)
=
lim
ϵ
→
0
∫
d
4
p
(
2
π
)
4
i
p
μ
p
μ
−
m
2
+
i
ϵ
e
−
i
p
μ
(
x
μ
−
y
μ
)
.
{\displaystyle \langle 0|T\{\phi (x)\phi (y)\}|0\rangle \equiv D_{F}(x-y)=\lim _{\epsilon \to 0}\int {\frac {d^{4}p}{(2\pi )^{4}}}{\frac {i}{p_{\mu }p^{\mu }-m^{2}+i\epsilon }}e^{-ip_{\mu }(x^{\mu }-y^{\mu })}.}
In an interacting theory, where the Lagrangian or Hamiltonian contains terms
L
I
(
t
)
{\displaystyle L_{I}(t)}
or
H
I
(
t
)
{\displaystyle H_{I}(t)}
that describe interactions, the two-point function is more difficult to define. However, through both the canonical quantization formulation and the path integral formulation, it is possible to express it through an infinite perturbation series of the free two-point function.
In canonical quantization, the two-point correlation function can be written as:: 87
⟨
Ω
|
T
{
ϕ
(
x
)
ϕ
(
y
)
}
|
Ω
⟩
=
lim
T
→
∞
(
1
−
i
ϵ
)
⟨
0
|
T
{
ϕ
I
(
x
)
ϕ
I
(
y
)
exp
[
−
i
∫
−
T
T
d
t
H
I
(
t
)
]
}
|
0
⟩
⟨
0
|
T
{
exp
[
−
i
∫
−
T
T
d
t
H
I
(
t
)
]
}
|
0
⟩
,
{\displaystyle \langle \Omega |T\{\phi (x)\phi (y)\}|\Omega \rangle =\lim _{T\to \infty (1-i\epsilon )}{\frac {\left\langle 0\left|T\left\{\phi _{I}(x)\phi _{I}(y)\exp \left[-i\int _{-T}^{T}dt\,H_{I}(t)\right]\right\}\right|0\right\rangle }{\left\langle 0\left|T\left\{\exp \left[-i\int _{-T}^{T}dt\,H_{I}(t)\right]\right\}\right|0\right\rangle }},}
where ε is an infinitesimal number and ϕI is the field operator under the free theory. Here, the exponential should be understood as its power series expansion. For example, in
ϕ
4
{\displaystyle \phi ^{4}}
-theory, the interacting term of the Hamiltonian is
H
I
(
t
)
=
∫
d
3
x
λ
4
!
ϕ
I
(
x
)
4
{\textstyle H_{I}(t)=\int d^{3}x\,{\frac {\lambda }{4!}}\phi _{I}(x)^{4}}
,: 84 and the expansion of the two-point correlator in terms of
λ
{\displaystyle \lambda }
becomes
⟨
Ω
|
T
{
ϕ
(
x
)
ϕ
(
y
)
}
|
Ω
⟩
=
∑
n
=
0
∞
(
−
i
λ
)
n
(
4
!
)
n
n
!
∫
d
4
z
1
⋯
∫
d
4
z
n
⟨
0
|
T
{
ϕ
I
(
x
)
ϕ
I
(
y
)
ϕ
I
(
z
1
)
4
⋯
ϕ
I
(
z
n
)
4
}
|
0
⟩
∑
n
=
0
∞
(
−
i
λ
)
n
(
4
!
)
n
n
!
∫
d
4
z
1
⋯
∫
d
4
z
n
⟨
0
|
T
{
ϕ
I
(
z
1
)
4
⋯
ϕ
I
(
z
n
)
4
}
|
0
⟩
.
{\displaystyle \langle \Omega |T\{\phi (x)\phi (y)\}|\Omega \rangle ={\frac {\displaystyle \sum _{n=0}^{\infty }{\frac {(-i\lambda )^{n}}{(4!)^{n}n!}}\int d^{4}z_{1}\cdots \int d^{4}z_{n}\langle 0|T\{\phi _{I}(x)\phi _{I}(y)\phi _{I}(z_{1})^{4}\cdots \phi _{I}(z_{n})^{4}\}|0\rangle }{\displaystyle \sum _{n=0}^{\infty }{\frac {(-i\lambda )^{n}}{(4!)^{n}n!}}\int d^{4}z_{1}\cdots \int d^{4}z_{n}\langle 0|T\{\phi _{I}(z_{1})^{4}\cdots \phi _{I}(z_{n})^{4}\}|0\rangle }}.}
This perturbation expansion expresses the interacting two-point function in terms of quantities
⟨
0
|
⋯
|
0
⟩
{\displaystyle \langle 0|\cdots |0\rangle }
that are evaluated in the free theory.
In the path integral formulation, the two-point correlation function can be written: 284
⟨
Ω
|
T
{
ϕ
(
x
)
ϕ
(
y
)
}
|
Ω
⟩
=
lim
T
→
∞
(
1
−
i
ϵ
)
∫
D
ϕ
ϕ
(
x
)
ϕ
(
y
)
exp
[
i
∫
−
T
T
d
4
z
L
]
∫
D
ϕ
exp
[
i
∫
−
T
T
d
4
z
L
]
,
{\displaystyle \langle \Omega |T\{\phi (x)\phi (y)\}|\Omega \rangle =\lim _{T\to \infty (1-i\epsilon )}{\frac {\int {\mathcal {D}}\phi \,\phi (x)\phi (y)\exp \left[i\int _{-T}^{T}d^{4}z\,{\mathcal {L}}\right]}{\int {\mathcal {D}}\phi \,\exp \left[i\int _{-T}^{T}d^{4}z\,{\mathcal {L}}\right]}},}
where
L
{\displaystyle {\mathcal {L}}}
is the Lagrangian density. As in the previous paragraph, the exponential can be expanded as a series in λ, reducing the interacting two-point function to quantities in the free theory.
Wick's theorem further reduce any n-point correlation function in the free theory to a sum of products of two-point correlation functions. For example,
⟨
0
|
T
{
ϕ
(
x
1
)
ϕ
(
x
2
)
ϕ
(
x
3
)
ϕ
(
x
4
)
}
|
0
⟩
=
⟨
0
|
T
{
ϕ
(
x
1
)
ϕ
(
x
2
)
}
|
0
⟩
⟨
0
|
T
{
ϕ
(
x
3
)
ϕ
(
x
4
)
}
|
0
⟩
+
⟨
0
|
T
{
ϕ
(
x
1
)
ϕ
(
x
3
)
}
|
0
⟩
⟨
0
|
T
{
ϕ
(
x
2
)
ϕ
(
x
4
)
}
|
0
⟩
+
⟨
0
|
T
{
ϕ
(
x
1
)
ϕ
(
x
4
)
}
|
0
⟩
⟨
0
|
T
{
ϕ
(
x
2
)
ϕ
(
x
3
)
}
|
0
⟩
.
{\displaystyle {\begin{aligned}\langle 0|T\{\phi (x_{1})\phi (x_{2})\phi (x_{3})\phi (x_{4})\}|0\rangle &=\langle 0|T\{\phi (x_{1})\phi (x_{2})\}|0\rangle \langle 0|T\{\phi (x_{3})\phi (x_{4})\}|0\rangle \\&+\langle 0|T\{\phi (x_{1})\phi (x_{3})\}|0\rangle \langle 0|T\{\phi (x_{2})\phi (x_{4})\}|0\rangle \\&+\langle 0|T\{\phi (x_{1})\phi (x_{4})\}|0\rangle \langle 0|T\{\phi (x_{2})\phi (x_{3})\}|0\rangle .\end{aligned}}}
Since interacting correlation functions can be expressed in terms of free correlation functions, only the latter need to be evaluated in order to calculate all physical quantities in the (perturbative) interacting theory.: 90 This makes the Feynman propagator one of the most important quantities in quantum field theory.
=== Feynman diagram ===
Correlation functions in the interacting theory can be written as a perturbation series. Each term in the series is a product of Feynman propagators in the free theory and can be represented visually by a Feynman diagram. For example, the λ1 term in the two-point correlation function in the ϕ4 theory is
−
i
λ
4
!
∫
d
4
z
⟨
0
|
T
{
ϕ
(
x
)
ϕ
(
y
)
ϕ
(
z
)
ϕ
(
z
)
ϕ
(
z
)
ϕ
(
z
)
}
|
0
⟩
.
{\displaystyle {\frac {-i\lambda }{4!}}\int d^{4}z\,\langle 0|T\{\phi (x)\phi (y)\phi (z)\phi (z)\phi (z)\phi (z)\}|0\rangle .}
After applying Wick's theorem, one of the terms is
12
⋅
−
i
λ
4
!
∫
d
4
z
D
F
(
x
−
z
)
D
F
(
y
−
z
)
D
F
(
z
−
z
)
.
{\displaystyle 12\cdot {\frac {-i\lambda }{4!}}\int d^{4}z\,D_{F}(x-z)D_{F}(y-z)D_{F}(z-z).}
This term can instead be obtained from the Feynman diagram
.
The diagram consists of
external vertices connected with one edge and represented by dots (here labeled
x
{\displaystyle x}
and
y
{\displaystyle y}
).
internal vertices connected with four edges and represented by dots (here labeled
z
{\displaystyle z}
).
edges connecting the vertices and represented by lines.
Every vertex corresponds to a single
ϕ
{\displaystyle \phi }
field factor at the corresponding point in spacetime, while the edges correspond to the propagators between the spacetime points. The term in the perturbation series corresponding to the diagram is obtained by writing down the expression that follows from the so-called Feynman rules:
For every internal vertex
z
i
{\displaystyle z_{i}}
, write down a factor
−
i
λ
∫
d
4
z
i
{\textstyle -i\lambda \int d^{4}z_{i}}
.
For every edge that connects two vertices
z
i
{\displaystyle z_{i}}
and
z
j
{\displaystyle z_{j}}
, write down a factor
D
F
(
z
i
−
z
j
)
{\displaystyle D_{F}(z_{i}-z_{j})}
.
Divide by the symmetry factor of the diagram.
With the symmetry factor
2
{\displaystyle 2}
, following these rules yields exactly the expression above. By Fourier transforming the propagator, the Feynman rules can be reformulated from position space into momentum space.: 91–94
In order to compute the n-point correlation function to the k-th order, list all valid Feynman diagrams with n external points and k or fewer vertices, and then use Feynman rules to obtain the expression for each term. To be precise,
⟨
Ω
|
T
{
ϕ
(
x
1
)
⋯
ϕ
(
x
n
)
}
|
Ω
⟩
{\displaystyle \langle \Omega |T\{\phi (x_{1})\cdots \phi (x_{n})\}|\Omega \rangle }
is equal to the sum of (expressions corresponding to) all connected diagrams with n external points. (Connected diagrams are those in which every vertex is connected to an external point through lines. Components that are totally disconnected from external lines are sometimes called "vacuum bubbles".) In the ϕ4 interaction theory discussed above, every vertex must have four legs.: 98
In realistic applications, the scattering amplitude of a certain interaction or the decay rate of a particle can be computed from the S-matrix, which itself can be found using the Feynman diagram method.: 102–115
Feynman diagrams devoid of "loops" are called tree-level diagrams, which describe the lowest-order interaction processes; those containing n loops are referred to as n-loop diagrams, which describe higher-order contributions, or radiative corrections, to the interaction.: 44 Lines whose end points are vertices can be thought of as the propagation of virtual particles.: 31
=== Renormalization ===
Feynman rules can be used to directly evaluate tree-level diagrams. However, naïve computation of loop diagrams such as the one shown above will result in divergent momentum integrals, which seems to imply that almost all terms in the perturbative expansion are infinite. The renormalisation procedure is a systematic process for removing such infinities.
Parameters appearing in the Lagrangian, such as the mass m and the coupling constant λ, have no physical meaning — m, λ, and the field strength ϕ are not experimentally measurable quantities and are referred to here as the bare mass, bare coupling constant, and bare field, respectively. The physical mass and coupling constant are measured in some interaction process and are generally different from the bare quantities. While computing physical quantities from this interaction process, one may limit the domain of divergent momentum integrals to be below some momentum cut-off Λ, obtain expressions for the physical quantities, and then take the limit Λ → ∞. This is an example of regularization, a class of methods to treat divergences in QFT, with Λ being the regulator.
The approach illustrated above is called bare perturbation theory, as calculations involve only the bare quantities such as mass and coupling constant. A different approach, called renormalized perturbation theory, is to use physically meaningful quantities from the very beginning. In the case of ϕ4 theory, the field strength is first redefined:
ϕ
=
Z
1
/
2
ϕ
r
,
{\displaystyle \phi =Z^{1/2}\phi _{r},}
where ϕ is the bare field, ϕr is the renormalized field, and Z is a constant to be determined. The Lagrangian density becomes:
L
=
1
2
(
∂
μ
ϕ
r
)
(
∂
μ
ϕ
r
)
−
1
2
m
r
2
ϕ
r
2
−
λ
r
4
!
ϕ
r
4
+
1
2
δ
Z
(
∂
μ
ϕ
r
)
(
∂
μ
ϕ
r
)
−
1
2
δ
m
ϕ
r
2
−
δ
λ
4
!
ϕ
r
4
,
{\displaystyle {\mathcal {L}}={\frac {1}{2}}(\partial _{\mu }\phi _{r})(\partial ^{\mu }\phi _{r})-{\frac {1}{2}}m_{r}^{2}\phi _{r}^{2}-{\frac {\lambda _{r}}{4!}}\phi _{r}^{4}+{\frac {1}{2}}\delta _{Z}(\partial _{\mu }\phi _{r})(\partial ^{\mu }\phi _{r})-{\frac {1}{2}}\delta _{m}\phi _{r}^{2}-{\frac {\delta _{\lambda }}{4!}}\phi _{r}^{4},}
where mr and λr are the experimentally measurable, renormalized, mass and coupling constant, respectively, and
δ
Z
=
Z
−
1
,
δ
m
=
m
2
Z
−
m
r
2
,
δ
λ
=
λ
Z
2
−
λ
r
{\displaystyle \delta _{Z}=Z-1,\quad \delta _{m}=m^{2}Z-m_{r}^{2},\quad \delta _{\lambda }=\lambda Z^{2}-\lambda _{r}}
are constants to be determined. The first three terms are the ϕ4 Lagrangian density written in terms of the renormalized quantities, while the latter three terms are referred to as "counterterms". As the Lagrangian now contains more terms, so the Feynman diagrams should include additional elements, each with their own Feynman rules. The procedure is outlined as follows. First select a regularization scheme (such as the cut-off regularization introduced above or dimensional regularization); call the regulator Λ. Compute Feynman diagrams, in which divergent terms will depend on Λ. Then, define δZ, δm, and δλ such that Feynman diagrams for the counterterms will exactly cancel the divergent terms in the normal Feynman diagrams when the limit Λ → ∞ is taken. In this way, meaningful finite quantities are obtained.: 323–326
It is only possible to eliminate all infinities to obtain a finite result in renormalizable theories, whereas in non-renormalizable theories infinities cannot be removed by the redefinition of a small number of parameters. The Standard Model of elementary particles is a renormalizable QFT,: 719–727 while quantum gravity is non-renormalizable.: 798 : 421
==== Renormalization group ====
The renormalization group, developed by Kenneth Wilson, is a mathematical apparatus used to study the changes in physical parameters (coefficients in the Lagrangian) as the system is viewed at different scales.: 393 The way in which each parameter changes with scale is described by its β function.: 417 Correlation functions, which underlie quantitative physical predictions, change with scale according to the Callan–Symanzik equation.: 410–411
As an example, the coupling constant in QED, namely the elementary charge e, has the following β function:
β
(
e
)
≡
1
Λ
d
e
d
Λ
=
e
3
12
π
2
+
O
(
e
5
)
,
{\displaystyle \beta (e)\equiv {\frac {1}{\Lambda }}{\frac {de}{d\Lambda }}={\frac {e^{3}}{12\pi ^{2}}}+O{\mathord {\left(e^{5}\right)}},}
where Λ is the energy scale under which the measurement of e is performed. This differential equation implies that the observed elementary charge increases as the scale increases. The renormalized coupling constant, which changes with the energy scale, is also called the running coupling constant.: 420
The coupling constant g in quantum chromodynamics, a non-Abelian gauge theory based on the symmetry group SU(3), has the following β function:
β
(
g
)
≡
1
Λ
d
g
d
Λ
=
g
3
16
π
2
(
−
11
+
2
3
N
f
)
+
O
(
g
5
)
,
{\displaystyle \beta (g)\equiv {\frac {1}{\Lambda }}{\frac {dg}{d\Lambda }}={\frac {g^{3}}{16\pi ^{2}}}\left(-11+{\frac {2}{3}}N_{f}\right)+O{\mathord {\left(g^{5}\right)}},}
where Nf is the number of quark flavours. In the case where Nf ≤ 16 (the Standard Model has Nf = 6), the coupling constant g decreases as the energy scale increases. Hence, while the strong interaction is strong at low energies, it becomes very weak in high-energy interactions, a phenomenon known as asymptotic freedom.: 531
Conformal field theories (CFTs) are special QFTs that admit conformal symmetry. They are insensitive to changes in the scale, as all their coupling constants have vanishing β function. (The converse is not true, however — the vanishing of all β functions does not imply conformal symmetry of the theory.) Examples include string theory and N = 4 supersymmetric Yang–Mills theory.
According to Wilson's picture, every QFT is fundamentally accompanied by its energy cut-off Λ, i.e. that the theory is no longer valid at energies higher than Λ, and all degrees of freedom above the scale Λ are to be omitted. For example, the cut-off could be the inverse of the atomic spacing in a condensed matter system, and in elementary particle physics it could be associated with the fundamental "graininess" of spacetime caused by quantum fluctuations in gravity. The cut-off scale of theories of particle interactions lies far beyond current experiments. Even if the theory were very complicated at that scale, as long as its couplings are sufficiently weak, it must be described at low energies by a renormalizable effective field theory.: 402–403 The difference between renormalizable and non-renormalizable theories is that the former are insensitive to details at high energies, whereas the latter do depend on them.: 2 According to this view, non-renormalizable theories are to be seen as low-energy effective theories of a more fundamental theory. The failure to remove the cut-off Λ from calculations in such a theory merely indicates that new physical phenomena appear at scales above Λ, where a new theory is necessary.: 156
=== Other theories ===
The quantization and renormalization procedures outlined in the preceding sections are performed for the free theory and ϕ4 theory of the real scalar field. A similar process can be done for other types of fields, including the complex scalar field, the vector field, and the Dirac field, as well as other types of interaction terms, including the electromagnetic interaction and the Yukawa interaction.
As an example, quantum electrodynamics contains a Dirac field ψ representing the electron field and a vector field Aμ representing the electromagnetic field (photon field). (Despite its name, the quantum electromagnetic "field" actually corresponds to the classical electromagnetic four-potential, rather than the classical electric and magnetic fields.) The full QED Lagrangian density is:
L
=
ψ
¯
(
i
γ
μ
∂
μ
−
m
)
ψ
−
1
4
F
μ
ν
F
μ
ν
−
e
ψ
¯
γ
μ
ψ
A
μ
,
{\displaystyle {\mathcal {L}}={\bar {\psi }}\left(i\gamma ^{\mu }\partial _{\mu }-m\right)\psi -{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }-e{\bar {\psi }}\gamma ^{\mu }\psi A_{\mu },}
where γμ are Dirac matrices,
ψ
¯
=
ψ
†
γ
0
{\displaystyle {\bar {\psi }}=\psi ^{\dagger }\gamma ^{0}}
, and
F
μ
ν
=
∂
μ
A
ν
−
∂
ν
A
μ
{\displaystyle F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }}
is the electromagnetic field strength. The parameters in this theory are the (bare) electron mass m and the (bare) elementary charge e. The first and second terms in the Lagrangian density correspond to the free Dirac field and free vector fields, respectively. The last term describes the interaction between the electron and photon fields, which is treated as a perturbation from the free theories.: 78
Shown above is an example of a tree-level Feynman diagram in QED. It describes an electron and a positron annihilating, creating an off-shell photon, and then decaying into a new pair of electron and positron. Time runs from left to right. Arrows pointing forward in time represent the propagation of electrons, while those pointing backward in time represent the propagation of positrons. A wavy line represents the propagation of a photon. Each vertex in QED Feynman diagrams must have an incoming and an outgoing fermion (positron/electron) leg as well as a photon leg.
==== Gauge symmetry ====
If the following transformation to the fields is performed at every spacetime point x (a local transformation), then the QED Lagrangian remains unchanged, or invariant:
ψ
(
x
)
→
e
i
α
(
x
)
ψ
(
x
)
,
A
μ
(
x
)
→
A
μ
(
x
)
+
i
e
−
1
e
−
i
α
(
x
)
∂
μ
e
i
α
(
x
)
,
{\displaystyle \psi (x)\to e^{i\alpha (x)}\psi (x),\quad A_{\mu }(x)\to A_{\mu }(x)+ie^{-1}e^{-i\alpha (x)}\partial _{\mu }e^{i\alpha (x)},}
where α(x) is any function of spacetime coordinates. If a theory's Lagrangian (or more precisely the action) is invariant under a certain local transformation, then the transformation is referred to as a gauge symmetry of the theory.: 482–483 Gauge symmetries form a group at every spacetime point. In the case of QED, the successive application of two different local symmetry transformations
e
i
α
(
x
)
{\displaystyle e^{i\alpha (x)}}
and
e
i
α
′
(
x
)
{\displaystyle e^{i\alpha '(x)}}
is yet another symmetry transformation
e
i
[
α
(
x
)
+
α
′
(
x
)
]
{\displaystyle e^{i[\alpha (x)+\alpha '(x)]}}
. For any α(x),
e
i
α
(
x
)
{\displaystyle e^{i\alpha (x)}}
is an element of the U(1) group, thus QED is said to have U(1) gauge symmetry.: 496 The photon field Aμ may be referred to as the U(1) gauge boson.
U(1) is an Abelian group, meaning that the result is the same regardless of the order in which its elements are applied. QFTs can also be built on non-Abelian groups, giving rise to non-Abelian gauge theories (also known as Yang–Mills theories).: 489 Quantum chromodynamics, which describes the strong interaction, is a non-Abelian gauge theory with an SU(3) gauge symmetry. It contains three Dirac fields ψi, i = 1,2,3 representing quark fields as well as eight vector fields Aa,μ, a = 1,...,8 representing gluon fields, which are the SU(3) gauge bosons.: 547 The QCD Lagrangian density is:: 490–491
L
=
i
ψ
¯
i
γ
μ
(
D
μ
)
i
j
ψ
j
−
1
4
F
μ
ν
a
F
a
,
μ
ν
−
m
ψ
¯
i
ψ
i
,
{\displaystyle {\mathcal {L}}=i{\bar {\psi }}^{i}\gamma ^{\mu }(D_{\mu })^{ij}\psi ^{j}-{\frac {1}{4}}F_{\mu \nu }^{a}F^{a,\mu \nu }-m{\bar {\psi }}^{i}\psi ^{i},}
where Dμ is the gauge covariant derivative:
D
μ
=
∂
μ
−
i
g
A
μ
a
t
a
,
{\displaystyle D_{\mu }=\partial _{\mu }-igA_{\mu }^{a}t^{a},}
where g is the coupling constant, ta are the eight generators of SU(3) in the fundamental representation (3×3 matrices),
F
μ
ν
a
=
∂
μ
A
ν
a
−
∂
ν
A
μ
a
+
g
f
a
b
c
A
μ
b
A
ν
c
,
{\displaystyle F_{\mu \nu }^{a}=\partial _{\mu }A_{\nu }^{a}-\partial _{\nu }A_{\mu }^{a}+gf^{abc}A_{\mu }^{b}A_{\nu }^{c},}
and fabc are the structure constants of SU(3). Repeated indices i,j,a are implicitly summed over following Einstein notation. This Lagrangian is invariant under the transformation:
ψ
i
(
x
)
→
U
i
j
(
x
)
ψ
j
(
x
)
,
A
μ
a
(
x
)
t
a
→
U
(
x
)
[
A
μ
a
(
x
)
t
a
+
i
g
−
1
∂
μ
]
U
†
(
x
)
,
{\displaystyle \psi ^{i}(x)\to U^{ij}(x)\psi ^{j}(x),\quad A_{\mu }^{a}(x)t^{a}\to U(x)\left[A_{\mu }^{a}(x)t^{a}+ig^{-1}\partial _{\mu }\right]U^{\dagger }(x),}
where U(x) is an element of SU(3) at every spacetime point x:
U
(
x
)
=
e
i
α
(
x
)
a
t
a
.
{\displaystyle U(x)=e^{i\alpha (x)^{a}t^{a}}.}
The preceding discussion of symmetries is on the level of the Lagrangian. In other words, these are "classical" symmetries. After quantization, some theories will no longer exhibit their classical symmetries, a phenomenon called anomaly. For instance, in the path integral formulation, despite the invariance of the Lagrangian density
L
[
ϕ
,
∂
μ
ϕ
]
{\displaystyle {\mathcal {L}}[\phi ,\partial _{\mu }\phi ]}
under a certain local transformation of the fields, the measure
∫
D
ϕ
{\textstyle \int {\mathcal {D}}\phi }
of the path integral may change.: 243 For a theory describing nature to be consistent, it must not contain any anomaly in its gauge symmetry. The Standard Model of elementary particles is a gauge theory based on the group SU(3) × SU(2) × U(1), in which all anomalies exactly cancel.: 705–707
The theoretical foundation of general relativity, the equivalence principle, can also be understood as a form of gauge symmetry, making general relativity a gauge theory based on the Lorentz group.
Noether's theorem states that every continuous symmetry, i.e. the parameter in the symmetry transformation being continuous rather than discrete, leads to a corresponding conservation law.: 17–18 : 73 For example, the U(1) symmetry of QED implies charge conservation.
Gauge-transformations do not relate distinct quantum states. Rather, it relates two equivalent mathematical descriptions of the same quantum state. As an example, the photon field Aμ, being a four-vector, has four apparent degrees of freedom, but the actual state of a photon is described by its two degrees of freedom corresponding to the polarization. The remaining two degrees of freedom are said to be "redundant" — apparently different ways of writing Aμ can be related to each other by a gauge transformation and in fact describe the same state of the photon field. In this sense, gauge invariance is not a "real" symmetry, but a reflection of the "redundancy" of the chosen mathematical description.: 168
To account for the gauge redundancy in the path integral formulation, one must perform the so-called Faddeev–Popov gauge fixing procedure. In non-Abelian gauge theories, such a procedure introduces new fields called "ghosts". Particles corresponding to the ghost fields are called ghost particles, which cannot be detected externally.: 512–515 A more rigorous generalization of the Faddeev–Popov procedure is given by BRST quantization.: 517
==== Spontaneous symmetry-breaking ====
Spontaneous symmetry breaking is a mechanism whereby the symmetry of the Lagrangian is violated by the system described by it.: 347
To illustrate the mechanism, consider a linear sigma model containing N real scalar fields, described by the Lagrangian density:
L
=
1
2
(
∂
μ
ϕ
i
)
(
∂
μ
ϕ
i
)
+
1
2
μ
2
ϕ
i
ϕ
i
−
λ
4
(
ϕ
i
ϕ
i
)
2
,
{\displaystyle {\mathcal {L}}={\frac {1}{2}}\left(\partial _{\mu }\phi ^{i}\right)\left(\partial ^{\mu }\phi ^{i}\right)+{\frac {1}{2}}\mu ^{2}\phi ^{i}\phi ^{i}-{\frac {\lambda }{4}}\left(\phi ^{i}\phi ^{i}\right)^{2},}
where μ and λ are real parameters. The theory admits an O(N) global symmetry:
ϕ
i
→
R
i
j
ϕ
j
,
R
∈
O
(
N
)
.
{\displaystyle \phi ^{i}\to R^{ij}\phi ^{j},\quad R\in \mathrm {O} (N).}
The lowest energy state (ground state or vacuum state) of the classical theory is any uniform field ϕ0 satisfying
ϕ
0
i
ϕ
0
i
=
μ
2
λ
.
{\displaystyle \phi _{0}^{i}\phi _{0}^{i}={\frac {\mu ^{2}}{\lambda }}.}
Without loss of generality, let the ground state be in the N-th direction:
ϕ
0
i
=
(
0
,
⋯
,
0
,
μ
λ
)
.
{\displaystyle \phi _{0}^{i}=\left(0,\cdots ,0,{\frac {\mu }{\sqrt {\lambda }}}\right).}
The original N fields can be rewritten as:
ϕ
i
(
x
)
=
(
π
1
(
x
)
,
⋯
,
π
N
−
1
(
x
)
,
μ
λ
+
σ
(
x
)
)
,
{\displaystyle \phi ^{i}(x)=\left(\pi ^{1}(x),\cdots ,\pi ^{N-1}(x),{\frac {\mu }{\sqrt {\lambda }}}+\sigma (x)\right),}
and the original Lagrangian density as:
L
=
1
2
(
∂
μ
π
k
)
(
∂
μ
π
k
)
+
1
2
(
∂
μ
σ
)
(
∂
μ
σ
)
−
1
2
(
2
μ
2
)
σ
2
−
λ
μ
σ
3
−
λ
μ
π
k
π
k
σ
−
λ
2
π
k
π
k
σ
2
−
λ
4
(
π
k
π
k
)
2
,
{\displaystyle {\mathcal {L}}={\frac {1}{2}}\left(\partial _{\mu }\pi ^{k}\right)\left(\partial ^{\mu }\pi ^{k}\right)+{\frac {1}{2}}\left(\partial _{\mu }\sigma \right)\left(\partial ^{\mu }\sigma \right)-{\frac {1}{2}}\left(2\mu ^{2}\right)\sigma ^{2}-{\sqrt {\lambda }}\mu \sigma ^{3}-{\sqrt {\lambda }}\mu \pi ^{k}\pi ^{k}\sigma -{\frac {\lambda }{2}}\pi ^{k}\pi ^{k}\sigma ^{2}-{\frac {\lambda }{4}}\left(\pi ^{k}\pi ^{k}\right)^{2},}
where k = 1, ..., N − 1. The original O(N) global symmetry is no longer manifest, leaving only the subgroup O(N − 1). The larger symmetry before spontaneous symmetry breaking is said to be "hidden" or spontaneously broken.: 349–350
Goldstone's theorem states that under spontaneous symmetry breaking, every broken continuous global symmetry leads to a massless field called the Goldstone boson. In the above example, O(N) has N(N − 1)/2 continuous symmetries (the dimension of its Lie algebra), while O(N − 1) has (N − 1)(N − 2)/2. The number of broken symmetries is their difference, N − 1, which corresponds to the N − 1 massless fields πk.: 351
On the other hand, when a gauge (as opposed to global) symmetry is spontaneously broken, the resulting Goldstone boson is "eaten" by the corresponding gauge boson by becoming an additional degree of freedom for the gauge boson. The Goldstone boson equivalence theorem states that at high energy, the amplitude for emission or absorption of a longitudinally polarized massive gauge boson becomes equal to the amplitude for emission or absorption of the Goldstone boson that was eaten by the gauge boson.: 743–744
In the QFT of ferromagnetism, spontaneous symmetry breaking can explain the alignment of magnetic dipoles at low temperatures.: 199 In the Standard Model of elementary particles, the W and Z bosons, which would otherwise be massless as a result of gauge symmetry, acquire mass through spontaneous symmetry breaking of the Higgs boson, a process called the Higgs mechanism.: 690
==== Supersymmetry ====
All experimentally known symmetries in nature relate bosons to bosons and fermions to fermions. Theorists have hypothesized the existence of a type of symmetry, called supersymmetry, that relates bosons and fermions.: 795 : 443
The Standard Model obeys Poincaré symmetry, whose generators are the spacetime translations Pμ and the Lorentz transformations Jμν.: 58–60 In addition to these generators, supersymmetry in (3+1)-dimensions includes additional generators Qα, called supercharges, which themselves transform as Weyl fermions.: 795 : 444 The symmetry group generated by all these generators is known as the super-Poincaré group. In general there can be more than one set of supersymmetry generators, QαI, I = 1, ..., N, which generate the corresponding N = 1 supersymmetry, N = 2 supersymmetry, and so on.: 795 : 450 Supersymmetry can also be constructed in other dimensions, most notably in (1+1) dimensions for its application in superstring theory.
The Lagrangian of a supersymmetric theory must be invariant under the action of the super-Poincaré group.: 448 Examples of such theories include: Minimal Supersymmetric Standard Model (MSSM), N = 4 supersymmetric Yang–Mills theory,: 450 and superstring theory. In a supersymmetric theory, every fermion has a bosonic superpartner and vice versa.: 444
If supersymmetry is promoted to a local symmetry, then the resultant gauge theory is an extension of general relativity called supergravity.
Supersymmetry is a potential solution to many current problems in physics. For example, the hierarchy problem of the Standard Model—why the mass of the Higgs boson is not radiatively corrected (under renormalization) to a very high scale such as the grand unified scale or the Planck scale—can be resolved by relating the Higgs field and its super-partner, the Higgsino. Radiative corrections due to Higgs boson loops in Feynman diagrams are cancelled by corresponding Higgsino loops. Supersymmetry also offers answers to the grand unification of all gauge coupling constants in the Standard Model as well as the nature of dark matter.: 796–797
Nevertheless, experiments have yet to provide evidence for the existence of supersymmetric particles. If supersymmetry were a true symmetry of nature, then it must be a broken symmetry, and the energy of symmetry breaking must be higher than those achievable by present-day experiments.: 797 : 443
==== Other spacetimes ====
The ϕ4 theory, QED, QCD, as well as the whole Standard Model all assume a (3+1)-dimensional Minkowski space (3 spatial and 1 time dimensions) as the background on which the quantum fields are defined. However, QFT a priori imposes no restriction on the number of dimensions nor the geometry of spacetime.
In condensed matter physics, QFT is used to describe (2+1)-dimensional electron gases. In high-energy physics, string theory is a type of (1+1)-dimensional QFT,: 452 while Kaluza–Klein theory uses gravity in extra dimensions to produce gauge theories in lower dimensions.: 428–429
In Minkowski space, the flat metric ημν is used to raise and lower spacetime indices in the Lagrangian, e.g.
A
μ
A
μ
=
η
μ
ν
A
μ
A
ν
,
∂
μ
ϕ
∂
μ
ϕ
=
η
μ
ν
∂
μ
ϕ
∂
ν
ϕ
,
{\displaystyle A_{\mu }A^{\mu }=\eta _{\mu \nu }A^{\mu }A^{\nu },\quad \partial _{\mu }\phi \partial ^{\mu }\phi =\eta ^{\mu \nu }\partial _{\mu }\phi \partial _{\nu }\phi ,}
where ημν is the inverse of ημν satisfying ημρηρν = δμν.
For QFTs in curved spacetime on the other hand, a general metric (such as the Schwarzschild metric describing a black hole) is used:
A
μ
A
μ
=
g
μ
ν
A
μ
A
ν
,
∂
μ
ϕ
∂
μ
ϕ
=
g
μ
ν
∂
μ
ϕ
∂
ν
ϕ
,
{\displaystyle A_{\mu }A^{\mu }=g_{\mu \nu }A^{\mu }A^{\nu },\quad \partial _{\mu }\phi \partial ^{\mu }\phi =g^{\mu \nu }\partial _{\mu }\phi \partial _{\nu }\phi ,}
where gμν is the inverse of gμν.
For a real scalar field, the Lagrangian density in a general spacetime background is
L
=
|
g
|
(
1
2
g
μ
ν
∇
μ
ϕ
∇
ν
ϕ
−
1
2
m
2
ϕ
2
)
,
{\displaystyle {\mathcal {L}}={\sqrt {|g|}}\left({\frac {1}{2}}g^{\mu \nu }\nabla _{\mu }\phi \nabla _{\nu }\phi -{\frac {1}{2}}m^{2}\phi ^{2}\right),}
where g = det(gμν), and ∇μ denotes the covariant derivative. The Lagrangian of a QFT, hence its calculational results and physical predictions, depends on the geometry of the spacetime background.
==== Topological quantum field theory ====
The correlation functions and physical predictions of a QFT depend on the spacetime metric gμν. For a special class of QFTs called topological quantum field theories (TQFTs), all correlation functions are independent of continuous changes in the spacetime metric.: 36 QFTs in curved spacetime generally change according to the geometry (local structure) of the spacetime background, while TQFTs are invariant under spacetime diffeomorphisms but are sensitive to the topology (global structure) of spacetime. This means that all calculational results of TQFTs are topological invariants of the underlying spacetime. Chern–Simons theory is an example of TQFT and has been used to construct models of quantum gravity. Applications of TQFT include the fractional quantum Hall effect and topological quantum computers.: 1–5 The world line trajectory of fractionalized particles (known as anyons) can form a link configuration in the spacetime, which relates the braiding statistics of anyons in physics to the
link invariants in mathematics. Topological quantum field theories (TQFTs) applicable to the frontier research of topological quantum matters include Chern-Simons-Witten gauge theories in 2+1 spacetime dimensions, other new exotic TQFTs in 3+1 spacetime dimensions and beyond.
=== Perturbative and non-perturbative methods ===
Using perturbation theory, the total effect of a small interaction term can be approximated order by order by a series expansion in the number of virtual particles participating in the interaction. Every term in the expansion may be understood as one possible way for (physical) particles to interact with each other via virtual particles, expressed visually using a Feynman diagram. The electromagnetic force between two electrons in QED is represented (to first order in perturbation theory) by the propagation of a virtual photon. In a similar manner, the W and Z bosons carry the weak interaction, while gluons carry the strong interaction. The interpretation of an interaction as a sum of intermediate states involving the exchange of various virtual particles only makes sense in the framework of perturbation theory. In contrast, non-perturbative methods in QFT treat the interacting Lagrangian as a whole without any series expansion. Instead of particles that carry interactions, these methods have spawned such concepts as 't Hooft–Polyakov monopole, domain wall, flux tube, and instanton. Examples of QFTs that are completely solvable non-perturbatively include minimal models of conformal field theory and the Thirring model.
== Mathematical rigor ==
In spite of its overwhelming success in particle physics and condensed matter physics, QFT itself lacks a formal mathematical foundation. For example, according to Haag's theorem, there does not exist a well-defined interaction picture for QFT, which implies that perturbation theory of QFT, which underlies the entire Feynman diagram method, is fundamentally ill-defined.
However, perturbative quantum field theory, which only requires that quantities be computable as a formal power series without any convergence requirements, can be given a rigorous mathematical treatment. In particular, Kevin Costello's monograph Renormalization and Effective Field Theory provides a rigorous formulation of perturbative renormalization that combines both the effective-field theory approaches of Kadanoff, Wilson, and Polchinski, together with the Batalin-Vilkovisky approach to quantizing gauge theories. Furthermore, perturbative path-integral methods, typically understood as formal computational methods inspired from finite-dimensional integration theory, can be given a sound mathematical interpretation from their finite-dimensional analogues.
Since the 1950s, theoretical physicists and mathematicians have attempted to organize all QFTs into a set of axioms, in order to establish the existence of concrete models of relativistic QFT in a mathematically rigorous way and to study their properties. This line of study is called constructive quantum field theory, a subfield of mathematical physics,: 2 which has led to such results as CPT theorem, spin–statistics theorem, and Goldstone's theorem, and also to mathematically rigorous constructions of many interacting QFTs in two and three spacetime dimensions, e.g. two-dimensional scalar field theories with arbitrary polynomial interactions, the three-dimensional scalar field theories with a quartic interaction, etc.
Compared to ordinary QFT, topological quantum field theory and conformal field theory are better supported mathematically — both can be classified in the framework of representations of cobordisms.
Algebraic quantum field theory is another approach to the axiomatization of QFT, in which the fundamental objects are local operators and the algebraic relations between them. Axiomatic systems following this approach include Wightman axioms and Haag–Kastler axioms.: 2–3 One way to construct theories satisfying Wightman axioms is to use Osterwalder–Schrader axioms, which give the necessary and sufficient conditions for a real time theory to be obtained from an imaginary time theory by analytic continuation (Wick rotation).: 10
Yang–Mills existence and mass gap, one of the Millennium Prize Problems, concerns the well-defined existence of Yang–Mills theories as set out by the above axioms. The full problem statement is as follows.
Prove that for any compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists on
R
4
{\displaystyle \mathbb {R} ^{4}}
and has a mass gap Δ > 0. Existence includes establishing axiomatic properties at least as strong as those cited in Streater & Wightman (1964), Osterwalder & Schrader (1973) and Osterwalder & Schrader (1975).
== See also ==
== References ==
Bibliography
Streater, R.; Wightman, A. (1964). PCT, Spin and Statistics and all That. W. A. Benjamin.
Osterwalder, K.; Schrader, R. (1973). "Axioms for Euclidean Green's functions". Communications in Mathematical Physics. 31 (2): 83–112. Bibcode:1973CMaPh..31...83O. doi:10.1007/BF01645738. S2CID 189829853.
Osterwalder, K.; Schrader, R. (1975). "Axioms for Euclidean Green's functions II". Communications in Mathematical Physics. 42 (3): 281–305. Bibcode:1975CMaPh..42..281O. doi:10.1007/BF01608978. S2CID 119389461.
== Further reading ==
General readers
Pais, A. (1994) [1986]. Inward Bound: Of Matter and Forces in the Physical World (reprint ed.). Oxford, New York, Toronto: Oxford University Press. ISBN 978-0198519973.
Schweber, S. S. (1994). QED and the Men Who Made It: Dyson, Feynman, Schwinger, and Tomonaga. Princeton University Press. ISBN 9780691033273.
Feynman, R.P. (2001) [1964]. The Character of Physical Law. MIT Press. ISBN 978-0-262-56003-0.
Feynman, R.P. (2006) [1985]. QED: The Strange Theory of Light and Matter. Princeton University Press. ISBN 978-0-691-12575-6.
Gribbin, J. (1998). Q is for Quantum: Particle Physics from A to Z. Weidenfeld & Nicolson. ISBN 978-0-297-81752-9.
Carroll, Sean (2024). The Biggest Ideas in the Universe : quanta and fields. Dutton. ISBN 978-0-593-18660-2.
Introductory text
Pierre van Baal (2016). A Course in Field Theory. CRC Press. ISBN 9780429073601.
McMahon, D. (2008). Quantum Field Theory. McGraw-Hill. ISBN 978-0-07-154382-8.
Bogolyubov, N.; Shirkov, D. (1982). Quantum Fields. Benjamin Cummings. ISBN 978-0-8053-0983-6.
Frampton, P.H. (2000). Gauge Field Theories. Frontiers in Physics (2nd ed.). Wiley.; Frampton, Paul H. (22 September 2008). 2008, 3rd edition. John Wiley & Sons. ISBN 978-3527408351.
Greiner, W.; Müller, B. (2000). Gauge Theory of Weak Interactions. Springer. ISBN 978-3-540-67672-0.
Itzykson, C.; Zuber, J.-B. (1980). Quantum Field Theory. McGraw-Hill. ISBN 978-0-07-032071-0.
Kane, G.L. (1987). Modern Elementary Particle Physics. Perseus Group. ISBN 978-0-201-11749-3.
Kleinert, H.; Schulte-Frohlinde, Verena (2001). Critical Properties of φ4-Theories. World Scientific. ISBN 978-981-02-4658-7.
Kleinert, H. (2008). Multivalued Fields in Condensed Matter, Electrodynamics, and Gravitation (PDF). World Scientific. ISBN 978-981-279-170-2.
Lancaster, Tom; Blundell, Stephen (2014). Quantum field theory for the gifted amateur. Oxford: Oxford University Press. ISBN 978-0-19-969933-9. OCLC 859651399.
Loudon, R. (1983). The Quantum Theory of Light. Oxford University Press. ISBN 978-0-19-851155-7.
Mandl, F.; Shaw, G. (1993). Quantum Field Theory. John Wiley & Sons. ISBN 978-0-471-94186-6.
Ryder, L.H. (1985). Quantum Field Theory. Cambridge University Press. ISBN 978-0-521-33859-2.
Schwartz, M.D. (2014). Quantum Field Theory and the Standard Model. Cambridge University Press. ISBN 978-1107034730. Archived from the original on 2018-03-22. Retrieved 2020-05-13.
Ynduráin, F.J. (1996). Relativistic Quantum Mechanics and Introduction to Field Theory (1st ed.). Springer. Bibcode:1996rqmi.book.....Y. doi:10.1007/978-3-642-61057-8. ISBN 978-3-540-60453-2.
Greiner, W.; Reinhardt, J. (1996). Field Quantization. Springer. ISBN 978-3-540-59179-5.
Peskin, M.; Schroeder, D. (1995). An Introduction to Quantum Field Theory. Westview Press. ISBN 978-0-201-50397-5.
Scharf, Günter (2014) [1989]. Finite Quantum Electrodynamics: The Causal Approach (third ed.). Dover Publications. ISBN 978-0486492735.
Srednicki, M. (2007). Quantum Field Theory. Cambridge University Press. ISBN 978-0521-8644-97.
Tong, David (2015). "Lectures on Quantum Field Theory". Retrieved 2016-02-09.
Williams, A.G. (2022). Introduction to Quantum Field Theory: Classical Mechanics to Gauge Field Theories. Cambridge University Press. ISBN 978-1108470902.
Zee, Anthony (2010). Quantum Field Theory in a Nutshell (2nd ed.). Princeton University Press. ISBN 978-0691140346.
Advanced texts
Heitler, W. (1953). The Quantum Theory of Radiation. Dover Publications, Inc. ISBN 0-486-64558-4.
Umezawa, H. (1956) Quantum Field Theory. North Holland Puplishing.
Barton, G. (1963). Introduction to Advanced Field Theory. Intescience Publishers.
Brown, Lowell S. (1994). Quantum Field Theory. Cambridge University Press. ISBN 978-0-521-46946-3.
Bogoliubov, N.; Logunov, A.A.; Oksak, A.I.; Todorov, I.T. (1990). General Principles of Quantum Field Theory. Kluwer Academic Publishers. ISBN 978-0-7923-0540-8.
Weinberg, S. (1995). The Quantum Theory of Fields. Vol. 1. Cambridge University Press. ISBN 978-0521550017.
== External links ==
Media related to Quantum field theory at Wikimedia Commons
"Quantum field theory", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Stanford Encyclopedia of Philosophy: "Quantum Field Theory", by Meinard Kuhlmann.
Siegel, Warren, 2005. Fields. arXiv:hep-th/9912205 .
Quantum Field Theory by P. J. Mulders | Wikipedia/Relativistic_quantum_theory |
The Bohr–Sommerfeld model (also known as the Sommerfeld model or Bohr–Sommerfeld theory) was an extension of the Bohr model to allow elliptical orbits of electrons around an atomic nucleus. Bohr–Sommerfeld theory is named after Danish physicist Niels Bohr and German physicist Arnold Sommerfeld. Sommerfeld showed that, if electronic orbits are elliptical instead of circular (as in Bohr's model of the atom), the fine-structure of the hydrogen atom can be described.
The Bohr–Sommerfeld model added to the quantized angular momentum condition of the Bohr model with a radial quantization (condition by William Wilson, the Wilson–Sommerfeld quantization condition):
∫
0
T
p
r
d
q
r
=
n
h
,
{\displaystyle \int _{0}^{T}p_{r}\,dq_{r}=nh,}
where pr is the radial momentum canonically conjugate to the coordinate q, which is the radial position, and T is one full orbital period. The integral is the action of action-angle coordinates. This condition, suggested by the correspondence principle, is the only one possible, since the quantum numbers are adiabatic invariants.
== History ==
In 1913, Niels Bohr displayed rudiments of the later defined correspondence principle and used it to formulate a model of the hydrogen atom which explained its line spectrum. In the next few years Arnold Sommerfeld extended the quantum rule to arbitrary integrable systems making use of the principle of adiabatic invariance of the quantum numbers introduced by Hendrik Lorentz and Albert Einstein. Sommerfeld made a crucial contribution by quantizing the z-component of the angular momentum, which in the old quantum era was called "space quantization" (German: Richtungsquantelung). This allowed the orbits of the electron to be ellipses instead of circles, and introduced the concept of quantum degeneracy. The theory would have correctly explained the Zeeman effect, except for the issue of electron spin. Sommerfeld's model was much closer to the modern quantum mechanical picture than Bohr's.
In the 1950s Joseph Keller updated Bohr–Sommerfeld quantization using Einstein's interpretation of 1917, now known as Einstein–Brillouin–Keller method. In 1971, Martin Gutzwiller took into account that this method only works for integrable systems and derived a semiclassical way of quantizing chaotic systems from path integrals.
== Predictions ==
The Sommerfeld model predicted that the magnetic moment of an atom measured along an axis will only take on discrete values, a result which seems to contradict rotational invariance but which was confirmed by the Stern–Gerlach experiment. This was a significant step in the development of quantum mechanics. It also described the possibility of atomic energy levels being split by a magnetic field (called the Zeeman effect). Walther Kossel worked with Bohr and Sommerfeld on the Bohr–Sommerfeld model of the atom introducing two electrons in the first shell and eight in the second.
== Issues ==
The Bohr–Sommerfeld model was fundamentally inconsistent and led to many paradoxes. The magnetic quantum number measured the tilt of the orbital plane relative to the xy plane, and it could only take a few discrete values. This contradicted the obvious fact that an atom could be turned this way and that relative to the coordinates without restriction. The Sommerfeld quantization can be performed in different canonical coordinates and sometimes gives different answers. The incorporation of radiation corrections was difficult, because it required finding action-angle coordinates for a combined radiation/atom system, which is difficult when the radiation is allowed to escape. The whole theory did not extend to non-integrable motions, which meant that many systems could not be treated even in principle. In the end, the model was replaced by the modern quantum-mechanical treatment of the hydrogen atom, which was first given by Wolfgang Pauli in 1925, using Heisenberg's matrix mechanics. The current picture of the hydrogen atom is based on the atomic orbitals of wave mechanics, which Erwin Schrödinger developed in 1926.
However, this is not to say that the Bohr–Sommerfeld model was without its successes. Calculations based on the Bohr–Sommerfeld model were able to accurately explain a number of more complex atomic spectral effects. For example, up to first-order perturbations, the Bohr model and quantum mechanics make the same predictions for the spectral line splitting in the Stark effect. At higher-order perturbations, however, the Bohr model and quantum mechanics differ, and measurements of the Stark effect under high field strengths helped confirm the correctness of quantum mechanics over the Bohr model. The prevailing theory behind this difference lies in the shapes of the orbitals of the electrons, which vary according to the energy state of the electron.
The Bohr–Sommerfeld quantization conditions lead to questions in modern mathematics. Consistent semiclassical quantization condition requires a certain type of structure on the phase space, which places topological limitations on the types of symplectic manifolds which can be quantized. In particular, the symplectic form should be the curvature form of a connection of a Hermitian line bundle, which is called a prequantization.
== Relativistic orbit ==
Arnold Sommerfeld derived the relativistic solution of atomic energy levels. We will start this derivation with the relativistic equation for energy in the electric potential
W
=
m
0
c
2
(
1
1
−
v
2
c
2
−
1
)
−
k
Z
e
2
r
{\displaystyle W={m_{\mathrm {0} }c^{2}}\left({\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}-1\right)-k{\frac {Ze^{2}}{r}}}
After substitution
u
=
1
r
{\displaystyle u={\frac {1}{r}}}
we get
1
1
−
v
2
c
2
=
1
+
W
m
0
c
2
+
k
Z
e
2
m
0
c
2
u
{\displaystyle {\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}=1+{\frac {W}{m_{\mathrm {0} }c^{2}}}+k{\frac {Ze^{2}}{m_{\mathrm {0} }c^{2}}}u}
For momentum
p
r
=
m
r
˙
{\displaystyle p_{\mathrm {r} }=m{\dot {r}}}
,
p
φ
=
m
r
2
φ
˙
{\displaystyle p_{\mathrm {\varphi } }=mr^{2}{\dot {\varphi }}}
and their ratio
p
r
p
φ
=
−
d
u
d
φ
{\displaystyle {\frac {p_{\mathrm {r} }}{p_{\mathrm {\varphi } }}}=-{\frac {du}{d\varphi }}}
the equation of motion is (see Binet equation)
d
2
u
d
φ
2
=
−
(
1
−
k
2
Z
2
e
4
c
2
p
φ
2
)
u
+
m
0
k
Z
e
2
p
φ
2
(
1
+
W
m
0
c
2
)
=
−
ω
0
2
u
+
K
{\displaystyle {\frac {d^{2}u}{d\varphi ^{2}}}=-\left(1-k^{2}{\frac {Z^{2}e^{4}}{c^{2}p_{\mathrm {\varphi } }^{2}}}\right)u+{\frac {m_{\mathrm {0} }kZe^{2}}{p_{\mathrm {\varphi } }^{2}}}\left(1+{\frac {W}{m_{\mathrm {0} }c^{2}}}\right)=-\omega _{\mathrm {0} }^{2}u+K}
with solution
u
=
1
r
=
K
+
A
cos
ω
0
φ
{\displaystyle u={\frac {1}{r}}=K+A\cos \omega _{\mathrm {0} }\varphi }
The angular shift of periapsis per revolution is given by
φ
s
=
2
π
(
1
ω
0
−
1
)
≈
4
π
3
k
2
Z
2
e
4
c
2
n
φ
2
h
2
{\displaystyle \varphi _{\mathrm {s} }=2\pi \left({\frac {1}{\omega _{\mathrm {0} }}}-1\right)\approx 4\pi ^{3}k^{2}{\frac {Z^{2}e^{4}}{c^{2}n_{\mathrm {\varphi } }^{2}h^{2}}}}
With the quantum conditions
∮
p
φ
d
φ
=
2
π
p
φ
=
n
φ
h
{\displaystyle \oint p_{\mathrm {\varphi } }\,d\varphi =2\pi p_{\mathrm {\varphi } }=n_{\mathrm {\varphi } }h}
and
∮
p
r
d
r
=
p
φ
∮
(
1
r
d
r
d
φ
)
2
d
φ
=
n
r
h
{\displaystyle \oint p_{\mathrm {r} }\,dr=p_{\mathrm {\varphi } }\oint \left({\frac {1}{r}}{\frac {dr}{d\varphi }}\right)^{2}\,d\varphi =n_{\mathrm {r} }h}
we will obtain energies
W
m
0
c
2
=
(
1
+
α
2
Z
2
(
n
r
+
n
φ
2
−
α
2
Z
2
)
2
)
−
1
/
2
−
1
{\displaystyle {\frac {W}{m_{\mathrm {0} }c^{2}}}=\left(1+{\frac {\alpha ^{2}Z^{2}}{\left(n_{\mathrm {r} }+{\sqrt {n_{\mathrm {\varphi } }^{2}-\alpha ^{2}Z^{2}}}\right)^{2}}}\right)^{-1/2}-1}
where
α
{\displaystyle \alpha }
is the fine-structure constant. This solution (using substitutions for quantum numbers) is equivalent to the solution of the Dirac equation. Nevertheless, both solutions fail to predict the Lamb shifts.
== See also ==
Bohr model
Old quantum theory
== References == | Wikipedia/Bohr-Sommerfeld_model |
Stochastic thermodynamics is an emergent field of research in statistical mechanics that uses stochastic variables to better understand the non-equilibrium dynamics present in many microscopic systems such as colloidal particles, biopolymers (e.g. DNA, RNA, and proteins), enzymes, and molecular motors.
== Overview ==
When a microscopic machine (e.g. a MEM) performs useful work it generates heat and entropy as a byproduct of the process, however it is also predicted that this machine will operate in "reverse" or "backwards" over appreciable short periods. That is, heat energy from the surroundings will be converted into useful work. For larger engines, this would be described as a violation of the second law of thermodynamics, as entropy is consumed rather than generated. Loschmidt's paradox states that in a time reversible system, for every trajectory there exists a time-reversed anti-trajectory. As the entropy production of a trajectory and its equal anti-trajectory are of identical magnitude but opposite sign, then, so the argument goes, one cannot prove that entropy production is positive.
For a long time, exact results in thermodynamics were only possible in linear systems capable of reaching equilibrium, leaving other questions like the Loschmidt paradox unsolved. During the last few decades fresh approaches have revealed general laws applicable to non-equilibrium system which are described by nonlinear equations, pushing the range of exact thermodynamic statements beyond the realm of traditional linear solutions. These exact results are particularly relevant for small systems where appreciable (typically non-Gaussian) fluctuations occur. Thanks to stochastic thermodynamics it is now possible to accurately predict distribution functions of thermodynamic quantities relating to exchanged heat, applied work or entropy production for these systems.
=== Fluctuation theorem ===
The mathematical resolution to Loschmidt's paradox is called the (steady state) fluctuation theorem (FT), which is a generalisation of the second law of thermodynamics. The FT shows that as a system gets larger or the trajectory duration becomes longer, entropy-consuming trajectories become more unlikely, and the expected second law behaviour is recovered.
The FT was first put forward by Evans et al. (1993) and much of the work done in developing and extending the theorem was accomplished by theoreticians and mathematicians interested in nonequilibrium statistical mechanics.
The first observation and experimental proof of Evan's fluctuation theorem (FT) was performed by Wang et al. (2002)
=== Jarzynski equality ===
Seifert writes:
Jarzynski (1997a, 1997b) proved a remarkable relation which allows to express the free energy difference between two equilibrium systems by a nonlinear average over the work required to drive the system in a non-equilibrium process from one state to the other. By comparing probability distributions for the work spent in the original process with the time-reversed one, Crooks found a “refinement” of the Jarzynski relation (JR), now called the Crooks fluctuation theorem. Both, this relation and another refinement of the JR, the Hummer-Szabo relation became particularly useful for determining free energy differences and landscapes of biomolecules. These relations are the most prominent ones within a class of exact results (some of which found even earlier and then rediscovered) valid for non-equilibrium systems driven by time-dependent forces. A close analogy to the JR, which relates different equilibrium states, is the Hatano-Sasa relation that applies to transitions between two different non-equilibrium steady states.
This is shown to be a special case of a more general relation.
=== Stochastic energetics ===
== History ==
Seifert writes:
Classical thermodynamics, at its heart, deals with general laws governing the transformations of a system, in particular, those involving the exchange of heat, work and matter with an environment. As a central result, total entropy production is identified that in any such process can never decrease, leading, inter alia, to fundamental limits on the efficiency of heat engines and refrigerators.
The thermodynamic characterisation of systems in equilibrium got its microscopic justification from equilibrium statistical mechanics which states that for a system in contact with a heat bath the probability to find it in any specific microstate is given by the Boltzmann factor. For small deviations from equilibrium, linear response theory allows to express transport properties caused by small external fields through equilibrium correlation functions. On a more phenomenological level, linear irreversible thermodynamics provides a relation between such transport coefficients and entropy production in terms of forces and fluxes. Beyond this linear response regime, for a long time, no universal exact results were available.
During the last 20 years fresh approaches have revealed general laws applicable to non-equilibrium system thus pushing the range of validity of exact thermodynamic statements beyond the realm of linear response deep into the genuine non-equilibrium region. These exact results, which become particularly relevant for small systems with appreciable (typically non-Gaussian) fluctuations, generically refer to distribution functions of thermodynamic quantities like exchanged heat, applied work or entropy production.
Stochastic thermodynamics combines the stochastic energetics introduced by Sekimoto (1998) with the idea that entropy can consistently be assigned to a single fluctuating trajectory.
== Open research ==
=== Quantum stochastic thermodynamics ===
Stochastic thermodynamics can be applied to driven (i.e. open) quantum systems whenever the effects of quantum coherence can be ignored. The dynamics of an open quantum system is then equivalent to a classical stochastic one. However, this is sometimes at the cost of requiring unrealistic measurements at the beginning and end of a process.
Understanding non-equilibrium quantum thermodynamics more broadly is an important and active area of research. The efficiency of some computing and information theory tasks can be greatly enhanced when using quantum correlated states; quantum correlations can be used not only as a valuable resource in quantum computation, but also in the realm of quantum thermodynamics. New types of quantum devices in non-equilibrium states function very differently to their classical counterparts. For example, it has been theoretically shown that non-equilibrium quantum ratchet systems function far more efficiently then that predicted by classical thermodynamics. It has also been shown that quantum coherence can be used to enhance the efficiency of systems beyond the classical Carnot limit. This is because it could be possible to extract work, in the form of photons, from a single heat bath. Quantum coherence can be used in effect to play the role of Maxwell's demon though the broader information theory based interpretation of the second law of thermodynamics is not violated.
Quantum versions of stochastic thermodynamics have been studied for some time and the past few years have seen a surge of interest in this topic. Quantum mechanics involves profound issues around the interpretation of reality (e.g. the Copenhagen interpretation, many-worlds, de Broglie-Bohm theory etc are all competing interpretations that try to explain the unintuitive results of quantum theory) . It is hoped that by trying to specify the quantum-mechanical definition of work, dealing with open quantum systems, analyzing exactly solvable models, or proposing and performing experiments to test non-equilibrium predictions, important insights into the interpretation of quantum mechanics and the true nature of reality will be gained.
Applications of non-equilibrium work relations, like the Jarzynski equality, have recently been proposed for the purposes of detecting quantum entanglement (Hide & Vedral 2010) and to improving optimization problems (minimize or maximize a function of multivariables called the cost function) via quantum annealing (Ohzeki & Nishimori 2011).
=== Active baths ===
Until recently thermodynamics has only considered systems coupled to a thermal bath and, therefore, satisfying Boltzmann statistics. However, some systems do not satisfy these conditions and are far from equilibrium such as living matter, for which fluctuations are expected to be non-Gaussian.
Active particle systems are able to take energy from their environment and drive themselves far from equilibrium. An important example of active matter is constituted by objects capable of self propulsion. Thanks to this property, they feature a series of novel behaviours that are not attainable by matter at thermal equilibrium, including, for example, swarming and the emergence of other collective properties. A passive particle is considered in an active bath when it is in an environment where a wealth of active particles are present. These particles will exert nonthermal forces on the passive object so that it will experience non-thermal fluctuations and will behave widely different from a passive Brownian particle in a thermal bath. The presence of an active bath can significantly influence the microscopic thermodynamics of a particle. Experiments have suggested that the Jarzynski equality does not hold in some cases due to the presence of non-Boltzmann statistics in active baths. This observation points towards a new direction in the study of non-equilibrium statistical physics and stochastic thermodynamics, where also the environment itself is far from equilibrium.
Active baths are a question of particular importance in biochemistry. For example, biomolecules within cells are coupled with an active bath due to the presence of molecular motors within the cytoplasm, which leads to striking and largely not yet understood phenomena such as the emergence of anomalous diffusion (Barkai et al., 2012). Also, protein folding might be facilitated by the presence of active fluctuations (Harder et al., 2014b) and active matter dynamics could play a central role in several biological functions (Mallory et al., 2015; Shin et al., 2015; Suzuki et al., 2015). It is an open question to what degree stochastic thermodynamics can be applied to systems coupled to active baths.
== References ==
=== Notes ===
=== Citations ===
=== Academic references ===
=== Press === | Wikipedia/Stochastic_thermodynamics |
Symmetries in quantum mechanics describe features of spacetime and particles which are unchanged under some transformation, in the context of quantum mechanics, relativistic quantum mechanics and quantum field theory, and with applications in the mathematical formulation of the standard model and condensed matter physics. In general, symmetry in physics, invariance, and conservation laws, are fundamentally important constraints for formulating physical theories and models. In practice, they are powerful methods for solving problems and predicting what can happen. While conservation laws do not always give the answer to the problem directly, they form the correct constraints and the first steps to solving a multitude of problems. In application, understanding symmetries can also provide insights on the eigenstates that can be expected. For example, the existence of degenerate states can be inferred by the presence of non commuting symmetry operators or that the non degenerate states are also eigenvectors of symmetry operators.
This article outlines the connection between the classical form of continuous symmetries as well as their quantum operators, and relates them to the Lie groups, and relativistic transformations in the Lorentz group and Poincaré group.
== Notation ==
The notational conventions used in this article are as follows. Boldface indicates vectors, four vectors, matrices, and vectorial operators, while quantum states use bra–ket notation. Wide hats are for operators, narrow hats are for unit vectors (including their components in tensor index notation). The summation convention on the repeated tensor indices is used, unless stated otherwise. The Minkowski metric signature is (+−−−).
== Symmetry transformations on the wavefunction in non-relativistic quantum mechanics ==
=== Continuous symmetries ===
Generally, the correspondence between continuous symmetries and conservation laws is given by Noether's theorem.
The form of the fundamental quantum operators, for example the energy operator as a partial time derivative and momentum operator as a spatial gradient, becomes clear when one considers the initial state, then changes one parameter of it slightly. This can be done for displacements (lengths), durations (time), and angles (rotations). Additionally, the invariance of certain quantities can be seen by making such changes in lengths and angles, illustrating conservation of these quantities.
In what follows, transformations on only one-particle wavefunctions in the form:
Ω
^
ψ
(
r
,
t
)
=
ψ
(
r
′
,
t
′
)
{\displaystyle {\widehat {\Omega }}\psi (\mathbf {r} ,t)=\psi (\mathbf {r} ',t')}
are considered, where
Ω
^
{\displaystyle {\widehat {\Omega }}}
denotes a unitary operator. Unitarity is generally required for operators representing transformations of space, time, and spin, since the norm of a state (representing the total probability of finding the particle somewhere with some spin) must be invariant under these transformations. The inverse is the Hermitian conjugate
Ω
^
−
1
=
Ω
^
†
{\displaystyle {\widehat {\Omega }}^{-1}={\widehat {\Omega }}^{\dagger }}
. The results can be extended to many-particle wavefunctions. Written in Dirac notation as standard, the transformations on quantum state vectors are:
Ω
^
|
r
(
t
)
⟩
=
|
r
′
(
t
′
)
⟩
{\displaystyle {\widehat {\Omega }}\left|\mathbf {r} (t)\right\rangle =\left|\mathbf {r} '(t')\right\rangle }
Now, the action of
Ω
^
{\displaystyle {\widehat {\Omega }}}
changes ψ(r, t) to ψ(r′, t′), so the inverse
Ω
^
−
1
=
Ω
^
†
{\displaystyle {\widehat {\Omega }}^{-1}={\widehat {\Omega }}^{\dagger }}
changes ψ(r′, t′) back to ψ(r, t). Thus, an operator
A
^
{\displaystyle {\widehat {A}}}
invariant under
Ω
^
{\displaystyle {\widehat {\Omega }}}
satisfies [I am sorry, but this is non-sequitor. You have not laid a foundation for this proposition]:
A
^
ψ
=
Ω
^
†
A
^
Ω
^
ψ
⇒
Ω
^
A
^
ψ
=
A
^
Ω
^
ψ
.
{\displaystyle {\widehat {A}}\psi ={\widehat {\Omega }}^{\dagger }{\widehat {A}}{\widehat {\Omega }}\psi \quad \Rightarrow \quad {\widehat {\Omega }}{\widehat {A}}\psi ={\widehat {A}}{\widehat {\Omega }}\psi .}
Concomitantly,
[
Ω
^
,
A
^
]
ψ
=
0
{\displaystyle [{\widehat {\Omega }},{\widehat {A}}]\psi =0}
for any state ψ. Quantum operators representing observables are also required to be Hermitian so that their eigenvalues are real numbers, i.e. the operator equals its Hermitian conjugate,
A
^
=
A
^
†
{\displaystyle {\widehat {A}}={\widehat {A}}^{\dagger }}
.
=== Overview of Lie group theory ===
Following are the key points of group theory relevant to quantum theory, examples are given throughout the article. For an alternative approach using matrix groups, see the books of Hall
Let G be a Lie group, which is a group that locally is parameterized by a finite number N of real continuously varying parameters ξ1, ξ2, ..., ξN. In more mathematical language, this means that G is a smooth manifold that is also a group, for which the group operations are smooth.
the dimension of the group, N, is the number of parameters it has.
the group elements, g, in G are functions of the parameters:
g
=
G
(
ξ
1
,
ξ
2
,
…
)
{\displaystyle g=G(\xi _{1},\xi _{2},\dots )}
and all parameters set to zero returns the identity element of the group:
I
=
G
(
0
,
0
,
…
)
{\displaystyle I=G(0,0,\dots )}
Group elements are often matrices which act on vectors, or transformations acting on functions.
The generators of the group are the partial derivatives of the group elements with respect to the group parameters with the result evaluated when the parameter is set to zero:
X
j
=
∂
g
∂
ξ
j
|
ξ
j
=
0
{\displaystyle X_{j}=\left.{\frac {\partial g}{\partial \xi _{j}}}\right|_{\xi _{j}=0}}
In the language of manifolds, the generators are the elements of the tangent space to G at the identity. The generators are also known as infinitesimal group elements or as the elements of the Lie algebra of G. (See the discussion below of the commutator.) One aspect of generators in theoretical physics is they can be constructed themselves as operators corresponding to symmetries, which may be written as matrices, or as differential operators. In quantum theory, for unitary representations of the group, the generators require a factor of i:
X
j
=
i
∂
g
∂
ξ
j
|
ξ
j
=
0
{\displaystyle X_{j}=i\left.{\frac {\partial g}{\partial \xi _{j}}}\right|_{\xi _{j}=0}}
The generators of the group form a vector space, which means linear combinations of generators also form a generator.
The generators (whether matrices or differential operators) satisfy the commutation relations:
[
X
a
,
X
b
]
=
i
f
a
b
c
X
c
{\displaystyle \left[X_{a},X_{b}\right]=if_{abc}X_{c}}
where fabc are the (basis dependent) structure constants of the group. This makes, together with the vector space property, the set of all generators of a group a Lie algebra. Due to the antisymmetry of the bracket, the structure constants of the group are antisymmetric in the first two indices.
The representations of the group then describe the ways that the group G (or its Lie algebra) can act on a vector space. (The vector space might be, for example, the space of eigenvectors for a Hamiltonian having G as its symmetry group.) We denote the representations using a capital D. One can then differentiate D to obtain a representation of the Lie algebra, often also denoted by D. These two representations are related as follows:
D
[
g
(
ξ
j
)
]
≡
D
(
ξ
j
)
=
e
i
ξ
j
D
(
X
j
)
{\displaystyle D[g(\xi _{j})]\equiv D(\xi _{j})=e^{i\xi _{j}D(X_{j})}}
without summation on the repeated index j. Representations are linear operators that take in group elements and preserve the composition rule:
D
(
ξ
a
)
D
(
ξ
b
)
=
D
(
ξ
a
ξ
b
)
.
{\displaystyle D(\xi _{a})D(\xi _{b})=D(\xi _{a}\xi _{b}).}
A representation which cannot be decomposed into a direct sum of other representations, is called irreducible. It is conventional to label irreducible representations by a superscripted number n in brackets, as in D(n), or if there is more than one number, we write D(n, m, ...).
There is an additional subtlety that arises in quantum theory, where two vectors that differ by multiplication by a scalar represent the same physical state. Here, the pertinent notion of representation is a projective representation, one that only satisfies the composition law up to a scalar. In the context of quantum mechanical spin, such representations are called spinorial.
=== Momentum and energy as generators of translation and time evolution, and rotation ===
The space translation operator
T
^
(
Δ
r
)
{\displaystyle {\widehat {T}}(\Delta \mathbf {r} )}
acts on a wavefunction to shift the space coordinates by an infinitesimal displacement Δr. The explicit expression
T
^
{\displaystyle {\widehat {T}}}
can be quickly determined by a Taylor expansion of ψ(r + Δr, t) about r, then (keeping the first order term and neglecting second and higher order terms), replace the space derivatives by the momentum operator
p
^
{\displaystyle {\widehat {\mathbf {p} }}}
. Similarly for the time translation operator acting on the time parameter, the Taylor expansion of ψ(r, t + Δt) is about t, and the time derivative replaced by the energy operator
E
^
{\displaystyle {\widehat {E}}}
.
The exponential functions arise by definition as those limits, due to Euler, and can be understood physically and mathematically as follows. A net translation can be composed of many small translations, so to obtain the translation operator for a finite increment, replace Δr by Δr/N and Δt by Δt/N, where N is a positive non-zero integer. Then as N increases, the magnitude of Δr and Δt become even smaller, while leaving the directions unchanged. Acting the infinitesimal operators on the wavefunction N times and taking the limit as N tends to infinity gives the finite operators.
Space and time translations commute, which means the operators and generators commute.
For a time-independent Hamiltonian, energy is conserved in time and quantum states are stationary states: the eigenstates of the Hamiltonian are the energy eigenvalues E:
U
^
(
t
)
=
exp
(
−
i
Δ
t
E
ℏ
)
{\displaystyle {\widehat {U}}(t)=\exp \left(-{\frac {i\Delta tE}{\hbar }}\right)}
and all stationary states have the form
ψ
(
r
,
t
+
t
0
)
=
U
^
(
t
−
t
0
)
ψ
(
r
,
t
0
)
{\displaystyle \psi (\mathbf {r} ,t+t_{0})={\widehat {U}}(t-t_{0})\psi (\mathbf {r} ,t_{0})}
where t0 is the initial time, usually set to zero since there is no loss of continuity when the initial time is set.
An alternative notation is
U
^
(
t
−
t
0
)
≡
U
(
t
,
t
0
)
{\displaystyle {\widehat {U}}(t-t_{0})\equiv U(t,t_{0})}
.
=== Angular momentum as the generator of rotations ===
==== Orbital angular momentum ====
The rotation operator,
R
^
{\displaystyle {\widehat {R}}}
, acts on a wavefunction to rotate the spatial coordinates of a particle by a constant angle Δθ:
R
^
(
Δ
θ
,
a
^
)
ψ
(
r
,
t
)
=
ψ
(
r
′
,
t
)
{\displaystyle {\widehat {R}}(\Delta \theta ,{\hat {\mathbf {a} }})\psi (\mathbf {r} ,t)=\psi (\mathbf {r} ',t)}
where r′ are the rotated coordinates about an axis defined by a unit vector
a
^
=
(
a
1
,
a
2
,
a
3
)
{\displaystyle {\hat {\mathbf {a} }}=(a_{1},a_{2},a_{3})}
through an angular increment Δθ, given by:
r
′
=
R
^
(
Δ
θ
,
a
^
)
r
.
{\displaystyle \mathbf {r} '={\widehat {R}}(\Delta \theta ,{\hat {\mathbf {a} }})\mathbf {r} \,.}
where
R
^
(
Δ
θ
,
a
^
)
{\displaystyle {\widehat {R}}(\Delta \theta ,{\hat {\mathbf {a} }})}
is a rotation matrix dependent on the axis and angle. In group theoretic language, the rotation matrices are group elements, and the angles and axis
Δ
θ
a
^
=
Δ
θ
(
a
1
,
a
2
,
a
3
)
{\displaystyle \Delta \theta {\hat {\mathbf {a} }}=\Delta \theta (a_{1},a_{2},a_{3})}
are the parameters, of the three-dimensional special orthogonal group, SO(3). The rotation matrices about the standard Cartesian basis vector
e
^
x
,
e
^
y
,
e
^
z
{\displaystyle {\hat {\mathbf {e} }}_{x},{\hat {\mathbf {e} }}_{y},{\hat {\mathbf {e} }}_{z}}
through angle Δθ, and the corresponding generators of rotations J = (Jx, Jy, Jz), are:
More generally for rotations about an axis defined by
a
^
{\displaystyle {\hat {\mathbf {a} }}}
, the rotation matrix elements are:
[
R
^
(
θ
,
a
^
)
]
i
j
=
(
δ
i
j
−
a
i
a
j
)
cos
θ
−
ε
i
j
k
a
k
sin
θ
+
a
i
a
j
{\displaystyle [{\widehat {R}}(\theta ,{\hat {\mathbf {a} }})]_{ij}=(\delta _{ij}-a_{i}a_{j})\cos \theta -\varepsilon _{ijk}a_{k}\sin \theta +a_{i}a_{j}}
where δij is the Kronecker delta, and εijk is the Levi-Civita symbol.
It is not as obvious how to determine the rotational operator compared to space and time translations. We may consider a special case (rotations about the x, y, or z-axis) then infer the general result, or use the general rotation matrix directly and tensor index notation with δij and εijk. To derive the infinitesimal rotation operator, which corresponds to small Δθ, we use the small angle approximations sin(Δθ) ≈ Δθ and cos(Δθ) ≈ 1, then Taylor expand about r or ri, keep the first order term, and substitute the angular momentum operator components.
The z-component of angular momentum can be replaced by the component along the axis defined by
a
^
{\displaystyle {\hat {\mathbf {a} }}}
, using the dot product
a
^
⋅
L
^
{\displaystyle {\hat {\mathbf {a} }}\cdot {\widehat {\mathbf {L} }}}
.
Again, a finite rotation can be made from many small rotations, replacing Δθ by Δθ/N and taking the limit as N tends to infinity gives the rotation operator for a finite rotation.
Rotations about the same axis do commute, for example a rotation through angles θ1 and θ2 about axis i can be written
R
(
θ
1
+
θ
2
,
e
i
)
=
R
(
θ
1
e
i
)
R
(
θ
2
e
i
)
,
[
R
(
θ
1
e
i
)
,
R
(
θ
2
e
i
)
]
=
0
.
{\displaystyle R(\theta _{1}+\theta _{2},\mathbf {e} _{i})=R(\theta _{1}\mathbf {e} _{i})R(\theta _{2}\mathbf {e} _{i})\,,\quad [R(\theta _{1}\mathbf {e} _{i}),R(\theta _{2}\mathbf {e} _{i})]=0\,.}
However, rotations about different axes do not commute. The general commutation rules are summarized by
[
L
i
,
L
j
]
=
i
ℏ
ε
i
j
k
L
k
.
{\displaystyle [L_{i},L_{j}]=i\hbar \varepsilon _{ijk}L_{k}.}
In this sense, orbital angular momentum has the common sense properties of rotations. Each of the above commutators can be easily demonstrated by holding an everyday object and rotating it through the same angle about any two different axes in both possible orderings; the final configurations are different.
In quantum mechanics, there is another form of rotation which mathematically appears similar to the orbital case, but has different properties, described next.
==== Spin angular momentum ====
All previous quantities have classical definitions. Spin is a quantity possessed by particles in quantum mechanics without any classical analogue, having the units of angular momentum. The spin vector operator is denoted
S
^
=
(
S
x
^
,
S
y
^
,
S
z
^
)
{\displaystyle {\widehat {\mathbf {S} }}=({\widehat {S_{x}}},{\widehat {S_{y}}},{\widehat {S_{z}}})}
. The eigenvalues of its components are the possible outcomes (in units of
ℏ
{\displaystyle \hbar }
) of a measurement of the spin projected onto one of the basis directions.
Rotations (of ordinary space) about an axis
a
^
{\displaystyle {\hat {\mathbf {a} }}}
through angle θ about the unit vector
a
^
{\displaystyle {\hat {a}}}
in space acting on a multicomponent wave function (spinor) at a point in space is represented by:
However, unlike orbital angular momentum in which the z-projection quantum number ℓ can only take positive or negative integer values (including zero), the z-projection spin quantum number s can take all positive and negative half-integer values. There are rotational matrices for each spin quantum number.
Evaluating the exponential for a given z-projection spin quantum number s gives a (2s + 1)-dimensional spin matrix. This can be used to define a spinor as a column vector of 2s + 1 components which transforms to a rotated coordinate system according to the spin matrix at a fixed point in space.
For the simplest non-trivial case of s = 1/2, the spin operator is given by
S
^
=
ℏ
2
σ
{\displaystyle {\widehat {\mathbf {S} }}={\frac {\hbar }{2}}{\boldsymbol {\sigma }}}
where the Pauli matrices in the standard representation are:
σ
1
=
σ
x
=
(
0
1
1
0
)
,
σ
2
=
σ
y
=
(
0
−
i
i
0
)
,
σ
3
=
σ
z
=
(
1
0
0
−
1
)
{\displaystyle \sigma _{1}=\sigma _{x}={\begin{pmatrix}0&1\\1&0\end{pmatrix}}\,,\quad \sigma _{2}=\sigma _{y}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}}\,,\quad \sigma _{3}=\sigma _{z}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}}
==== Total angular momentum ====
The total angular momentum operator is the sum of the orbital and spin
J
^
=
L
^
+
S
^
{\displaystyle {\widehat {\mathbf {J} }}={\widehat {\mathbf {L} }}+{\widehat {\mathbf {S} }}}
and is an important quantity for multi-particle systems, especially in nuclear physics and the quantum chemistry of multi-electron atoms and molecules.
We have a similar rotation matrix:
J
^
(
θ
,
a
^
)
=
exp
(
−
i
ℏ
θ
a
^
⋅
J
^
)
{\displaystyle {\widehat {J}}(\theta ,{\hat {\mathbf {a} }})=\exp \left(-{\frac {i}{\hbar }}\theta {\hat {\mathbf {a} }}\cdot {\widehat {\mathbf {J} }}\right)}
=== Conserved quantities in the quantum harmonic oscillator ===
The dynamical symmetry group of the n dimensional quantum harmonic oscillator is the special unitary group SU(n). As an example, the number of infinitesimal generators of the corresponding Lie algebras of SU(2) and SU(3) are three and eight respectively. This leads to exactly three and eight independent conserved quantities (other than the Hamiltonian) in these systems.
The two dimensional quantum harmonic oscillator has the expected conserved quantities of the Hamiltonian and the angular momentum, but has additional hidden conserved quantities of energy level difference and another form of angular momentum.
== Lorentz group in relativistic quantum mechanics ==
Following is an overview of the Lorentz group; a treatment of boosts and rotations in spacetime. Throughout this section, see (for example) T. Ohlsson (2011) and E. Abers (2004).
Lorentz transformations can be parametrized by rapidity φ for a boost in the direction of a three-dimensional unit vector
n
^
=
(
n
1
,
n
2
,
n
3
)
{\displaystyle {\hat {\mathbf {n} }}=(n_{1},n_{2},n_{3})}
, and a rotation angle θ about a three-dimensional unit vector
a
^
=
(
a
1
,
a
2
,
a
3
)
{\displaystyle {\hat {\mathbf {a} }}=(a_{1},a_{2},a_{3})}
defining an axis, so
φ
n
^
=
φ
(
n
1
,
n
2
,
n
3
)
{\displaystyle \varphi {\hat {\mathbf {n} }}=\varphi (n_{1},n_{2},n_{3})}
and
θ
a
^
=
θ
(
a
1
,
a
2
,
a
3
)
{\displaystyle \theta {\hat {\mathbf {a} }}=\theta (a_{1},a_{2},a_{3})}
are together six parameters of the Lorentz group (three for rotations and three for boosts). The Lorentz group is 6-dimensional.
=== Pure rotations in spacetime ===
The rotation matrices and rotation generators considered above form the spacelike part of a four-dimensional matrix, representing pure-rotation Lorentz transformations. Three of the Lorentz group elements
R
^
x
,
R
^
y
,
R
^
z
{\displaystyle {\widehat {R}}_{x},{\widehat {R}}_{y},{\widehat {R}}_{z}}
and generators J = (J1, J2, J3) for pure rotations are:
The rotation matrices act on any four vector A = (A0, A1, A2, A3) and rotate the space-like components according to
A
′
=
R
^
(
Δ
θ
,
n
^
)
A
{\displaystyle \mathbf {A} '={\widehat {R}}(\Delta \theta ,{\hat {\mathbf {n} }})\mathbf {A} }
leaving the time-like coordinate unchanged. In matrix expressions, A is treated as a column vector.
=== Pure boosts in spacetime ===
A boost with velocity ctanhφ in the x, y, or z directions given by the standard Cartesian basis vector
e
^
x
,
e
^
y
,
e
^
z
{\displaystyle {\hat {\mathbf {e} }}_{x},{\hat {\mathbf {e} }}_{y},{\hat {\mathbf {e} }}_{z}}
, are the boost transformation matrices. These matrices
B
^
x
,
B
^
y
,
B
^
z
{\displaystyle {\widehat {B}}_{x},{\widehat {B}}_{y},{\widehat {B}}_{z}}
and the corresponding generators K = (K1, K2, K3) are the remaining three group elements and generators of the Lorentz group:
The boost matrices act on any four vector A = (A0, A1, A2, A3) and mix the time-like and the space-like components, according to:
A
′
=
B
^
(
φ
,
n
^
)
A
{\displaystyle \mathbf {A} '={\widehat {B}}(\varphi ,{\hat {\mathbf {n} }})\mathbf {A} }
The term "boost" refers to the relative velocity between two frames, and is not to be conflated with momentum as the generator of translations, as explained below.
=== Combining boosts and rotations ===
Products of rotations give another rotation (a frequent exemplification of a subgroup), while products of boosts and boosts or of rotations and boosts cannot be expressed as pure boosts or pure rotations. In general, any Lorentz transformation can be expressed as a product of a pure rotation and a pure boost. For more background see (for example) B.R. Durney (2011) and H.L. Berk et al. and references therein.
The boost and rotation generators have representations denoted D(K) and D(J) respectively, the capital D in this context indicates a group representation.
For the Lorentz group, the representations D(K) and D(J) of the generators K and J fulfill the following commutation rules.
In all commutators, the boost entities mixed with those for rotations, although rotations alone simply give another rotation. Exponentiating the generators gives the boost and rotation operators which combine into the general Lorentz transformation, under which the spacetime coordinates transform from one rest frame to another boosted and/or rotating frame. Likewise, exponentiating the representations of the generators gives the representations of the boost and rotation operators, under which a particle's spinor field transforms.
In the literature, the boost generators K and rotation generators J are sometimes combined into one generator for Lorentz transformations M, an antisymmetric four-dimensional matrix with entries:
M
0
a
=
−
M
a
0
=
K
a
,
M
a
b
=
ε
a
b
c
J
c
.
{\displaystyle M^{0a}=-M^{a0}=K_{a}\,,\quad M^{ab}=\varepsilon _{abc}J_{c}\,.}
and correspondingly, the boost and rotation parameters are collected into another antisymmetric four-dimensional matrix ω, with entries:
ω
0
a
=
−
ω
a
0
=
φ
n
a
,
ω
a
b
=
θ
ε
a
b
c
a
c
,
{\displaystyle \omega _{0a}=-\omega _{a0}=\varphi n_{a}\,,\quad \omega _{ab}=\theta \varepsilon _{abc}a_{c}\,,}
The general Lorentz transformation is then:
Λ
(
φ
,
n
^
,
θ
,
a
^
)
=
exp
(
−
i
2
ω
α
β
M
α
β
)
=
exp
[
−
i
2
(
φ
n
^
⋅
K
+
θ
a
^
⋅
J
)
]
{\displaystyle \Lambda (\varphi ,{\hat {\mathbf {n} }},\theta ,{\hat {\mathbf {a} }})=\exp \left(-{\frac {i}{2}}\omega _{\alpha \beta }M^{\alpha \beta }\right)=\exp \left[-{\frac {i}{2}}\left(\varphi {\hat {\mathbf {n} }}\cdot \mathbf {K} +\theta {\hat {\mathbf {a} }}\cdot \mathbf {J} \right)\right]}
with summation over repeated matrix indices α and β. The Λ matrices act on any four vector A = (A0, A1, A2, A3) and mix the time-like and the space-like components, according to:
A
′
=
Λ
(
φ
,
n
^
,
θ
,
a
^
)
A
{\displaystyle \mathbf {A} '=\Lambda (\varphi ,{\hat {\mathbf {n} }},\theta ,{\hat {\mathbf {a} }})\mathbf {A} }
=== Transformations of spinor wavefunctions in relativistic quantum mechanics ===
In relativistic quantum mechanics, wavefunctions are no longer single-component scalar fields, but now 2(2s + 1) component spinor fields, where s is the spin of the particle. The transformations of these functions in spacetime are given below.
Under a proper orthochronous Lorentz transformation (r, t) → Λ(r, t) in Minkowski space, all one-particle quantum states ψσ locally transform under some representation D of the Lorentz group:
ψ
σ
(
r
,
t
)
→
D
(
Λ
)
ψ
σ
(
Λ
−
1
(
r
,
t
)
)
{\displaystyle \psi _{\sigma }(\mathbf {r} ,t)\rightarrow D(\Lambda )\psi _{\sigma }(\Lambda ^{-1}(\mathbf {r} ,t))}
where D(Λ) is a finite-dimensional representation, in other words a (2s + 1)×(2s + 1) dimensional square matrix, and ψ is thought of as a column vector containing components with the (2s + 1) allowed values of σ:
ψ
(
r
,
t
)
=
[
ψ
σ
=
s
(
r
,
t
)
ψ
σ
=
s
−
1
(
r
,
t
)
⋮
ψ
σ
=
−
s
+
1
(
r
,
t
)
ψ
σ
=
−
s
(
r
,
t
)
]
⇌
ψ
(
r
,
t
)
†
=
[
ψ
σ
=
s
(
r
,
t
)
⋆
ψ
σ
=
s
−
1
(
r
,
t
)
⋆
⋯
ψ
σ
=
−
s
+
1
(
r
,
t
)
⋆
ψ
σ
=
−
s
(
r
,
t
)
⋆
]
{\displaystyle \psi (\mathbf {r} ,t)={\begin{bmatrix}\psi _{\sigma =s}(\mathbf {r} ,t)\\\psi _{\sigma =s-1}(\mathbf {r} ,t)\\\vdots \\\psi _{\sigma =-s+1}(\mathbf {r} ,t)\\\psi _{\sigma =-s}(\mathbf {r} ,t)\end{bmatrix}}\quad \rightleftharpoons \quad {\psi (\mathbf {r} ,t)}^{\dagger }={\begin{bmatrix}{\psi _{\sigma =s}(\mathbf {r} ,t)}^{\star }&{\psi _{\sigma =s-1}(\mathbf {r} ,t)}^{\star }&\cdots &{\psi _{\sigma =-s+1}(\mathbf {r} ,t)}^{\star }&{\psi _{\sigma =-s}(\mathbf {r} ,t)}^{\star }\end{bmatrix}}}
=== Real irreducible representations and spin ===
The irreducible representations of D(K) and D(J), in short "irreps", can be used to build to spin representations of the Lorentz group. Defining new operators:
A
=
J
+
i
K
2
,
B
=
J
−
i
K
2
,
{\displaystyle \mathbf {A} ={\frac {\mathbf {J} +i\mathbf {K} }{2}}\,,\quad \mathbf {B} ={\frac {\mathbf {J} -i\mathbf {K} }{2}}\,,}
so A and B are simply complex conjugates of each other, it follows they satisfy the symmetrically formed commutators:
[
A
i
,
A
j
]
=
ε
i
j
k
A
k
,
[
B
i
,
B
j
]
=
ε
i
j
k
B
k
,
[
A
i
,
B
j
]
=
0
,
{\displaystyle \left[A_{i},A_{j}\right]=\varepsilon _{ijk}A_{k}\,,\quad \left[B_{i},B_{j}\right]=\varepsilon _{ijk}B_{k}\,,\quad \left[A_{i},B_{j}\right]=0\,,}
and these are essentially the commutators the orbital and spin angular momentum operators satisfy. Therefore, A and B form operator algebras analogous to angular momentum; same ladder operators, z-projections, etc., independently of each other as each of their components mutually commute. By the analogy to the spin quantum number, we can introduce positive integers or half integers, a, b, with corresponding sets of values m = a, a − 1, ... −a + 1, −a and n = b, b − 1, ... −b + 1, −b. The matrices satisfying the above commutation relations are the same as for spins a and b have components given by multiplying Kronecker delta values with angular momentum matrix elements:
(
A
x
)
m
′
n
′
,
m
n
=
δ
n
′
n
(
J
x
(
m
)
)
m
′
m
(
B
x
)
m
′
n
′
,
m
n
=
δ
m
′
m
(
J
x
(
n
)
)
n
′
n
{\displaystyle \left(A_{x}\right)_{m'n',mn}=\delta _{n'n}\left(J_{x}^{(m)}\right)_{m'm}\,\quad \left(B_{x}\right)_{m'n',mn}=\delta _{m'm}\left(J_{x}^{(n)}\right)_{n'n}}
(
A
y
)
m
′
n
′
,
m
n
=
δ
n
′
n
(
J
y
(
m
)
)
m
′
m
(
B
y
)
m
′
n
′
,
m
n
=
δ
m
′
m
(
J
y
(
n
)
)
n
′
n
{\displaystyle \left(A_{y}\right)_{m'n',mn}=\delta _{n'n}\left(J_{y}^{(m)}\right)_{m'm}\,\quad \left(B_{y}\right)_{m'n',mn}=\delta _{m'm}\left(J_{y}^{(n)}\right)_{n'n}}
(
A
z
)
m
′
n
′
,
m
n
=
δ
n
′
n
(
J
z
(
m
)
)
m
′
m
(
B
z
)
m
′
n
′
,
m
n
=
δ
m
′
m
(
J
z
(
n
)
)
n
′
n
{\displaystyle \left(A_{z}\right)_{m'n',mn}=\delta _{n'n}\left(J_{z}^{(m)}\right)_{m'm}\,\quad \left(B_{z}\right)_{m'n',mn}=\delta _{m'm}\left(J_{z}^{(n)}\right)_{n'n}}
where in each case the row number m′n′ and column number mn are separated by a comma, and in turn:
(
J
z
(
m
)
)
m
′
m
=
m
δ
m
′
m
(
J
x
(
m
)
±
i
J
y
(
m
)
)
m
′
m
=
m
δ
a
′
,
a
±
1
(
a
∓
m
)
(
a
±
m
+
1
)
{\displaystyle \left(J_{z}^{(m)}\right)_{m'm}=m\delta _{m'm}\,\quad \left(J_{x}^{(m)}\pm iJ_{y}^{(m)}\right)_{m'm}=m\delta _{a',a\pm 1}{\sqrt {(a\mp m)(a\pm m+1)}}}
and similarly for J(n). The three J(m) matrices are each (2m + 1)×(2m + 1) square matrices, and the three J(n) are each (2n + 1)×(2n + 1) square matrices. The integers or half-integers m and n numerate all the irreducible representations by, in equivalent notations used by authors: D(m, n) ≡ (m, n) ≡ D(m) ⊗ D(n), which are each [(2m + 1)(2n + 1)]×[(2m + 1)(2n + 1)] square matrices.
Applying this to particles with spin s;
left-handed (2s + 1)-component spinors transform under the real irreps D(s, 0),
right-handed (2s + 1)-component spinors transform under the real irreps D(0, s),
taking direct sums symbolized by ⊕ (see direct sum of matrices for the simpler matrix concept), one obtains the representations under which 2(2s + 1)-component spinors transform: D(m, n) ⊕ D(n, m) where m + n = s. These are also real irreps, but as shown above, they split into complex conjugates.
In these cases the D refers to any of D(J), D(K), or a full Lorentz transformation D(Λ).
=== Relativistic wave equations ===
In the context of the Dirac equation and Weyl equation, the Weyl spinors satisfying the Weyl equation transform under the simplest irreducible spin representations of the Lorentz group, since the spin quantum number in this case is the smallest non-zero number allowed: 1/2. The 2-component left-handed Weyl spinor transforms under D(1/2, 0) and the 2-component right-handed Weyl spinor transforms under D(0, 1/2). Dirac spinors satisfying the Dirac equation transform under the representation D(1/2, 0) ⊕ D(0, 1/2), the direct sum of the irreps for the Weyl spinors.
== The Poincaré group in relativistic quantum mechanics and field theory ==
Space translations, time translations, rotations, and boosts, all taken together, constitute the Poincaré group. The group elements are the three rotation matrices and three boost matrices (as in the Lorentz group), and one for time translations and three for space translations in spacetime. There is a generator for each. Therefore, the Poincaré group is 10-dimensional.
In special relativity, space and time can be collected into a four-position vector X = (ct, −r), and in parallel so can energy and momentum which combine into a four-momentum vector P = (E/c, −p). With relativistic quantum mechanics in mind, the time duration and spatial displacement parameters (four in total, one for time and three for space) combine into a spacetime displacement ΔX = (cΔt, −Δr), and the energy and momentum operators are inserted in the four-momentum to obtain a four-momentum operator,
P
^
=
(
E
^
c
,
−
p
^
)
=
i
ℏ
(
1
c
∂
∂
t
,
∇
)
,
{\displaystyle {\widehat {\mathbf {P} }}=\left({\frac {\widehat {E}}{c}},-{\widehat {\mathbf {p} }}\right)=i\hbar \left({\frac {1}{c}}{\frac {\partial }{\partial t}},\nabla \right)\,,}
which are the generators of spacetime translations (four in total, one time and three space):
X
^
(
Δ
X
)
=
exp
(
−
i
ℏ
Δ
X
⋅
P
^
)
=
exp
[
−
i
ℏ
(
Δ
t
E
^
+
Δ
r
⋅
p
^
)
]
.
{\displaystyle {\widehat {X}}(\Delta \mathbf {X} )=\exp \left(-{\frac {i}{\hbar }}\Delta \mathbf {X} \cdot {\widehat {\mathbf {P} }}\right)=\exp \left[-{\frac {i}{\hbar }}\left(\Delta t{\widehat {E}}+\Delta \mathbf {r} \cdot {\widehat {\mathbf {p} }}\right)\right]\,.}
There are commutation relations between the components four-momentum P (generators of spacetime translations), and angular momentum M (generators of Lorentz transformations), that define the Poincaré algebra:
[
P
μ
,
P
ν
]
=
0
{\displaystyle [P_{\mu },P_{\nu }]=0\,}
1
i
[
M
μ
ν
,
P
ρ
]
=
η
μ
ρ
P
ν
−
η
ν
ρ
P
μ
{\displaystyle {\frac {1}{i}}[M_{\mu \nu },P_{\rho }]=\eta _{\mu \rho }P_{\nu }-\eta _{\nu \rho }P_{\mu }\,}
1
i
[
M
μ
ν
,
M
ρ
σ
]
=
η
μ
ρ
M
ν
σ
−
η
μ
σ
M
ν
ρ
−
η
ν
ρ
M
μ
σ
+
η
ν
σ
M
μ
ρ
{\displaystyle {\frac {1}{i}}[M_{\mu \nu },M_{\rho \sigma }]=\eta _{\mu \rho }M_{\nu \sigma }-\eta _{\mu \sigma }M_{\nu \rho }-\eta _{\nu \rho }M_{\mu \sigma }+\eta _{\nu \sigma }M_{\mu \rho }\,}
where η is the Minkowski metric tensor. (It is common to drop any hats for the four-momentum operators in the commutation relations). These equations are an expression of the fundamental properties of space and time as far as they are known today. They have a classical counterpart where the commutators are replaced by Poisson brackets.
To describe spin in relativistic quantum mechanics, the Pauli–Lubanski pseudovector
W
μ
=
1
2
ε
μ
ν
ρ
σ
J
ν
ρ
P
σ
,
{\displaystyle W_{\mu }={\frac {1}{2}}\varepsilon _{\mu \nu \rho \sigma }J^{\nu \rho }P^{\sigma },}
a Casimir operator, is the constant spin contribution to the total angular momentum, and there are commutation relations between P and W and between M and W:
[
P
μ
,
W
ν
]
=
0
,
{\displaystyle \left[P^{\mu },W^{\nu }\right]=0\,,}
[
J
μ
ν
,
W
ρ
]
=
i
(
η
ρ
ν
W
μ
−
η
ρ
μ
W
ν
)
,
{\displaystyle \left[J^{\mu \nu },W^{\rho }\right]=i\left(\eta ^{\rho \nu }W^{\mu }-\eta ^{\rho \mu }W^{\nu }\right)\,,}
[
W
μ
,
W
ν
]
=
−
i
ϵ
μ
ν
ρ
σ
W
ρ
P
σ
.
{\displaystyle \left[W_{\mu },W_{\nu }\right]=-i\epsilon _{\mu \nu \rho \sigma }W^{\rho }P^{\sigma }\,.}
Invariants constructed from W, instances of Casimir invariants can be used to classify irreducible representations of the Lorentz group.
== Symmetries in quantum field theory and particle physics ==
=== Unitary groups in quantum field theory ===
Group theory is an abstract way of mathematically analyzing symmetries. Unitary operators are paramount to quantum theory, so unitary groups are important in particle physics. The group of N dimensional unitary square matrices is denoted U(N). Unitary operators preserve inner products which means probabilities are also preserved, so the quantum mechanics of the system is invariant under unitary transformations. Let
U
^
{\displaystyle {\widehat {U}}}
be a unitary operator, so the inverse is the Hermitian adjoint
U
^
−
1
=
U
^
†
{\displaystyle {\widehat {U}}^{-1}={\widehat {U}}^{\dagger }}
, which commutes with the Hamiltonian:
[
U
^
,
H
^
]
=
0
{\displaystyle \left[{\widehat {U}},{\widehat {H}}\right]=0}
then the observable corresponding to the operator
U
^
{\displaystyle {\widehat {U}}}
is conserved, and the Hamiltonian is invariant under the transformation
U
^
{\displaystyle {\widehat {U}}}
.
Since the predictions of quantum mechanics should be invariant under the action of a group, physicists look for unitary transformations to represent the group.
Important subgroups of each U(N) are those unitary matrices which have unit determinant (or are "unimodular"): these are called the special unitary groups and are denoted SU(N).
==== U(1) ====
The simplest unitary group is U(1), which is just the complex numbers of modulus 1. This one-dimensional matrix entry is of the form:
U
=
e
−
i
θ
{\displaystyle U=e^{-i\theta }}
in which θ is the parameter of the group, and the group is Abelian since one-dimensional matrices always commute under matrix multiplication. Lagrangians in quantum field theory for complex scalar fields are often invariant under U(1) transformations. If there is a quantum number a associated with the U(1) symmetry, for example baryon and the three lepton numbers in electromagnetic interactions, we have:
U
=
e
−
i
a
θ
{\displaystyle U=e^{-ia\theta }}
==== U(2) and SU(2) ====
The general form of an element of a U(2) element is parametrized by two complex numbers a and b:
U
=
(
a
b
−
b
⋆
a
⋆
)
{\displaystyle U={\begin{pmatrix}a&b\\-b^{\star }&a^{\star }\\\end{pmatrix}}}
and for SU(2), the determinant is restricted to 1:
det
(
U
)
=
a
a
⋆
+
b
b
⋆
=
|
a
|
2
+
|
b
|
2
=
1
{\displaystyle \det(U)=aa^{\star }+bb^{\star }={|a|}^{2}+{|b|}^{2}=1}
In group theoretic language, the Pauli matrices are the generators of the special unitary group in two dimensions, denoted SU(2). Their commutation relation is the same as for orbital angular momentum, aside from a factor of 2:
[
σ
a
,
σ
b
]
=
2
i
ℏ
ε
a
b
c
σ
c
{\displaystyle [\sigma _{a},\sigma _{b}]=2i\hbar \varepsilon _{abc}\sigma _{c}}
A group element of SU(2) can be written:
U
(
θ
,
e
^
j
)
=
e
i
θ
σ
j
/
2
{\displaystyle U(\theta ,{\hat {\mathbf {e} }}_{j})=e^{i\theta \sigma _{j}/2}}
where σj is a Pauli matrix, and the group parameters are the angles turned through about an axis.
The two-dimensional isotropic quantum harmonic oscillator has symmetry group SU(2), while the symmetry algebra of the rational anisotropic oscillator is a nonlinear extension of u(2).
==== U(3) and SU(3) ====
The eight Gell-Mann matrices λn (see article for them and the structure constants) are important for quantum chromodynamics. They originally arose in the theory SU(3) of flavor which is still of practical importance in nuclear physics. They are the generators for the SU(3) group, so an element of SU(3) can be written analogously to an element of SU(2):
U
(
θ
,
e
^
j
)
=
exp
(
−
i
2
∑
n
=
1
8
θ
n
λ
n
)
{\displaystyle U(\theta ,{\hat {\mathbf {e} }}_{j})=\exp \left(-{\frac {i}{2}}\sum _{n=1}^{8}\theta _{n}\lambda _{n}\right)}
where θn are eight independent parameters. The λn matrices satisfy the commutator:
[
λ
a
,
λ
b
]
=
2
i
f
a
b
c
λ
c
{\displaystyle \left[\lambda _{a},\lambda _{b}\right]=2if_{abc}\lambda _{c}}
where the indices a, b, c take the values 1, 2, 3, ..., 8. The structure constants fabc are totally antisymmetric in all indices analogous to those of SU(2). In the standard colour charge basis (r for red, g for green, b for blue):
|
r
⟩
=
(
1
0
0
)
,
|
g
⟩
=
(
0
1
0
)
,
|
b
⟩
=
(
0
0
1
)
{\displaystyle |r\rangle ={\begin{pmatrix}1\\0\\0\end{pmatrix}}\,,\quad |g\rangle ={\begin{pmatrix}0\\1\\0\end{pmatrix}}\,,\quad |b\rangle ={\begin{pmatrix}0\\0\\1\end{pmatrix}}}
the colour states are eigenstates of the λ3 and λ8 matrices, while the other matrices mix colour states together.
The eight gluons states (8-dimensional column vectors) are simultaneous eigenstates of the adjoint representation of SU(3), the 8-dimensional representation acting on its own Lie algebra su(3), for the λ3 and λ8 matrices. By forming tensor products of representations (the standard representation and its dual) and taking appropriate quotients, protons and neutrons, and other hadrons are eigenstates of various representations of SU(3) of color. The representations of SU(3) can be described by a "theorem of the highest weight".
=== Matter and antimatter ===
In relativistic quantum mechanics, relativistic wave equations predict a remarkable symmetry of nature: that every particle has a corresponding antiparticle. This is mathematically contained in the spinor fields which are the solutions of the relativistic wave equations.
Charge conjugation switches particles and antiparticles. Physical laws and interactions unchanged by this operation have C symmetry.
=== Discrete spacetime symmetries ===
Parity mirrors the orientation of the spatial coordinates from left-handed to right-handed. Informally, space is "reflected" into its mirror image. Physical laws and interactions unchanged by this operation have P symmetry.
Time reversal flips the time coordinate, which amounts to time running from future to past. A curious property of time, which space does not have, is that it is unidirectional: particles traveling forwards in time are equivalent to antiparticles traveling back in time. Physical laws and interactions unchanged by this operation have T symmetry.
=== C, P, T symmetries ===
Parity (physics) § Molecules
CPT theorem
CP violation
PT symmetry
Lorentz violation
=== Gauge theory ===
In quantum electrodynamics, the local symmetry group is U(1) and is abelian. In quantum chromodynamics, the local symmetry group is SU(3) and is non-abelian.
The electromagnetic interaction is mediated by photons, which have no electric charge. The electromagnetic tensor has an electromagnetic four-potential field possessing gauge symmetry.
The strong (color) interaction is mediated by gluons, which can have eight color charges. There are eight gluon field strength tensors with corresponding gluon four potentials field, each possessing gauge symmetry.
=== The strong (color) interaction ===
==== Color charge ====
Analogous to the spin operator, there are color charge operators in terms of the Gell-Mann matrices λj:
F
^
j
=
1
2
λ
j
{\displaystyle {\hat {F}}_{j}={\frac {1}{2}}\lambda _{j}}
and since color charge is a conserved charge, all color charge operators must commute with the Hamiltonian:
[
F
^
j
,
H
^
]
=
0
{\displaystyle \left[{\hat {F}}_{j},{\hat {H}}\right]=0}
==== Isospin ====
Isospin is conserved in strong interactions.
=== The weak and electromagnetic interactions ===
==== Duality transformation ====
Magnetic monopoles can be theoretically realized, although current observations and theory are consistent with them existing or not existing. Electric and magnetic charges can effectively be "rotated into one another" by a duality transformation.
==== Electroweak symmetry ====
Electroweak symmetry
Electroweak symmetry breaking
=== Supersymmetry ===
A Lie superalgebra is an algebra in which (suitable) basis elements either have a commutation relation or have an anticommutation relation. Symmetries have been proposed to the effect that all fermionic particles have bosonic analogues, and vice versa. These symmetry have theoretical appeal in that no extra assumptions (such as existence of strings) barring symmetries are made. In addition, by assuming supersymmetry, a number of puzzling issues can be resolved. These symmetries, which are represented by Lie superalgebras, have not been confirmed experimentally. It is now believed that they are broken symmetries, if they exist. But it has been speculated that dark matter is constitutes gravitinos, a spin 3/2 particle with mass, its supersymmetric partner being the graviton.
== Exchange symmetry ==
The concept of exchange symmetry is derived from a fundamental postulate of quantum statistics, which states that no observable physical quantity should change after exchanging two identical particles. It states that because all observables are proportional to
|
ψ
|
2
{\displaystyle \left|\psi \right|^{2}}
for a system of identical particles, the wave function
ψ
{\displaystyle \psi }
must either remain the same or change sign upon such an exchange. More generally, for a system of n identical particles the wave function
ψ
{\displaystyle \psi }
must transform as an irreducible representation of the finite symmetric group Sn. It turns out that, according to the spin-statistics theorem, fermion states transform as the antisymmetric irreducible representation of Sn and boson states as the symmetric irreducible representation.
Because the exchange of two identical particles is mathematically equivalent to the rotation of each particle by 180 degrees (and so to the rotation of one particle's frame by 360 degrees), the symmetric nature of the wave function depends on the particle's spin after the rotation operator is applied to it. Integer spin particles do not change the sign of their wave function upon a 360 degree rotation—therefore the sign of the wave function of the entire system does not change. Semi-integer spin particles change the sign of their wave function upon a 360 degree rotation (see more in spin–statistics theorem).
Particles for which the wave function does not change sign upon exchange are called bosons, or particles with a symmetric wave function. The particles for which the wave function of the system changes sign are called fermions, or particles with an antisymmetric wave function.
Fermions therefore obey different statistics (called Fermi–Dirac statistics) than bosons (which obey Bose–Einstein statistics). One of the consequences of Fermi–Dirac statistics is the exclusion principle for fermions—no two identical fermions can share the same quantum state (in other words, the wave function of two identical fermions in the same state is zero). This in turn results in degeneracy pressure for fermions—the strong resistance of fermions to compression into smaller volume. This resistance gives rise to the “stiffness” or “rigidity” of ordinary atomic matter (as atoms contain electrons which are fermions).
== See also ==
== Footnotes ==
== References ==
== Further reading ==
== External links ==
The molecular symmetry group[1] @ The University of Western Ontario
(2010) Irreducible Tensor Operators and the Wigner-Eckart Theorem Archived 2014-07-20 at the Wayback Machine
Reece, R.D. (2006). "A Derivation of the Quantum Mechanical Momentum Operator in the Position Representation".
Soper, D.E. (2011). "Position and momentum in quantum mechanics" (PDF).
Lie groups
Porter, F. (2009). "Lie Groups and Lie Algebras" (PDF). Archived from the original (PDF) on 2017-03-29. Retrieved 2013-06-05.
Continuous Groups, Lie Groups, and Lie Algebras Archived 2016-03-04 at the Wayback Machine
Mulders, P.J. (November 2011). "Quantum field theory" (PDF). Department of Theoretical Physics, VU University. 6.04.
Hall, B.C. (2000). "An Elementary Introduction to Groups and Representations". arXiv:math-ph/0005032. | Wikipedia/Symmetries_in_quantum_mechanics |
In physics, the Yang–Baxter equation (or star–triangle relation) is a consistency equation which was first introduced in the field of statistical mechanics. It depends on the idea that in some scattering situations, particles may preserve their momentum while changing their quantum internal states. It states that a matrix
R
{\displaystyle R}
, acting on two out of three objects, satisfies
(
R
ˇ
⊗
1
)
(
1
⊗
R
ˇ
)
(
R
ˇ
⊗
1
)
=
(
1
⊗
R
ˇ
)
(
R
ˇ
⊗
1
)
(
1
⊗
R
ˇ
)
,
{\displaystyle ({\check {R}}\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}})({\check {R}}\otimes \mathbf {1} )=(\mathbf {1} \otimes {\check {R}})({\check {R}}\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}}),}
where
R
ˇ
{\displaystyle {\check {R}}}
is
R
{\displaystyle R}
followed by a swap of the two objects. In one-dimensional quantum systems,
R
{\displaystyle R}
is the scattering matrix and if it satisfies the Yang–Baxter equation then the system is integrable. The Yang–Baxter equation also shows up when discussing knot theory and the braid groups where
R
{\displaystyle R}
corresponds to swapping two strands. Since one can swap three strands in two different ways, the Yang–Baxter equation enforces that both paths are the same.
== History ==
According to Michio Jimbo, the Yang–Baxter equation (YBE) manifested itself in the works of J. B. McGuire in 1964 and C. N. Yang in 1967. They considered a quantum mechanical many-body problem on a line having
c
∑
i
<
j
δ
(
x
i
−
x
j
)
{\displaystyle c\sum _{i<j}\delta (x_{i}-x_{j})}
as the potential. Using the Bethe ansatz techniques, they found that the scattering matrix factorized to that of the two-body problem, and determined it exactly. Here YBE arises as the consistency condition for the factorization.
In statistical mechanics, the source of YBE probably goes back to Onsager's star-triangle relation, briefly mentioned in the introduction to his solution of the Ising model in 1944. The hunt for solvable lattice models has been actively pursued since then, culminating in Rodney Baxter's solution of the eight vertex model in 1972.
Another line of development was the theory of factorized S-matrix in two dimensional quantum field theory. Alexander B. Zamolodchikov pointed out that the algebraic mechanics working here is the same as that in the Baxter's and others' works.
The YBE has also manifested itself in a study of Young operators in the group algebra
C
[
S
n
]
{\displaystyle \mathbb {C} [S_{n}]}
of the symmetric group in the work of A. A. Jucys in 1966.
== General form of the parameter-dependent Yang–Baxter equation ==
Let
A
{\displaystyle A}
be a unital associative algebra. In its most general form, the parameter-dependent Yang–Baxter equation is an equation for
R
(
u
,
u
′
)
{\displaystyle R(u,u')}
, a parameter-dependent element of the tensor product
A
⊗
A
{\displaystyle A\otimes A}
(here,
u
{\displaystyle u}
and
u
′
{\displaystyle u'}
are the parameters, which usually range over the real numbers ℝ in the case of an additive parameter, or over positive real numbers ℝ+ in the case of a multiplicative parameter).
Let
R
i
j
(
u
,
u
′
)
=
ϕ
i
j
(
R
(
u
,
u
′
)
)
{\displaystyle R_{ij}(u,u')=\phi _{ij}(R(u,u'))}
for
1
≤
i
<
j
≤
3
{\displaystyle 1\leq i<j\leq 3}
, with algebra homomorphisms
ϕ
i
j
:
A
⊗
A
→
A
⊗
A
⊗
A
{\displaystyle \phi _{ij}:A\otimes A\to A\otimes A\otimes A}
determined by
ϕ
12
(
a
⊗
b
)
=
a
⊗
b
⊗
1
,
{\displaystyle \phi _{12}(a\otimes b)=a\otimes b\otimes 1,}
ϕ
13
(
a
⊗
b
)
=
a
⊗
1
⊗
b
,
{\displaystyle \phi _{13}(a\otimes b)=a\otimes 1\otimes b,}
ϕ
23
(
a
⊗
b
)
=
1
⊗
a
⊗
b
.
{\displaystyle \phi _{23}(a\otimes b)=1\otimes a\otimes b.}
The general form of the Yang–Baxter equation is
R
12
(
u
1
,
u
2
)
R
13
(
u
1
,
u
3
)
R
23
(
u
2
,
u
3
)
=
R
23
(
u
2
,
u
3
)
R
13
(
u
1
,
u
3
)
R
12
(
u
1
,
u
2
)
,
{\displaystyle R_{12}(u_{1},u_{2})\ R_{13}(u_{1},u_{3})\ R_{23}(u_{2},u_{3})=R_{23}(u_{2},u_{3})\ R_{13}(u_{1},u_{3})\ R_{12}(u_{1},u_{2}),}
for all values of
u
1
{\displaystyle u_{1}}
,
u
2
{\displaystyle u_{2}}
and
u
3
{\displaystyle u_{3}}
.
=== Parameter-independent form ===
Let
A
{\displaystyle A}
be a unital associative algebra. The parameter-independent Yang–Baxter equation is an equation for
R
{\displaystyle R}
, an invertible element of the tensor product
A
⊗
A
{\displaystyle A\otimes A}
. The Yang–Baxter equation is
R
12
R
13
R
23
=
R
23
R
13
R
12
,
{\displaystyle R_{12}\ R_{13}\ R_{23}=R_{23}\ R_{13}\ R_{12},}
where
R
12
=
ϕ
12
(
R
)
{\displaystyle R_{12}=\phi _{12}(R)}
,
R
13
=
ϕ
13
(
R
)
{\displaystyle R_{13}=\phi _{13}(R)}
, and
R
23
=
ϕ
23
(
R
)
{\displaystyle R_{23}=\phi _{23}(R)}
.
=== With respect to a basis ===
Often the unital associative algebra is the algebra of endomorphisms of a vector space
V
{\displaystyle V}
over a field
k
{\displaystyle k}
, that is,
A
=
End
(
V
)
{\displaystyle A={\text{End}}(V)}
. With respect to a basis
{
e
i
}
{\displaystyle \{e_{i}\}}
of
V
{\displaystyle V}
, the components of the matrices
R
∈
End
(
V
)
⊗
End
(
V
)
≅
End
(
V
⊗
V
)
{\displaystyle R\in {\text{End}}(V)\otimes {\text{End}}(V)\cong {\text{End}}(V\otimes V)}
are written
R
i
j
k
l
{\displaystyle R_{ij}^{kl}}
, which is the component associated to the map
e
i
⊗
e
j
↦
e
k
⊗
e
l
{\displaystyle e_{i}\otimes e_{j}\mapsto e_{k}\otimes e_{l}}
. Omitting parameter dependence, the component of the Yang–Baxter equation associated to the map
e
a
⊗
e
b
⊗
e
c
↦
e
d
⊗
e
e
⊗
e
f
{\displaystyle e_{a}\otimes e_{b}\otimes e_{c}\mapsto e_{d}\otimes e_{e}\otimes e_{f}}
reads
(
R
12
)
i
j
d
e
(
R
13
)
a
k
i
f
(
R
23
)
b
c
j
k
=
(
R
23
)
j
k
e
f
(
R
13
)
i
c
d
k
(
R
12
)
a
b
i
j
.
{\displaystyle (R_{12})_{ij}^{de}(R_{13})_{ak}^{if}(R_{23})_{bc}^{jk}=(R_{23})_{jk}^{ef}(R_{13})_{ic}^{dk}(R_{12})_{ab}^{ij}.}
== Alternate form and representations of the braid group ==
Let
V
{\displaystyle V}
be a module of
A
{\displaystyle A}
, and
P
i
j
=
ϕ
i
j
(
P
)
{\displaystyle P_{ij}=\phi _{ij}(P)}
. Let
P
:
V
⊗
V
→
V
⊗
V
{\displaystyle P:V\otimes V\to V\otimes V}
be the linear map satisfying
P
(
x
⊗
y
)
=
y
⊗
x
{\displaystyle P(x\otimes y)=y\otimes x}
for all
x
,
y
∈
V
{\displaystyle x,y\in V}
. The Yang–Baxter equation then has the following alternate form in terms of
R
ˇ
(
u
,
u
′
)
=
P
∘
R
(
u
,
u
′
)
{\displaystyle {\check {R}}(u,u')=P\circ R(u,u')}
on
V
⊗
V
{\displaystyle V\otimes V}
.
(
1
⊗
R
ˇ
(
u
1
,
u
2
)
)
(
R
ˇ
(
u
1
,
u
3
)
⊗
1
)
(
1
⊗
R
ˇ
(
u
2
,
u
3
)
)
=
(
R
ˇ
(
u
2
,
u
3
)
⊗
1
)
(
1
⊗
R
ˇ
(
u
1
,
u
3
)
)
(
R
ˇ
(
u
1
,
u
2
)
⊗
1
)
{\displaystyle (\mathbf {1} \otimes {\check {R}}(u_{1},u_{2}))({\check {R}}(u_{1},u_{3})\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}}(u_{2},u_{3}))=({\check {R}}(u_{2},u_{3})\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}}(u_{1},u_{3}))({\check {R}}(u_{1},u_{2})\otimes \mathbf {1} )}
.
Alternatively, we can express it in the same notation as above, defining
R
ˇ
i
j
(
u
,
u
′
)
=
ϕ
i
j
(
R
ˇ
(
u
,
u
′
)
)
{\displaystyle {\check {R}}_{ij}(u,u')=\phi _{ij}({\check {R}}(u,u'))}
, in which case the alternate form is
R
ˇ
23
(
u
1
,
u
2
)
R
ˇ
12
(
u
1
,
u
3
)
R
ˇ
23
(
u
2
,
u
3
)
=
R
ˇ
12
(
u
2
,
u
3
)
R
ˇ
23
(
u
1
,
u
3
)
R
ˇ
12
(
u
1
,
u
2
)
.
{\displaystyle {\check {R}}_{23}(u_{1},u_{2})\ {\check {R}}_{12}(u_{1},u_{3})\ {\check {R}}_{23}(u_{2},u_{3})={\check {R}}_{12}(u_{2},u_{3})\ {\check {R}}_{23}(u_{1},u_{3})\ {\check {R}}_{12}(u_{1},u_{2}).}
In the parameter-independent special case where
R
ˇ
{\displaystyle {\check {R}}}
does not depend on parameters, the equation reduces to
(
1
⊗
R
ˇ
)
(
R
ˇ
⊗
1
)
(
1
⊗
R
ˇ
)
=
(
R
ˇ
⊗
1
)
(
1
⊗
R
ˇ
)
(
R
ˇ
⊗
1
)
{\displaystyle (\mathbf {1} \otimes {\check {R}})({\check {R}}\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}})=({\check {R}}\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}})({\check {R}}\otimes \mathbf {1} )}
,
and (if
R
{\displaystyle R}
is invertible) a representation of the braid group,
B
n
{\displaystyle B_{n}}
, can be constructed on
V
⊗
n
{\displaystyle V^{\otimes n}}
by
σ
i
=
1
⊗
i
−
1
⊗
R
ˇ
⊗
1
⊗
n
−
i
−
1
{\displaystyle \sigma _{i}=1^{\otimes i-1}\otimes {\check {R}}\otimes 1^{\otimes n-i-1}}
for
i
=
1
,
…
,
n
−
1
{\displaystyle i=1,\dots ,n-1}
. This representation can be used to determine quasi-invariants of braids, knots and links.
== Symmetry ==
Solutions to the Yang–Baxter equation are often constrained by requiring the
R
{\displaystyle R}
matrix to be invariant under the action of a Lie group
G
{\displaystyle G}
. For example, in the case
G
=
G
L
(
V
)
{\displaystyle G=GL(V)}
and
R
(
u
,
u
′
)
∈
End
(
V
⊗
V
)
{\displaystyle R(u,u')\in {\text{End}}(V\otimes V)}
, the only
G
{\displaystyle G}
-invariant maps in
End
(
V
⊗
V
)
{\displaystyle {\text{End}}(V\otimes V)}
are the identity
I
{\displaystyle I}
and the permutation map
P
{\displaystyle P}
. The general form of the
R
{\displaystyle R}
-matrix is then
R
(
u
,
u
′
)
=
A
(
u
,
u
′
)
I
+
B
(
u
,
u
′
)
P
{\displaystyle R(u,u')=A(u,u')I+B(u,u')P}
for scalar functions
A
,
B
{\displaystyle A,B}
.
The Yang–Baxter equation is homogeneous in parameter dependence in the sense that if one defines
R
′
(
u
i
,
u
j
)
=
f
(
u
i
,
u
j
)
R
(
u
i
,
u
j
)
{\displaystyle R'(u_{i},u_{j})=f(u_{i},u_{j})R(u_{i},u_{j})}
, where
f
{\displaystyle f}
is a scalar function, then
R
′
{\displaystyle R'}
also satisfies the Yang–Baxter equation.
The argument space itself may have symmetry. For example translation invariance enforces that the dependence on the arguments
(
u
,
u
′
)
{\displaystyle (u,u')}
must be dependent only on the translation-invariant difference
u
−
u
′
{\displaystyle u-u'}
, while scale invariance enforces that
R
{\displaystyle R}
is a function of the scale-invariant ratio
u
/
u
′
{\displaystyle u/u'}
.
== Parametrizations and example solutions ==
A common ansatz for computing solutions is the difference property,
R
(
u
,
u
′
)
=
R
(
u
−
u
′
)
{\displaystyle R(u,u')=R(u-u')}
, where R depends only on a single (additive) parameter. Equivalently, taking logarithms, we may choose the parametrization
R
(
u
,
u
′
)
=
R
(
u
/
u
′
)
{\displaystyle R(u,u')=R(u/u')}
, in which case R is said to depend on a multiplicative parameter. In those cases, we may reduce the YBE to two free parameters in a form that facilitates computations:
R
12
(
u
)
R
13
(
u
+
v
)
R
23
(
v
)
=
R
23
(
v
)
R
13
(
u
+
v
)
R
12
(
u
)
,
{\displaystyle R_{12}(u)\ R_{13}(u+v)\ R_{23}(v)=R_{23}(v)\ R_{13}(u+v)\ R_{12}(u),}
for all values of
u
{\displaystyle u}
and
v
{\displaystyle v}
. For a multiplicative parameter, the Yang–Baxter equation is
R
12
(
u
)
R
13
(
u
v
)
R
23
(
v
)
=
R
23
(
v
)
R
13
(
u
v
)
R
12
(
u
)
,
{\displaystyle R_{12}(u)\ R_{13}(uv)\ R_{23}(v)=R_{23}(v)\ R_{13}(uv)\ R_{12}(u),}
for all values of
u
{\displaystyle u}
and
v
{\displaystyle v}
.
The braided forms read as:
(
1
⊗
R
ˇ
(
u
)
)
(
R
ˇ
(
u
+
v
)
⊗
1
)
(
1
⊗
R
ˇ
(
v
)
)
=
(
R
ˇ
(
v
)
⊗
1
)
(
1
⊗
R
ˇ
(
u
+
v
)
)
(
R
ˇ
(
u
)
⊗
1
)
{\displaystyle (\mathbf {1} \otimes {\check {R}}(u))({\check {R}}(u+v)\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}}(v))=({\check {R}}(v)\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}}(u+v))({\check {R}}(u)\otimes \mathbf {1} )}
(
1
⊗
R
ˇ
(
u
)
)
(
R
ˇ
(
u
v
)
⊗
1
)
(
1
⊗
R
ˇ
(
v
)
)
=
(
R
ˇ
(
v
)
⊗
1
)
(
1
⊗
R
ˇ
(
u
v
)
)
(
R
ˇ
(
u
)
⊗
1
)
{\displaystyle (\mathbf {1} \otimes {\check {R}}(u))({\check {R}}(uv)\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}}(v))=({\check {R}}(v)\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}}(uv))({\check {R}}(u)\otimes \mathbf {1} )}
In some cases, the determinant of
R
(
u
)
{\displaystyle R(u)}
can vanish at specific values of the spectral parameter
u
=
u
0
{\displaystyle u=u_{0}}
. Some
R
{\displaystyle R}
matrices turn into a one dimensional projector at
u
=
u
0
{\displaystyle u=u_{0}}
. In this case a quantum determinant can be defined .
=== Example solutions of the parameter-dependent YBE ===
A particularly simple class of parameter-dependent solutions can be obtained from solutions of the parameter-independent YBE satisfying
R
ˇ
2
=
1
{\displaystyle {\check {R}}^{2}=\mathbf {1} }
, where the corresponding braid group representation is a permutation group representation. In this case,
R
ˇ
(
u
)
=
1
+
u
R
ˇ
{\displaystyle {\check {R}}(u)=\mathbf {1} +u{\check {R}}}
(equivalently,
R
(
u
)
=
P
+
u
P
∘
R
ˇ
{\displaystyle R(u)=P+uP\circ {\check {R}}}
) is a solution of the (additive) parameter-dependent YBE. In the case where
R
ˇ
=
P
{\displaystyle {\check {R}}=P}
and
R
(
u
)
=
P
+
u
1
{\displaystyle R(u)=P+u\mathbf {1} }
, this gives the scattering matrix of the Heisenberg XXX spin chain.
The
R
{\displaystyle R}
-matrices of the evaluation modules of the quantum group
U
q
(
s
l
^
(
2
)
)
{\displaystyle U_{q}({\widehat {sl}}(2))}
are given explicitly by the matrix
R
ˇ
(
z
)
=
(
q
z
−
q
−
1
z
−
1
q
−
q
−
1
z
−
z
−
1
z
−
z
−
1
q
−
q
−
1
q
z
−
q
−
1
z
−
1
)
.
{\displaystyle {\check {R}}(z)={\begin{pmatrix}qz-q^{-1}z^{-1}&&&\\&q-q^{-1}&z-z^{-1}&\\&z-z^{-1}&q-q^{-1}&\\&&&qz-q^{-1}z^{-1}\end{pmatrix}}.}
Then the parametrized Yang-Baxter equation (in braided form) with the multiplicative parameter is satisfied:
(
1
⊗
R
ˇ
(
z
)
)
(
R
ˇ
(
z
z
′
)
⊗
1
)
(
1
⊗
R
ˇ
(
z
′
)
)
=
(
R
ˇ
(
z
′
)
⊗
1
)
(
1
⊗
R
ˇ
(
z
z
′
)
)
(
R
ˇ
(
z
)
⊗
1
)
{\displaystyle (\mathbf {1} \otimes {\check {R}}(z))({\check {R}}(zz')\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}}(z'))=({\check {R}}(z')\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}}(zz'))({\check {R}}(z)\otimes \mathbf {1} )}
=== Classification of solutions ===
There are broadly speaking three classes of solutions: rational, trigonometric and elliptic. These are related to quantum groups known as the Yangian, affine quantum groups and elliptic algebras respectively.
== Set-theoretic Yang–Baxter equation ==
Set-theoretic solutions were studied by Drinfeld. In this case, there is an
R
{\displaystyle R}
-matrix invariant basis
X
{\displaystyle X}
for the vector space
V
{\displaystyle V}
in the sense that the
R
{\displaystyle R}
-matrix maps the induced basis of
V
⊗
V
{\displaystyle V\otimes V}
to itself. This then induces a map
r
:
X
×
X
→
X
×
X
{\displaystyle r:X\times X\rightarrow X\times X}
given by restriction of the
R
{\displaystyle R}
-matrix to the basis. The set-theoretic Yang–Baxter equation is then defined using the 'twisted' alternate form above, asserting
(
i
d
×
r
)
(
r
×
i
d
)
(
i
d
×
r
)
=
(
r
×
i
d
)
(
i
d
×
r
)
(
r
×
i
d
)
{\displaystyle (id\times r)(r\times id)(id\times r)=(r\times id)(id\times r)(r\times id)}
as maps on
X
×
X
×
X
{\displaystyle X\times X\times X}
. The equation can then be considered purely as an equation in the category of sets.
=== Examples ===
R
=
i
d
{\displaystyle R=id}
.
R
=
τ
{\displaystyle R=\tau }
where
τ
(
u
⊗
v
)
=
v
⊗
u
{\displaystyle \tau (u\otimes v)=v\otimes u}
, the transposition map.
If
(
X
,
◃
)
{\displaystyle (X,\triangleleft )}
is a (right) shelf, then
r
(
x
,
y
)
=
(
y
,
x
◃
y
)
{\displaystyle r(x,y)=(y,x\triangleleft y)}
is a set-theoretic solution to the YBE.
== Classical Yang–Baxter equation ==
Solutions to the classical YBE were studied and to some extent classified by Belavin and Drinfeld. Given a 'classical
r
{\displaystyle r}
-matrix'
r
:
V
⊗
V
→
V
⊗
V
{\displaystyle r:V\otimes V\rightarrow V\otimes V}
, which may also depend on a pair of arguments
(
u
,
v
)
{\displaystyle (u,v)}
, the classical YBE is (suppressing parameters)
[
r
12
,
r
13
]
+
[
r
12
,
r
23
]
+
[
r
13
,
r
23
]
=
0.
{\displaystyle [r_{12},r_{13}]+[r_{12},r_{23}]+[r_{13},r_{23}]=0.}
This is quadratic in the
r
{\displaystyle r}
-matrix, unlike the usual quantum YBE which is cubic in
R
{\displaystyle R}
.
This equation emerges from so called quasi-classical solutions to the quantum YBE, in which the
R
{\displaystyle R}
-matrix admits an asymptotic expansion in terms of an expansion parameter
ℏ
,
{\displaystyle \hbar ,}
R
ℏ
=
I
+
ℏ
r
+
O
(
ℏ
2
)
.
{\displaystyle R_{\hbar }=I+\hbar r+{\mathcal {O}}(\hbar ^{2}).}
The classical YBE then comes from reading off the
ℏ
2
{\displaystyle \hbar ^{2}}
coefficient of the quantum YBE (and the equation trivially holds at orders
ℏ
0
,
ℏ
{\displaystyle \hbar ^{0},\hbar }
).
== See also ==
Lie bialgebra
Yangian
Reidemeister move
Quasitriangular Hopf algebra
Yang–Baxter operator
== References ==
H.-D. Doebner, J.-D. Hennig, eds, Quantum groups, Proceedings of the 8th International Workshop on Mathematical Physics, Arnold Sommerfeld Institute, Clausthal, FRG, 1989, Springer-Verlag Berlin, ISBN 3-540-53503-9.
Vyjayanthi Chari and Andrew Pressley, A Guide to Quantum Groups, (1994), Cambridge University Press, Cambridge ISBN 0-521-55884-0.
Jacques H.H. Perk and Helen Au-Yang, "Yang–Baxter Equations", (2006), arXiv:math-ph/0606053 .
== External links ==
"Yang-Baxter equation", Encyclopedia of Mathematics, EMS Press, 2001 [1994] | Wikipedia/Yang–Baxter_equation |
Hydrogen is a chemical element; it has symbol H and atomic number 1. It is the lightest and most abundant chemical element in the universe, constituting about 75% of all normal matter. Under standard conditions, hydrogen is a gas of diatomic molecules with the formula H2, called dihydrogen, or sometimes hydrogen gas, molecular hydrogen, or simply hydrogen. Dihydrogen is colorless, odorless, non-toxic, and highly combustible. Stars, including the Sun, mainly consist of hydrogen in a plasma state, while on Earth, hydrogen is found as the gas H2 (dihydrogen) and in molecular forms, such as in water and organic compounds. The most common isotope of hydrogen (1H) consists of one proton, one electron, and no neutrons.
Hydrogen gas was first produced artificially in the 17th century by the reaction of acids with metals. Henry Cavendish, in 1766–1781, identified hydrogen gas as a distinct substance and discovered its property of producing water when burned; hence its name means 'water-former' in Greek. Understanding the colors of light absorbed and emitted by hydrogen was a crucial part of developing quantum mechanics.
Hydrogen, typically nonmetallic except under extreme pressure, readily forms covalent bonds with most nonmetals, contributing to the formation of compounds like water and various organic substances. Its role is crucial in acid-base reactions, which mainly involve proton exchange among soluble molecules. In ionic compounds, hydrogen can take the form of either a negatively charged anion, where it is known as hydride, or as a positively charged cation, H+, called a proton. Although tightly bonded to water molecules, protons strongly affect the behavior of aqueous solutions, as reflected in the importance of pH. Hydride, on the other hand, is rarely observed because it tends to deprotonate solvents, yielding H2.
In the early universe, neutral hydrogen atoms formed about 370,000 years after the Big Bang as the universe expanded and plasma had cooled enough for electrons to remain bound to protons. Once stars formed most of the atoms in the intergalactic medium re-ionized.
Nearly all hydrogen production is done by transforming fossil fuels, particularly steam reforming of natural gas. It can also be produced from water or saline by electrolysis, but this process is more expensive. Its main industrial uses include fossil fuel processing and ammonia production for fertilizer. Emerging uses for hydrogen include the use of fuel cells to generate electricity.
== Properties ==
=== Atomic hydrogen ===
==== Electron energy levels ====
The ground state energy level of the electron in a hydrogen atom is −13.6 eV, equivalent to an ultraviolet photon of roughly 91 nm wavelength. The energy levels of hydrogen are referred to by consecutive quantum numbers, with
n
=
1
{\displaystyle n=1}
being the ground state. The hydrogen spectral series corresponds to emission of light due to transitions from higher to lower energy levels.: 105 Each energy level is further split by spin interactions between the electron and proton into 4 hyperfine levels.
High precision values for the hydrogen atom energy levels are required for definitions of physical constants. Quantum calculations have identified 9 contributions to the energy levels. The eigenvalue from the Dirac equation is the largest contribution. Other terms include relativistic recoil, the self-energy, and the vacuum polarization terms.
==== Isotopes ====
Hydrogen has three naturally occurring isotopes, denoted 1H, 2H and 3H. Other, highly unstable nuclei (4H to 7H) have been synthesized in the laboratory but not observed in nature.
1H is the most common hydrogen isotope, with an abundance of >99.98%. Because the nucleus of this isotope consists of only a single proton, it is given the descriptive but rarely used formal name protium. It is the only stable isotope with no neutrons; see diproton for a discussion of why others do not exist.
2H, the other stable hydrogen isotope, is known as deuterium and contains one proton and one neutron in the nucleus. Nearly all deuterium nuclei in the universe is thought to have been produced at the time of the Big Bang, and has endured since then.: 24.2 Deuterium is not radioactive, and is not a significant toxicity hazard. Water enriched in molecules that include deuterium instead of normal hydrogen is called heavy water. Deuterium and its compounds are used as a non-radioactive label in chemical experiments and in solvents for 1H-NMR spectroscopy. Heavy water is used as a neutron moderator and coolant for nuclear reactors. Deuterium is also a potential fuel for commercial nuclear fusion.
3H is known as tritium and contains one proton and two neutrons in its nucleus. It is radioactive, decaying into helium-3 through beta decay with a half-life of 12.32 years. It is radioactive enough to be used in luminous paint to enhance the visibility of data displays, such as for painting the hands and dial-markers of watches. The watch glass prevents the small amount of radiation from escaping the case. Small amounts of tritium are produced naturally by cosmic rays striking atmospheric gases; tritium has also been released in nuclear weapons tests. It is used in nuclear fusion, as a tracer in isotope geochemistry, and in specialized self-powered lighting devices. Tritium has also been used in chemical and biological labeling experiments as a radiolabel.
Unique among the elements, distinct names are assigned to its isotopes in common use. During the early study of radioactivity, heavy radioisotopes were given their own names, but these are mostly no longer used. The symbols D and T (instead of 2H and 3H) are sometimes used for deuterium and tritium, but the symbol P was already used for phosphorus and thus was not available for protium. In its nomenclatural guidelines, the International Union of Pure and Applied Chemistry (IUPAC) allows any of D, T, 2H, and 3H to be used, though 2H and 3H are preferred.
Antihydrogen (H) is the antimatter counterpart to hydrogen. It consists of an antiproton with a positron. Antihydrogen is the only type of antimatter atom to have been produced as of 2015. The exotic atom muonium (symbol Mu), composed of an antimuon and an electron, is analogous hydrogen and IUPAC nomenclature incorporates such hypothetical compounds as muonium chloride (MuCl) and sodium muonide (NaMu), analogous to hydrogen chloride and sodium hydride respectively.
=== Dihydrogen ===
Under standard conditions, hydrogen is a gas of diatomic molecules with the formula H2, officially called "dihydrogen",: 308 but also called "molecular hydrogen", or simply hydrogen. Dihydrogen is a colorless, odorless, flammable gas.
==== Combustion ====
Hydrogen gas is highly flammable, reacting with oxygen in air, to produce liquid water:
2 H2(g) + O2(g) → 2 H2O(l)
The amount of heat released per mole of hydrogen is −286 kJ or 141.865 MJ for a kilogram mass.
Hydrogen gas forms explosive mixtures with air in concentrations from 4–74% and with chlorine at 5–95%. The hydrogen autoignition temperature, the temperature of spontaneous ignition in air, is 500 °C (932 °F). In a high-pressure hydrogen leak, the shock wave from the leak itself can heat air to the autoignition temperature, leading to flaming and possibly explosion.
Hydrogen flames emit faint blue and ultraviolet light. Flame detectors are used to detect hydrogen fires as they are nearly invisible to the naked eye in daylight.
==== Spin isomers ====
Molecular H2 exists as two nuclear isomers that differ in the spin states of their nuclei. In the orthohydrogen form, the spins of the two nuclei are parallel, forming a spin triplet state having a total molecular spin
S
=
1
{\displaystyle S=1}
; in the parahydrogen form the spins are antiparallel and form a spin singlet state having spin
S
=
0
{\displaystyle S=0}
. The equilibrium ratio of ortho- to para-hydrogen depends on temperature. At room temperature or warmer, equilibrium hydrogen gas contains about 25% of the para form and 75% of the ortho form. The ortho form is an excited state, having higher energy than the para form by 1.455 kJ/mol, and it converts to the para form over the course of several minutes when cooled to low temperature. The thermal properties of these isomers differ because each has distinct rotational quantum states.
The ortho-to-para ratio in H2 is an important consideration in the liquefaction and storage of liquid hydrogen: the conversion from ortho to para is exothermic and produces sufficient heat to evaporate most of the liquid if not converted first to parahydrogen during the cooling process. Catalysts for the ortho-para interconversion, such as ferric oxide and activated carbon compounds, are used during hydrogen cooling to avoid this loss of liquid.
==== Phases ====
Liquid hydrogen can exist at temperatures below hydrogen's critical point of 33 K. However, for it to be in a fully liquid state at atmospheric pressure, H2 needs to be cooled to 20.28 K (−252.87 °C; −423.17 °F). Hydrogen was liquefied by James Dewar in 1898 by using regenerative cooling and his invention, the vacuum flask.
Liquid hydrogen becomes solid hydrogen at standard pressure below hydrogen's melting point of 14.01 K (−259.14 °C; −434.45 °F). Distinct solid phases exist, known as Phase I through Phase V, each exhibiting a characteristic molecular arrangement. Liquid and solid phases can exist in combination at the triple point, a substance known as slush hydrogen.
Metallic hydrogen, a phase obtained at extremely high pressures (in excess of 400 gigapascals (3,900,000 atm; 58,000,000 psi)), is an electrical conductor. It is believed to exist deep within giant planets like Jupiter.
When ionized, hydrogen becomes a plasma. This is the form in which hydrogen exists within stars.
==== Thermal and physical properties ====
== History ==
=== 18th century ===
In 1671, Irish scientist Robert Boyle discovered and described the reaction between iron filings and dilute acids, which results in the production of hydrogen gas.
Boyle did not note that the gas was inflammable, but hydrogen would play a key role in overturning the phlogiston theory of combustion.
In 1766, Henry Cavendish was the first to recognize hydrogen gas as a discrete substance, by naming the gas from a metal-acid reaction "inflammable air". He speculated that "inflammable air" was in fact identical to the hypothetical substance "phlogiston" and further finding in 1781 that the gas produces water when burned. He is usually given credit for the discovery of hydrogen as an element.
In 1783, Antoine Lavoisier identified the element that came to be known as hydrogen when he and Laplace reproduced Cavendish's finding that water is produced when hydrogen is burned. Lavoisier produced hydrogen for his experiments on mass conservation by treating metallic iron with a stream of H2O through an incandescent iron tube heated in a fire. Anaerobic oxidation of iron by the protons of water at high temperature can be schematically represented by the set of following reactions:
Fe + H2O → FeO + H2
2Fe + 3 H2O → Fe2O3 + 3 H2
3Fe + 4 H2O → Fe3O4 + 4 H2
Many metals react similarly with water leading to the production of hydrogen. In some situations, this H2-producing process is problematic as is the case of zirconium cladding on nuclear fuel rods.
=== 19th century ===
By 1806 hydrogen was used to fill balloons.
François Isaac de Rivaz built the first de Rivaz engine, an internal combustion engine powered by a mixture of hydrogen and oxygen in 1806. Edward Daniel Clarke invented the hydrogen gas blowpipe in 1819. The Döbereiner's lamp and limelight were invented in 1823. Hydrogen was liquefied for the first time by James Dewar in 1898 by using regenerative cooling and his invention, the vacuum flask. He produced solid hydrogen the next year.
One of the first quantum effects to be explicitly noticed (but not understood at the time) was James Clerk Maxwell's observation that the specific heat capacity of H2 unaccountably departs from that of a diatomic gas below room temperature and begins to increasingly resemble that of a monatomic gas at cryogenic temperatures. According to quantum theory, this behavior arises from the spacing of the (quantized) rotational energy levels, which are particularly wide-spaced in H2 because of its low mass. These widely spaced levels inhibit equal partition of heat energy into rotational motion in hydrogen at low temperatures. Diatomic gases composed of heavier atoms do not have such widely spaced levels and do not exhibit the same effect.
=== 20th century ===
The existence of the hydride anion was suggested by Gilbert N. Lewis in 1916 for group 1 and 2 salt-like compounds. In 1920, Moers electrolyzed molten lithium hydride (LiH), producing a stoichiometric quantity of hydrogen at the anode.
Because of its simple atomic structure, consisting only of a proton and an electron, the hydrogen atom, together with the spectrum of light produced from it or absorbed by it, has been central to the development of the theory of atomic structure. The energy levels of hydrogen can be calculated fairly accurately using the Bohr model of the atom, in which the electron "orbits" the proton, like how Earth orbits the Sun. However, the electron and proton are held together by electrostatic attraction, while planets and celestial objects are held by gravity. Due to the discretization of angular momentum postulated in early quantum mechanics by Bohr, the electron in the Bohr model can only occupy certain allowed distances from the proton, and therefore only certain allowed energies.
Hydrogen's unique position as the only neutral atom for which the Schrödinger equation can be directly solved, has significantly contributed to the understanding of quantum mechanics through the exploration of its energetics. Furthermore, study of the corresponding simplicity of the hydrogen molecule and the corresponding cation H+2 brought understanding of the nature of the chemical bond, which followed shortly after the quantum mechanical treatment of the hydrogen atom had been developed in the mid-1920s.
==== Hydrogen-lifted airship ====
Because H2 is only 7% the density of air, it was once widely used as a lifting gas in balloons and airships. The first hydrogen-filled balloon was invented by Jacques Charles in 1783. Hydrogen provided the lift for the first reliable form of air-travel following the 1852 invention of the first hydrogen-lifted airship by Henri Giffard. German count Ferdinand von Zeppelin promoted the idea of rigid airships lifted by hydrogen that later were called Zeppelins; the first of which had its maiden flight in 1900. Regularly scheduled flights started in 1910 and by the outbreak of World War I in August 1914, they had carried 35,000 passengers without a serious incident. Hydrogen-lifted airships in the form of blimps were used as observation platforms and bombers during the War II, especially on the US Eastern seaboard.
The first non-stop transatlantic crossing was made by the British airship R34 in 1919 and regular passenger service resumed in the 1920s. Hydrogen was used in the Hindenburg airship, which caught fire over New Jersey on 6 May 1937. The hydrogen that filled the airship was ignited, possibly by static electricity, and burst into flames. Following this Hindenburg disaster, commercial hydrogen airship travel ceased. Hydrogen is still used, in preference to non-flammable but more expensive helium, as a lifting gas for weather balloons.
==== Deuterium and tritium ====
Deuterium was discovered in December 1931 by Harold Urey, and tritium was prepared in 1934 by Ernest Rutherford, Mark Oliphant, and Paul Harteck. Heavy water, which consists of deuterium in the place of regular hydrogen, was discovered by Urey's group in 1932.
== Chemistry ==
=== Reactions of H2 ===
H2 is relatively unreactive. The thermodynamic basis of this low reactivity is the very strong H–H bond, with a bond dissociation energy of 435.7 kJ/mol. It does form coordination complexes called dihydrogen complexes. These species provide insights into the early steps in the interactions of hydrogen with metal catalysts. According to neutron diffraction, the metal and two H atoms form a triangle in these complexes. The H-H bond remains intact but is elongated. They are acidic.
Although exotic on Earth, the H+3 ion is common in the universe. It is a triangular species, like the aforementioned dihydrogen complexes. It is known as protonated molecular hydrogen or the trihydrogen cation.
Hydrogen reacts with chlorine to produce HCl and with bromine to produce HBr by a chain reaction. The reaction requires initiation. For example in the case of Br2, the diatomic molecule is broken into atoms, Br2 + (UV light) → 2Br•. Propagating reactions consume hydrogen molecules and produce HBr, as well as Br and H atoms:
Br• + H2 → HBr + H
H + Br2 → HBr +Br
Finally the terminating reaction:
H + HBr → H2 + Br•
2Br• → Br2.
consumes the remaining atoms.: 289
The addition of H2 to unsaturated organic compounds, such as alkenes and alkynes, is called hydrogenation. Even if the reaction is energetically favorable, it does not take place even at higher temperatures. In the presence of a catalyst like finely divided platinum or nickel, the reaction proceeds at room temperature.: 477
=== Hydrogen-containing compounds ===
Hydrogen can exist in both +1 and −1 oxidation states, forming compounds through ionic and covalent bonding. It is a part of a wide range of substances, including water, hydrocarbons, and numerous other organic compounds. The H+ ion—commonly referred to as a proton due to its single proton and absence of electrons—is central to acid–base chemistry, although the proton does not move freely. In the Brønsted–Lowry framework, acids are defined by their ability to donate H+ ions to bases.
Hydrogen forms a vast variety of compounds with carbon known as hydrocarbons, and an even greater diversity with other elements (heteroatoms), giving rise to the broad class of organic compounds often associated with living organisms.
Hydrogen compounds with hydrogen in the oxidation state −1 are known as hydrides, which are usually formed between hydrogen and metals. The hydrides can be ionic (aka saline), covalent, nor metallic. With heating, H2 reacts efficiently with the alkali and alkaline earth metals to give the ionic hydrides of the formula MH and MH2, respectively. These salt-like crystalline compounds have high melting points and all react with water to liberate hydrogen. Covalent hydrides are include boranes and polymeric aluminium hydride. Transition metals form metal hydrides via continuous dissolution of hydrogen into the metal. A well known hydride is lithium aluminium hydride, the [AlH4]− anion carries hydridic centers firmly attached to the Al(III). Perhaps the most extensive series of hydrides are the boranes, compounds consisting only of boron and hydrogen.
Hydrides can bond to these electropositive elements not only as a terminal ligand but also as bridging ligands. In diborane (B2H6), four H's are terminal and two bridge between the two B atoms.
=== Hydrogen bonding ===
When bonded to a more electronegative element, particularly fluorine, oxygen, or nitrogen, hydrogen can participate in a form of medium-strength noncovalent bonding with another electronegative element with a lone pair like oxygen or nitrogen, a phenomenon called hydrogen bonding that is critical to the stability of many biological molecules.: 375 Hydrogen bonding alters molecule structures, viscosity, solubility, as well as melting and boiling points even protein folding dynamics.
=== Protons and acids ===
In water, hydrogen bonding plays an important role in reaction thermodynamics. A hydrogen bond can shift over to proton transfer.
Under the Brønsted–Lowry acid–base theory, acids are proton donors, while bases are proton acceptors.: 28
A bare proton, H+ essentially cannot exist in anything other than a vacuum. Otherwise it attaches to other atoms, ions, or molecules. Even species as inert as methane can be protonated. The term 'proton' is used loosely and metaphorically to refer to refer to solvated H+" without any implication that any single protons exist freely as a species. To avoid the implication of the naked proton in solution, acidic aqueous solutions are sometimes considered to contain the "hydronium ion" ([H3O]+) or still more accurately, [H9O4]+. Other oxonium ions are found when water is in acidic solution with other solvents.
The concentration of these solvated protons determines the pH of a solution, a logarithmic scale that reflects its acidity or basicity. Lower pH values indicate higher concentrations of hydronium ions, corresponding to more acidic conditions.
== Occurrence ==
=== Cosmic ===
Hydrogen, as atomic H, is the most abundant chemical element in the universe, making up 75% of normal matter by mass and >90% by number of atoms. In the early universe, the protons formed in the first second after the Big Bang; neutral hydrogen atoms formed about 370,000 years later during the recombination epoch as the universe expanded and plasma had cooled enough for electrons to remain bound to protons.
In astrophysics, neutral hydrogen in the interstellar medium is called H I and ionized hydrogen is called H II. Radiation from stars ionizes H I to H II, creating spheres of ionized H II around stars. In the chronology of the universe neutral hydrogen dominated until the birth of stars during the era of reionization led to bubbles of ionized hydrogen that grew and merged over 500 million of years.
They are the source of the 21-cm hydrogen line at 1420 MHz that is detected in order to probe primordial hydrogen. The large amount of neutral hydrogen found in the damped Lyman-alpha systems is thought to dominate the cosmological baryonic density of the universe up to a redshift of z = 4.
Hydrogen is found in great abundance in stars and gas giant planets. Molecular clouds of H2 are associated with star formation. Hydrogen plays a vital role in powering stars through the proton-proton reaction in lower-mass stars, and through the CNO cycle of nuclear fusion in case of stars more massive than the Sun.
A molecular form called protonated molecular hydrogen (H+3) is found in the interstellar medium, where it is generated by ionization of molecular hydrogen from cosmic rays. This ion has also been observed in the upper atmosphere of Jupiter. The ion is long-lived in outer space due to the low temperature and density. H+3 is one of the most abundant ions in the universe, and it plays a notable role in the chemistry of the interstellar medium. Neutral triatomic hydrogen H3 can exist only in an excited form and is unstable.
=== Terrestrial ===
Hydrogen is the third most abundant element on the Earth's surface, mostly in the form of chemical compounds such as hydrocarbons and water. Elemental hydrogen is normally in the form of a gas, H2. It is present in a very low concentration in Earth's atmosphere (around 0.53 ppm on a molar basis) because of its light weight, which enables it to escape the atmosphere more rapidly than heavier gases. Despite its low concentration in our atmosphere, terrestrial hydrogen is sufficiently abundant to support the metabolism of several bacteria.
Large underground deposits of hydrogen gas have been discovered in several countries including Mali, France and Australia. As of 2024, it is uncertain how much underground hydrogen can be extracted economically.
== Production and storage ==
=== Industrial routes ===
Nearly all of the world's current supply of hydrogen gas (H2) is created from fossil fuels.: 1 Many methods exist for producing H2, but three dominate commercially: steam reforming often coupled to water-gas shift, partial oxidation of hydrocarbons, and water electrolysis.
==== Steam reforming ====
Hydrogen is mainly produced by steam methane reforming (SMR), the reaction of water and methane. Thus, at high temperature (1000–1400 K, 700–1100 °C or 1300–2000 °F), steam (water vapor) reacts with methane to yield carbon monoxide and H2.
CH4 + H2O → CO + 3 H2
Producing one tonne of hydrogen through this process emits 6.6–9.3 tonnes of carbon dioxide. The production of natural gas feedstock also produces emissions such as vented and fugitive methane, which further contributes to the overall carbon footprint of hydrogen.
This reaction is favored at low pressures, Nonetheless, conducted at high pressures (2.0 MPa, 20 atm or 600 inHg) because high-pressure H2 is the most marketable product, and pressure swing adsorption (PSA) purification systems work better at higher pressures. The product mixture is known as "synthesis gas" because it is often used directly for the production of methanol and many other compounds. Hydrocarbons other than methane can be used to produce synthesis gas with varying product ratios. One of the many complications to this highly optimized technology is the formation of coke or carbon:
CH4 → C + 2 H2
Therefore, steam reforming typically employs an excess of H2O. Additional hydrogen can be recovered from the steam by using carbon monoxide through the water gas shift reaction (WGS). This process requires an iron oxide catalyst:
CO + H2O → CO2 + H2
Hydrogen is sometimes produced and consumed in the same industrial process, without being separated. In the Haber process for ammonia production, hydrogen is generated from natural gas.
==== Partial oxidation of hydrocarbons ====
Other methods for CO and H2 production include partial oxidation of hydrocarbons:
2 CH4 + O2 → 2 CO + 4 H2
Although less important commercially, coal can serve as a prelude to the shift reaction above:
C + H2O → CO + H2
Olefin production units may produce substantial quantities of byproduct hydrogen particularly from cracking light feedstocks like ethane or propane.
==== Water electrolysis ====
Electrolysis of water is a conceptually simple method of producing hydrogen.
2 H2O(l) → 2 H2(g) + O2(g)
Commercial electrolyzers use nickel-based catalysts in strongly alkaline solution. Platinum is a better catalyst but is expensive. The hydrogen created through electrolysis using renewable energy is commonly referred to as "green hydrogen".
Electrolysis of brine to yield chlorine also produces high purity hydrogen as a co-product, which is used for a variety of transformations such as hydrogenations.
The electrolysis process is more expensive than producing hydrogen from methane without carbon capture and storage.
Innovation in hydrogen electrolyzers could make large-scale production of hydrogen from electricity more cost-competitive.
==== Methane pyrolysis ====
Hydrogen can be produced by pyrolysis of natural gas (methane), producing hydrogen gas and solid carbon with the aid a catalyst and 74 kJ/mol input heat:
CH4(g) → C(s) + 2 H2(g) (ΔH° = 74 kJ/mol)
The carbon may be sold as a manufacturing feedstock or fuel, or landfilled.
This route could have a lower carbon footprint than existing hydrogen production processes, but mechanisms for removing the carbon and preventing it from reacting with the catalyst remain obstacles for industrial scale use.: 17
==== Thermochemical ====
Water splitting is the process by which water is decomposed into its components. Relevant to the biological scenario is this simple equation:
2 H2O → 4 H+ + O2 + 4e−
The reaction occurs in the light reactions in all photosynthetic organisms. A few organisms, including the alga Chlamydomonas reinhardtii and cyanobacteria, have evolved a second step in the dark reactions in which protons and electrons are reduced to form H2 gas by specialized hydrogenases in the chloroplast.
Efforts have been undertaken to genetically modify cyanobacterial hydrogenases to more efficiently generate H2 gas even in the presence of oxygen. Efforts have also been undertaken with genetically modified alga in a bioreactor.
Relevant to the thermal water-splitting scenario is this simple equation:
2 H2O → 2 H2 + O2
More than 200 thermochemical cycles can be used for water splitting. Many of these cycles such as the iron oxide cycle, cerium(IV) oxide–cerium(III) oxide cycle, zinc zinc-oxide cycle, sulfur-iodine cycle, copper-chlorine cycle and hybrid sulfur cycle have been evaluated for their commercial potential to produce hydrogen and oxygen from water and heat without using electricity. A number of labs (including in France, Germany, Greece, Japan, and the United States) are developing thermochemical methods to produce hydrogen from solar energy and water.
=== Natural routes ===
==== Biohydrogen ====
H2 is produced by enzymes called hydrogenases. This process allows the host organism to use fermentation as a source of energy. These same enzymes also can oxidize H2, such that the host organisms can subsist by reducing oxidized substrates using electrons extracted from H2.
The hydrogenase enzyme feature iron or nickel-iron centers at their active sites. The natural cycle of hydrogen production and consumption by organisms is called the hydrogen cycle.
Some bacteria such as Mycobacterium smegmatis can use the small amount of hydrogen in the atmosphere as a source of energy when other sources are lacking. Their hydrogenase are designed with small channels that exclude oxygen and so permits the reaction to occur even though the hydrogen concentration is very low and the oxygen concentration is as in normal air.
Confirming the existence of hydrogenases in the human gut, H2 occurs in human breath. The concentration in the breath of fasting people at rest is typically less than 5 parts per million (ppm) but can be 50 ppm when people with intestinal disorders consume molecules they cannot absorb during diagnostic hydrogen breath tests.
==== Serpentinization ====
Serpentinization is a geological mechanism that produce highly reducing conditions. Under these conditions, water is capable of oxidizing ferrous (Fe2+) ions in fayalite, generating hydrogen gas:
Fe2SiO4 + H2O → 2 Fe3O4 + SiO2 + H2
Closely related to this geological process is the Schikorr reaction:
3 Fe(OH)2 → Fe3O4 + 2 H2O + H2
This process also is relevant to the corrosion of iron and steel in oxygen-free groundwater and in reducing soils below the water table.
=== Laboratory syntheses ===
H2 is produced in laboratory settings, such as in the small-scale electrolysis of water using metal electrodes and water containing an electrolyte, which liberates hydrogen gas at the cathode:
2H+(aq) + 2e− → H2(g)
Hydrogen is also often a by-product of other reactions. Many metals react with water to produce H2, but the rate of hydrogen evolution depends on the metal, the pH, and the presence of alloying agents. Most often, hydrogen evolution is induced by acids. The alkali and alkaline earth metals, aluminium, zinc, manganese, and iron react readily with aqueous acids.
Zn + 2 H+ → Zn2+ + H2
Many metals, such as aluminium, are slow to react with water because they form passivated oxide coatings of oxides. An alloy of aluminium and gallium, however, does react with water. At high pH, aluminium can produce H2:
2 Al + 6 H2O + 2 OH− → 2 [Al(OH)4]− + 3 H2
=== Storage ===
If H2 is to be used as an energy source, its storage is important. It dissolves only poorly in solvents. For example, at room temperature and 0.1 Mpascal, ca. 0.05 moles dissolves in one kilogram of diethyl ether. The H2 can be stored in compressed form, although compressing costs energy. Liquifaction is impractical given its low critical temperature. In contrast, ammonia and many hydrocarbons can be liquified at room temperature under pressure. For these reasons, hydrogen carriers - materials that reversibly bind H2 - have attracted much attention. The key question is then the weight percent of H2-equivalents within the carrier material. For example, hydrogen can be reversibly absorbed into many rare earth and transition metals and is soluble in both nanocrystalline and amorphous metals. Hydrogen solubility in metals is influenced by local distortions or impurities in the crystal lattice. These properties may be useful when hydrogen is purified by passage through hot palladium disks, but the gas's high solubility is also a metallurgical problem, contributing to the embrittlement of many metals, complicating the design of pipelines and storage tanks.
The most problematic aspect of metal hydrides for storage is their modest H2 content, often on the order of 1%. For this reason, there is interest in storage of H2 in compounds of low molecular weight. For example, ammonia borane (H3N−BH3) contains 19.8 weight percent of H2. The problem with this material is that after release of H2, the resulting boron nitride does not re-add H2, i.e. ammonia borane is an irreversible hydrogen carrier. More attractive, somewhat ironically, are hydrocarbons such as tetrahydroquinoline, which reversibly release some H2 when heated in the presence of a catalyst:
C9H10NH ⇌ C9H7N + 2H2
== Applications ==
=== Petrochemical industry ===
Large quantities of H2 are used in the "upgrading" of fossil fuels. Key consumers of H2 include hydrodesulfurization, and hydrocracking. Many of these reactions can be classified as hydrogenolysis, i.e., the cleavage of bonds by hydrogen. Illustrative is the separation of sulfur from liquid fossil fuels:
R2S + 2 H2 → H2S + 2 RH
=== Hydrogenation ===
Hydrogenation, the addition of H2 to various substrates, is done on a large scale. Hydrogenation of N2 produces ammonia by the Haber process:
N2 + 3 H2 → 2 NH3
This process consumes a few percent of the energy budget in the entire industry and is the biggest consumer of hydrogen. The resulting ammonia is used in fertilizers critical to the supply of protein consumed by humans. Hydrogenation is also used to convert unsaturated fats and oils to saturated fats and oils. The major application is the production of margarine. Methanol is produced by hydrogenation of carbon dioxide; the mixture of hydrogen and carbon dioxide used for this process is known as syngas. It is similarly the source of hydrogen in the manufacture of hydrochloric acid. H2 is also used as a reducing agent for the conversion of some ores to the metals.
=== Fuel ===
The potential for using hydrogen (H2) as a fuel has been widely discussed. Hydrogen can be used in fuel cells to produce electricity, or burned to generate heat. When hydrogen is consumed in fuel cells, the only emission at the point of use is water vapor. When burned, hydrogen produces relatively little pollution at the point of combustion, but can lead to thermal formation of harmful nitrogen oxides.
If hydrogen is produced with low or zero greenhouse gas emissions (green hydrogen), it can play a significant role in decarbonizing energy systems where there are challenges and limitations to replacing fossil fuels with direct use of electricity.
Hydrogen fuel can produce the intense heat required for industrial production of steel, cement, glass, and chemicals, thus contributing to the decarbonization of industry alongside other technologies, such as electric arc furnaces for steelmaking. However, it is likely to play a larger role in providing industrial feedstock for cleaner production of ammonia and organic chemicals. For example, in steelmaking, hydrogen could function as a clean fuel and also as a low-carbon catalyst, replacing coal-derived coke (carbon):
2FeO + C → 2Fe + CO2
vs
FeO + H2 → Fe + H2O
Hydrogen used to decarbonize transportation is likely to find its largest applications in shipping, aviation and, to a lesser extent, heavy goods vehicles, through the use of hydrogen-derived synthetic fuels such as ammonia and methanol and fuel cell technology. For light-duty vehicles including cars, hydrogen is far behind other alternative fuel vehicles, especially compared with the rate of adoption of battery electric vehicles, and may not play a significant role in future.
Liquid hydrogen and liquid oxygen together serve as cryogenic propellants in liquid-propellant rockets, as in the Space Shuttle main engines. NASA has investigated the use of rocket propellant made from atomic hydrogen, boron or carbon that is frozen into solid molecular hydrogen particles suspended in liquid helium. Upon warming, the mixture vaporizes to allow the atomic species to recombine, heating the mixture to high temperature.
Hydrogen produced when there is a surplus of variable renewable electricity could in principle be stored and later used to generate heat or to re-generate electricity. It can be further transformed into synthetic fuels such as ammonia and methanol. Disadvantages of hydrogen fuel include high costs of storage and distribution due to hydrogen's explosivity, its large volume compared to other fuels, and its tendency to make pipes brittle.
=== Nickel–hydrogen battery ===
The very long-lived, rechargeable nickel–hydrogen battery developed for satellite power systems uses pressurized gaseous H2. The International Space Station, Mars Odyssey and the Mars Global Surveyor are equipped with nickel-hydrogen batteries. In the dark part of its orbit, the Hubble Space Telescope is also powered by nickel-hydrogen batteries, which were finally replaced in May 2009, more than 19 years after launch and 13 years beyond their design life.
=== Semiconductor industry ===
Hydrogen is employed to saturate broken ("dangling") bonds of amorphous silicon and amorphous carbon that helps stabilizing material properties. Hydrogen, introduced as a unintended side-effect of production, acts as a shallow electron donor leading to n-type conductivity in ZnO, with important uses in transducers and phosphors. Detailed analysis of ZnO and of MgO show evidence of four and six-fold hydrogen multicentre bonds.
The doping behavior of hydrogen varies with the material.
=== Niche and evolving uses ===
Other than the uses mentioned above, hydrogen is also used in smaller scales in the following applications:
Shielding gas: Hydrogen is used as a shielding gas in welding methods such as atomic hydrogen welding.
Coolant: Hydrogen is used as a coolant in large power stations generators due to its high thermal conductivity and low density. The first hydrogen-cooled turbogenerator went into service using gaseous hydrogen as a coolant in the rotor and the stator in 1937 at Dayton, Ohio.
Cryogenic research: Liquid H2 is used in cryogenic research, including superconductivity studies.
Leak detection: Pure or mixed with nitrogen (sometimes called forming gas), hydrogen is a tracer gas for detection of minute leaks. Applications can be found in the automotive, chemical, power generation, aerospace, and telecommunications industries. Hydrogen is an authorized food additive (E 949) that allows food package leak testing, as well as having anti-oxidizing properties.
Neutron moderation: Deuterium (hydrogen-2) is used in nuclear fission applications as a moderator to slow neutrons.
Nuclear fusion fuel: Deuterium is used in nuclear fusion reactions.
Isotopic labeling: Deuterium compounds have applications in chemistry and biology in studies of isotope effects on reaction rates.
Tritium uses: Tritium (hydrogen-3), produced in nuclear reactors, is used in the production of hydrogen bombs, as an isotopic label in the biosciences, and as a source of beta radiation in radioluminescent paint for instrument dials and emergency signage.
== Safety and precautions ==
In hydrogen pipelines and steel storage vessels, hydrogen molecules are prone to react with metals, causing hydrogen embrittlement and leaks in the pipeline or storage vessel. Since it is lighter than air, hydrogen does not easily accumulate to form a combustible gas mixture. However, even without ignition sources, high-pressure hydrogen leakage may cause spontaneous combustion and detonation.
Hydrogen is flammable when mixed even in small amounts with air. Ignition can occur at a volumetric ratio of hydrogen to air as low as 4%. In approximately 70% of hydrogen ignition accidents, the ignition source cannot be found, and it is widely believed by scholars that spontaneous ignition of hydrogen occurs.
Hydrogen fire, while being extremely hot, is almost invisible, and thus can lead to accidental burns. Hydrogen is non-toxic, but like most gases it can cause asphyxiation in the absence of adequate ventilation.
== See also ==
== References ==
== Further reading ==
Chart of the Nuclides (17th ed.). Knolls Atomic Power Laboratory. 2010. ISBN 978-0-9843653-0-2.
Newton, David E. (1994). The Chemical Elements. New York: Franklin Watts. ISBN 978-0-531-12501-4.
Rigden, John S. (2002). Hydrogen: The Essential Element. Cambridge, Massachusetts: Harvard University Press. ISBN 978-0-531-12501-4.
Romm, Joseph J. (2004). The Hype about Hydrogen, Fact and Fiction in the Race to Save the Climate. Island Press. ISBN 978-1-55963-703-9.
Scerri, Eric (2007). The Periodic System, Its Story and Its Significance. New York: Oxford University Press. ISBN 978-0-19-530573-9.
== External links ==
Basic Hydrogen Calculations of Quantum Mechanics
Hydrogen at The Periodic Table of Videos (University of Nottingham)
High temperature hydrogen phase diagram
Wavefunction of hydrogen | Wikipedia/Hydrogen_molecule |
In NMR spectroscopy, the Solomon equations describe the dipolar relaxation process of a system consisting of two spins. They take the form of the following differential equations:
d
I
1
z
d
t
=
−
R
z
1
(
I
1
z
−
I
1
z
0
)
−
σ
12
(
I
2
z
−
I
2
z
0
)
{\displaystyle {d{I_{1z}} \over dt}=-R_{z}^{1}(I_{1z}-I_{1z}^{0})-\sigma _{12}(I_{2z}-I_{2z}^{0})}
d
I
2
z
d
t
=
−
R
z
2
(
I
2
z
−
I
2
z
0
)
−
σ
12
(
I
1
z
−
I
1
z
0
)
{\displaystyle {d{I_{2z}} \over dt}=-R_{z}^{2}(I_{2z}-I_{2z}^{0})-\sigma _{12}(I_{1z}-I_{1z}^{0})}
d
I
1
z
I
2
z
d
t
=
−
R
z
12
2
I
1
z
I
2
z
{\displaystyle {d{I_{1z}I_{2z}} \over dt}=-R_{z}^{12}2I_{1z}I_{2z}}
These equations, so named after physicist Ionel Solomon, describe how the population of the different spin states changes in relation to the strength of the self-relaxation rate constant R and
σ
12
{\displaystyle \sigma _{12}}
, which accounts instead for cross-relaxation. The latter is the important term which is responsible for transferring magnetization from one spin to the other and gives rise to the nuclear Overhauser effect.
In an NOE experiment, the magnetization on one of the spins, say spin 2, is reversed by applying a selective pulse sequence. At short times then, the resulting magnetization on spin 1 will be given by
d
I
1
z
d
t
=
−
R
z
1
(
I
1
z
0
−
I
1
z
0
)
−
σ
12
(
−
I
2
z
0
−
I
2
z
0
)
=
2
σ
12
I
2
z
0
{\displaystyle {d{I_{1z}} \over dt}=-R_{z}^{1}(I_{1z}^{0}-I_{1z}^{0})-\sigma _{12}(-I_{2z}^{0}-I_{2z}^{0})=2\sigma _{12}I_{2z}^{0}}
since there is no time for a significant change in the populations of the energy levels. Integrating with respect to time gives:
I
1
z
(
t
)
=
2
σ
12
t
I
2
z
0
+
I
1
z
0
{\displaystyle I_{1z}(t)=2\sigma _{12}tI_{2z}^{0}+I_{1z}^{0}}
which results in an enhancement of the signal of spin 1 on the spectrum. Typically, another spectrum is recorded without applying the reversal of magnetization on spin 2 and the signals from the two experiments are then subtracted. In the final spectrum, only peaks which have an nOe enhancement show up, demonstrating which spins are in spatial proximity in the molecule under study: only those will in fact have a significant
σ
12
{\displaystyle \sigma _{12}}
cross relaxation factor.
== References == | Wikipedia/Solomon_equations |
In mathematics, a Hopf algebra, H, is quasitriangular if there exists an invertible element, R, of
H
⊗
H
{\displaystyle H\otimes H}
such that
R
Δ
(
x
)
R
−
1
=
(
T
∘
Δ
)
(
x
)
{\displaystyle R\ \Delta (x)R^{-1}=(T\circ \Delta )(x)}
for all
x
∈
H
{\displaystyle x\in H}
, where
Δ
{\displaystyle \Delta }
is the coproduct on H, and the linear map
T
:
H
⊗
H
→
H
⊗
H
{\displaystyle T:H\otimes H\to H\otimes H}
is given by
T
(
x
⊗
y
)
=
y
⊗
x
{\displaystyle T(x\otimes y)=y\otimes x}
,
(
Δ
⊗
1
)
(
R
)
=
R
13
R
23
{\displaystyle (\Delta \otimes 1)(R)=R_{13}\ R_{23}}
,
(
1
⊗
Δ
)
(
R
)
=
R
13
R
12
{\displaystyle (1\otimes \Delta )(R)=R_{13}\ R_{12}}
,
where
R
12
=
ϕ
12
(
R
)
{\displaystyle R_{12}=\phi _{12}(R)}
,
R
13
=
ϕ
13
(
R
)
{\displaystyle R_{13}=\phi _{13}(R)}
, and
R
23
=
ϕ
23
(
R
)
{\displaystyle R_{23}=\phi _{23}(R)}
, where
ϕ
12
:
H
⊗
H
→
H
⊗
H
⊗
H
{\displaystyle \phi _{12}:H\otimes H\to H\otimes H\otimes H}
,
ϕ
13
:
H
⊗
H
→
H
⊗
H
⊗
H
{\displaystyle \phi _{13}:H\otimes H\to H\otimes H\otimes H}
, and
ϕ
23
:
H
⊗
H
→
H
⊗
H
⊗
H
{\displaystyle \phi _{23}:H\otimes H\to H\otimes H\otimes H}
, are algebra morphisms determined by
ϕ
12
(
a
⊗
b
)
=
a
⊗
b
⊗
1
,
{\displaystyle \phi _{12}(a\otimes b)=a\otimes b\otimes 1,}
ϕ
13
(
a
⊗
b
)
=
a
⊗
1
⊗
b
,
{\displaystyle \phi _{13}(a\otimes b)=a\otimes 1\otimes b,}
ϕ
23
(
a
⊗
b
)
=
1
⊗
a
⊗
b
.
{\displaystyle \phi _{23}(a\otimes b)=1\otimes a\otimes b.}
R is called the R-matrix.
As a consequence of the properties of quasitriangularity, the R-matrix, R, is a solution of the Yang–Baxter equation (and so a module V of H can be used to determine quasi-invariants of braids, knots and links). Also as a consequence of the properties of quasitriangularity,
(
ϵ
⊗
1
)
R
=
(
1
⊗
ϵ
)
R
=
1
∈
H
{\displaystyle (\epsilon \otimes 1)R=(1\otimes \epsilon )R=1\in H}
; moreover
R
−
1
=
(
S
⊗
1
)
(
R
)
{\displaystyle R^{-1}=(S\otimes 1)(R)}
,
R
=
(
1
⊗
S
)
(
R
−
1
)
{\displaystyle R=(1\otimes S)(R^{-1})}
, and
(
S
⊗
S
)
(
R
)
=
R
{\displaystyle (S\otimes S)(R)=R}
. One may further show that the
antipode S must be a linear isomorphism, and thus S2 is an automorphism. In fact, S2 is given by conjugating by an invertible element:
S
2
(
x
)
=
u
x
u
−
1
{\displaystyle S^{2}(x)=uxu^{-1}}
where
u
:=
m
(
(
S
⊗
1
)
∘
T
)
R
{\displaystyle u:=m((S\otimes 1)\circ T)R}
(cf. Ribbon Hopf algebras).
It is possible to construct a quasitriangular Hopf algebra from a Hopf algebra and its dual, using the Drinfeld quantum double construction.
If the Hopf algebra H is quasitriangular, then the category of modules over H is braided with braiding
c
U
,
V
(
u
⊗
v
)
=
T
(
R
⋅
(
u
⊗
v
)
)
=
T
(
R
1
u
⊗
R
2
v
)
{\displaystyle c_{U,V}(u\otimes v)=T\left(R\cdot (u\otimes v)\right)=T\left(R_{1}u\otimes R_{2}v\right)}
.
== Twisting ==
The property of being a quasi-triangular Hopf algebra is preserved by twisting via an invertible element
F
=
∑
i
f
i
⊗
f
i
∈
A
⊗
A
{\displaystyle F=\sum _{i}f^{i}\otimes f_{i}\in {\mathcal {A\otimes A}}}
such that
(
ε
⊗
i
d
)
F
=
(
i
d
⊗
ε
)
F
=
1
{\displaystyle (\varepsilon \otimes id)F=(id\otimes \varepsilon )F=1}
and satisfying the cocycle condition
(
F
⊗
1
)
⋅
(
Δ
⊗
i
d
)
(
F
)
=
(
1
⊗
F
)
⋅
(
i
d
⊗
Δ
)
(
F
)
{\displaystyle (F\otimes 1)\cdot (\Delta \otimes id)(F)=(1\otimes F)\cdot (id\otimes \Delta )(F)}
Furthermore,
u
=
∑
i
f
i
S
(
f
i
)
{\displaystyle u=\sum _{i}f^{i}S(f_{i})}
is invertible and the twisted antipode is given by
S
′
(
a
)
=
u
S
(
a
)
u
−
1
{\displaystyle S'(a)=uS(a)u^{-1}}
, with the twisted comultiplication, R-matrix and co-unit change according to those defined for the quasi-triangular quasi-Hopf algebra. Such a twist is known as an admissible (or Drinfeld) twist.
== See also ==
Quasi-triangular quasi-Hopf algebra
Ribbon Hopf algebra
== Notes ==
== References ==
Montgomery, Susan (1993). Hopf algebras and their actions on rings. Regional Conference Series in Mathematics. Vol. 82. Providence, RI: American Mathematical Society. ISBN 0-8218-0738-2. Zbl 0793.16029.
Montgomery, Susan; Schneider, Hans-Jürgen (2002). New directions in Hopf algebras. Mathematical Sciences Research Institute Publications. Vol. 43. Cambridge University Press. ISBN 978-0-521-81512-3. Zbl 0990.00022. | Wikipedia/Quasitriangular_Hopf_algebra |
A quasi-Hopf algebra is a generalization of a Hopf algebra, which was defined by the Russian mathematician Vladimir Drinfeld in 1989.
A quasi-Hopf algebra is a quasi-bialgebra
B
A
=
(
A
,
Δ
,
ε
,
Φ
)
{\displaystyle {\mathcal {B_{A}}}=({\mathcal {A}},\Delta ,\varepsilon ,\Phi )}
for which there exist
α
,
β
∈
A
{\displaystyle \alpha ,\beta \in {\mathcal {A}}}
and a bijective antihomomorphism S (antipode) of
A
{\displaystyle {\mathcal {A}}}
such that
∑
i
S
(
b
i
)
α
c
i
=
ε
(
a
)
α
{\displaystyle \sum _{i}S(b_{i})\alpha c_{i}=\varepsilon (a)\alpha }
∑
i
b
i
β
S
(
c
i
)
=
ε
(
a
)
β
{\displaystyle \sum _{i}b_{i}\beta S(c_{i})=\varepsilon (a)\beta }
for all
a
∈
A
{\displaystyle a\in {\mathcal {A}}}
and where
Δ
(
a
)
=
∑
i
b
i
⊗
c
i
{\displaystyle \Delta (a)=\sum _{i}b_{i}\otimes c_{i}}
and
∑
i
X
i
β
S
(
Y
i
)
α
Z
i
=
I
,
{\displaystyle \sum _{i}X_{i}\beta S(Y_{i})\alpha Z_{i}=\mathbb {I} ,}
∑
j
S
(
P
j
)
α
Q
j
β
S
(
R
j
)
=
I
.
{\displaystyle \sum _{j}S(P_{j})\alpha Q_{j}\beta S(R_{j})=\mathbb {I} .}
where the expansions for the quantities
Φ
{\displaystyle \Phi }
and
Φ
−
1
{\displaystyle \Phi ^{-1}}
are given by
Φ
=
∑
i
X
i
⊗
Y
i
⊗
Z
i
{\displaystyle \Phi =\sum _{i}X_{i}\otimes Y_{i}\otimes Z_{i}}
and
Φ
−
1
=
∑
j
P
j
⊗
Q
j
⊗
R
j
.
{\displaystyle \Phi ^{-1}=\sum _{j}P_{j}\otimes Q_{j}\otimes R_{j}.}
As for a quasi-bialgebra, the property of being quasi-Hopf is preserved under twisting.
== Usage ==
Quasi-Hopf algebras form the basis of the study of Drinfeld twists and the representations in terms of F-matrices associated with finite-dimensional irreducible representations of quantum affine algebra. F-matrices can be used to factorize the corresponding R-matrix. This leads to applications in Statistical mechanics, as quantum affine algebras, and their representations give rise to solutions of the Yang–Baxter equation, a solvability condition for various statistical models, allowing characteristics of the model to be deduced from its corresponding quantum affine algebra. The study of F-matrices has been applied to models such as the Heisenberg XXZ model in the framework of the algebraic Bethe ansatz. It provides a framework for solving two-dimensional integrable models by using the quantum inverse scattering method.
== See also ==
Quasitriangular Hopf algebra
Quasi-triangular quasi-Hopf algebra
Ribbon Hopf algebra
== References ==
Vladimir Drinfeld, "Quasi-Hopf algebras", Leningrad Math J. 1 (1989), 1419-1457
J. M. Maillet and J. Sanchez de Santos, Drinfeld Twists and Algebraic Bethe Ansatz, Amer. Math. Soc. Transl. (2) Vol. 201, 2000 | Wikipedia/Quasi-Hopf_algebra |
In physics, the eightfold way is an organizational scheme for a class of subatomic particles known as hadrons that led to the development of the quark model. Both the American physicist Murray Gell-Mann and the Israeli physicist Yuval Ne'eman independently and simultaneously proposed the idea in 1961.
The name comes from Gell-Mann's (1961) paper and is an allusion to the Noble Eightfold Path of Buddhism.
== Background ==
By 1947, physicists believed that they had a good understanding of what the smallest bits of matter were. There were electrons, protons, neutrons, and photons (the components that make up the vast part of everyday experience such as visible matter and light) along with a handful of unstable (i.e., they undergo radioactive decay) exotic particles needed to explain cosmic rays observations such as pions, muons and the hypothesized neutrinos. In addition, the discovery of the positron suggested there could be anti-particles for each of them. It was known a "strong interaction" must exist to overcome electrostatic repulsion in atomic nuclei. Not all particles are influenced by this strong force; but those that are, are dubbed "hadrons"; these are now further classified as mesons (from the Greek for "intermediate") and baryons (from the Greek for "heavy").
But the discovery of the neutral kaon in late 1947 and the subsequent discovery of a positively charged kaon in 1949 extended the meson family in an unexpected way, and in 1950 the lambda particle did the same thing for the baryon family. These particles decay much more slowly than they are produced, a hint that there are two different physical processes involved. This was first suggested by Abraham Pais in 1952. In 1953, Murray Gell-Mann and a collaboration in Japan, Tadao Nakano with Kazuhiko Nishijima, independently suggested a new conserved value now known as "strangeness" during their attempts to understand the growing collection of known particles.
The discovery of new mesons and baryons continued through the 1950s; the number of known "elementary" particles ballooned. Physicists were interested in understanding hadron-hadron interactions via the strong interaction. The concept of isospin, introduced in 1932 by Werner Heisenberg shortly after the discovery of the neutron, was used to group some hadrons together into "multiplets" but no successful scientific theory as yet covered the hadrons as a whole. This was the beginning of a chaotic period in particle physics that has become known as the "particle zoo" era. The eightfold way represented a step out of this confusion and towards the quark model, which proved to be the solution.
== Organization ==
Group representation theory is the mathematical underpinning of the eightfold way, but that rather technical mathematics is not needed to understand how it helps organize particles. Particles are sorted into groups as mesons or baryons. Within each group, they are further separated by their spin angular momentum. Symmetrical patterns appear when these groups of particles have their strangeness plotted against their electric charge. (This is the most common way to make these plots today, but originally physicists used an equivalent pair of properties called hypercharge and isotopic spin, the latter of which is now known as isospin.) The symmetry in these patterns is a hint of the underlying symmetry of the strong interaction between the particles themselves. In the plots below, points representing particles that lie along the same horizontal line share the same strangeness, s, while those on the same left-leaning diagonals share the same electric charge, q (given as multiples of the elementary charge).
=== Mesons ===
In the original eightfold way, the mesons were organized into octets and singlets. This is one of the finer points of differences between the eightfold way and the quark model it inspired, which suggests the mesons should be grouped into nonets (groups of nine).
==== Meson octet ====
The eightfold way organizes eight of the lowest spin-0 mesons into an octet. They are:
K0, K+, K− and K0 kaons
π+, π0, and π− pions
η, the eta meson
Diametrically opposite particles in the diagram are anti-particles of one another, while particles in the center are their own anti-particle.
==== Meson singlet ====
The chargeless, strangeless eta prime meson was originally classified by itself as a singlet:
η′
Under the quark model later developed, it is better viewed as part of a meson nonet, as previously mentioned.
=== Baryons ===
==== Baryon octet ====
The eightfold way organizes the spin-1/ 2 baryons into an octet. They consist of
neutron (n) and proton (p)
Σ−, Σ0, and Σ+ sigma baryons
Λ0, the strange lambda baryon
Ξ− and Ξ0 xi baryons
==== Baryon decuplet ====
The organizational principles of the eightfold way also apply to the spin-3/2 baryons, forming a decuplet.
Δ−, Δ0, Δ+, and Δ++ delta baryons
Σ∗−, Σ∗0, and Σ∗+ sigma baryons
Ξ∗− and Ξ∗0 xi baryons
Ω− omega baryon
However, one of the particles of this decuplet had never been previously observed when the eightfold way was proposed. Gell-Mann called this particle the Ω− and predicted in 1962 that it would have a strangeness −3, electric charge −1 and a mass near 1680 MeV/c2. In 1964, a particle closely matching these predictions was discovered by a particle accelerator group at Brookhaven. Gell-Mann received the 1969 Nobel Prize in Physics for his work on the theory of elementary particles.
== Historical development ==
=== Development ===
Historically, quarks were motivated by an understanding of flavour symmetry. First, it was noticed (1961) that groups of particles were related to each other in a way that matched the representation theory of SU(3). From that, it was inferred that there is an approximate symmetry of the universe which is represented by the group SU(3). Finally (1964), this led to the discovery of three light quarks (up, down, and strange) interchanged by these SU(3) transformations.
=== Modern interpretation ===
The eightfold way may be understood in modern terms as a consequence of flavor symmetries between various kinds of quarks. Since the strong nuclear force affects quarks the same way regardless of their flavor, replacing one flavor of quark with another in a hadron should not alter its mass very much, provided the respective quark masses are smaller than the strong interaction scale—which holds for the three light quarks. Mathematically, this replacement may be described by elements of the SU(3) group. The octets and other hadron arrangements are representations of this group.
== Flavor symmetry ==
=== SU(3) ===
There is an abstract three-dimensional vector space:
up quark
→
(
1
0
0
)
,
down quark
→
(
0
1
0
)
,
strange quark
→
(
0
0
1
)
,
{\displaystyle {\text{up quark}}\to {\begin{pmatrix}1\\0\\0\end{pmatrix}},\qquad {\text{down quark}}\to {\begin{pmatrix}0\\1\\0\end{pmatrix}},\qquad {\text{strange quark}}\to {\begin{pmatrix}0\\0\\1\end{pmatrix}},}
and the laws of physics are approximately invariant under a determinant-1 unitary transformation to this space (sometimes called a flavour rotation):
(
x
y
z
)
↦
A
(
x
y
z
)
,
where
A
is in
S
U
(
3
)
.
{\displaystyle {\begin{pmatrix}x\\y\\z\end{pmatrix}}\mapsto A{\begin{pmatrix}x\\y\\z\end{pmatrix}},\quad {\text{where}}\ A\ {\text{is in}}\ SU(3).}
Here, SU(3) refers to the Lie group of 3×3 unitary matrices with determinant 1 (special unitary group). For example, the flavour rotation
A
=
(
−
0
1
0
−
1
0
0
−
0
0
1
)
{\displaystyle A={\begin{pmatrix}{\phantom {-}}0&1&0\\-1&0&0\\{\phantom {-}}0&0&1\end{pmatrix}}}
is a transformation that simultaneously turns all the up quarks in the universe into down quarks and conversely. More specifically, these flavour rotations are exact symmetries if only strong force interactions are looked at, but they are not truly exact symmetries of the universe because the three quarks have different masses and different electroweak interactions.
This approximate symmetry is called flavour symmetry, or more specifically flavour SU(3) symmetry.
=== Connection to representation theory ===
Assume we have a certain particle—for example, a proton—in a quantum state
|
ψ
⟩
{\displaystyle |\psi \rangle }
. If we apply one of the flavour rotations A to our particle, it enters a new quantum state which we can call
A
|
ψ
⟩
{\displaystyle A|\psi \rangle }
. Depending on A, this new state might be a proton, or a neutron, or a superposition of a proton and a neutron, or various other possibilities. The set of all possible quantum states spans a vector space.
Representation theory is a mathematical theory that describes the situation where elements of a group (here, the flavour rotations A in the group SU(3)) are automorphisms of a vector space (here, the set of all possible quantum states that you get from flavour-rotating a proton). Therefore, by studying the representation theory of SU(3), we can learn the possibilities for what the vector space is and how it is affected by flavour symmetry.
Since the flavour rotations A are approximate, not exact, symmetries, each orthogonal state in the vector space corresponds to a different particle species. In the example above, when a proton is transformed by every possible flavour rotation A, it turns out that it moves around an 8 dimensional vector space. Those 8 dimensions correspond to the 8 particles in the so-called "baryon octet" (proton, neutron, Σ+, Σ0, Σ−, Ξ−, Ξ0, Λ). This corresponds to an 8-dimensional ("octet") representation of the group SU(3). Since A is an approximate symmetry, all the particles in this octet have similar mass.
Every Lie group has a corresponding Lie algebra, and each group representation of the Lie group can be mapped to a corresponding Lie algebra representation on the same vector space. The Lie algebra
s
u
{\displaystyle {\mathfrak {su}}}
(3) can be written as the set of 3×3 traceless Hermitian matrices. Physicists generally discuss the representation theory of the Lie algebra
s
u
{\displaystyle {\mathfrak {su}}}
(3) instead of the Lie group SU(3), since the former is simpler and the two are ultimately equivalent.
== Notes ==
== References ==
== Further reading ==
M. Gell-Mann; Y. Ne'eman, eds. (1964). The Eightfold Way. W. A. Benjamin. LCCN 65013009. (contains most historical papers on the eightfold way and related topics, including the Gell-Mann–Okubo mass formula.) | Wikipedia/Eightfold_Way_(physics) |
The Hückel method or Hückel molecular orbital theory, proposed by Erich Hückel in 1930, is a simple method for calculating molecular orbitals as linear combinations of atomic orbitals. The theory predicts the molecular orbitals for π-electrons in π-delocalized molecules, such as ethylene, benzene, butadiene, and pyridine. It provides the theoretical basis for Hückel's rule that cyclic, planar molecules or ions with
4
n
+
2
{\displaystyle 4n+2}
π-electrons are aromatic. It was later extended to conjugated molecules such as pyridine, pyrrole and furan that contain atoms other than carbon and hydrogen (heteroatoms). A more dramatic extension of the method to include σ-electrons, known as the extended Hückel method (EHM), was developed by Roald Hoffmann. The extended Hückel method gives some degree of quantitative accuracy for organic molecules in general (not just planar systems) and was used to provide computational justification for the Woodward–Hoffmann rules. To distinguish the original approach from Hoffmann's extension, the Hückel method is also known as the simple Hückel method (SHM).
Although undeniably a cornerstone of organic chemistry, Hückel's concepts were undeservedly unrecognized for two decades. Pauling and Wheland characterized his approach as "cumbersome" at the time, and their competing resonance theory was relatively easier to understand for chemists without fundamental physics background, even if they couldn't grasp the concept of quantum superposition and confused it with tautomerism. His lack of communication skills contributed: when Robert Robinson sent him a friendly request, he responded arrogantly that he is not interested in organic chemistry.
In spite of its simplicity, the Hückel method in its original form makes qualitatively accurate and chemically useful predictions for many common molecules and is therefore a powerful and widely taught educational tool. It is described in many introductory quantum chemistry and physical organic chemistry textbooks, and organic chemists in particular still routinely apply Hückel theory to obtain a very approximate, back-of-the-envelope understanding of π-bonding.
== Hückel characteristics ==
The method has several characteristics:
It limits itself to conjugated molecules.
Only π electron molecular orbitals are included because these determine much of the chemical and spectral properties of these molecules. The σ electrons are assumed to form the framework of the molecule and σ connectivity is used to determine whether two π orbitals interact. However, the orbitals formed by σ electrons are ignored and assumed not to interact with π electrons. This is referred to as σ-π separability. It is justified by the orthogonality of σ and π orbitals in planar molecules. For this reason, the Hückel method is limited to systems that are planar or nearly so.
The method is based on applying the variational method to linear combination of atomic orbitals and making simplifying assumptions regarding the overlap, resonance and Coulomb integrals of these atomic orbitals. It does not attempt to solve the Schrödinger equation, and neither the functional form of the basis atomic orbitals nor details of the Hamiltonian are involved.
For hydrocarbons, the method takes atomic connectivity as the only input; empirical parameters are only needed when heteroatoms are introduced.
The method predicts how many energy levels exist for a given molecule, which levels are degenerate and it expresses the molecular orbital energies in terms of two parameters, called α, the energy of an electron in a 2p orbital, and β, the interaction energy between two 2p orbitals (the extent to which an electron is stabilized by allowing it to delocalize between two orbitals). The usual sign convention is to let both α and β be negative numbers. To understand and compare systems in a qualitative or even semi-quantitative sense, explicit numerical values for these parameters are typically not required.
In addition the method also enables calculation of charge density for each atom in the π framework, the fractional bond order between any two atoms, and the overall molecular dipole moment.
== Hückel results ==
=== Results for simple molecules and general results for cyclic and linear systems ===
The results for a few simple molecules are tabulated below:
The theory predicts two energy levels for ethylene with its two π electrons filling the low-energy HOMO and the high energy LUMO remaining empty. In butadiene the 4 π-electrons occupy 2 low energy molecular orbitals, out of a total of 4, and for benzene 6 energy levels are predicted, two of them degenerate.
For linear and cyclic systems (with N atoms), general solutions exist:
Linear system (polyene/polyenyl):
E
k
=
α
+
2
β
cos
(
k
+
1
)
π
N
+
1
(
k
=
0
,
1
,
…
,
N
−
1
)
{\displaystyle E_{k}=\alpha +2\beta \cos {\frac {(k+1)\pi }{N+1}}\quad (k=0,1,\ldots ,N-1)}
.
Energy levels are all distinct.
Cyclic system, Hückel topology (annulene/annulenyl):
E
k
=
α
+
2
β
cos
2
k
π
N
(
k
=
0
,
1
,
…
,
⌊
N
/
2
⌋
)
{\displaystyle E_{k}=\alpha +2\beta \cos {\frac {2k\pi }{N}}\quad (k=0,1,\ldots ,\lfloor N/2\rfloor )}
.
Energy levels
k
=
1
,
…
,
⌈
N
/
2
⌉
−
1
{\displaystyle k=1,\ldots ,\lceil N/2\rceil -1}
are each doubly degenerate.
Cyclic system, Möbius topology (hypothetical for N < 8):
E
k
=
α
+
2
β
′
cos
(
2
k
+
1
)
π
N
,
β
′
=
β
cos
(
π
/
N
)
(
k
=
0
,
1
,
…
,
⌈
N
/
2
⌉
−
1
)
{\displaystyle E_{k}=\alpha +2\beta '\cos {\frac {(2k+1)\pi }{N}},\ \beta '=\beta \cos(\pi /N)\quad (k=0,1,\ldots ,\lceil N/2\rceil -1)}
.
Energy levels
k
=
0
,
…
,
⌊
N
/
2
⌋
−
1
{\displaystyle k=0,\ldots ,\lfloor N/2\rfloor -1}
are each doubly degenerate.
The energy levels for cyclic systems can be predicted using the Frost circle mnemonic (named after the American chemist Arthur Atwater Frost). A circle centered at α with radius 2β is inscribed with a regular N-gon with one vertex pointing down; the y-coordinate of the vertices of the polygon then represent the orbital energies of the [N]annulene/annulenyl system. Related mnemonics exists for linear and Möbius systems.
=== The values of α and β ===
The value of α is the energy of an electron in a 2p orbital, relative to an unbound electron at infinity. This quantity is negative, since the electron is stabilized by being electrostatically bound to the positively charged nucleus. For carbon this value is known to be approximately –11.4 eV. Since Hückel theory is generally only interested in energies relative to a reference localized system, the value of α is often immaterial and can be set to zero without affecting any conclusions.
Roughly speaking, β physically represents the energy of stabilization experienced by an electron allowed to delocalize in a π molecular orbital formed from the 2p orbitals of adjacent atoms, compared to being localized in an isolated 2p atomic orbital. As such, it is also a negative number, although it is often spoken of in terms of its absolute value. The value for |β| in Hückel theory is roughly constant for structurally similar compounds, but not surprisingly, structurally dissimilar compounds will give very different values for |β|. For example, using the π bond energy of ethylene (65 kcal/mole) and comparing the energy of a doubly-occupied π orbital (2α + 2β) with the energy of electrons in two isolated p orbitals (2α), a value of |β| = 32.5 kcal/mole can be inferred. On the other hand, using the resonance energy of benzene (36 kcal/mole, derived from heats of hydrogenation) and comparing benzene (6α + 8β) with a hypothetical "non-aromatic 1,3,5-cyclohexatriene" (6α + 6β), a much smaller value of |β| = 18 kcal/mole emerges. These differences are not surprising, given the substantially shorter bond length of ethylene (1.33 Å) compared to benzene (1.40 Å). The shorter distance between the interacting p orbitals accounts for the greater energy of interaction, which is reflected by a higher value of |β|. Nevertheless, heat of hydrogenation measurements of various polycyclic aromatic hydrocarbons like naphthalene and anthracene all imply values of |β| between 17 and 20 kcal/mol.
However, even for the same compound, the correct assignment of |β| can be controversial. For instance, it is argued that the resonance energy measured experimentally via heats of hydrogenation is diminished by the distortions in bond lengths that must take place going from the single and double bonds of "non-aromatic 1,3,5-cyclohexatriene" to the delocalized bonds of benzene. Taking this distortion energy into account, the value of |β| for delocalization without geometric change (called the "vertical resonance energy") for benzene is found to be around 37 kcal/mole. On the other hand, experimental measurements of electronic spectra have given a value of |β| (called the "spectroscopic resonance energy") as high as 3 eV (~70 kcal/mole) for benzene. Given these subtleties, qualifications, and ambiguities, Hückel theory should not be called upon to provide accurate quantitative predictions – only semi-quantitative or qualitative trends and comparisons are reliable and robust.
=== Other successful predictions ===
With this caveat in mind, many predictions of the theory have been experimentally verified:
The HOMO–LUMO gap, in terms of the β constant, correlates directly with the respective molecular electronic transitions observed with UV/VIS spectroscopy. For linear polyenes, the energy gap is given as:
Δ
E
=
−
4
β
sin
π
2
(
n
+
1
)
{\displaystyle \Delta E=-4\beta \sin {\frac {\pi }{2(n+1)}}}
from which a value for β can be obtained between −60 and −70 kcal/mol (−250 to −290 kJ/mol).
The predicted molecular orbital energies as stipulated by Koopmans' theorem correlate with photoelectron spectroscopy.
The Hückel delocalization energy correlates with the experimental heat of combustion. This energy is defined as the difference between the total predicted π energy (in benzene 8β) and a hypothetical π energy in which all ethylene units are assumed isolated, each contributing 2β (making benzene 3 × 2β = 6β).
Molecules with molecular orbitals paired up such that only the sign differs (for example α ± β) are called alternant hydrocarbons and have in common small molecular dipole moments. This is in contrast to non-alternant hydrocarbons, such as azulene and fulvene that have large dipole moments. The Hückel theory is more accurate for alternant hydrocarbons.
For cyclobutadiene the theory predicts that the two high-energy electrons occupy a degenerate pair of molecular orbitals (following from Hund's rules) that are neither stabilized nor destabilized. Hence the square molecule would be a very reactive triplet diradical (the ground state is actually rectangular without degenerate orbitals). In fact, all cyclic conjugated hydrocarbons with a total of 4n π-electrons share this molecular orbital pattern, and this forms the basis of Hückel's rule.
Dewar reactivity numbers deriving from the Hückel approach correctly predict the reactivity of aromatic systems with nucleophiles and electrophiles.
The benzyl cation and anion serve as simple models for arenes with electron-withdrawing and electron-donating groups, respectively. The π-electron population correctly implies the meta- and ortho-/para-selectivity for electrophilic aromatic substitution of π electron-poor and π electron-rich arenes, respectively.
=== Application in optical activity analysis ===
The analysis of the optical activity of a molecule depends to a certain extent on the study of its chiral characteristics. However, for achiral molecules applying pesudoscalars to simplify the calculations of optical activity cannot be achieved due to the lack of spatial average.
Instead of traditional chiroptical solution measurements, Hückel theory helps focus on oriented π systems by separating from σ electrons especially in the planar,
C
2
v
{\displaystyle C_{\mathrm {2v} }}
-symmetric cases. Transition dipole moments derived by multiplying each wavefunction of individual planar molecule one by one, contribute to the directions of the most optical activity, where sit at the bisectors of two orthogonal ones. Despite the zero value for the trace of the tensor, cis-butadiene shows considerable off diagonal component which was computed as the first optical activity evaluation of achiral molecule.
Consider 3,5-dimethylene-1-cyclopentene as an example. Transition electric dipole, magnetic dipole and electric quadrupole moments interactions result in optical rotation(OR), which can be described by both tensor components and chemical geometries. The in phase overlap of two molecular orbitals yield negative charge while depleting charge out of phase. The movement can be interpreted quantitatively by corresponding π and π* orbitals coefficients.
== Delocalization energy, π-bond orders, and π-electron populations ==
The delocalization energy, π-bond orders, and π-electron population are chemically significant parameters that can be gleaned from the orbital energies and coefficients that are the direct outputs of Hückel theory. These are quantities strictly derived from theory, as opposed to measurable physical properties, though they correlate with measurable qualitative and quantitative properties of the chemical species. Delocalization energy is defined as the difference in energy between that of the most stable localized Lewis structure and the energy of the molecule computed from Hückel theory orbital energies and occupancies. Since all energies are relative, we set
α
=
0
{\displaystyle \alpha =0}
without loss of generality to simplify discussion. The energy of the localized structure is then set to be 2β for every two-electron localized π-bond. The Hückel energy of the molecule is
∑
i
n
i
E
i
{\displaystyle \sum _{i}n_{i}E_{i}}
, where the sum is over all Hückel orbitals,
n
i
{\displaystyle n_{i}}
is the occupancy of orbital i, set to be 2 for doubly-occupied orbitals, 1 for singly-occupied orbitals, and 0 for unoccupied orbitals, and
E
i
{\displaystyle E_{i}}
is the energy of orbital i. Thus, the delocalization energy, conventionally a positive number, is defined as
E
d
e
l
o
c
.
=
|
(
∑
i
n
i
E
i
)
−
2
β
×
(
#
o
f
l
o
c
a
l
i
z
e
d
π
b
o
n
d
s
)
|
{\displaystyle E_{\mathrm {deloc.} }={\Bigg |}{\Big (}\sum _{i}n_{i}E_{i}{\Big )}-2\beta \times (\#\ \mathrm {of} \ \mathrm {localized} \ \pi \ \mathrm {bonds} ){\Bigg |}}
.
In the case of benzene, the occupied orbitals have energies (again setting
α
=
0
{\displaystyle \alpha =0}
) 2β, β, and β. This gives the Hückel energy of benzene as
2
×
2
β
+
2
×
β
+
2
×
β
=
8
β
{\displaystyle 2\times 2\beta +2\times \beta +2\times \beta =8\beta }
. Each Kekulé structure of benzene has three double bonds, so the localized structure is assigned an energy of
2
β
×
3
=
6
β
{\displaystyle 2\beta \times 3=6\beta }
. The delocalization energy, measured in units of
|
β
|
{\displaystyle |\beta |}
, is then
|
8
β
−
6
β
|
=
2
|
β
|
{\displaystyle |8\beta -6\beta |=2|\beta |}
.
The π-bond orders derived from Hückel theory are defined using the orbital coefficients of the Hückel MOs. The π-bond order between atoms j and k is defined as
B
O
π
(
j
,
k
)
=
∑
i
n
i
c
j
(
i
)
c
k
(
i
)
{\displaystyle \mathrm {BO} _{\pi }(j,k)=\sum _{i}n_{i}c_{j}^{(i)}c_{k}^{(i)}}
,
where
n
i
{\displaystyle n_{i}}
is again the orbital occupancy of orbital i and
c
j
(
i
)
{\displaystyle c_{j}^{(i)}}
and
c
k
(
i
)
{\displaystyle c_{k}^{(i)}}
are the coefficients on atoms j and k, respectively, for orbital i. For benzene, the three occupied MOs, expressed as linear combinations of AOs
ϕ
i
{\displaystyle \phi _{i}}
, are:
Ψ
(
A
2
u
)
=
1
6
(
ϕ
1
+
ϕ
2
+
ϕ
3
+
ϕ
4
+
ϕ
5
+
ϕ
6
)
{\displaystyle \Psi (A_{2u})={\frac {1}{\sqrt {6}}}(\phi _{1}+\phi _{2}+\phi _{3}+\phi _{4}+\phi _{5}+\phi _{6})}
, [
E
=
2
β
{\displaystyle E=2\beta }
];
Ψ
(
E
1
g
(
x
)
)
=
1
12
(
2
ϕ
1
+
ϕ
2
−
ϕ
3
−
2
ϕ
4
−
ϕ
5
+
ϕ
6
)
{\displaystyle \Psi (E_{1g}^{(x)})={\frac {1}{\sqrt {12}}}(2\phi _{1}+\phi _{2}-\phi _{3}-2\phi _{4}-\phi _{5}+\phi _{6})}
, [
E
=
β
{\displaystyle E=\beta }
];
Ψ
(
E
1
g
(
y
)
)
=
1
2
(
ϕ
2
+
ϕ
3
−
ϕ
5
−
ϕ
6
)
{\displaystyle \Psi (E_{1g}^{(y)})={\frac {1}{2}}(\phi _{2}+\phi _{3}-\phi _{5}-\phi _{6})}
, [
E
=
β
{\displaystyle E=\beta }
].
Perhaps surprisingly, the π-bond order formula gives a bond order of
2
(
1
6
)
(
1
6
)
+
2
(
2
12
)
(
1
12
)
+
2
(
0
)
(
1
2
)
=
2
3
{\displaystyle 2{\Bigg (}{\frac {1}{\sqrt {6}}}{\Bigg )}{\Bigg (}{\frac {1}{\sqrt {6}}}{\Bigg )}+2{\Bigg (}{\frac {2}{\sqrt {12}}}{\Bigg )}{\Bigg (}{\frac {1}{\sqrt {12}}}{\Bigg )}+2(0){\Big (}{\frac {1}{2}}{\Big )}={\frac {2}{3}}}
for the bond between carbons 1 and 2. The resulting total (σ + π) bond order of
1
2
3
{\displaystyle 1{\frac {2}{3}}}
is the same between any other pair of adjacent carbon atoms. This is more than the naive π-bond order of
1
2
{\displaystyle {\frac {1}{2}}}
(for a total bond order of
1
1
2
{\displaystyle 1{\frac {1}{2}}}
) that one might guess when simply considering the Kekulé structures and the usual definition of bond order in valence bond theory. The Hückel definition of bond order attempts to quantify any additional stabilization that the system enjoys resulting from delocalization. In a sense, the Hückel bond order suggests that there are four π-bonds in benzene instead of the three that are implied by the Kekulé-type Lewis structures. The "extra" bond is attributed to the additional stabilization that results from the aromaticity of the benzene molecule. (This is only one of several definitions for non-integral bond orders, and other definitions will lead to different values that fall between 1 and 2.)
The π-electron population is calculated in a very similar way to the bond order using the orbital coefficients of the Hückel MOs. The π-electron population on atom j is defined as
n
π
(
j
)
=
∑
i
n
i
[
c
j
(
i
)
]
2
{\displaystyle n_{\pi }(j)=\sum _{i}n_{i}[c_{j}^{(i)}]^{2}}
.
The associated Hückel Coulomb charge is defined as
q
j
=
N
π
(
j
)
−
n
π
(
j
)
{\displaystyle q_{j}=N_{\pi }(j)-n_{\pi }(j)}
, where
N
π
(
j
)
{\displaystyle N_{\pi }(j)}
is the number of π-electrons contributed by a neutral, sp2-hybridized atom j (we always have
N
π
=
1
{\displaystyle N_{\pi }=1}
for carbon).
For carbon 1 on benzene, this yields a π-electron population of
2
(
1
6
)
(
1
6
)
+
2
(
2
12
)
(
2
12
)
+
2
(
0
)
(
0
)
=
1
{\displaystyle 2{\Bigg (}{\frac {1}{\sqrt {6}}}{\Bigg )}{\Bigg (}{\frac {1}{\sqrt {6}}}{\Bigg )}+2{\Bigg (}{\frac {2}{\sqrt {12}}}{\Bigg )}{\Bigg (}{\frac {2}{\sqrt {12}}}{\Bigg )}+2(0)(0)=1}
.
Since each carbon atom contributes one π-electron to the molecule, this gives a Coulomb charge of 0 for carbon 1 (and all other carbon atoms), as expected.
In the cases of benzyl cation and benzyl anion shown above,
q
j
(
C
H
2
+
)
=
N
π
(
j
)
−
n
π
(
j
)
=
1
−
0.43
=
+
0.57
{\displaystyle q_{j}(\mathrm {CH} _{2}^{+})=N_{\pi }(j)-n_{\pi }(j)=1-0.43=+0.57}
and
q
j
(
C
H
2
−
)
=
N
π
(
j
)
−
n
π
(
j
)
=
1
−
1.57
=
−
0.57
{\displaystyle q_{j}(\mathrm {CH} _{2}^{-})=N_{\pi }(j)-n_{\pi }(j)=1-1.57=-0.57}
,
q
j
(
C
o
,
p
+
)
=
N
π
(
j
)
−
n
π
(
j
)
=
1
−
0.86
=
+
0.14
{\displaystyle q_{j}(\mathrm {C} _{o,p}^{+})=N_{\pi }(j)-n_{\pi }(j)=1-0.86=+0.14}
and
q
j
(
C
o
,
p
−
)
=
N
π
(
j
)
−
n
π
(
j
)
=
1
−
1.14
=
−
0.14
{\displaystyle q_{j}(\mathrm {C} _{o,p}^{-})=N_{\pi }(j)-n_{\pi }(j)=1-1.14=-0.14}
.
== Mathematics behind the Hückel method ==
The mathematics of the Hückel method is based on the Ritz method. In short, given a basis set of n normalized atomic orbitals
{
ϕ
i
}
i
=
1
n
{\displaystyle \{\phi _{i}\}_{i=1}^{n}}
, an ansatz molecular orbital
ψ
g
=
N
(
c
1
ϕ
1
+
⋯
+
c
n
ϕ
n
)
{\displaystyle \psi _{g}=N(c_{1}\phi _{1}+\cdots +c_{n}\phi _{n})}
is written down, with normalization constant N and coefficients
c
i
{\displaystyle c_{i}}
which are to be determined. In other words, we are assuming that the molecular orbital (MO) can be written as a linear combination of atomic orbitals, a conceptually intuitive and convenient approximation (the linear combination of atomic orbitals or LCAO approximation). The variational theorem states that given an eigenvalue problem
H
^
|
ψ
(
i
)
⟩
=
E
(
i
)
|
ψ
(
i
)
⟩
{\displaystyle {\hat {H}}|\psi ^{(i)}\rangle =E^{(i)}|\psi ^{(i)}\rangle }
with smallest eigenvalue
E
(
0
)
{\displaystyle E^{(0)}}
and corresponding wavefunction
ψ
(
0
)
{\displaystyle \psi ^{(0)}}
, any normalized trial wavefunction
ψ
g
{\displaystyle \psi _{g}}
(i.e.,
⟨
ψ
g
|
ψ
g
⟩
=
∫
R
3
ψ
g
∗
ψ
g
d
V
=
1
{\textstyle \langle \psi _{g}|\psi _{g}\rangle =\int _{\mathbb {R} ^{3}}\psi _{g}^{*}\,\psi _{g}\,dV=1}
holds) will satisfy
E
[
ψ
g
]
=
⟨
ψ
g
|
H
^
|
ψ
g
⟩
=
∫
R
3
ψ
g
∗
H
^
ψ
g
d
V
≥
E
(
0
)
{\displaystyle {\mathcal {E}}[\psi _{g}]=\langle \psi _{g}|{\hat {H}}|\psi _{g}\rangle =\int _{\mathbb {R} ^{3}}\psi _{g}^{*}\,{\hat {H}}\psi _{g}\,dV\geq E^{(0)}}
,
with equality holding if and only if
ψ
g
=
ψ
(
0
)
{\displaystyle \psi _{g}=\psi ^{(0)}}
. Thus, by minimizing
E
(
c
1
,
…
,
c
n
)
=
E
[
ψ
g
]
{\displaystyle E(c_{1},\ldots ,c_{n})={\mathcal {E}}[\psi _{g}]}
with respect to coefficients
c
i
{\displaystyle c_{i}}
for normalized trial wavefunctions
ψ
g
(
c
1
,
…
,
c
n
)
{\displaystyle \psi _{g}(c_{1},\ldots ,c_{n})}
, we obtain a closer approximation of the true ground-state wavefunction and its energy.
To start, we apply the normalization condition to the ansatz and expand to get an expression for N in terms of the
c
i
{\displaystyle c_{i}}
. Then, we substitute the ansatz into the expression for E and expand, yielding
E
(
c
1
,
…
,
c
n
)
=
N
2
[
∑
i
=
1
n
c
i
2
H
i
i
+
∑
1
≤
i
≠
j
≤
n
c
i
c
j
H
i
j
]
{\displaystyle E(c_{1},\ldots ,c_{n})=N^{2}{\Big [}\sum _{i=1}^{n}c_{i}^{2}H_{ii}+\sum _{1\leq i\neq j\leq n}c_{i}c_{j}H_{ij}{\Big ]}}
, where
N
=
[
∑
i
=
1
n
c
i
2
S
i
i
+
∑
1
≤
i
≠
j
≤
n
c
i
c
j
S
i
j
]
−
1
/
2
{\displaystyle N={\Big [}\sum _{i=1}^{n}c_{i}^{2}S_{ii}+\sum _{1\leq i\neq j\leq n}c_{i}c_{j}S_{ij}{\Big ]}^{-1/2}}
,
S
i
j
=
⟨
ϕ
i
|
ϕ
j
⟩
=
∫
R
3
ϕ
i
∗
ϕ
j
d
V
{\displaystyle S_{ij}=\langle \phi _{i}|\phi _{j}\rangle =\int _{\mathbb {R} ^{3}}\phi _{i}^{*}\,\phi _{j}\,dV}
, and
H
i
j
=
⟨
ϕ
i
|
H
^
|
ϕ
j
⟩
=
∫
R
3
ϕ
i
∗
H
^
ϕ
j
d
V
{\displaystyle H_{ij}=\langle \phi _{i}|{\hat {H}}|\phi _{j}\rangle =\int _{\mathbb {R} ^{3}}\phi _{i}^{*}\,{\hat {H}}\phi _{j}\,dV}
.
In the remainder of the derivation, we will assume that the atomic orbitals are real. (For the simple case of the Hückel theory, they will be the 2pz orbitals on carbon.) Thus,
S
i
j
=
S
j
i
∗
=
S
j
i
{\displaystyle S_{ij}=S_{ji}^{*}=S_{ji}}
, and because the Hamiltonian operator is hermitian,
H
i
j
=
H
j
i
∗
=
H
j
i
{\displaystyle H_{ij}=H_{ji}^{*}=H_{ji}}
. Setting
∂
E
/
∂
c
i
=
0
{\displaystyle \partial {E}/\partial {c_{i}}=0}
for
i
=
1
,
…
,
n
{\displaystyle i=1,\ldots ,n}
to minimize E and collecting terms, we obtain a system of n simultaneous equations
∑
j
=
1
n
c
j
(
H
i
j
−
E
S
i
j
)
=
0
(
i
=
1
,
⋯
,
n
)
{\displaystyle \sum _{j=1}^{n}c_{j}(H_{ij}-ES_{ij})=0\quad (i=1,\cdots ,n)}
.
When
i
≠
j
{\displaystyle i\neq j}
,
S
i
j
{\displaystyle S_{ij}}
and
H
i
j
{\displaystyle H_{ij}}
are called the overlap and resonance (or exchange) integrals, respectively, while
H
i
i
{\displaystyle H_{ii}}
is called the Coulomb integral, and
S
i
i
=
1
{\displaystyle S_{ii}=1}
simply expresses the fact that the
ϕ
i
{\displaystyle \phi _{i}}
are normalized. The n × n matrices
[
S
i
j
]
{\displaystyle [S_{ij}]}
and
[
H
i
j
]
{\displaystyle [H_{ij}]}
are known as the overlap and Hamiltonian matrices, respectively.
By a well-known result from linear algebra, nontrivial solutions
(
c
1
,
c
2
,
…
,
c
n
)
{\displaystyle (c_{1},c_{2},\ldots ,c_{n})}
to the above system of linear equations can only exist if the coefficient matrix
[
H
i
j
−
E
S
i
j
]
{\displaystyle [H_{ij}-ES_{ij}]}
is singular. Hence,
E
{\displaystyle E}
must have a value such that the determinant of the coefficient matrix vanishes:
d
e
t
(
[
H
i
j
−
E
S
i
j
]
)
=
0
{\displaystyle \mathrm {det} ([H_{ij}-ES_{ij}])=0}
. (*)
This determinant expression is known as the secular determinant and gives rise to a generalized eigenvalue problem. The variational theorem guarantees that the lowest value of
E
{\displaystyle E}
that gives rise to a nontrivial (that is, not all zero) solution vector
(
c
1
,
c
2
,
…
,
c
n
)
{\displaystyle (c_{1},c_{2},\ldots ,c_{n})}
represents the best LCAO approximation of the energy of the most stable π orbital; higher values of
E
{\displaystyle E}
with nontrivial solution vectors represent reasonable estimates of the energies of the remaining π orbitals.
The Hückel method makes a few further simplifying assumptions concerning the values of the
S
i
j
{\displaystyle S_{ij}}
and
H
i
j
{\displaystyle H_{ij}}
. In particular, it is first assumed that distinct
ϕ
i
{\displaystyle \phi _{i}}
have zero overlap. Together with the assumption that
ϕ
i
{\displaystyle \phi _{i}}
are normalized, this means that the overlap matrix is the n × n identity matrix:
[
S
i
j
]
=
I
n
{\displaystyle [S_{ij}]=\mathbf {I} _{n}}
. Solving for E in (*) then reduces to finding the eigenvalues of the Hamiltonian matrix.
Second, in the simplest case of a planar, unsaturated hydrocarbon, the Hamiltonian matrix
H
=
[
H
i
j
]
{\displaystyle \mathbf {H} =[H_{ij}]}
is parameterized in the following way:
H
i
j
=
{
α
,
i
=
j
;
β
,
i
,
j
adjacent
;
0
,
otherwise
.
{\displaystyle H_{ij}={\begin{cases}\alpha ,&i=j;\\\beta ,&i,j\ \ {\text{adjacent}};\\0,&{\text{otherwise}}.\end{cases}}}
(**)
To summarize, we are assuming that: (1) the energy of an electron in an isolated C(2pz) orbital is
H
i
i
=
α
{\displaystyle H_{ii}=\alpha }
; (2) the energy of interaction between C(2pz) orbitals on adjacent carbons i and j (i.e., i and j are connected by a σ-bond) is
H
i
j
=
β
{\displaystyle H_{ij}=\beta }
; (3) orbitals on carbons not joined in this way are assumed not to interact, so
H
i
j
=
0
{\displaystyle H_{ij}=0}
for nonadjacent i and j; and, as mentioned above, (4) the spatial overlap of electron density between different orbitals, represented by non-diagonal elements of the overlap matrix, is ignored by setting
S
i
j
=
0
(
i
≠
j
)
{\displaystyle S_{ij}=0\ \ (i\neq j)}
, even when the orbitals are adjacent.
This neglect of orbital overlap is an especially severe approximation. In actuality, orbital overlap is a prerequisite for orbital interaction, and it is impossible to have
H
i
j
=
β
{\displaystyle H_{ij}=\beta }
while
S
i
j
=
0
{\displaystyle S_{ij}=0}
. For typical bond distances (1.40 Å) as might be found in benzene, for example, the true value of the overlap for C(2pz) orbitals on adjacent atoms i and j is about
S
i
j
=
0.21
{\displaystyle S_{ij}=0.21}
; even larger values are found when the bond distance is shorter (e.g.,
S
i
j
=
0.27
{\displaystyle S_{ij}=0.27}
ethylene). A major consequence of having nonzero overlap integrals is the fact that, compared to non-interacting isolated orbitals, bonding orbitals are not energetically stabilized by nearly as much as antibonding orbitals are destabilized. The orbital energies derived from the Hückel treatment do not account for this asymmetry (see Hückel solution for ethylene (below) for details).
The eigenvalues of
H
{\displaystyle \mathbf {H} }
are the Hückel molecular orbital energies
E
1
,
…
,
E
n
{\displaystyle E_{1},\ldots ,E_{n}}
, expressed in terms of
α
{\displaystyle \alpha }
and
β
{\displaystyle \beta }
, while the eigenvectors are the Hückel MOs
Ψ
1
,
…
,
Ψ
n
{\displaystyle \Psi _{1},\ldots ,\Psi _{n}}
, expressed as linear combinations of the atomic orbitals
ϕ
i
{\displaystyle \phi _{i}}
. Using the expression for the normalization constant N and the fact that
[
S
i
j
]
=
I
n
{\displaystyle [S_{ij}]=\mathbf {I} _{n}}
, we can find the normalized MOs by incorporating the additional condition
∑
i
=
1
n
c
i
2
=
1
{\displaystyle \sum _{i=1}^{n}c_{i}^{2}=1}
.
The Hückel MOs are thus uniquely determined when eigenvalues are all distinct. When an eigenvalue is degenerate (two or more of the
E
i
{\displaystyle E_{i}}
are equal), the eigenspace corresponding to the degenerate energy level has dimension greater than 1, and the normalized MOs at that energy level are then not uniquely determined. When that happens, further assumptions pertaining to the coefficients of the degenerate orbitals (usually ones that make the MOs orthogonal and mathematically convenient) have to be made in order to generate a concrete set of molecular orbital functions.
If the substance is a planar, unsaturated hydrocarbon, the coefficients of the MOs can be found without appeal to empirical parameters, while orbital energies are given in terms of only
α
{\displaystyle \alpha }
and
β
{\displaystyle \beta }
. On the other hand, for systems containing heteroatoms, such as pyridine or formaldehyde, values of correction constants
h
X
{\displaystyle h_{\mathrm {X} }}
and
k
X
−
Y
{\displaystyle k_{\mathrm {X-Y} }}
have to be specified for the atoms and bonds in question, and
α
{\displaystyle \alpha }
and
β
{\displaystyle \beta }
in (**) are replaced by
α
+
h
X
β
{\displaystyle \alpha +h_{\mathrm {X} }\beta }
and
k
X
−
Y
β
{\displaystyle k_{\mathrm {X-Y} }\beta }
, respectively.
== Hückel solution for ethylene in detail ==
In the Hückel treatment for ethylene, we write the Hückel MOs
Ψ
{\displaystyle \Psi \,}
as a linear combination of the atomic orbitals (2p orbitals) on each of the carbon atoms:
Ψ
=
c
1
ϕ
1
+
c
2
ϕ
2
{\displaystyle \ \Psi =c_{1}\phi _{1}+c_{2}\phi _{2}}
.
Applying the result obtained by the Ritz method, we have the system of equations
[
H
11
−
E
S
11
H
12
−
E
S
12
H
21
−
E
S
21
H
22
−
E
S
22
]
[
c
1
c
2
]
=
0
{\displaystyle {\begin{bmatrix}H_{11}-ES_{11}&H_{12}-ES_{12}\\H_{21}-ES_{21}&H_{22}-ES_{22}\\\end{bmatrix}}{\begin{bmatrix}c_{1}\\c_{2}\\\end{bmatrix}}=0}
,
where:
H
i
j
=
⟨
ϕ
i
|
H
^
|
ϕ
j
⟩
{\displaystyle H_{ij}=\langle \phi _{i}|{\hat {H}}|\phi _{j}\rangle }
and
S
i
j
=
⟨
ϕ
i
|
ϕ
j
⟩
{\displaystyle S_{ij}=\langle \phi _{i}|\phi _{j}\rangle }
.
(Since 2pz atomic orbital can be expressed as a pure real function, the * representing complex conjugation can be dropped.) The Hückel method assumes that all overlap integrals (including the normalization integrals) equal the Kronecker delta,
S
i
j
=
δ
i
j
{\displaystyle S_{ij}=\delta _{ij}\,}
, all Coulomb integrals
H
i
i
{\displaystyle H_{ii}\,}
are equal, and the resonance integral
H
i
j
{\displaystyle H_{ij}\,}
is nonzero when the atoms i and j are bonded. Using the standard Hückel variable names, we set
H
11
=
H
22
=
α
{\displaystyle H_{11}=H_{22}=\alpha \,}
,
H
12
=
H
21
=
β
{\displaystyle H_{12}=H_{21}=\beta \,}
,
S
11
=
S
22
=
1
{\displaystyle S_{11}=S_{22}=1\,}
, and
S
12
=
S
21
=
0
{\displaystyle S_{12}=S_{21}=0\,}
.
The Hamiltonian matrix is
H
=
[
α
β
β
α
]
{\displaystyle \mathbf {H} ={\begin{bmatrix}\alpha &\beta \\\beta &\alpha \\\end{bmatrix}}}
.
The matrix equation that needs to be solved is then
[
α
−
E
β
β
α
−
E
]
[
c
1
c
2
]
=
0
{\displaystyle {\begin{bmatrix}\alpha -E&\beta \\\beta &\alpha -E\\\end{bmatrix}}{\begin{bmatrix}c_{1}\\c_{2}\\\end{bmatrix}}=0}
,
or, dividing by
β
{\displaystyle \beta }
,
[
α
−
E
β
1
1
α
−
E
β
]
[
c
1
c
2
]
=
0
{\displaystyle {\begin{bmatrix}{\frac {\alpha -E}{\beta }}&1\\1&{\frac {\alpha -E}{\beta }}\\\end{bmatrix}}{\begin{bmatrix}c_{1}\\c_{2}\\\end{bmatrix}}=0}
.
Setting
x
:=
α
−
E
β
{\displaystyle x:={\frac {\alpha -E}{\beta }}}
, we obtain
[
x
1
1
x
]
[
c
1
c
2
]
=
0
{\displaystyle {\begin{bmatrix}x&1\\1&x\\\end{bmatrix}}{\begin{bmatrix}c_{1}\\c_{2}\\\end{bmatrix}}=0}
. (***)
This homogeneous system of equations has nontrivial solutions for
c
1
,
c
2
{\displaystyle c_{1},c_{2}}
(solutions besides the physically meaningless
c
1
=
c
2
=
0
{\displaystyle c_{1}=c_{2}=0}
) iff the matrix is singular and the determinant is zero:
|
x
1
1
x
|
=
0
{\displaystyle {\begin{vmatrix}x&1\\1&x\\\end{vmatrix}}=0}
.
Solving for
x
{\displaystyle x}
,
x
2
−
1
=
0
{\displaystyle x^{2}-1=0\,}
, or
x
=
±
1
{\displaystyle x=\pm 1\,}
.
Since
E
=
α
−
x
β
{\displaystyle E=\alpha -x\beta }
, the energy levels are
E
=
α
−
±
1
×
β
{\displaystyle E=\alpha -\pm 1\times \beta }
, or
E
=
α
∓
β
{\displaystyle E=\alpha \mp \beta }
.
The coefficients can then be found by expanding (***):
c
2
=
−
x
c
1
{\displaystyle c_{2}=-xc_{1}\,}
and
c
1
=
−
x
c
2
{\displaystyle c_{1}=-xc_{2}\,}
.
Since the matrix is singular, the two equations are linearly dependent, and the solution set is not uniquely determined until we apply the normalization condition. We can only solve for
c
2
{\displaystyle c_{2}}
in terms of
c
1
{\displaystyle c_{1}}
:
c
2
=
−
±
1
×
c
1
{\displaystyle c_{2}=-\pm 1\times c_{1}\,}
, or
c
2
=
∓
c
1
{\displaystyle c_{2}=\mp c_{1}\,}
.
After normalization with
c
1
2
+
c
2
2
=
1
{\displaystyle c_{1}^{2}+c_{2}^{2}=1}
, the numerical values of
c
1
{\displaystyle c_{1}}
and
c
2
{\displaystyle c_{2}}
can be found:
c
1
=
1
2
{\displaystyle c_{1}={\frac {1}{\sqrt {2}}}}
and
c
2
=
∓
1
2
{\displaystyle c_{2}=\mp {\frac {1}{\sqrt {2}}}}
.
Finally, the Hückel molecular orbitals are
Ψ
∓
=
c
1
ϕ
1
+
c
2
ϕ
2
=
1
2
ϕ
1
∓
1
2
ϕ
2
=
ϕ
1
∓
ϕ
2
2
{\displaystyle \Psi _{\mp }=c_{1}\phi _{1}+c_{2}\phi _{2}={\frac {1}{\sqrt {2}}}\phi _{1}\mp {\frac {1}{\sqrt {2}}}\phi _{2}={\frac {\phi _{1}\mp \phi _{2}}{\sqrt {2}}}\,}
.
The constant β in the energy term is negative; therefore,
E
+
=
α
+
β
{\displaystyle E_{+}=\alpha +\beta }
with
Ψ
+
=
1
2
(
ϕ
1
+
ϕ
2
)
{\textstyle \Psi _{+}={\frac {1}{\sqrt {2}}}(\phi _{1}+\phi _{2})\,}
is the lower energy corresponding to the HOMO energy and
E
−
=
α
−
β
{\displaystyle E_{-}=\alpha -\beta }
with
Ψ
−
=
1
2
(
ϕ
1
−
ϕ
2
)
{\textstyle \Psi _{-}={\frac {1}{\sqrt {2}}}(\phi _{1}-\phi _{2})\,}
is the LUMO energy.
If, contrary to the Hückel treatment, a positive value for
S
:=
S
12
=
S
21
{\displaystyle S:=S_{12}=S_{21}}
were included, the energies would instead be
E
±
=
α
±
β
1
±
S
{\displaystyle E_{\pm }={\frac {\alpha \pm \beta }{1\pm S}}}
,
while the corresponding orbitals would take the form
Ψ
±
=
1
2
±
2
S
ϕ
1
±
1
2
±
2
S
ϕ
2
{\displaystyle \Psi _{\pm }={\sqrt {\frac {1}{2\pm 2S}}}\phi _{1}\pm {\sqrt {\frac {1}{2\pm 2S}}}\phi _{2}}
.
An important consequence of setting
S
>
0
{\displaystyle S>0}
is that the bonding (in-phase) combination is always stabilized to a lesser extent than the antibonding (out-of-phase) combination is destabilized, relative to the energy of the free 2p orbital. Thus, in general, 2-center 4-electron interactions, where both the bonding and antibonding orbitals are occupied, are destabilizing overall. This asymmetry is ignored by Hückel theory. In general, for the orbital energies derived from Hückel theory, the sum of stabilization energies for the bonding orbitals is equal to the sum of destabilization energies for the antibonding orbitals, as in the simplest case of ethylene shown here and the case of butadiene shown below.
== Hückel solution for 1,3-butadiene ==
The Hückel MO theory treatment of 1,3-butadiene is largely analogous to the treatment of ethylene, shown in detail above, though we must now find the eigenvalues and eigenvectors of a 4 × 4 Hamiltonian matrix. We first write the molecular orbital
Ψ
{\displaystyle \Psi \,}
as a linear combination of the four atomic orbitals
ϕ
i
{\displaystyle \phi _{i}}
(carbon 2p orbitals) with coefficients
c
i
{\displaystyle c_{i}}
:
Ψ
=
c
1
ϕ
1
+
c
2
ϕ
2
+
c
3
ϕ
3
+
c
4
ϕ
4
{\displaystyle \ \Psi =c_{1}\phi _{1}+c_{2}\phi _{2}+c_{3}\phi _{3}+c_{4}\phi _{4}}
.
The Hamiltonian matrix is
H
=
[
α
β
0
0
β
α
β
0
0
β
α
β
0
0
β
α
]
{\displaystyle \mathbf {H} ={\begin{bmatrix}\alpha &\beta &0&0\\\beta &\alpha &\beta &0\\0&\beta &\alpha &\beta \\0&0&\beta &\alpha \\\end{bmatrix}}}
.
In the same way, we write the secular equations in matrix form as
[
α
−
E
β
0
0
β
α
−
E
β
0
0
β
α
−
E
β
0
0
β
α
−
E
]
[
c
1
c
2
c
3
c
4
]
=
0
{\displaystyle {\begin{bmatrix}\alpha -E&\beta &0&0\\\beta &\alpha -E&\beta &0\\0&\beta &\alpha -E&\beta \\0&0&\beta &\alpha -E\\\end{bmatrix}}{\begin{bmatrix}c_{1}\\c_{2}\\c_{3}\\c_{4}\\\end{bmatrix}}=0}
,
which leads to
(
α
−
E
β
)
4
−
3
(
α
−
E
β
)
2
+
1
=
0
{\displaystyle {\Big (}{\frac {\alpha -E}{\beta }}{\Big )}^{4}-3{\Big (}{\frac {\alpha -E}{\beta }}{\Big )}^{2}+1=0}
and
E
1
,
2
,
3
,
4
=
α
+
5
±
1
2
β
,
α
−
5
∓
1
2
β
{\displaystyle E_{1,2,3,4}=\alpha +{\frac {{\sqrt {5}}\pm 1}{2}}\beta ,\alpha -{\frac {{\sqrt {5}}\mp 1}{2}}\beta }
, or approximately,
E
1
,
2
,
3
,
4
≈
α
+
1.618
β
,
α
+
0.618
β
,
α
−
0.618
β
,
α
−
1.618
β
{\displaystyle E_{1,2,3,4}\approx \alpha +1.618\beta ,\alpha +0.618\beta ,\alpha -0.618\beta ,\alpha -1.618\beta }
, where 1.618... and 0.618... are the golden ratios
φ
{\displaystyle \varphi }
and
1
/
φ
{\displaystyle 1/\varphi }
.
The orbitals are given by
Ψ
1
≈
0.372
ϕ
1
+
0.602
ϕ
2
+
0.602
ϕ
3
+
0.372
ϕ
4
{\displaystyle \Psi _{1}\approx 0.372\phi _{1}+0.602\phi _{2}+0.602\phi _{3}+0.372\phi _{4}}
,
Ψ
2
≈
0.602
ϕ
1
+
0.372
ϕ
2
−
0.372
ϕ
3
−
0.602
ϕ
4
{\displaystyle \Psi _{2}\approx 0.602\phi _{1}+0.372\phi _{2}-0.372\phi _{3}-0.602\phi _{4}}
,
Ψ
3
≈
0.602
ϕ
1
−
0.372
ϕ
2
−
0.372
ϕ
3
+
0.602
ϕ
4
{\displaystyle \Psi _{3}\approx 0.602\phi _{1}-0.372\phi _{2}-0.372\phi _{3}+0.602\phi _{4}}
, and
Ψ
4
≈
0.372
ϕ
1
−
0.602
ϕ
2
+
0.602
ϕ
3
−
0.372
ϕ
4
{\displaystyle \Psi _{4}\approx 0.372\phi _{1}-0.602\phi _{2}+0.602\phi _{3}-0.372\phi _{4}}
.
== See also ==
Möbius–Hückel concept
Möbius aromaticity
Tight Binding
== External links ==
"Hückel method" at chem.swin.edu.au, webpage: mod3-huckel.
N. Goudard; Y. Carissan; D. Hagebaum-Reignier; S. Humbel (2008). "HuLiS : Java Applet – Simple Hückel Theory and Mesomery – program logiciel software" (in French). Retrieved 19 August 2010.
Rauk, Arvi. SHMO, Simple Hückel Molecular Orbital Theory Calculator. Java Applet (downloadable) Archived 2018-06-22 at the Wayback Machine.
== Further reading ==
The HMO-Model and its applications: Basis and Manipulation, E. Heilbronner and H. Bock, English translation, 1976, Verlag Chemie.
The HMO-Model and its applications: Problems with Solutions, E. Heilbronner and H. Bock, English translation, 1976, Verlag Chemie.
The HMO-Model and its applications: Tables of Hückel Molecular Orbitals, E. Heilbronner and H. Bock, English translation, 1976, Verlag Chemie.
== References == | Wikipedia/Hückel_molecular_orbital_method |
In physics, mass–energy equivalence is the relationship between mass and energy in a system's rest frame. The two differ only by a multiplicative constant and the units of measurement. The principle is described by the physicist Albert Einstein's formula:
E
=
m
c
2
{\displaystyle E=mc^{2}}
. In a reference frame where the system is moving, its relativistic energy and relativistic mass (instead of rest mass) obey the same formula.
The formula defines the energy (E) of a particle in its rest frame as the product of mass (m) with the speed of light squared (c2). Because the speed of light is a large number in everyday units (approximately 300000 km/s or 186000 mi/s), the formula implies that a small amount of mass corresponds to an enormous amount of energy.
Rest mass, also called invariant mass, is a fundamental physical property of matter, independent of velocity. Massless particles such as photons have zero invariant mass, but massless free particles have both momentum and energy.
The equivalence principle implies that when mass is lost in chemical reactions or nuclear reactions, a corresponding amount of energy will be released. The energy can be released to the environment (outside of the system being considered) as radiant energy, such as light, or as thermal energy. The principle is fundamental to many fields of physics, including nuclear and particle physics.
Mass–energy equivalence arose from special relativity as a paradox described by the French polymath Henri Poincaré (1854–1912). Einstein was the first to propose the equivalence of mass and energy as a general principle and a consequence of the symmetries of space and time. The principle first appeared in "Does the inertia of a body depend upon its energy-content?", one of his annus mirabilis papers, published on 21 November 1905. The formula and its relationship to momentum, as described by the energy–momentum relation, were later developed by other physicists.
== Description ==
Mass–energy equivalence states that all objects having mass, or massive objects, have a corresponding intrinsic energy, even when they are stationary. In the rest frame of an object, where by definition it is motionless and so has no momentum, the mass and energy are equal or they differ only by a constant factor, the speed of light squared (c2). In Newtonian mechanics, a motionless body has no kinetic energy, and it may or may not have other amounts of internal stored energy, like chemical energy or thermal energy, in addition to any potential energy it may have from its position in a field of force. These energies tend to be much smaller than the mass of the object multiplied by c2, which is on the order of 1017 joules for a mass of one kilogram. Due to this principle, the mass of the atoms that come out of a nuclear reaction is less than the mass of the atoms that go in, and the difference in mass shows up as heat and light with the same equivalent energy as the difference. In analyzing these extreme events, Einstein's formula can be used with E as the energy released (removed), and m as the change in mass.
In relativity, all the energy that moves with an object (i.e., the energy as measured in the object's rest frame) contributes to the total mass of the body, which measures how much it resists acceleration. If an isolated box of ideal mirrors could contain light, the individually massless photons would contribute to the total mass of the box by the amount equal to their energy divided by c2. For an observer in the rest frame, removing energy is the same as removing mass and the formula m = E/c2 indicates how much mass is lost when energy is removed. In the same way, when any energy is added to an isolated system, the increase in the mass is equal to the added energy divided by c2.
== Mass in special relativity ==
An object moves at different speeds in different frames of reference, depending on the motion of the observer. This implies the kinetic energy, in both Newtonian mechanics and relativity, is 'frame dependent', so that the amount of relativistic energy that an object is measured to have depends on the observer. The relativistic mass of an object is given by the relativistic energy divided by c2. Because the relativistic mass is exactly proportional to the relativistic energy, relativistic mass and relativistic energy are nearly synonymous; the only difference between them is the units. The rest mass or invariant mass of an object is defined as the mass an object has in its rest frame, when it is not moving with respect to the observer. The rest mass is the same for all inertial frames, as it is independent of the motion of the observer, it is the smallest possible value of the relativistic mass of the object. Because of the attraction between components of a system, which results in potential energy, the rest mass is almost never additive; in general, the mass of an object is not the sum of the masses of its parts. The rest mass of an object is the total energy of all the parts, including kinetic energy, as observed from the center of momentum frame, and potential energy. The masses add up only if the constituents are at rest (as observed from the center of momentum frame) and do not attract or repel, so that they do not have any extra kinetic or potential energy. Massless particles are particles with no rest mass, and therefore have no intrinsic energy; their energy is due only to their momentum.
=== Relativistic mass ===
Relativistic mass depends on the motion of the object, so that different observers in relative motion see different values for it. The relativistic mass of a moving object is larger than the relativistic mass of an object at rest, because a moving object has kinetic energy. If the object moves slowly, the relativistic mass is nearly equal to the rest mass and both are nearly equal to the classical inertial mass (as it appears in Newton's laws of motion). If the object moves quickly, the relativistic mass is greater than the rest mass by an amount equal to the mass associated with the kinetic energy of the object. Massless particles also have relativistic mass derived from their kinetic energy, equal to their relativistic energy divided by c2, or mrel = E/c2. The speed of light is one in a system where length and time are measured in natural units and the relativistic mass and energy would be equal in value and dimension. As it is just another name for the energy, the use of the term relativistic mass is redundant and physicists generally reserve mass to refer to rest mass, or invariant mass, as opposed to relativistic mass. A consequence of this terminology is that the mass is not conserved in special relativity, whereas the conservation of momentum and conservation of energy are both fundamental laws.
=== Conservation of mass and energy ===
Conservation of energy is a universal principle in physics and holds for any interaction, along with the conservation of momentum. The classical conservation of mass, in contrast, is violated in certain relativistic settings. This concept has been experimentally proven in a number of ways, including the conversion of mass into kinetic energy in nuclear reactions and other interactions between elementary particles. While modern physics has discarded the expression 'conservation of mass', in older terminology a relativistic mass can also be defined to be equivalent to the energy of a moving system, allowing for a conservation of relativistic mass. Mass conservation breaks down when the energy associated with the mass of a particle is converted into other forms of energy, such as kinetic energy, thermal energy, or radiant energy.
=== Massless particles ===
Massless particles have zero rest mass. The Planck–Einstein relation for the energy for photons is given by the equation E = hf, where h is the Planck constant and f is the photon frequency. This frequency and thus the relativistic energy are frame-dependent. If an observer runs away from a photon in the direction the photon travels from a source, and it catches up with the observer, the observer sees it as having less energy than it had at the source. The faster the observer is traveling with regard to the source when the photon catches up, the less energy the photon would be seen to have. As an observer approaches the speed of light with regard to the source, the redshift of the photon increases, according to the relativistic Doppler effect. The energy of the photon is reduced and as the wavelength becomes arbitrarily large, the photon's energy approaches zero, because of the massless nature of photons, which does not permit any intrinsic energy.
=== Composite systems ===
For closed systems made up of many parts, like an atomic nucleus, planet, or star, the relativistic energy is given by the sum of the relativistic energies of each of the parts, because energies are additive in these systems. If a system is bound by attractive forces, and the energy gained in excess of the work done is removed from the system, then mass is lost with this removed energy. The mass of an atomic nucleus is less than the total mass of the protons and neutrons that make it up. This mass decrease is also equivalent to the energy required to break up the nucleus into individual protons and neutrons. This effect can be understood by looking at the potential energy of the individual components. The individual particles have a force attracting them together, and forcing them apart increases the potential energy of the particles in the same way that lifting an object up on earth does. This energy is equal to the work required to split the particles apart. The mass of the Solar System is slightly less than the sum of its individual masses.
For an isolated system of particles moving in different directions, the invariant mass of the system is the analog of the rest mass, and is the same for all observers, even those in relative motion. It is defined as the total energy (divided by c2) in the center of momentum frame. The center of momentum frame is defined so that the system has zero total momentum; the term center of mass frame is also sometimes used, where the center of mass frame is a special case of the center of momentum frame where the center of mass is put at the origin. A simple example of an object with moving parts but zero total momentum is a container of gas. In this case, the mass of the container is given by its total energy (including the kinetic energy of the gas molecules), since the system's total energy and invariant mass are the same in any reference frame where the momentum is zero, and such a reference frame is also the only frame in which the object can be weighed. In a similar way, the theory of special relativity posits that the thermal energy in all objects, including solids, contributes to their total masses, even though this energy is present as the kinetic and potential energies of the atoms in the object, and it (in a similar way to the gas) is not seen in the rest masses of the atoms that make up the object. Similarly, even photons, if trapped in an isolated container, would contribute their energy to the mass of the container. Such extra mass, in theory, could be weighed in the same way as any other type of rest mass, even though individually photons have no rest mass. The property that trapped energy in any form adds weighable mass to systems that have no net momentum is one of the consequences of relativity. It has no counterpart in classical Newtonian physics, where energy never exhibits weighable mass.
=== Relation to gravity ===
Physics has two concepts of mass, the gravitational mass and the inertial mass. The gravitational mass is the quantity that determines the strength of the gravitational field generated by an object, as well as the gravitational force acting on the object when it is immersed in a gravitational field produced by other bodies. The inertial mass, on the other hand, quantifies how much an object accelerates if a given force is applied to it. The mass–energy equivalence in special relativity refers to the inertial mass. However, already in the context of Newtonian gravity, the weak equivalence principle is postulated: the gravitational and the inertial mass of every object are the same. Thus, the mass–energy equivalence, combined with the weak equivalence principle, results in the prediction that all forms of energy contribute to the gravitational field generated by an object. This observation is one of the pillars of the general theory of relativity.
The prediction that all forms of energy interact gravitationally has been subject to experimental tests. One of the first observations testing this prediction, called the Eddington experiment, was made during the solar eclipse of May 29, 1919. During the eclipse, the English astronomer and physicist Arthur Eddington observed that the light from stars passing close to the Sun was bent. The effect is due to the gravitational attraction of light by the Sun. The observation confirmed that the energy carried by light indeed is equivalent to a gravitational mass. Another seminal experiment, the Pound–Rebka experiment, was performed in 1960. In this test a beam of light was emitted from the top of a tower and detected at the bottom. The frequency of the light detected was higher than the light emitted. This result confirms that the energy of photons increases when they fall in the gravitational field of the Earth. The energy, and therefore the gravitational mass, of photons is proportional to their frequency as stated by the Planck's relation.
== Efficiency ==
In some reactions, matter particles can be destroyed and their associated energy released to the environment as other forms of energy, such as light and heat. One example of such a conversion takes place in elementary particle interactions, where the rest energy is transformed into kinetic energy. Such conversions between types of energy happen in nuclear weapons, in which the protons and neutrons in atomic nuclei lose a small fraction of their original mass, though the mass lost is not due to the destruction of any smaller constituents. Nuclear fission allows a tiny fraction of the energy associated with the mass to be converted into usable energy such as radiation; in the decay of the uranium, for instance, about 0.1% of the mass of the original atom is lost. In theory, it should be possible to destroy matter and convert all of the rest-energy associated with matter into heat and light, but none of the theoretically known methods are practical. One way to harness all the energy associated with mass is to annihilate matter with antimatter. Antimatter is rare in the universe, however, and the known mechanisms of production require more usable energy than would be released in annihilation. CERN estimated in 2011 that over a billion times more energy is required to make and store antimatter than could be released in its annihilation.
As most of the mass which comprises ordinary objects resides in protons and neutrons, converting all the energy of ordinary matter into more useful forms requires that the protons and neutrons be converted to lighter particles, or particles with no mass at all. In the Standard Model of particle physics, the number of protons plus neutrons is nearly exactly conserved. Despite this, Gerard 't Hooft showed that there is a process that converts protons and neutrons to antielectrons and neutrinos. This is the weak SU(2) instanton proposed by the physicists Alexander Belavin, Alexander Markovich Polyakov, Albert Schwarz, and Yu. S. Tyupkin. This process, can in principle destroy matter and convert all the energy of matter into neutrinos and usable energy, but it is normally extraordinarily slow. It was later shown that the process occurs rapidly at extremely high temperatures that would only have been reached shortly after the Big Bang.
Many extensions of the standard model contain magnetic monopoles, and in some models of grand unification, these monopoles catalyze proton decay, a process known as the Callan–Rubakov effect. This process would be an efficient mass–energy conversion at ordinary temperatures, but it requires making monopoles and anti-monopoles, whose production is expected to be inefficient. Another method of completely annihilating matter uses the gravitational field of black holes. The British theoretical physicist Stephen Hawking theorized it is possible to throw matter into a black hole and use the emitted heat to generate power. According to the theory of Hawking radiation, however, larger black holes radiate less than smaller ones, so that usable power can only be produced by small black holes.
== Extension for systems in motion ==
Unlike a system's energy in an inertial frame, the relativistic energy (
E
r
e
l
{\displaystyle E_{\rm {rel}}}
) of a system depends on both the rest mass (
m
0
{\displaystyle m_{0}}
) and the total momentum of the system. The extension of Einstein's equation to these systems is given by:
E
r
e
l
2
−
|
p
|
2
c
2
=
m
0
2
c
4
{\displaystyle {\begin{aligned}E_{\rm {rel}}^{2}-|\mathbf {p} |^{2}c^{2}&=m_{0}^{2}c^{4}\\\end{aligned}}}
or
E
r
e
l
2
−
(
p
c
)
2
=
(
m
0
c
2
)
2
{\displaystyle {\begin{aligned}E_{\rm {rel}}^{2}-(pc)^{2}&=(m_{0}c^{2})^{2}\\\end{aligned}}}
or
E
r
e
l
=
(
m
0
c
2
)
2
+
(
p
c
)
2
{\displaystyle {\begin{aligned}E_{\rm {rel}}={\sqrt {(m_{0}c^{2})^{2}+(pc)^{2}}}\,\!\end{aligned}}}
where the
(
p
c
)
2
{\displaystyle (pc)^{2}}
term represents the square of the Euclidean norm (total vector length) of the various momentum vectors in the system, which reduces to the square of the simple momentum magnitude, if only a single particle is considered. This equation is called the energy–momentum relation and reduces to
E
r
e
l
=
m
c
2
{\displaystyle E_{\rm {rel}}=mc^{2}}
when the momentum term is zero. For photons where
m
0
=
0
{\displaystyle m_{0}=0}
, the equation reduces to
E
r
e
l
=
p
c
{\displaystyle E_{\rm {rel}}=pc}
.
== Low-speed approximation ==
Using the Lorentz factor, γ, the energy–momentum can be rewritten as E = γmc2 and expanded as a power series:
E
=
m
0
c
2
[
1
+
1
2
(
v
c
)
2
+
3
8
(
v
c
)
4
+
5
16
(
v
c
)
6
+
…
]
.
{\displaystyle E=m_{0}c^{2}\left[1+{\frac {1}{2}}\left({\frac {v}{c}}\right)^{2}+{\frac {3}{8}}\left({\frac {v}{c}}\right)^{4}+{\frac {5}{16}}\left({\frac {v}{c}}\right)^{6}+\ldots \right].}
For speeds much smaller than the speed of light, higher-order terms in this expression get smaller and smaller because v/c is small. For low speeds, all but the first two terms can be ignored:
E
≈
m
0
c
2
+
1
2
m
0
v
2
.
{\displaystyle E\approx m_{0}c^{2}+{\frac {1}{2}}m_{0}v^{2}.}
In classical mechanics, both the m0c2 term and the high-speed corrections are ignored. The initial value of the energy is arbitrary, as only the change in energy can be measured and so the m0c2 term is ignored in classical physics. While the higher-order terms become important at higher speeds, the Newtonian equation is a highly accurate low-speed approximation; adding in the third term yields:
E
≈
m
0
c
2
+
1
2
m
0
v
2
(
1
+
3
v
2
4
c
2
)
{\displaystyle E\approx m_{0}c^{2}+{\frac {1}{2}}m_{0}v^{2}\left(1+{\frac {3v^{2}}{4c^{2}}}\right)}
.
The difference between the two approximations is given by
3
v
2
4
c
2
{\displaystyle {\tfrac {3v^{2}}{4c^{2}}}}
, a number very small for everyday objects. In 2018 NASA announced the Parker Solar Probe was the fastest ever, with a speed of 153,454 miles per hour (68,600 m/s). The difference between the approximations for the Parker Solar Probe in 2018 is
3
v
2
4
c
2
≈
3.9
×
10
−
8
{\displaystyle {\tfrac {3v^{2}}{4c^{2}}}\approx 3.9\times 10^{-8}}
, which accounts for an energy correction of four parts per hundred million. The gravitational constant, in contrast, has a standard relative uncertainty of about
2.2
×
10
−
5
{\displaystyle 2.2\times 10^{-5}}
.
== Applications ==
=== Application to nuclear physics ===
The nuclear binding energy is the minimum energy that is required to disassemble the nucleus of an atom into its component parts. The mass of an atom is less than the sum of the masses of its constituents due to the attraction of the strong nuclear force. The difference between the two masses is called the mass defect and is related to the binding energy through Einstein's formula. The principle is used in modeling nuclear fission reactions, and it implies that a great amount of energy can be released by the nuclear fission chain reactions used in both nuclear weapons and nuclear power.
A water molecule weighs a little less than two free hydrogen atoms and an oxygen atom. The minuscule mass difference is the energy needed to split the molecule into three individual atoms (divided by c2), which was given off as heat when the molecule formed (this heat had mass). Similarly, a stick of dynamite in theory weighs a little bit more than the fragments after the explosion; in this case the mass difference is the energy and heat that is released when the dynamite explodes. Such a change in mass may only happen when the system is open, and the energy and mass are allowed to escape. Thus, if a stick of dynamite is detonated in a hermetically sealed chamber, the mass of the chamber and fragments, the heat, sound, and light would still be equal to the original mass of the chamber and dynamite. If sitting on a scale, the weight and mass would not change. This would in theory also happen even with a nuclear bomb, if it could be kept in an ideal box of infinite strength, which did not rupture or pass radiation. Thus, a 21.5 kiloton (9×1013 joule) nuclear bomb produces about one gram of heat and electromagnetic radiation, but the mass of this energy would not be detectable in an exploded bomb in an ideal box sitting on a scale; instead, the contents of the box would be heated to millions of degrees without changing total mass and weight. If a transparent window passing only electromagnetic radiation were opened in such an ideal box after the explosion, and a beam of X-rays and other lower-energy light allowed to escape the box, it would eventually be found to weigh one gram less than it had before the explosion. This weight loss and mass loss would happen as the box was cooled by this process, to room temperature. However, any surrounding mass that absorbed the X-rays (and other "heat") would gain this gram of mass from the resulting heating, thus, in this case, the mass "loss" would represent merely its relocation.
=== Practical examples ===
Einstein used the centimetre–gram–second system of units (cgs), but the formula is independent of the system of units. In natural units, the numerical value of the speed of light is set to equal 1, and the formula expresses an equality of numerical values: E = m. In the SI system (expressing the ratio E/m in joules per kilogram using the value of c in metres per second):
E/m = c2 = (299792458 m/s)2 = 89875517873681764 J/kg (≈ 9.0 × 1016 joules per kilogram).
So the energy equivalent of one kilogram of mass is
89.9 petajoules
25.0 billion kilowatt-hours (or 25,000 GW·h)
21.5 trillion kilocalories (or 21.5 Pcal)
85.2 trillion BTUs (or 0.0852 quads)
or the energy released by combustion of any of the following:
21 500 kilotons of TNT-equivalent energy (or 21.5 Mt)
2630000000 litres or 695000000 US gallons of automotive gasoline
Any time energy is released, the process can be evaluated from an E = mc2 perspective. For instance, the "gadget"-style bomb used in the Trinity test and the bombing of Nagasaki had an explosive yield equivalent to 21 kt of TNT. About 1 kg of the approximately 6.15 kg of plutonium in each of these bombs fissioned into lighter elements totaling almost exactly one gram less, after cooling. The electromagnetic radiation and kinetic energy (thermal and blast energy) released in this explosion carried the missing gram of mass.
Whenever energy is added to a system, the system gains mass, as shown when the equation is rearranged:
A spring's mass increases whenever it is put into compression or tension. Its mass increase arises from the increased potential energy stored within it, which is bound in the stretched chemical (electron) bonds linking the atoms within the spring.
Raising the temperature of an object (increasing its thermal energy) increases its mass. For example, consider the world's primary mass standard for the kilogram, made of platinum and iridium. If its temperature is allowed to change by 1 °C, its mass changes by 1.5 picograms (1 pg = 1×10−12 g).
A spinning ball has greater mass than when it is not spinning. Its increase of mass is exactly the equivalent of the mass of energy of rotation, which is itself the sum of the kinetic energies of all the moving parts of the ball. For example, the Earth itself is more massive due to its rotation, than it would be with no rotation. The rotational energy of the Earth is greater than 1024 Joules, which is over 107 kg.
== History ==
While Einstein was the first to have correctly deduced the mass–energy equivalence formula, he was not the first to have related energy with mass, though nearly all previous authors thought that the energy that contributes to mass comes only from electromagnetic fields. Once discovered, Einstein's formula was initially written in many different notations, and its interpretation and justification was further developed in several steps.
=== Developments prior to Einstein ===
Eighteenth century theories on the correlation of mass and energy included that devised by the English scientist Isaac Newton in 1717, who speculated that light particles and matter particles were interconvertible in "Query 30" of the Opticks, where he asks: "Are not the gross bodies and light convertible into one another, and may not bodies receive much of their activity from the particles of light which enter their composition?" Swedish scientist and theologian Emanuel Swedenborg, in his Principia of 1734 theorized that all matter is ultimately composed of dimensionless points of "pure and total motion". He described this motion as being without force, direction or speed, but having the potential for force, direction and speed everywhere within it.
During the nineteenth century there were several speculative attempts to show that mass and energy were proportional in various ether theories. In 1873 the Russian physicist and mathematician Nikolay Umov pointed out a relation between mass and energy for ether in the form of Е = kmc2, where 0.5 ≤ k ≤ 1. English engineer Samuel Tolver Preston in 1875 and the Italian industrialist and geologist Olinto De Pretto in 1903, following physicist Georges-Louis Le Sage, imagined that the universe was filled with an ether of tiny particles that always move at speed c. Each of these particles has a kinetic energy of mc2 up to a small numerical factor, giving a mass–energy relation.
In 1905, independently of Einstein, French polymath Gustave Le Bon speculated that atoms could release large amounts of latent energy, reasoning from an all-encompassing qualitative philosophy of physics.
==== Electromagnetic mass ====
There were many attempts in the 19th and the beginning of the 20th century—like those of British physicists J. J. Thomson in 1881 and Oliver Heaviside in 1889, and George Frederick Charles Searle in 1897, German physicists Wilhelm Wien in 1900 and Max Abraham in 1902, and the Dutch physicist Hendrik Antoon Lorentz in 1904—to understand how the mass of a charged object depends on the electrostatic field. This concept was called electromagnetic mass, and was considered as being dependent on velocity and direction as well. Lorentz in 1904 gave the following expressions for longitudinal and transverse electromagnetic mass:
m
L
=
m
0
(
1
−
v
2
c
2
)
3
,
m
T
=
m
0
1
−
v
2
c
2
{\displaystyle m_{L}={\frac {m_{0}}{\left({\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\right)^{3}}},\quad m_{T}={\frac {m_{0}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}
,
where
m
0
=
4
3
E
e
m
c
2
{\displaystyle m_{0}={\frac {4}{3}}{\frac {E_{em}}{c^{2}}}}
Another way of deriving a type of electromagnetic mass was based on the concept of radiation pressure. In 1900, French polymath Henri Poincaré associated electromagnetic radiation energy with a "fictitious fluid" having momentum and mass
m
e
m
=
E
e
m
c
2
.
{\displaystyle m_{em}={\frac {E_{em}}{c^{2}}}\,.}
By that, Poincaré tried to save the center of mass theorem in Lorentz's theory, though his treatment led to radiation paradoxes.
Austrian physicist Friedrich Hasenöhrl showed in 1904 that electromagnetic cavity radiation contributes the "apparent mass"
m
0
=
4
3
E
e
m
c
2
{\displaystyle m_{0}={\frac {4}{3}}{\frac {E_{em}}{c^{2}}}}
to the cavity's mass. He argued that this implies mass dependence on temperature as well.
=== Einstein: mass–energy equivalence ===
Einstein did not write the exact formula E = mc2 in his 1905 Annus Mirabilis paper "Does the Inertia of an object Depend Upon Its Energy Content?"; rather, the paper states that if a body gives off the energy L by emitting light, its mass diminishes by L/c2. This formulation relates only a change Δm in mass to a change L in energy without requiring the absolute relationship. The relationship convinced him that mass and energy can be seen as two names for the same underlying, conserved physical quantity. He has stated that the laws of conservation of energy and conservation of mass are "one and the same". Einstein elaborated in a 1946 essay that "the principle of the conservation of mass… proved inadequate in the face of the special theory of relativity. It was therefore merged with the energy conservation principle—just as, about 60 years before, the principle of the conservation of mechanical energy had been combined with the principle of the conservation of heat [thermal energy]. We might say that the principle of the conservation of energy, having previously swallowed up that of the conservation of heat, now proceeded to swallow that of the conservation of mass—and holds the field alone."
==== Mass–velocity relationship ====
In developing special relativity, Einstein found that the kinetic energy of a moving body is
E
k
=
m
0
c
2
(
γ
−
1
)
=
m
0
c
2
(
1
1
−
v
2
c
2
−
1
)
,
{\displaystyle E_{k}=m_{0}c^{2}(\gamma -1)=m_{0}c^{2}\left({\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}-1\right),}
with v the velocity, m0 the rest mass, and γ the Lorentz factor.
He included the second term on the right to make sure that for small velocities the energy would be the same as in classical mechanics, thus satisfying the correspondence principle:
E
k
=
1
2
m
0
v
2
+
⋯
{\displaystyle E_{k}={\frac {1}{2}}m_{0}v^{2}+\cdots }
Without this second term, there would be an additional contribution in the energy when the particle is not moving.
==== Einstein's view on mass ====
Einstein, following Lorentz and Abraham, used velocity- and direction-dependent mass concepts in his 1905 electrodynamics paper and in another paper in 1906. In Einstein's first 1905 paper on E = mc2, he treated m as what would now be called the rest mass, and it has been noted that in his later years he did not like the idea of "relativistic mass".
In modern physics terminology, relativistic energy is used in lieu of relativistic mass and the term "mass" is reserved for the rest mass. Historically, there has been considerable debate over the use of the concept of "relativistic mass" and the connection of "mass" in relativity to "mass" in Newtonian dynamics. One view is that only rest mass is a viable concept and is a property of the particle; while relativistic mass is a conglomeration of particle properties and properties of spacetime. Another view, attributed to Norwegian physicist Kjell Vøyenli, is that the Newtonian concept of mass as a particle property and the relativistic concept of mass have to be viewed as embedded in their own theories and as having no precise connection.
==== Einstein's 1905 derivation ====
Already in his relativity paper "On the electrodynamics of moving bodies", Einstein derived the correct expression for the kinetic energy of particles:
E
k
=
m
c
2
(
1
1
−
v
2
c
2
−
1
)
{\displaystyle E_{k}=mc^{2}\left({\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}-1\right)}
.
Now the question remained open as to which formulation applies to bodies at rest. This was tackled by Einstein in his paper "Does the inertia of a body depend upon its energy content?", one of his Annus Mirabilis papers. Here, Einstein used V to represent the speed of light in vacuum and L to represent the energy lost by a body in the form of radiation. Consequently, the equation E = mc2 was not originally written as a formula but as a sentence in German saying that "if a body gives off the energy L in the form of radiation, its mass diminishes by L/V2." A remark placed above it informed that the equation was approximated by neglecting "magnitudes of fourth and higher orders" of a series expansion. Einstein used a body emitting two light pulses in opposite directions, having energies of E0 before and E1 after the emission as seen in its rest frame. As seen from a moving frame, E0 becomes H0 and E1 becomes H1. Einstein obtained, in modern notation:
(
H
0
−
E
0
)
−
(
H
1
−
E
1
)
=
E
(
1
1
−
v
2
c
2
−
1
)
{\displaystyle \left(H_{0}-E_{0}\right)-\left(H_{1}-E_{1}\right)=E\left({\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}-1\right)}
.
He then argued that H − E can only differ from the kinetic energy K by an additive constant, which gives
K
0
−
K
1
=
E
(
1
1
−
v
2
c
2
−
1
)
{\displaystyle K_{0}-K_{1}=E\left({\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}-1\right)}
.
Neglecting effects higher than third order in v/c after a Taylor series expansion of the right side of this yields:
K
0
−
K
1
=
E
c
2
v
2
2
.
{\displaystyle K_{0}-K_{1}={\frac {E}{c^{2}}}{\frac {v^{2}}{2}}.}
Einstein concluded that the emission reduces the body's mass by E/c2, and that the mass of a body is a measure of its energy content.
The correctness of Einstein's 1905 derivation of E = mc2 was criticized by German theoretical physicist Max Planck in 1907, who argued that it is only valid to first approximation. Another criticism was formulated by American physicist Herbert Ives in 1952 and the Israeli physicist Max Jammer in 1961, asserting that Einstein's derivation is based on begging the question. Other scholars, such as American and Chilean philosophers John Stachel and Roberto Torretti, have argued that Ives' criticism was wrong, and that Einstein's derivation was correct. American physics writer Hans Ohanian, in 2008, agreed with Stachel/Torretti's criticism of Ives, though he argued that Einstein's derivation was wrong for other reasons.
==== Relativistic center-of-mass theorem of 1906 ====
Like Poincaré, Einstein concluded in 1906 that the inertia of electromagnetic energy is a necessary condition for the center-of-mass theorem to hold. On this occasion, Einstein referred to Poincaré's 1900 paper and wrote: "Although the merely formal considerations, which we will need for the proof, are already mostly contained in a work by H. Poincaré2, for the sake of clarity I will not rely on that work." In Einstein's more physical, as opposed to formal or mathematical, point of view, there was no need for fictitious masses. He could avoid the perpetual motion problem because, on the basis of the mass–energy equivalence, he could show that the transport of inertia that accompanies the emission and absorption of radiation solves the problem. Poincaré's rejection of the principle of action–reaction can be avoided through Einstein's E = mc2, because mass conservation appears as a special case of the energy conservation law.
==== Further developments ====
There were several further developments in the first decade of the twentieth century. In May 1907, Einstein explained that the expression for energy ε of a moving mass point assumes the simplest form when its expression for the state of rest is chosen to be ε0 = μV2 (where μ is the mass), which is in agreement with the "principle of the equivalence of mass and energy". In addition, Einstein used the formula μ = E0/V2, with E0 being the energy of a system of mass points, to describe the energy and mass increase of that system when the velocity of the differently moving mass points is increased. Max Planck rewrote Einstein's mass–energy relationship as M = E0 + pV0/c2 in June 1907, where p is the pressure and V0 the volume to express the relation between mass, its latent energy, and thermodynamic energy within the body. Subsequently, in October 1907, this was rewritten as M0 = E0/c2 and given a quantum interpretation by German physicist Johannes Stark, who assumed its validity and correctness. In December 1907, Einstein expressed the equivalence in the form M = μ + E0/c2 and concluded: "A mass μ is equivalent, as regards inertia, to a quantity of energy μc2. […] It appears far more natural to consider every inertial mass as a store of energy." American physical chemists Gilbert N. Lewis and Richard C. Tolman used two variations of the formula in 1909: m = E/c2 and m0 = E0/c2, with E being the relativistic energy (the energy of an object when the object is moving), E0 is the rest energy (the energy when not moving), m is the relativistic mass (the rest mass and the extra mass gained when moving), and m0 is the rest mass. The same relations in different notation were used by Lorentz in 1913 and 1914, though he placed the energy on the left-hand side: ε = Mc2 and ε0 = mc2, with ε being the total energy (rest energy plus kinetic energy) of a moving material point, ε0 its rest energy, M the relativistic mass, and m the invariant mass.
In 1911, German physicist Max von Laue gave a more comprehensive proof of M0 = E0/c2 from the stress–energy tensor, which was later generalized by German mathematician Felix Klein in 1918.
Einstein returned to the topic once again after World War II and this time he wrote E = mc2 in the title of his article intended as an explanation for a general reader by analogy.
==== Alternative version ====
An alternative version of Einstein's thought experiment was proposed by American theoretical physicist Fritz Rohrlich in 1990, who based his reasoning on the Doppler effect. Like Einstein, he considered a body at rest with mass M. If the body is examined in a frame moving with nonrelativistic velocity v, it is no longer at rest and in the moving frame it has momentum P = Mv. Then he supposed the body emits two pulses of light to the left and to the right, each carrying an equal amount of energy E/2. In its rest frame, the object remains at rest after the emission since the two beams are equal in strength and carry opposite momentum. However, if the same process is considered in a frame that moves with velocity v to the left, the pulse moving to the left is redshifted, while the pulse moving to the right is blue shifted. The blue light carries more momentum than the red light, so that the momentum of the light in the moving frame is not balanced: the light is carrying some net momentum to the right. The object has not changed its velocity before or after the emission. Yet in this frame it has lost some right-momentum to the light. The only way it could have lost momentum is by losing mass. This also solves Poincaré's radiation paradox. The velocity is small, so the right-moving light is blueshifted by an amount equal to the nonrelativistic Doppler shift factor 1 − v/c. The momentum of the light is its energy divided by c, and it is increased by a factor of v/c. So the right-moving light is carrying an extra momentum ΔP given by:
Δ
P
=
v
c
E
2
c
.
{\displaystyle \Delta P={v \over c}{E \over 2c}.}
The left-moving light carries a little less momentum, by the same amount ΔP. So the total right-momentum in both light pulses is twice ΔP. This is the right-momentum that the object lost.
2
Δ
P
=
v
E
c
2
.
{\displaystyle 2\Delta P=v{E \over c^{2}}.}
The momentum of the object in the moving frame after the emission is reduced to this amount:
P
′
=
M
v
−
2
Δ
P
=
(
M
−
E
c
2
)
v
.
{\displaystyle P'=Mv-2\Delta P=\left(M-{E \over c^{2}}\right)v.}
So the change in the object's mass is equal to the total energy lost divided by c2. Since any emission of energy can be carried out by a two-step process, where first the energy is emitted as light and then the light is converted to some other form of energy, any emission of energy is accompanied by a loss of mass. Similarly, by considering absorption, a gain in energy is accompanied by a gain in mass.
=== Radioactivity and nuclear energy ===
It was quickly noted after the discovery of radioactivity in 1897 that the total energy due to radioactive processes is about one million times greater than that involved in any known molecular change, raising the question of where the energy comes from. After eliminating the idea of absorption and emission of some sort of Lesagian ether particles, the existence of a huge amount of latent energy, stored within matter, was proposed by New Zealand physicist Ernest Rutherford and British radiochemist Frederick Soddy in 1903. Rutherford also suggested that this internal energy is stored within normal matter as well. He went on to speculate in 1904: "If it were ever found possible to control at will the rate of disintegration of the radio-elements, an enormous amount of energy could be obtained from a small quantity of matter."
Einstein's equation does not explain the large energies released in radioactive decay, but can be used to quantify them. The theoretical explanation for radioactive decay is given by the nuclear forces responsible for holding atoms together, though these forces were still unknown in 1905. The enormous energy released from radioactive decay had previously been measured by Rutherford and was much more easily measured than the small change in the gross mass of materials as a result. Einstein's equation, by theory, can give these energies by measuring mass differences before and after reactions, but in practice, these mass differences in 1905 were still too small to be measured in bulk. Prior to this, the ease of measuring radioactive decay energies with a calorimeter was thought possibly likely to allow measurement of changes in mass difference, as a check on Einstein's equation itself. Einstein mentions in his 1905 paper that mass–energy equivalence might perhaps be tested with radioactive decay, which was known by then to release enough energy to possibly be "weighed," when missing from the system. However, radioactivity seemed to proceed at its own unalterable pace, and even when simple nuclear reactions became possible using proton bombardment, the idea that these great amounts of usable energy could be liberated at will with any practicality, proved difficult to substantiate. Rutherford was reported in 1933 to have declared that this energy could not be exploited efficiently: "Anyone who expects a source of power from the transformation of the atom is talking moonshine."
This outlook changed dramatically in 1932 with the discovery of the neutron and its mass, allowing mass differences for single nuclides and their reactions to be calculated directly, and compared with the sum of masses for the particles that made up their composition. In 1933, the energy released from the reaction of lithium-7 plus protons giving rise to two alpha particles, allowed Einstein's equation to be tested to an error of ±0.5%. However, scientists still did not see such reactions as a practical source of power, due to the energy cost of accelerating reaction particles. After the very public demonstration of huge energies released from nuclear fission after the atomic bombings of Hiroshima and Nagasaki in 1945, the equation E = mc2 became directly linked in the public eye with the power and peril of nuclear weapons. The equation was featured on page 2 of the Smyth Report, the official 1945 release by the US government on the development of the atomic bomb, and by 1946 the equation was linked closely enough with Einstein's work that the cover of Time magazine prominently featured a picture of Einstein next to an image of a mushroom cloud emblazoned with the equation. Einstein himself had only a minor role in the Manhattan Project: he had cosigned a letter to the U.S. president in 1939 urging funding for research into atomic energy, warning that an atomic bomb was theoretically possible. The letter persuaded Roosevelt to devote a significant portion of the wartime budget to atomic research. Without a security clearance, Einstein's only scientific contribution was an analysis of an isotope separation method in theoretical terms. It was inconsequential, on account of Einstein not being given sufficient information to fully work on the problem.
While E = mc2 is useful for understanding the amount of energy potentially released in a fission reaction, it was not strictly necessary to develop the weapon, once the fission process was known, and its energy measured at 200 MeV (which was directly possible, using a quantitative Geiger counter, at that time). The physicist and Manhattan Project participant Robert Serber noted that somehow "the popular notion took hold long ago that Einstein's theory of relativity, in particular his equation E = mc2, plays some essential role in the theory of fission. Einstein had a part in alerting the United States government to the possibility of building an atomic bomb, but his theory of relativity is not required in discussing fission. The theory of fission is what physicists call a non-relativistic theory, meaning that relativistic effects are too small to affect the dynamics of the fission process significantly." There are other views on the equation's importance to nuclear reactions. In late 1938, the Austrian-Swedish and British physicists Lise Meitner and Otto Robert Frisch—while on a winter walk during which they solved the meaning of Hahn's experimental results and introduced the idea that would be called atomic fission—directly used Einstein's equation to help them understand the quantitative energetics of the reaction that overcame the "surface tension-like" forces that hold the nucleus together, and allowed the fission fragments to separate to a configuration from which their charges could force them into an energetic fission. To do this, they used packing fraction, or nuclear binding energy values for elements. These, together with use of E = mc2 allowed them to realize on the spot that the basic fission process was energetically possible.
=== Einstein's equation written ===
According to the Einstein Papers Project at the California Institute of Technology and Hebrew University of Jerusalem, there remain only four known copies of this equation as written by Einstein. One of these is a letter written in German to Ludwik Silberstein, which was in Silberstein's archives, and sold at auction for $1.2 million, RR Auction of Boston, Massachusetts said on May 21, 2021.
== See also ==
== Notes ==
== References ==
== External links ==
Einstein on the Inertia of Energy – MathPages
Einstein-on film explaining a mass energy equivalence
Mass and Energy – Conversations About Science with Theoretical Physicist Matt Strassler
The Equivalence of Mass and Energy – Entry in the Stanford Encyclopedia of Philosophy
Merrifield, Michael; Copeland, Ed; Bowley, Roger. "E=mc2 – Mass–Energy Equivalence". Sixty Symbols. Brady Haran for the University of Nottingham. | Wikipedia/Equivalence_of_matter_and_energy |
General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. General relativity generalizes special relativity and refines Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time, or four-dimensional spacetime. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever is
present, including matter and radiation. The relation is specified by the Einstein field equations, a system of second-order partial differential equations.
Newton's law of universal gravitation, which describes gravity in classical mechanics, can be seen as a prediction of general relativity for the almost flat spacetime geometry around stationary mass distributions. Some predictions of general relativity, however, are beyond Newton's law of universal gravitation in classical physics. These predictions concern the passage of time, the geometry of space, the motion of bodies in free fall, and the propagation of light, and include gravitational time dilation, gravitational lensing, the gravitational redshift of light, the Shapiro time delay and singularities/black holes. So far, all tests of general relativity have been shown to be in agreement with the theory. The time-dependent solutions of general relativity enable us to talk about the history of the universe and have provided the modern framework for cosmology, thus leading to the discovery of the Big Bang and cosmic microwave background radiation. Despite the introduction of a number of alternative theories, general relativity continues to be the simplest theory consistent with experimental data.
Reconciliation of general relativity with the laws of quantum physics remains a problem, however, as there is a lack of a self-consistent theory of quantum gravity. It is not yet known how gravity can be unified with the three non-gravitational forces: strong, weak and electromagnetic.
Einstein's theory has astrophysical implications, including the prediction of black holes—regions of space in which space and time are distorted in such a way that nothing, not even light, can escape from them. Black holes are the end-state for massive stars. Microquasars and active galactic nuclei are believed to be stellar black holes and supermassive black holes. It also predicts gravitational lensing, where the bending of light results in multiple images of the same distant astronomical phenomenon. Other predictions include the existence of gravitational waves, which have been observed directly by the physics collaboration LIGO and other observatories. In addition, general relativity has provided the base of cosmological models of an expanding universe.
Widely acknowledged as a theory of extraordinary beauty, general relativity has often been described as the most beautiful of all existing physical theories.
== History ==
Henri Poincaré's 1905 theory of the dynamics of the electron was a relativistic theory which he applied to all forces, including gravity. While others thought that gravity was instantaneous or of electromagnetic origin, he suggested that relativity was "something due to our methods of measurement". In his theory, he showed that gravitational waves propagate at the speed of light. Soon afterwards, Einstein started thinking about how to incorporate gravity into his relativistic framework. In 1907, beginning with a simple thought experiment involving an observer in free fall (FFO), he embarked on what would be an eight-year search for a relativistic theory of gravity. After numerous detours and false starts, his work culminated in the presentation to the Prussian Academy of Science in November 1915 of what are now known as the Einstein field equations, which form the core of Einstein's general theory of relativity. These equations specify how the geometry of space and time is influenced by whatever matter and radiation are present. A version of non-Euclidean geometry, called Riemannian geometry, enabled Einstein to develop general relativity by providing the key mathematical framework on which he fit his physical ideas of gravity. This idea was pointed out by mathematician Marcel Grossmann and published by Grossmann and Einstein in 1913.
The Einstein field equations are nonlinear and considered difficult to solve. Einstein used approximation methods in working out initial predictions of the theory. But in 1916, the astrophysicist Karl Schwarzschild found the first non-trivial exact solution to the Einstein field equations, the Schwarzschild metric. This solution laid the groundwork for the description of the final stages of gravitational collapse, and the objects known today as black holes. In the same year, the first steps towards generalizing Schwarzschild's solution to electrically charged objects were taken, eventually resulting in the Reissner–Nordström solution, which is now associated with electrically charged black holes. In 1917, Einstein applied his theory to the universe as a whole, initiating the field of relativistic cosmology. In line with contemporary thinking, he assumed a static universe, adding a new parameter to his original field equations—the cosmological constant—to match that observational presumption. By 1929, however, the work of Hubble and others had shown that the universe is expanding. This is readily described by the expanding cosmological solutions found by Friedmann in 1922, which do not require a cosmological constant. Lemaître used these solutions to formulate the earliest version of the Big Bang models, in which the universe has evolved from an extremely hot and dense earlier state. Einstein later declared the cosmological constant the biggest blunder of his life.
During that period, general relativity remained something of a curiosity among physical theories. It was clearly superior to Newtonian gravity, being consistent with special relativity and accounting for several effects unexplained by the Newtonian theory. Einstein showed in 1915 how his theory explained the anomalous perihelion advance of the planet Mercury without any arbitrary parameters ("fudge factors"), and in 1919 an expedition led by Eddington confirmed general relativity's prediction for the deflection of starlight by the Sun during the total solar eclipse of 29 May 1919, instantly making Einstein famous. Yet the theory remained outside the mainstream of theoretical physics and astrophysics until developments between approximately 1960 and 1975, now known as the golden age of general relativity. Physicists began to understand the concept of a black hole, and to identify quasars as one of these objects' astrophysical manifestations. Ever more precise solar system tests confirmed the theory's predictive power, and relativistic cosmology also became amenable to direct observational tests.
General relativity has acquired a reputation as a theory of extraordinary beauty. Subrahmanyan Chandrasekhar has noted that at multiple levels, general relativity exhibits what Francis Bacon has termed a "strangeness in the proportion" (i.e. elements that excite wonderment and surprise). It juxtaposes fundamental concepts (space and time versus matter and motion) which had previously been considered as entirely independent. Chandrasekhar also noted that Einstein's only guides in his search for an exact theory were the principle of equivalence and his sense that a proper description of gravity should be geometrical at its basis, so that there was an "element of revelation" in the manner in which Einstein arrived at his theory. Other elements of beauty associated with the general theory of relativity are its simplicity and symmetry, the manner in which it incorporates invariance and unification, and its perfect logical consistency.
In the preface to Relativity: The Special and the General Theory, Einstein said "The present book is intended, as far as possible, to give an exact insight into the theory of Relativity to those readers who, from a general scientific and philosophical point of view, are interested in the theory, but who are not conversant with the mathematical apparatus of theoretical physics. The work presumes a standard of education corresponding to that of a university matriculation examination, and, despite the shortness of the book, a fair amount of patience and force of will on the part of the reader. The author has spared himself no pains in his endeavour to present the main ideas in the simplest and most intelligible form, and on the whole, in the sequence and connection in which they actually originated."
== From classical mechanics to general relativity ==
General relativity can be understood by examining its similarities with and departures from classical physics. The first step is the realization that classical mechanics and Newton's law of gravity admit a geometric description. The combination of this description with the laws of special relativity results in a heuristic derivation of general relativity.
=== Geometry of Newtonian gravity ===
At the base of classical mechanics is the notion that a body's motion can be described as a combination of free (or inertial) motion, and deviations from this free motion. Such deviations are caused by external forces acting on a body in accordance with Newton's second law of motion, which states that the net force acting on a body is equal to that body's (inertial) mass multiplied by its acceleration. The preferred inertial motions are related to the geometry of space and time: in the standard reference frames of classical mechanics, objects in free motion move along straight lines at constant speed. In modern parlance, their paths are geodesics, straight world lines in curved spacetime.
Conversely, one might expect that inertial motions, once identified by observing the actual motions of bodies and making allowances for the external forces (such as electromagnetism or friction), can be used to define the geometry of space, as well as a time coordinate. However, there is an ambiguity once gravity comes into play. According to Newton's law of gravity, and independently verified by experiments such as that of Eötvös and its successors (see Eötvös experiment), there is a universality of free fall (also known as the weak equivalence principle, or the universal equality of inertial and passive-gravitational mass): the trajectory of a test body in free fall depends only on its position and initial speed, but not on any of its material properties. A simplified version of this is embodied in Einstein's elevator experiment, illustrated in the figure on the right: for an observer in an enclosed room, it is impossible to decide, by mapping the trajectory of bodies such as a dropped ball, whether the room is stationary in a gravitational field and the ball accelerating, or in free space aboard a rocket that is accelerating at a rate equal to that of the gravitational field versus the ball which upon release has nil acceleration.
Given the universality of free fall, there is no observable distinction between inertial motion and motion under the influence of the gravitational force. This suggests the definition of a new class of inertial motion, namely that of objects in free fall under the influence of gravity. This new class of preferred motions, too, defines a geometry of space and time—in mathematical terms, it is the geodesic motion associated with a specific connection which depends on the gradient of the gravitational potential. Space, in this construction, still has the ordinary Euclidean geometry. However, spacetime as a whole is more complicated. As can be shown using simple thought experiments following the free-fall trajectories of different test particles, the result of transporting spacetime vectors that can denote a particle's velocity (time-like vectors) will vary with the particle's trajectory; mathematically speaking, the Newtonian connection is not integrable. From this, one can deduce that spacetime is curved. The resulting Newton–Cartan theory is a geometric formulation of Newtonian gravity using only covariant concepts, i.e. a description which is valid in any desired coordinate system. In this geometric description, tidal effects—the relative acceleration of bodies in free fall—are related to the derivative of the connection, showing how the modified geometry is caused by the presence of mass.
=== Relativistic generalization ===
As intriguing as geometric Newtonian gravity may be, its basis, classical mechanics, is merely a limiting case of (special) relativistic mechanics. In the language of symmetry: where gravity can be neglected, physics is Lorentz invariant as in special relativity rather than Galilei invariant as in classical mechanics. (The defining symmetry of special relativity is the Poincaré group, which includes translations, rotations, boosts and reflections.) The differences between the two become significant when dealing with speeds approaching the speed of light, and with high-energy phenomena.
With Lorentz symmetry, additional structures come into play. They are defined by the set of light cones (see image). The light-cones define a causal structure: for each event A, there is a set of events that can, in principle, either influence or be influenced by A via signals or interactions that do not need to travel faster than light (such as event B in the image), and a set of events for which such an influence is impossible (such as event C in the image). These sets are observer-independent. In conjunction with the world-lines of freely falling particles, the light-cones can be used to reconstruct the spacetime's semi-Riemannian metric, at least up to a positive scalar factor. In mathematical terms, this defines a conformal structure or conformal geometry.
Special relativity is defined in the absence of gravity. For practical applications, it is a suitable model whenever gravity can be neglected. Bringing gravity into play, and assuming the universality of free fall motion, an analogous reasoning as in the previous section applies: there are no global inertial frames. Instead there are approximate inertial frames moving alongside freely falling particles. Translated into the language of spacetime: the straight time-like lines that define a gravity-free inertial frame are deformed to lines that are curved relative to each other, suggesting that the inclusion of gravity necessitates a change in spacetime geometry.
A priori, it is not clear whether the new local frames in free fall coincide with the reference frames in which the laws of special relativity hold—that theory is based on the propagation of light, and thus on electromagnetism, which could have a different set of preferred frames. But using different assumptions about the special-relativistic frames (such as their being earth-fixed, or in free fall), one can derive different predictions for the gravitational redshift, that is, the way in which the frequency of light shifts as the light propagates through a gravitational field (cf. below). The actual measurements show that free-falling frames are the ones in which light propagates as it does in special relativity. The generalization of this statement, namely that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames, is known as the Einstein equivalence principle, a crucial guiding principle for generalizing special-relativistic physics to include gravity.
The same experimental data shows that time as measured by clocks in a gravitational field—proper time, to give the technical term—does not follow the rules of special relativity. In the language of spacetime geometry, it is not measured by the Minkowski metric. As in the Newtonian case, this is suggestive of a more general geometry. At small scales, all reference frames that are in free fall are equivalent, and approximately Minkowskian. Consequently, we are now dealing with a curved generalization of Minkowski space. The metric tensor that defines the geometry—in particular, how lengths and angles are measured—is not the Minkowski metric of special relativity, it is a generalization known as a semi- or pseudo-Riemannian metric. Furthermore, each Riemannian metric is naturally associated with one particular kind of connection, the Levi-Civita connection, and this is, in fact, the connection that satisfies the equivalence principle and makes space locally Minkowskian (that is, in suitable locally inertial coordinates, the metric is Minkowskian, and its first partial derivatives and the connection coefficients vanish).
=== Einstein's equations ===
Having formulated the relativistic, geometric version of the effects of gravity, the question of gravity's source remains. In Newtonian gravity, the source is mass. In special relativity, mass turns out to be part of a more general quantity called the energy–momentum tensor, which includes both energy and momentum densities as well as stress: pressure and shear. Using the equivalence principle, this tensor is readily generalized to curved spacetime. Drawing further upon the analogy with geometric Newtonian gravity, it is natural to assume that the field equation for gravity relates this tensor and the Ricci tensor, which describes a particular class of tidal effects: the change in volume for a small cloud of test particles that are initially at rest, and then fall freely. In special relativity, conservation of energy–momentum corresponds to the statement that the energy–momentum tensor is divergence-free. This formula, too, is readily generalized to curved spacetime by replacing partial derivatives with their curved-manifold counterparts, covariant derivatives studied in differential geometry. With this additional condition—the covariant divergence of the energy–momentum tensor, and hence of whatever is on the other side of the equation, is zero—the simplest nontrivial set of equations are what are called Einstein's (field) equations:
On the left-hand side is the Einstein tensor,
G
μ
ν
{\displaystyle G_{\mu \nu }}
, which is symmetric and a specific divergence-free combination of the Ricci tensor
R
μ
ν
{\displaystyle R_{\mu \nu }}
and the metric. In particular,
R
=
g
μ
ν
R
μ
ν
{\displaystyle R=g^{\mu \nu }R_{\mu \nu }}
is the curvature scalar. The Ricci tensor itself is related to the more general Riemann curvature tensor as
R
μ
ν
=
R
α
μ
α
ν
.
{\displaystyle R_{\mu \nu }={R^{\alpha }}_{\mu \alpha \nu }.}
On the right-hand side,
κ
{\displaystyle \kappa }
is a constant and
T
μ
ν
{\displaystyle T_{\mu \nu }}
is the energy–momentum tensor. All tensors are written in abstract index notation. Matching the theory's prediction to observational results for planetary orbits or, equivalently, assuring that the weak-gravity, low-speed limit is Newtonian mechanics, the proportionality constant
κ
{\displaystyle \kappa }
is found to be
κ
=
8
π
G
c
4
{\textstyle \kappa ={\frac {8\pi G}{c^{4}}}}
, where
G
{\displaystyle G}
is the Newtonian constant of gravitation and
c
{\displaystyle c}
the speed of light in vacuum. When there is no matter present, so that the energy–momentum tensor vanishes, the results are the vacuum Einstein equations,
R
μ
ν
=
0.
{\displaystyle R_{\mu \nu }=0.}
In general relativity, the world line of a particle free from all external, non-gravitational force is a particular type of geodesic in curved spacetime. In other words, a freely moving or falling particle always moves along a geodesic.
The geodesic equation is:
d
2
x
μ
d
s
2
+
Γ
μ
α
β
d
x
α
d
s
d
x
β
d
s
=
0
,
{\displaystyle {d^{2}x^{\mu } \over ds^{2}}+\Gamma ^{\mu }{}_{\alpha \beta }{dx^{\alpha } \over ds}{dx^{\beta } \over ds}=0,}
where
s
{\displaystyle s}
is a scalar parameter of motion (e.g. the proper time), and
Γ
μ
α
β
{\displaystyle \Gamma ^{\mu }{}_{\alpha \beta }}
are Christoffel symbols (sometimes called the affine connection coefficients or Levi-Civita connection coefficients) which is symmetric in the two lower indices. Greek indices may take the values: 0, 1, 2, 3 and the summation convention is used for repeated indices
α
{\displaystyle \alpha }
and
β
{\displaystyle \beta }
. The quantity on the left-hand-side of this equation is the acceleration of a particle, and so this equation is analogous to Newton's laws of motion which likewise provide formulae for the acceleration of a particle. This equation of motion employs the Einstein notation, meaning that repeated indices are summed (i.e. from zero to three). The Christoffel symbols are functions of the four spacetime coordinates, and so are independent of the velocity or acceleration or other characteristics of a test particle whose motion is described by the geodesic equation.
=== Total force in general relativity ===
In general relativity, the effective gravitational potential energy of an object of mass m revolving around a massive central body M is given by
U
f
(
r
)
=
−
G
M
m
r
+
L
2
2
m
r
2
−
G
M
L
2
m
c
2
r
3
{\displaystyle U_{f}(r)=-{\frac {GMm}{r}}+{\frac {L^{2}}{2mr^{2}}}-{\frac {GML^{2}}{mc^{2}r^{3}}}}
A conservative total force can then be obtained as its negative gradient
F
f
(
r
)
=
−
G
M
m
r
2
+
L
2
m
r
3
−
3
G
M
L
2
m
c
2
r
4
{\displaystyle F_{f}(r)=-{\frac {GMm}{r^{2}}}+{\frac {L^{2}}{mr^{3}}}-{\frac {3GML^{2}}{mc^{2}r^{4}}}}
where L is the angular momentum. The first term represents the force of Newtonian gravity, which is described by the inverse-square law. The second term represents the centrifugal force in the circular motion. The third term represents the relativistic effect.
=== Alternatives to general relativity ===
There are alternatives to general relativity built upon the same premises, which include additional rules and/or constraints, leading to different field equations. Examples are Whitehead's theory, Brans–Dicke theory, teleparallelism, f(R) gravity and Einstein–Cartan theory.
== Definition and basic applications ==
The derivation outlined in the previous section contains all the information needed to define general relativity, describe its key properties, and address a question of crucial importance in physics, namely how the theory can be used for model-building.
=== Definition and basic properties ===
General relativity is a metric theory of gravitation. At its core are Einstein's equations, which describe the relation between the geometry of a four-dimensional pseudo-Riemannian manifold representing spacetime, and the energy–momentum contained in that spacetime. Phenomena that in classical mechanics are ascribed to the action of the force of gravity (such as free-fall, orbital motion, and spacecraft trajectories), correspond to inertial motion within a curved geometry of spacetime in general relativity; there is no gravitational force deflecting objects from their natural, straight paths. Instead, gravity corresponds to changes in the properties of space and time, which in turn changes the straightest-possible paths that objects will naturally follow. The curvature is, in turn, caused by the energy–momentum of matter. Paraphrasing the relativist John Archibald Wheeler, spacetime tells matter how to move; matter tells spacetime how to curve.
While general relativity replaces the scalar gravitational potential of classical physics by a symmetric rank-two tensor, the latter reduces to the former in certain limiting cases. For weak gravitational fields and slow speed relative to the speed of light, the theory's predictions converge on those of Newton's law of universal gravitation.
As it is constructed using tensors, general relativity exhibits general covariance: its laws—and further laws formulated within the general relativistic framework—take on the same form in all coordinate systems. Furthermore, the theory does not contain any invariant geometric background structures, i.e. it is background independent. It thus satisfies a more stringent general principle of relativity, namely that the laws of physics are the same for all observers. Locally, as expressed in the equivalence principle, spacetime is Minkowskian, and the laws of physics exhibit local Lorentz invariance.
=== Model-building ===
The core concept of general-relativistic model-building is that of a solution of Einstein's equations. Given both Einstein's equations and suitable equations for the properties of matter, such a solution consists of a specific semi-Riemannian manifold (usually defined by giving the metric in specific coordinates), and specific matter fields defined on that manifold. Matter and geometry must satisfy Einstein's equations, so in particular, the matter's energy–momentum tensor must be divergence-free. The matter must, of course, also satisfy whatever additional equations were imposed on its properties. In short, such a solution is a model universe that satisfies the laws of general relativity, and possibly additional laws governing whatever matter might be present.
Einstein's equations are nonlinear partial differential equations and, as such, difficult to solve exactly. Nevertheless, a number of exact solutions are known, although only a few have direct physical applications. The best-known exact solutions, and also those most interesting from a physics point of view, are the Schwarzschild solution, the Reissner–Nordström solution and the Kerr metric, each corresponding to a certain type of black hole in an otherwise empty universe, and the Friedmann–Lemaître–Robertson–Walker and de Sitter universes, each describing an expanding cosmos. Exact solutions of great theoretical interest include the Gödel universe (which opens up the intriguing possibility of time travel in curved spacetimes), the Taub–NUT solution (a model universe that is homogeneous, but anisotropic), and anti-de Sitter space (which has recently come to prominence in the context of what is called the Maldacena conjecture).
Given the difficulty of finding exact solutions, Einstein's field equations are also solved frequently by numerical integration on a computer, or by considering small perturbations of exact solutions. In the field of numerical relativity, powerful computers are employed to simulate the geometry of spacetime and to solve Einstein's equations for interesting situations such as two colliding black holes. In principle, such methods may be applied to any system, given sufficient computer resources, and may address fundamental questions such as naked singularities. Approximate solutions may also be found by perturbation theories such as linearized gravity and its generalization, the post-Newtonian expansion, both of which were developed by Einstein. The latter provides a systematic approach to solving for the geometry of a spacetime that contains a distribution of matter that moves slowly compared with the speed of light. The expansion involves a series of terms; the first terms represent Newtonian gravity, whereas the later terms represent ever smaller corrections to Newton's theory due to general relativity. An extension of this expansion is the parametrized post-Newtonian (PPN) formalism, which allows quantitative comparisons between the predictions of general relativity and alternative theories.
== Consequences of Einstein's theory ==
General relativity has a number of physical consequences. Some follow directly from the theory's axioms, whereas others have become clear only in the course of many years of research that followed Einstein's initial publication.
=== Gravitational time dilation and frequency shift ===
Assuming that the equivalence principle holds, gravity influences the passage of time. Light sent down into a gravity well is blueshifted, whereas light sent in the opposite direction (i.e., climbing out of the gravity well) is redshifted; collectively, these two effects are known as the gravitational frequency shift. More generally, processes close to a massive body run more slowly when compared with processes taking place farther away; this effect is known as gravitational time dilation.
Gravitational redshift has been measured in the laboratory and using astronomical observations. Gravitational time dilation in the Earth's gravitational field has been measured numerous times using atomic clocks, while ongoing validation is provided as a side effect of the operation of the Global Positioning System (GPS). Tests in stronger gravitational fields are provided by the observation of binary pulsars. All results are in agreement with general relativity. However, at the current level of accuracy, these observations cannot distinguish between general relativity and other theories in which the equivalence principle is valid.
=== Light deflection and gravitational time delay ===
General relativity predicts that the path of light will follow the curvature of spacetime as it passes near a massive object. This effect was initially confirmed by observing the light of stars or distant quasars being deflected as it passes the Sun.
This and related predictions follow from the fact that light follows what is called a light-like or null geodesic—a generalization of the straight lines along which light travels in classical physics. Such geodesics are the generalization of the invariance of lightspeed in special relativity. As one examines suitable model spacetimes (either the exterior Schwarzschild solution or, for more than a single mass, the post-Newtonian expansion), several effects of gravity on light propagation emerge. Although the bending of light can also be derived by extending the universality of free fall to light, the angle of deflection resulting from such calculations is only half the value given by general relativity.
Closely related to light deflection is the Shapiro Time Delay, the phenomenon that light signals take longer to move through a gravitational field than they would in the absence of that field. There have been numerous successful tests of this prediction. In the parameterized post-Newtonian formalism (PPN), measurements of both the deflection of light and the gravitational time delay determine a parameter called γ, which encodes the influence of gravity on the geometry of space.
=== Gravitational waves ===
Predicted in 1916 by Albert Einstein, there are gravitational waves: ripples in the metric of spacetime that propagate at the speed of light. These are one of several analogies between weak-field gravity and electromagnetism in that, they are analogous to electromagnetic waves. On 11 February 2016, the Advanced LIGO team announced that they had directly detected gravitational waves from a pair of black holes merging.
The simplest type of such a wave can be visualized by its action on a ring of freely floating particles. A sine wave propagating through such a ring towards the reader distorts the ring in a characteristic, rhythmic fashion (animated image to the right). Since Einstein's equations are non-linear, arbitrarily strong gravitational waves do not obey linear superposition, making their description difficult. However, linear approximations of gravitational waves are sufficiently accurate to describe the exceedingly weak waves that are expected to arrive here on Earth from far-off cosmic events, which typically result in relative distances increasing and decreasing by
10
−
21
{\displaystyle 10^{-21}}
or less. Data analysis methods routinely make use of the fact that these linearized waves can be Fourier decomposed.
Some exact solutions describe gravitational waves without any approximation, e.g., a wave train traveling through empty space or Gowdy universes, varieties of an expanding cosmos filled with gravitational waves. But for gravitational waves produced in astrophysically relevant situations, such as the merger of two black holes, numerical methods are presently the only way to construct appropriate models.
=== Orbital effects and the relativity of direction ===
General relativity differs from classical mechanics in a number of predictions concerning orbiting bodies. It predicts an overall rotation (precession) of planetary orbits, as well as orbital decay caused by the emission of gravitational waves and effects related to the relativity of direction.
==== Precession of apsides ====
In general relativity, the apsides of any orbit (the point of the orbiting body's closest approach to the system's center of mass) will precess; the orbit is not an ellipse, but akin to an ellipse that rotates on its focus, resulting in a rose curve-like shape (see image). Einstein first derived this result by using an approximate metric representing the Newtonian limit and treating the orbiting body as a test particle. For him, the fact that his theory gave a straightforward explanation of Mercury's anomalous perihelion shift, discovered earlier by Urbain Le Verrier in 1859, was important evidence that he had at last identified the correct form of the gravitational field equations.
The effect can also be derived by using either the exact Schwarzschild metric (describing spacetime around a spherical mass) or the much more general post-Newtonian formalism. It is due to the influence of gravity on the geometry of space and to the contribution of self-energy to a body's gravity (encoded in the nonlinearity of Einstein's equations). Relativistic precession has been observed for all planets that allow for accurate precession measurements (Mercury, Venus, and Earth), as well as in binary pulsar systems, where it is larger by five orders of magnitude.
In general relativity the perihelion shift
σ
{\displaystyle \sigma }
, expressed in radians per revolution, is approximately given by:
σ
=
24
π
3
L
2
T
2
c
2
(
1
−
e
2
)
,
{\displaystyle \sigma ={\frac {24\pi ^{3}L^{2}}{T^{2}c^{2}(1-e^{2})}}\ ,}
where:
L
{\displaystyle L}
is the semi-major axis
T
{\displaystyle T}
is the orbital period
c
{\displaystyle c}
is the speed of light in vacuum
e
{\displaystyle e}
is the orbital eccentricity
==== Orbital decay ====
According to general relativity, a binary system will emit gravitational waves, thereby losing energy. Due to this loss, the distance between the two orbiting bodies decreases, and so does their orbital period. Within the Solar System or for ordinary double stars, the effect is too small to be observable. This is not the case for a close binary pulsar, a system of two orbiting neutron stars, one of which is a pulsar: from the pulsar, observers on Earth receive a regular series of radio pulses that can serve as a highly accurate clock, which allows precise measurements of the orbital period. Because neutron stars are immensely compact, significant amounts of energy are emitted in the form of gravitational radiation.
The first observation of a decrease in orbital period due to the emission of gravitational waves was made by Hulse and Taylor, using the binary pulsar PSR1913+16 they had discovered in 1974. This was the first detection of gravitational waves, albeit indirect, for which they were awarded the 1993 Nobel Prize in physics. Since then, several other binary pulsars have been found, in particular the double pulsar PSR J0737−3039, where both stars are pulsars and which was last reported to also be in agreement with general relativity in 2021 after 16 years of observations.
==== Geodetic precession and frame-dragging ====
Several relativistic effects are directly related to the relativity of direction. One is geodetic precession: the axis direction of a gyroscope in free fall in curved spacetime will change when compared, for instance, with the direction of light received from distant stars—even though such a gyroscope represents the way of keeping a direction as stable as possible ("parallel transport"). For the Moon–Earth system, this effect has been measured with the help of lunar laser ranging. More recently, it has been measured for test masses aboard the satellite Gravity Probe B to a precision of better than 0.3%.
Near a rotating mass, there are gravitomagnetic or frame-dragging effects. A distant observer will determine that objects close to the mass get "dragged around". This is most extreme for rotating black holes where, for any object entering a zone known as the ergosphere, rotation is inevitable. Such effects can again be tested through their influence on the orientation of gyroscopes in free fall. Somewhat controversial tests have been performed using the LAGEOS satellites, confirming the relativistic prediction. Also the Mars Global Surveyor probe around Mars has been used.
== Astrophysical applications ==
=== Gravitational lensing ===
The deflection of light by gravity is responsible for a new class of astronomical phenomena. If a massive object is situated between the astronomer and a distant target object with appropriate mass and relative distances, the astronomer will see multiple distorted images of the target. Such effects are known as gravitational lensing. Depending on the configuration, scale, and mass distribution, there can be two or more images, a bright ring known as an Einstein ring, or partial rings called arcs.
The earliest example was discovered in 1979; since then, more than a hundred gravitational lenses have been observed. Even if the multiple images are too close to each other to be resolved, the effect can still be measured, e.g., as an overall brightening of the target object; a number of such "microlensing events" have been observed.
Gravitational lensing has developed into a tool of observational astronomy. It is used to detect the presence and distribution of dark matter, provide a "natural telescope" for observing distant galaxies, and to obtain an independent estimate of the Hubble constant. Statistical evaluations of lensing data provide valuable insight into the structural evolution of galaxies.
=== Gravitational-wave astronomy ===
Observations of binary pulsars provide strong indirect evidence for the existence of gravitational waves (see Orbital decay, above). Detection of these waves is a major goal of current relativity-related research. Several land-based gravitational wave detectors are currently in operation, most notably the interferometric detectors GEO 600, LIGO (two detectors), TAMA 300 and VIRGO. Various pulsar timing arrays are using millisecond pulsars to detect gravitational waves in the 10−9 to 10−6 hertz frequency range, which originate from binary supermassive blackholes. A European space-based detector, eLISA / NGO, is currently under development, with a precursor mission (LISA Pathfinder) having launched in December 2015.
Observations of gravitational waves promise to complement observations in the electromagnetic spectrum. They are expected to yield information about black holes and other dense objects such as neutron stars and white dwarfs, about certain kinds of supernova implosions, and about processes in the very early universe, including the signature of certain types of hypothetical cosmic string. In February 2016, the Advanced LIGO team announced that they had detected gravitational waves from a black hole merger.
=== Black holes and other compact objects ===
Whenever the ratio of an object's mass to its radius becomes sufficiently large, general relativity predicts the formation of a black hole, a region of space from which nothing, not even light, can escape. In the currently accepted models of stellar evolution, neutron stars of around 1.4 solar masses, and stellar black holes with a few to a few dozen solar masses, are thought to be the final state for the evolution of massive stars. Usually a galaxy has one supermassive black hole with a few million to a few billion solar masses in its center, and its presence is thought to have played an important role in the formation of the galaxy and larger cosmic structures.
Astronomically, the most important property of compact objects is that they provide a supremely efficient mechanism for converting gravitational energy into electromagnetic radiation. Accretion, the falling of dust or gaseous matter onto stellar or supermassive black holes, is thought to be responsible for some spectacularly luminous astronomical objects, notably diverse kinds of active galactic nuclei on galactic scales and stellar-size objects such as microquasars. In particular, accretion can lead to relativistic jets, focused beams of highly energetic particles that are being flung into space at almost light speed.
General relativity plays a central role in modelling all these phenomena, and observations provide strong evidence for the existence of black holes with the properties predicted by the theory.
Black holes are also sought-after targets in the search for gravitational waves (cf. Gravitational waves, above). Merging black hole binaries should lead to some of the strongest gravitational wave signals reaching detectors here on Earth, and the phase directly before the merger ("chirp") could be used as a "standard candle" to deduce the distance to the merger events–and hence serve as a probe of cosmic expansion at large distances. The gravitational waves produced as a stellar black hole plunges into a supermassive one should provide direct information about the supermassive black hole's geometry.
=== Cosmology ===
The current models of cosmology are based on Einstein's field equations, which include the cosmological constant
Λ
{\displaystyle \Lambda }
since it has important influence on the large-scale dynamics of the cosmos,
R
μ
ν
−
1
2
R
g
μ
ν
+
Λ
g
μ
ν
=
8
π
G
c
4
T
μ
ν
{\displaystyle R_{\mu \nu }-{\textstyle 1 \over 2}R\,g_{\mu \nu }+\Lambda \ g_{\mu \nu }={\frac {8\pi G}{c^{4}}}\,T_{\mu \nu }}
where
g
μ
ν
{\displaystyle g_{\mu \nu }}
is the spacetime metric. Isotropic and homogeneous solutions of these enhanced equations, the Friedmann–Lemaître–Robertson–Walker solutions, allow physicists to model a universe that has evolved over the past 14 billion years from a hot, early Big Bang phase. Once a small number of parameters (for example the universe's mean matter density) have been fixed by astronomical observation, further observational data can be used to put the models to the test. Predictions, all successful, include the initial abundance of chemical elements formed in a period of primordial nucleosynthesis, the large-scale structure of the universe, and the existence and properties of a "thermal echo" from the early cosmos, the cosmic background radiation.
Astronomical observations of the cosmological expansion rate allow the total amount of matter in the universe to be estimated, although the nature of that matter remains mysterious in part. About 90% of all matter appears to be dark matter, which has mass (or, equivalently, gravitational influence), but does not interact electromagnetically and, hence, cannot be observed directly. There is no generally accepted description of this new kind of matter, within the framework of known particle physics or otherwise. Observational evidence from redshift surveys of distant supernovae and measurements of the cosmic background radiation also show that the evolution of our universe is significantly influenced by a cosmological constant resulting in an acceleration of cosmic expansion or, equivalently, by a form of energy with an unusual equation of state, known as dark energy, the nature of which remains unclear.
An inflationary phase, an additional phase of strongly accelerated expansion at cosmic times of around 10−33 seconds, was hypothesized in 1980 to account for several puzzling observations that were unexplained by classical cosmological models, such as the nearly perfect homogeneity of the cosmic background radiation. Recent measurements of the cosmic background radiation have resulted in the first evidence for this scenario. However, there is a bewildering variety of possible inflationary scenarios, which cannot be restricted by current observations. An even larger question is the physics of the earliest universe, prior to the inflationary phase and close to where the classical models predict the big bang singularity. An authoritative answer would require a complete theory of quantum gravity, which has not yet been developed (cf. the section on quantum gravity, below).
=== Exotic solutions: time travel, warp drives ===
Kurt Gödel showed that solutions to Einstein's equations exist that contain closed timelike curves (CTCs), which allow for loops in time. The solutions require extreme physical conditions unlikely ever to occur in practice, and it remains an open question whether further laws of physics will eliminate them completely. Since then, other—similarly impractical—GR solutions containing CTCs have been found, such as the Tipler cylinder and traversable wormholes. Stephen Hawking introduced chronology protection conjecture, which is an assumption beyond those of standard general relativity to prevent time travel.
Some exact solutions in general relativity such as Alcubierre drive present examples of warp drive but these solutions requires exotic matter distribution, and generally suffers from semiclassical instability.
== Advanced concepts ==
=== Asymptotic symmetries ===
The spacetime symmetry group for special relativity is the Poincaré group, which is a ten-dimensional group of three Lorentz boosts, three rotations, and four spacetime translations. It is logical to ask what symmetries, if any, might apply in General Relativity. A tractable case might be to consider the symmetries of spacetime as seen by observers located far away from all sources of the gravitational field. The naive expectation for asymptotically flat spacetime symmetries might be simply to extend and reproduce the symmetries of flat spacetime of special relativity, viz., the Poincaré group.
In 1962 Hermann Bondi, M. G. van der Burg, A. W. Metzner and Rainer K. Sachs addressed this asymptotic symmetry problem in order to investigate the flow of energy at infinity due to propagating gravitational waves. Their first step was to decide on some physically sensible boundary conditions to place on the gravitational field at light-like infinity to characterize what it means to say a metric is asymptotically flat, making no a priori assumptions about the nature of the asymptotic symmetry group—not even the assumption that such a group exists. Then after designing what they considered to be the most sensible boundary conditions, they investigated the nature of the resulting asymptotic symmetry transformations that leave invariant the form of the boundary conditions appropriate for asymptotically flat gravitational fields. What they found was that the asymptotic symmetry transformations actually do form a group and the structure of this group does not depend on the particular gravitational field that happens to be present. This means that, as expected, one can separate the kinematics of spacetime from the dynamics of the gravitational field at least at spatial infinity. The puzzling surprise in 1962 was their discovery of a rich infinite-dimensional group (the so-called BMS group) as the asymptotic symmetry group, instead of the finite-dimensional Poincaré group, which is a subgroup of the BMS group. Not only are the Lorentz transformations asymptotic symmetry transformations, there are also additional transformations that are not Lorentz transformations but are asymptotic symmetry transformations. In fact, they found an additional infinity of transformation generators known as supertranslations. This implies the conclusion that General Relativity (GR) does not reduce to special relativity in the case of weak fields at long distances. It turns out that the BMS symmetry, suitably modified, could be seen as a restatement of the universal soft graviton theorem in quantum field theory (QFT), which relates universal infrared (soft) QFT with GR asymptotic spacetime symmetries.
=== Causal structure and global geometry ===
In general relativity, no material body can catch up with or overtake a light pulse. No influence from an event A can reach any other location X before light sent out at A to X. In consequence, an exploration of all light worldlines (null geodesics) yields key information about the spacetime's causal structure. This structure can be displayed using Penrose–Carter diagrams in which infinitely large regions of space and infinite time intervals are shrunk ("compactified") so as to fit onto a finite map, while light still travels along diagonals as in standard spacetime diagrams.
Aware of the importance of causal structure, Roger Penrose and others developed what is known as global geometry. In global geometry, the object of study is not one particular solution (or family of solutions) to Einstein's equations. Rather, relations that hold true for all geodesics, such as the Raychaudhuri equation, and additional non-specific assumptions about the nature of matter (usually in the form of energy conditions) are used to derive general results.
=== Horizons ===
Using global geometry, some spacetimes can be shown to contain boundaries called horizons, which demarcate one region from the rest of spacetime. The best-known examples are black holes: if mass is compressed into a sufficiently compact region of space (as specified in the hoop conjecture, the relevant length scale is the Schwarzschild radius), no light from inside can escape to the outside. Since no object can overtake a light pulse, all interior matter is imprisoned as well. Passage from the exterior to the interior is still possible, showing that the boundary, the black hole's horizon, is not a physical barrier.
Early studies of black holes relied on explicit solutions of Einstein's equations, notably the spherically symmetric Schwarzschild solution (used to describe a static black hole) and the axisymmetric Kerr solution (used to describe a rotating, stationary black hole, and introducing interesting features such as the ergosphere). Using global geometry, later studies have revealed more general properties of black holes. With time they become rather simple objects characterized by eleven parameters specifying: electric charge, mass–energy, linear momentum, angular momentum, and location at a specified time. This is stated by the black hole uniqueness theorem: "black holes have no hair", that is, no distinguishing marks like the hairstyles of humans. Irrespective of the complexity of a gravitating object collapsing to form a black hole, the object that results (having emitted gravitational waves) is very simple.
Even more remarkably, there is a general set of laws known as black hole mechanics, which is analogous to the laws of thermodynamics. For instance, by the second law of black hole mechanics, the area of the event horizon of a general black hole will never decrease with time, analogous to the entropy of a thermodynamic system. This limits the energy that can be extracted by classical means from a rotating black hole (e.g. by the Penrose process). There is strong evidence that the laws of black hole mechanics are, in fact, a subset of the laws of thermodynamics, and that the black hole area is proportional to its entropy. This leads to a modification of the original laws of black hole mechanics: for instance, as the second law of black hole mechanics becomes part of the second law of thermodynamics, it is possible for the black hole area to decrease as long as other processes ensure that entropy increases overall. As thermodynamical objects with nonzero temperature, black holes should emit thermal radiation. Semiclassical calculations indicate that indeed they do, with the surface gravity playing the role of temperature in Planck's law. This radiation is known as Hawking radiation (cf. the quantum theory section, below).
There are many other types of horizons. In an expanding universe, an observer may find that some regions of the past cannot be observed ("particle horizon"), and some regions of the future cannot be influenced (event horizon). Even in flat Minkowski space, when described by an accelerated observer (Rindler space), there will be horizons associated with a semiclassical radiation known as Unruh radiation.
=== Singularities ===
Another general feature of general relativity is the appearance of spacetime boundaries known as singularities. Spacetime can be explored by following up on timelike and lightlike geodesics—all possible ways that light and particles in free fall can travel. But some solutions of Einstein's equations have "ragged edges"—regions known as spacetime singularities, where the paths of light and falling particles come to an abrupt end, and geometry becomes ill-defined. In the more interesting cases, these are "curvature singularities", where geometrical quantities characterizing spacetime curvature, such as the Ricci scalar, take on infinite values. Well-known examples of spacetimes with future singularities—where worldlines end—are the Schwarzschild solution, which describes a singularity inside an eternal static black hole, or the Kerr solution with its ring-shaped singularity inside an eternal rotating black hole. The Friedmann–Lemaître–Robertson–Walker solutions and other spacetimes describing universes have past singularities on which worldlines begin, namely Big Bang singularities, and some have future singularities (Big Crunch) as well.
Given that these examples are all highly symmetric—and thus simplified—it is tempting to conclude that the occurrence of singularities is an artifact of idealization. The famous singularity theorems, proved using the methods of global geometry, say otherwise: singularities are a generic feature of general relativity, and unavoidable once the collapse of an object with realistic matter properties has proceeded beyond a certain stage and also at the beginning of a wide class of expanding universes. However, the theorems say little about the properties of singularities, and much of current research is devoted to characterizing these entities' generic structure (hypothesized e.g. by the BKL conjecture). The cosmic censorship hypothesis states that all realistic future singularities (no perfect symmetries, matter with realistic properties) are safely hidden away behind a horizon, and thus invisible to all distant observers. While no formal proof yet exists, numerical simulations offer supporting evidence of its validity.
=== Evolution equations ===
Each solution of Einstein's equation encompasses the whole history of a universe—it is not just some snapshot of how things are, but a whole, possibly matter-filled, spacetime. It describes the state of matter and geometry everywhere and at every moment in that particular universe. Due to its general covariance, Einstein's theory is not sufficient by itself to determine the time evolution of the metric tensor. It must be combined with a coordinate condition, which is analogous to gauge fixing in other field theories.
To understand Einstein's equations as partial differential equations, it is helpful to formulate them in a way that describes the evolution of the universe over time. This is done in "3+1" formulations, where spacetime is split into three space dimensions and one time dimension. The best-known example is the ADM formalism. These decompositions show that the spacetime evolution equations of general relativity are well-behaved: solutions always exist, and are uniquely defined, once suitable initial conditions have been specified. Such formulations of Einstein's field equations are the basis of numerical relativity.
=== Global and quasi-local quantities ===
The notion of evolution equations is intimately tied in with another aspect of general relativistic physics. In Einstein's theory, it turns out to be impossible to find a general definition for a seemingly simple property such as a system's total mass (or energy). The main reason is that the gravitational field—like any physical field—must be ascribed a certain energy, but that it proves to be fundamentally impossible to localize that energy.
Nevertheless, there are possibilities to define a system's total mass, either using a hypothetical "infinitely distant observer" (ADM mass) or suitable symmetries (Komar mass). If one excludes from the system's total mass the energy being carried away to infinity by gravitational waves, the result is the Bondi mass at null infinity. Just as in classical physics, it can be shown that these masses are positive. Corresponding global definitions exist for momentum and angular momentum. There have also been a number of attempts to define quasi-local quantities, such as the mass of an isolated system formulated using only quantities defined within a finite region of space containing that system. The hope is to obtain a quantity useful for general statements about isolated systems, such as a more precise formulation of the hoop conjecture.
== Relationship with quantum theory ==
If general relativity were considered to be one of the two pillars of modern physics, then quantum theory, the basis of understanding matter from elementary particles to solid-state physics, would be the other. However, how to reconcile quantum theory with general relativity is still an open question.
=== Quantum field theory in curved spacetime ===
Ordinary quantum field theories, which form the basis of modern elementary particle physics, are defined in flat Minkowski space, which is an excellent approximation when it comes to describing the behavior of microscopic particles in weak gravitational fields like those found on Earth. In order to describe situations in which gravity is strong enough to influence (quantum) matter, yet not strong enough to require quantization itself, physicists have formulated quantum field theories in curved spacetime. These theories rely on general relativity to describe a curved background spacetime, and define a generalized quantum field theory to describe the behavior of quantum matter within that spacetime. Using this formalism, it can be shown that black holes emit a blackbody spectrum of particles known as Hawking radiation leading to the possibility that they evaporate over time. As briefly mentioned above, this radiation plays an important role for the thermodynamics of black holes.
=== Quantum gravity ===
The demand for consistency between a quantum description of matter and a geometric description of spacetime, as well as the appearance of singularities (where curvature length scales become microscopic), indicate the need for a full theory of quantum gravity: for an adequate description of the interior of black holes, and of the very early universe, a theory is required in which gravity and the associated geometry of spacetime are described in the language of quantum physics. Despite major efforts, no complete and consistent theory of quantum gravity is currently known, even though a number of promising candidates exist.
Attempts to generalize ordinary quantum field theories, used in elementary particle physics to describe fundamental interactions, so as to include gravity have led to serious problems. Some have argued that at low energies, this approach proves successful, in that it results in an acceptable effective (quantum) field theory of gravity. At very high energies, however, the perturbative results are badly divergent and lead to models devoid of predictive power ("perturbative non-renormalizability").
One attempt to overcome these limitations is string theory, a quantum theory not of point particles, but of minute one-dimensional extended objects. The theory promises to be a unified description of all particles and interactions, including gravity; the price to pay is unusual features such as six extra dimensions of space in addition to the usual three. In what is called the second superstring revolution, it was conjectured that both string theory and a unification of general relativity and supersymmetry known as supergravity form part of a hypothesized eleven-dimensional model known as M-theory, which would constitute a uniquely defined and consistent theory of quantum gravity.
Another approach starts with the canonical quantization procedures of quantum theory. Using the initial-value-formulation of general relativity (cf. evolution equations above), the result is the Wheeler–deWitt equation (an analogue of the Schrödinger equation) which, regrettably, turns out to be ill-defined without a proper ultraviolet (lattice) cutoff. However, with the introduction of what are now known as Ashtekar variables, this leads to a promising model known as loop quantum gravity. Space is represented by a web-like structure called a spin network, evolving over time in discrete steps.
Depending on which features of general relativity and quantum theory are accepted unchanged, and on what level changes are introduced, there are numerous other attempts to arrive at a viable theory of quantum gravity, some examples being the lattice theory of gravity based on the Feynman Path Integral approach and Regge calculus, dynamical triangulations, causal sets, twistor models or the path integral based models of quantum cosmology.
All candidate theories still have major formal and conceptual problems to overcome. They also face the common problem that, as yet, there is no way to put quantum gravity predictions to experimental tests (and thus to decide between the candidates where their predictions vary), although there is hope for this to change as future data from cosmological observations and particle physics experiments becomes available.
== Current status ==
General relativity has emerged as a highly successful model of gravitation and cosmology, which has so far passed many unambiguous observational and experimental tests. However, there are strong indications that the theory is incomplete. The problem of quantum gravity and the question of the reality of spacetime singularities remain open. Observational data that is taken as evidence for dark energy and dark matter could indicate the need for new physics.
Even taken as is, general relativity is rich with possibilities for further exploration. Mathematical relativists seek to understand the nature of singularities and the fundamental properties of Einstein's equations, while numerical relativists run increasingly powerful computer simulations (such as those describing merging black holes). In February 2016, it was announced that the existence of gravitational waves was directly detected by the Advanced LIGO team on 14 September 2015. A century after its introduction, general relativity remains a highly active area of research.
== See also ==
Alcubierre drive – Hypothetical FTL transportation by warping space (warp drive)
Alternatives to general relativity – Proposed theories of gravity
Contributors to general relativity
Derivations of the Lorentz transformations
Ehrenfest paradox – Paradox in special relativity
Einstein–Hilbert action – Concept in general relativity
Einstein's thought experiments – Albert Einstein's hypothetical situations to argue scientific points
General relativity priority dispute – Debate about credit for general relativity
Introduction to the mathematics of general relativity
Nordström's theory of gravitation – Predecessor to the theory of relativity
Ricci calculus – Tensor index notation for tensor-based calculations
Timeline of gravitational physics and relativity
== References ==
== Bibliography ==
== Further reading ==
=== Popular books ===
Einstein, A. (1916), Relativity: The Special and the General Theory, Berlin, ISBN 978-3-528-06059-6 {{citation}}: ISBN / Date incompatibility (help)CS1 maint: location missing publisher (link)
Geroch, R. (1981), General Relativity from A to B, Chicago: University of Chicago Press, ISBN 978-0-226-28864-2
Lieber, Lillian (2008), The Einstein Theory of Relativity: A Trip to the Fourth Dimension, Philadelphia: Paul Dry Books, Inc., ISBN 978-1-58988-044-3
Schutz, Bernard F. (2001), "Gravitational radiation", in Murdin, Paul (ed.), Encyclopedia of Astronomy and Astrophysics, Institute of Physics Pub., ISBN 978-1-56159-268-5
Thorne, Kip; Hawking, Stephen (1994). Black Holes and Time Warps: Einstein's Outrageous Legacy. New York: W. W. Norton. ISBN 0-393-03505-0.
Wald, Robert M. (1992), Space, Time, and Gravity: the Theory of the Big Bang and Black Holes, Chicago: University of Chicago Press, ISBN 978-0-226-87029-8
Wheeler, John; Ford, Kenneth (1998), Geons, Black Holes, & Quantum Foam: a life in physics, New York: W. W. Norton, ISBN 978-0-393-31991-0
=== Beginning undergraduate textbooks ===
Yvonne Choquet-Bruhat (2014). Introduction to General Relativity, Black Holes, and Cosmology. Oxford University Press. ISBN 9780191936500.
Taylor, Edwin F.; Wheeler, John Archibald (2000), Exploring Black Holes: Introduction to General Relativity, Addison Wesley, ISBN 978-0-201-38423-9
=== Advanced undergraduate textbooks ===
Crowell, Ben (2020). General Relativity.
Dirac, Paul (1996), General Theory of Relativity, Princeton University Press, ISBN 978-0-691-01146-2
Gron, O.; Hervik, S. (2007), Einstein's General theory of Relativity, Springer, ISBN 978-0-387-69199-2
Hartle, James B. (2003), Gravity: an Introduction to Einstein's General Relativity, San Francisco: Addison-Wesley, ISBN 978-0-8053-8662-2
Hughston, L. P.; Tod, K. P. (1991), Introduction to General Relativity, Cambridge: Cambridge University Press, ISBN 978-0-521-33943-8
d'Inverno, Ray (1992), Introducing Einstein's Relativity, Oxford: Oxford University Press, ISBN 978-0-19-859686-8
Ludyk, Günter (2013). Einstein in Matrix Form (1st ed.). Berlin: Springer. ISBN 978-3-642-35797-8.
Møller, Christian (1955) [1952], The Theory of Relativity, Oxford University Press, OCLC 7644624
Moore, Thomas A (2012), A General Relativity Workbook, University Science Books, ISBN 978-1-891389-82-5
Schutz, B. F. (2009), A First Course in General Relativity (Second ed.), Cambridge University Press, Bibcode:2009fcgr.book.....S, ISBN 978-0-521-88705-2
=== Graduate textbooks ===
Carroll, Sean M. (2004), Spacetime and Geometry: An Introduction to General Relativity, San Francisco: Addison-Wesley, Bibcode:2004sgig.book.....C, ISBN 978-0-8053-8732-2
Grøn, Øyvind; Hervik, Sigbjørn (2007), Einstein's General Theory of Relativity, New York: Springer, ISBN 978-0-387-69199-2
Landau, Lev D.; Lifshitz, Evgeny F. (1980), The Classical Theory of Fields (4th ed.), London: Butterworth-Heinemann, ISBN 978-0-7506-2768-9
Landsman, Klaas (2021). Foundations of General Relativity: From Einstein to Black Holes. Radboud University Press. ISBN 9789083178929.
Stephani, Hans (1990), General Relativity: An Introduction to the Theory of the Gravitational Field, Cambridge: Cambridge University Press, Bibcode:1990grit.book.....S, ISBN 978-0-521-37941-0
Charles W. Misner; Kip S. Thorne; John Archibald Wheeler (1973), Gravitation, W. H. Freeman, Princeton University Press, ISBN 0-7167-0344-0
R.K. Sachs; H. Wu (1977), General Relativity for Mathematicians, Springer-Verlag, Bibcode:1977grm..book.....S, ISBN 1-4612-9905-5
Wald, Robert M. (1984). General Relativity. Chicago: University of Chicago Press. ISBN 0-226-87032-4. OCLC 10018614.
=== Specialists' books ===
Hawking, Stephen; Ellis, George (1975). The Large Scale Structure of Space-time. Cambridge University Press. ISBN 978-0-521-09906-6.
Poisson, Eric (2007). A Relativist's Toolkit: The Mathematics of Black-Hole Mechanics. Cambridge University Press. ISBN 978-0-521-53780-3.
=== Journal articles ===
Einstein, Albert (1916), "Die Grundlage der allgemeinen Relativitätstheorie", Annalen der Physik, 49 (7): 769–822, Bibcode:1916AnP...354..769E, doi:10.1002/andp.19163540702 See also English translation at Einstein Papers Project
Flanagan, Éanna É.; Hughes, Scott A. (2005), "The basics of gravitational wave theory", New J. Phys., 7 (1): 204, arXiv:gr-qc/0501041, Bibcode:2005NJPh....7..204F, doi:10.1088/1367-2630/7/1/204
Landgraf, M.; Hechler, M.; Kemble, S. (2005), "Mission design for LISA Pathfinder", Class. Quantum Grav., 22 (10): S487 – S492, arXiv:gr-qc/0411071, Bibcode:2005CQGra..22S.487L, doi:10.1088/0264-9381/22/10/048, S2CID 119476595
Nieto, Michael Martin (2006), "The quest to understand the Pioneer anomaly" (PDF), Europhysics News, 37 (6): 30–34, arXiv:gr-qc/0702017, Bibcode:2006ENews..37f..30N, doi:10.1051/epn:2006604, archived (PDF) from the original on 24 September 2015
Shapiro, I. I.; Pettengill, Gordon; Ash, Michael; Stone, Melvin; Smith, William; Ingalls, Richard; Brockelman, Richard (1968), "Fourth test of general relativity: preliminary results", Phys. Rev. Lett., 20 (22): 1265–1269, Bibcode:1968PhRvL..20.1265S, doi:10.1103/PhysRevLett.20.1265
Valtonen, M. J.; Lehto, H. J.; Nilsson, K.; Heidt, J.; Takalo, L. O.; Sillanpää, A.; Villforth, C.; Kidger, M.; et al. (2008), "A massive binary black-hole system in OJ 287 and a test of general relativity", Nature, 452 (7189): 851–853, arXiv:0809.1280, Bibcode:2008Natur.452..851V, doi:10.1038/nature06896, PMID 18421348, S2CID 4412396
== External links ==
Einstein Online Archived 1 June 2014 at the Wayback Machine – Articles on a variety of aspects of relativistic physics for a general audience; hosted by the Max Planck Institute for Gravitational Physics
GEO600 home page, the official website of the GEO600 project.
LIGO Laboratory
NCSA Spacetime Wrinkles – produced by the numerical relativity group at the NCSA, with an elementary introduction to general relativity
Einstein's General Theory of Relativity on YouTube (lecture by Leonard Susskind recorded 22 September 2008 at Stanford University).
Series of lectures on General Relativity given in 2006 at the Institut Henri Poincaré (introductory/advanced).
General Relativity Tutorials by John Baez.
Brown, Kevin. "Reflections on relativity". Mathpages.com. Archived from the original on 18 December 2015. Retrieved 29 May 2005.
Carroll, Sean M. (1997). "Lecture Notes on General Relativity". arXiv:gr-qc/9712019.
Moor, Rafi. "Understanding General Relativity". Retrieved 11 July 2006.
Waner, Stefan. "Introduction to Differential Geometry and General Relativity". Retrieved 5 April 2015.
The Feynman Lectures on Physics Vol. II Ch. 42: Curved Space | Wikipedia/General_Theory_of_Relativity |
Relativity may refer to:
== Physics ==
Galilean relativity, Galileo's conception of relativity
Numerical relativity, a subfield of computational physics that aims to establish numerical solutions to Einstein's field equations in general relativity
Principle of relativity, used in Einstein's theories and derived from Galileo's principle
Theory of relativity, a general treatment that refers to both special relativity and general relativity
General relativity, Albert Einstein's theory of gravitation
Special relativity, a theory formulated by Albert Einstein, Henri Poincaré, and Hendrik Lorentz
Relativity: The Special and the General Theory, a 1920 book by Albert Einstein
== Social sciences ==
Linguistic relativity
Cultural relativity
Moral relativity
== Arts and entertainment ==
=== Music ===
Relativity Music Group, a Universal subsidiary record label for releasing film soundtracks
Relativity Records, an American record label
Relativity (band), a Scots-Irish traditional music quartet 1985–1987
Relativity (Emarosa album), 2008
Relativity (Indecent Obsession album), 1993
Relativity (Walt Dickerson album) or the title song, 1962
Relativity, an EP by Grafton Primary, 2007
=== Television ===
Relativity (TV series), a 1996–1997 American drama series
"Relativity" (Farscape), an episode
"Relativity" (Star Trek: Voyager), an episode
=== Other ===
Relativity (M. C. Escher), a 1953 lithograph print by M. C. Escher
Relativity Media, an American film production company
== Business ==
Relativity Space, an American aerospace manufacturing company
== See also ==
Relative (disambiguation)
Relativism, a family of philosophical, religious, and social views | Wikipedia/relativity |
In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation is then parameterized by the negative of this velocity. The transformations are named after the Dutch physicist Hendrik Lorentz.
The most common form of the transformation, parametrized by the real constant
v
,
{\displaystyle v,}
representing a velocity confined to the x-direction, is expressed as
t
′
=
γ
(
t
−
v
x
c
2
)
x
′
=
γ
(
x
−
v
t
)
y
′
=
y
z
′
=
z
{\displaystyle {\begin{aligned}t'&=\gamma \left(t-{\frac {vx}{c^{2}}}\right)\\x'&=\gamma \left(x-vt\right)\\y'&=y\\z'&=z\end{aligned}}}
where (t, x, y, z) and (t′, x′, y′, z′) are the coordinates of an event in two frames with the spatial origins coinciding at t = t′ = 0, where the primed frame is seen from the unprimed frame as moving with speed v along the x-axis, where c is the speed of light, and
γ
=
1
1
−
v
2
/
c
2
{\displaystyle \gamma ={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}}
is the Lorentz factor. When speed v is much smaller than c, the Lorentz factor is negligibly different from 1, but as v approaches c,
γ
{\displaystyle \gamma }
grows without bound. The value of v must be smaller than c for the transformation to make sense.
Expressing the speed as a fraction of the speed of light,
β
=
v
/
c
,
{\textstyle \beta =v/c,}
an equivalent form of the transformation is
c
t
′
=
γ
(
c
t
−
β
x
)
x
′
=
γ
(
x
−
β
c
t
)
y
′
=
y
z
′
=
z
.
{\displaystyle {\begin{aligned}ct'&=\gamma \left(ct-\beta x\right)\\x'&=\gamma \left(x-\beta ct\right)\\y'&=y\\z'&=z.\end{aligned}}}
Frames of reference can be divided into two groups: inertial (relative motion with constant velocity) and non-inertial (accelerating, moving in curved paths, rotational motion with constant angular velocity, etc.). The term "Lorentz transformations" only refers to transformations between inertial frames, usually in the context of special relativity.
In each reference frame, an observer can use a local coordinate system (usually Cartesian coordinates in this context) to measure lengths, and a clock to measure time intervals. An event is something that happens at a point in space at an instant of time, or more formally a point in spacetime. The transformations connect the space and time coordinates of an event as measured by an observer in each frame.
They supersede the Galilean transformation of Newtonian physics, which assumes an absolute space and time (see Galilean relativity). The Galilean transformation is a good approximation only at relative speeds much less than the speed of light. Lorentz transformations have a number of unintuitive features that do not appear in Galilean transformations. For example, they reflect the fact that observers moving at different velocities may measure different distances, elapsed times, and even different orderings of events, but always such that the speed of light is the same in all inertial reference frames. The invariance of light speed is one of the postulates of special relativity.
Historically, the transformations were the result of attempts by Lorentz and others to explain how the speed of light was observed to be independent of the reference frame, and to understand the symmetries of the laws of electromagnetism. The transformations later became a cornerstone for special relativity.
The Lorentz transformation is a linear transformation. It may include a rotation of space; a rotation-free Lorentz transformation is called a Lorentz boost. In Minkowski space—the mathematical model of spacetime in special relativity—the Lorentz transformations preserve the spacetime interval between any two events. They describe only the transformations in which the spacetime event at the origin is left fixed. They can be considered as a hyperbolic rotation of Minkowski space. The more general set of transformations that also includes translations is known as the Poincaré group.
== History ==
Many physicists—including Woldemar Voigt, George FitzGerald, Joseph Larmor, and Hendrik Lorentz himself—had been discussing the physics implied by these equations since 1887. Early in 1889, Oliver Heaviside had shown from Maxwell's equations that the electric field surrounding a spherical distribution of charge should cease to have spherical symmetry once the charge is in motion relative to the luminiferous aether. FitzGerald then conjectured that Heaviside's distortion result might be applied to a theory of intermolecular forces. Some months later, FitzGerald published the conjecture that bodies in motion are being contracted, in order to explain the baffling outcome of the 1887 aether-wind experiment of Michelson and Morley. In 1892, Lorentz independently presented the same idea in a more detailed manner, which was subsequently called FitzGerald–Lorentz contraction hypothesis. Their explanation was widely known before 1905.
Lorentz (1892–1904) and Larmor (1897–1900), who believed the luminiferous aether hypothesis, also looked for the transformation under which Maxwell's equations are invariant when transformed from the aether to a moving frame. They extended the FitzGerald–Lorentz contraction hypothesis and found out that the time coordinate has to be modified as well ("local time"). Henri Poincaré gave a physical interpretation to local time (to first order in v/c, the relative velocity of the two reference frames normalized to the speed of light) as the consequence of clock synchronization, under the assumption that the speed of light is constant in moving frames. Larmor is credited to have been the first to understand the crucial time dilation property inherent in his equations.
In 1905, Poincaré was the first to recognize that the transformation has the properties of a mathematical group,
and he named it after Lorentz.
Later in the same year Albert Einstein published what is now called special relativity, by deriving the Lorentz transformation under the assumptions of the principle of relativity and the constancy of the speed of light in any inertial reference frame, and by abandoning the mechanistic aether as unnecessary.
== Derivation of the group of Lorentz transformations ==
An event is something that happens at a certain point in spacetime, or more generally, the point in spacetime itself. In any inertial frame an event is specified by a time coordinate ct and a set of Cartesian coordinates x, y, z to specify position in space in that frame. Subscripts label individual events.
From Einstein's second postulate of relativity (invariance of c) it follows that:
in all inertial frames for events connected by light signals. The quantity on the left is called the spacetime interval between events a1 = (t1, x1, y1, z1) and a2 = (t2, x2, y2, z2). The interval between any two events, not necessarily separated by light signals, is in fact invariant, i.e., independent of the state of relative motion of observers in different inertial frames, as is shown using homogeneity and isotropy of space. The transformation sought after thus must possess the property that:
where (t, x, y, z) are the spacetime coordinates used to define events in one frame, and (t′, x′, y′, z′) are the coordinates in another frame. First one observes that (D2) is satisfied if an arbitrary 4-tuple b of numbers are added to events a1 and a2. Such transformations are called spacetime translations and are not dealt with further here. Then one observes that a linear solution preserving the origin of the simpler problem solves the general problem too:
(a solution satisfying the first formula automatically satisfies the second one as well; see polarization identity). Finding the solution to the simpler problem is just a matter of look-up in the theory of classical groups that preserve bilinear forms of various signature. First equation in (D3) can be written more compactly as:
where (·, ·) refers to the bilinear form of signature (1, 3) on R4 exposed by the right hand side formula in (D3). The alternative notation defined on the right is referred to as the relativistic dot product. Spacetime mathematically viewed as R4 endowed with this bilinear form is known as Minkowski space M. The Lorentz transformation is thus an element of the group O(1, 3), the Lorentz group or, for those that prefer the other metric signature, O(3, 1) (also called the Lorentz group). One has:
which is precisely preservation of the bilinear form (D3) which implies (by linearity of Λ and bilinearity of the form) that (D2) is satisfied. The elements of the Lorentz group are rotations and boosts and mixes thereof. If the spacetime translations are included, then one obtains the inhomogeneous Lorentz group or the Poincaré group.
== Generalities ==
The relations between the primed and unprimed spacetime coordinates are the Lorentz transformations, each coordinate in one frame is a linear function of all the coordinates in the other frame, and the inverse functions are the inverse transformation. Depending on how the frames move relative to each other, and how they are oriented in space relative to each other, other parameters that describe direction, speed, and orientation enter the transformation equations.
Transformations describing relative motion with constant (uniform) velocity and without rotation of the space coordinate axes are called Lorentz boosts or simply boosts, and the relative velocity between the frames is the parameter of the transformation. The other basic type of Lorentz transformation is rotation in the spatial coordinates only, these like boosts are inertial transformations since there is no relative motion, the frames are simply tilted (and not continuously rotating), and in this case quantities defining the rotation are the parameters of the transformation (e.g., axis–angle representation, or Euler angles, etc.). A combination of a rotation and boost is a homogeneous transformation, which transforms the origin back to the origin.
The full Lorentz group O(3, 1) also contains special transformations that are neither rotations nor boosts, but rather reflections in a plane through the origin. Two of these can be singled out; spatial inversion in which the spatial coordinates of all events are reversed in sign and temporal inversion in which the time coordinate for each event gets its sign reversed.
Boosts should not be conflated with mere displacements in spacetime; in this case, the coordinate systems are simply shifted and there is no relative motion. However, these also count as symmetries forced by special relativity since they leave the spacetime interval invariant. A combination of a rotation with a boost, followed by a shift in spacetime, is an inhomogeneous Lorentz transformation, an element of the Poincaré group, which is also called the inhomogeneous Lorentz group.
== Physical formulation of Lorentz boosts ==
=== Coordinate transformation ===
A "stationary" observer in frame F defines events with coordinates t, x, y, z. Another frame F′ moves with velocity v relative to F, and an observer in this "moving" frame F′ defines events using the coordinates t′, x′, y′, z′.
The coordinate axes in each frame are parallel (the x and x′ axes are parallel, the y and y′ axes are parallel, and the z and z′ axes are parallel), remain mutually perpendicular, and relative motion is along the coincident xx′ axes. At t = t′ = 0, the origins of both coordinate systems are the same, (x, y, z) = (x′, y′, z′) = (0, 0, 0). In other words, the times and positions are coincident at this event. If all these hold, then the coordinate systems are said to be in standard configuration, or synchronized.
If an observer in F records an event t, x, y, z, then an observer in F′ records the same event with coordinates
where v is the relative velocity between frames in the x-direction, c is the speed of light, and
γ
=
1
1
−
v
2
c
2
{\displaystyle \gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}
(lowercase gamma) is the Lorentz factor.
Here, v is the parameter of the transformation, for a given boost it is a constant number, but can take a continuous range of values. In the setup used here, positive relative velocity v > 0 is motion along the positive directions of the xx′ axes, zero relative velocity v = 0 is no relative motion, while negative relative velocity v < 0 is relative motion along the negative directions of the xx′ axes. The magnitude of relative velocity v cannot equal or exceed c, so only subluminal speeds −c < v < c are allowed. The corresponding range of γ is 1 ≤ γ < ∞.
The transformations are not defined if v is outside these limits. At the speed of light (v = c) γ is infinite, and faster than light (v > c) γ is a complex number, each of which make the transformations unphysical. The space and time coordinates are measurable quantities and numerically must be real numbers.
As an active transformation, an observer in F′ notices the coordinates of the event to be "boosted" in the negative directions of the xx′ axes, because of the −v in the transformations. This has the equivalent effect of the coordinate system F′ boosted in the positive directions of the xx′ axes, while the event does not change and is simply represented in another coordinate system, a passive transformation.
The inverse relations (t, x, y, z in terms of t′, x′, y′, z′) can be found by algebraically solving the original set of equations. A more efficient way is to use physical principles. Here F′ is the "stationary" frame while F is the "moving" frame. According to the principle of relativity, there is no privileged frame of reference, so the transformations from F′ to F must take exactly the same form as the transformations from F to F′. The only difference is F moves with velocity −v relative to F′ (i.e., the relative velocity has the same magnitude but is oppositely directed). Thus if an observer in F′ notes an event t′, x′, y′, z′, then an observer in F notes the same event with coordinates
and the value of γ remains unchanged. This "trick" of simply reversing the direction of relative velocity while preserving its magnitude, and exchanging primed and unprimed variables, always applies to finding the inverse transformation of every boost in any direction.
Sometimes it is more convenient to use β = v/c (lowercase beta) instead of v, so that
c
t
′
=
γ
(
c
t
−
β
x
)
,
x
′
=
γ
(
x
−
β
c
t
)
,
{\displaystyle {\begin{aligned}ct'&=\gamma \left(ct-\beta x\right)\,,\\x'&=\gamma \left(x-\beta ct\right)\,,\\\end{aligned}}}
which shows much more clearly the symmetry in the transformation. From the allowed ranges of v and the definition of β, it follows −1 < β < 1. The use of β and γ is standard throughout the literature. In the case of three spatial dimensions [ct,x,y,z], where the boost
β
{\displaystyle \beta }
is in the x direction, the eigenstates of the transformation are [1,1,0,0] with eigenvalue
(
1
−
β
)
/
(
1
+
β
)
{\displaystyle {\sqrt {(1-\beta )/(1+\beta )}}}
, [1, −1,0,0] with eigenvalue
(
1
+
β
)
/
(
1
−
β
)
{\displaystyle {\sqrt {(1+\beta )/(1-\beta )}}}
, and [0,0,1,0] and [0,0,0,1], the latter two with eigenvalue 1.
When the boost velocity
v
{\displaystyle {\boldsymbol {v}}}
is in an arbitrary vector direction with the boost vector
β
=
v
/
c
{\displaystyle {\boldsymbol {\beta }}={\boldsymbol {v}}/c}
, then the transformation from an unprimed spacetime coordinate system to a primed coordinate system is given by
[
c
t
′
−
γ
β
x
x
′
1
+
γ
2
1
+
γ
β
x
2
y
′
γ
2
1
+
γ
β
x
β
y
z
′
γ
2
1
+
γ
β
y
β
z
]
=
[
γ
−
γ
β
x
−
γ
β
y
−
γ
β
z
−
γ
β
x
1
+
γ
2
1
+
γ
β
x
2
γ
2
1
+
γ
β
x
β
y
γ
2
1
+
γ
β
x
β
z
−
γ
β
y
γ
2
1
+
γ
β
x
β
y
1
+
γ
2
1
+
γ
β
y
2
γ
2
1
+
γ
β
y
β
z
−
γ
β
z
γ
2
1
+
γ
β
x
β
z
γ
2
1
+
γ
β
y
β
z
1
+
γ
2
1
+
γ
β
z
2
]
[
c
t
−
γ
β
x
x
1
+
γ
2
1
+
γ
β
x
2
y
γ
2
1
+
γ
β
x
β
y
z
γ
2
1
+
γ
β
y
β
z
]
,
{\displaystyle {\begin{bmatrix}ct'{\vphantom {-\gamma \beta _{x}}}\\x'{\vphantom {1+{\frac {\gamma ^{2}}{1+\gamma }}\beta _{x}^{2}}}\\y'{\vphantom {{\frac {\gamma ^{2}}{1+\gamma }}\beta _{x}\beta _{y}}}\\z'{\vphantom {{\frac {\gamma ^{2}}{1+\gamma }}\beta _{y}\beta _{z}}}\end{bmatrix}}={\begin{bmatrix}\gamma &-\gamma \beta _{x}&-\gamma \beta _{y}&-\gamma \beta _{z}\\-\gamma \beta _{x}&1+{\frac {\gamma ^{2}}{1+\gamma }}\beta _{x}^{2}&{\frac {\gamma ^{2}}{1+\gamma }}\beta _{x}\beta _{y}&{\frac {\gamma ^{2}}{1+\gamma }}\beta _{x}\beta _{z}\\-\gamma \beta _{y}&{\frac {\gamma ^{2}}{1+\gamma }}\beta _{x}\beta _{y}&1+{\frac {\gamma ^{2}}{1+\gamma }}\beta _{y}^{2}&{\frac {\gamma ^{2}}{1+\gamma }}\beta _{y}\beta _{z}\\-\gamma \beta _{z}&{\frac {\gamma ^{2}}{1+\gamma }}\beta _{x}\beta _{z}&{\frac {\gamma ^{2}}{1+\gamma }}\beta _{y}\beta _{z}&1+{\frac {\gamma ^{2}}{1+\gamma }}\beta _{z}^{2}\\\end{bmatrix}}{\begin{bmatrix}ct{\vphantom {-\gamma \beta _{x}}}\\x{\vphantom {1+{\frac {\gamma ^{2}}{1+\gamma }}\beta _{x}^{2}}}\\y{\vphantom {{\frac {\gamma ^{2}}{1+\gamma }}\beta _{x}\beta _{y}}}\\z{\vphantom {{\frac {\gamma ^{2}}{1+\gamma }}\beta _{y}\beta _{z}}}\end{bmatrix}},}
where the Lorentz factor is
γ
=
1
/
1
−
β
2
{\displaystyle \gamma =1/{\sqrt {1-{\boldsymbol {\beta }}^{2}}}}
. The determinant of the transformation matrix is +1 and its trace is
2
(
1
+
γ
)
{\displaystyle 2(1+\gamma )}
. The inverse of the transformation is given by reversing the sign of
β
{\displaystyle {\boldsymbol {\beta }}}
. The quantity
c
2
t
2
−
x
2
−
y
2
−
z
2
{\displaystyle c^{2}t^{2}-x^{2}-y^{2}-z^{2}}
is invariant under the transformation: namely
(
c
t
′
2
−
x
′
2
−
y
′
2
−
z
′
2
)
=
(
c
t
2
−
x
2
−
y
2
−
z
2
)
{\displaystyle (ct'^{2}-x'^{2}-y'^{2}-z'^{2})=(ct^{2}-x^{2}-y^{2}-z^{2})}
.
The Lorentz transformations can also be derived in a way that resembles circular rotations in 3-dimensional space using the hyperbolic functions. For the boost in the x direction, the results are
where ζ (lowercase zeta) is a parameter called rapidity (many other symbols are used, including θ, ϕ, φ, η, ψ, ξ). Given the strong resemblance to rotations of spatial coordinates in 3-dimensional space in the Cartesian xy, yz, and zx planes, a Lorentz boost can be thought of as a hyperbolic rotation of spacetime coordinates in the xt, yt, and zt Cartesian-time planes of 4-dimensional Minkowski space. The parameter ζ is the hyperbolic angle of rotation, analogous to the ordinary angle for circular rotations. This transformation can be illustrated with a Minkowski diagram.
The hyperbolic functions arise from the difference between the squares of the time and spatial coordinates in the spacetime interval, rather than a sum. The geometric significance of the hyperbolic functions can be visualized by taking x = 0 or ct = 0 in the transformations. Squaring and subtracting the results, one can derive hyperbolic curves of constant coordinate values but varying ζ, which parametrizes the curves according to the identity
cosh
2
ζ
−
sinh
2
ζ
=
1
.
{\displaystyle \cosh ^{2}\zeta -\sinh ^{2}\zeta =1\,.}
Conversely the ct and x axes can be constructed for varying coordinates but constant ζ. The definition
tanh
ζ
=
sinh
ζ
cosh
ζ
,
{\displaystyle \tanh \zeta ={\frac {\sinh \zeta }{\cosh \zeta }}\,,}
provides the link between a constant value of rapidity, and the slope of the ct axis in spacetime. A consequence these two hyperbolic formulae is an identity that matches the Lorentz factor
cosh
ζ
=
1
1
−
tanh
2
ζ
.
{\displaystyle \cosh \zeta ={\frac {1}{\sqrt {1-\tanh ^{2}\zeta }}}\,.}
Comparing the Lorentz transformations in terms of the relative velocity and rapidity, or using the above formulae, the connections between β, γ, and ζ are
β
=
tanh
ζ
,
γ
=
cosh
ζ
,
β
γ
=
sinh
ζ
.
{\displaystyle {\begin{aligned}\beta &=\tanh \zeta \,,\\\gamma &=\cosh \zeta \,,\\\beta \gamma &=\sinh \zeta \,.\end{aligned}}}
Taking the inverse hyperbolic tangent gives the rapidity
ζ
=
tanh
−
1
β
.
{\displaystyle \zeta =\tanh ^{-1}\beta \,.}
Since −1 < β < 1, it follows −∞ < ζ < ∞. From the relation between ζ and β, positive rapidity ζ > 0 is motion along the positive directions of the xx′ axes, zero rapidity ζ = 0 is no relative motion, while negative rapidity ζ < 0 is relative motion along the negative directions of the xx′ axes.
The inverse transformations are obtained by exchanging primed and unprimed quantities to switch the coordinate frames, and negating rapidity ζ → −ζ since this is equivalent to negating the relative velocity. Therefore,
The inverse transformations can be similarly visualized by considering the cases when x′ = 0 and ct′ = 0.
So far the Lorentz transformations have been applied to one event. If there are two events, there is a spatial separation and time interval between them. It follows from the linearity of the Lorentz transformations that two values of space and time coordinates can be chosen, the Lorentz transformations can be applied to each, then subtracted to get the Lorentz transformations of the differences;
Δ
t
′
=
γ
(
Δ
t
−
v
Δ
x
c
2
)
,
Δ
x
′
=
γ
(
Δ
x
−
v
Δ
t
)
,
{\displaystyle {\begin{aligned}\Delta t'&=\gamma \left(\Delta t-{\frac {v\,\Delta x}{c^{2}}}\right)\,,\\\Delta x'&=\gamma \left(\Delta x-v\,\Delta t\right)\,,\end{aligned}}}
with inverse relations
Δ
t
=
γ
(
Δ
t
′
+
v
Δ
x
′
c
2
)
,
Δ
x
=
γ
(
Δ
x
′
+
v
Δ
t
′
)
.
{\displaystyle {\begin{aligned}\Delta t&=\gamma \left(\Delta t'+{\frac {v\,\Delta x'}{c^{2}}}\right)\,,\\\Delta x&=\gamma \left(\Delta x'+v\,\Delta t'\right)\,.\end{aligned}}}
where Δ (uppercase delta) indicates a difference of quantities; e.g., Δx = x2 − x1 for two values of x coordinates, and so on.
These transformations on differences rather than spatial points or instants of time are useful for a number of reasons:
in calculations and experiments, it is lengths between two points or time intervals that are measured or of interest (e.g., the length of a moving vehicle, or time duration it takes to travel from one place to another),
the transformations of velocity can be readily derived by making the difference infinitesimally small and dividing the equations, and the process repeated for the transformation of acceleration,
if the coordinate systems are never coincident (i.e., not in standard configuration), and if both observers can agree on an event t0, x0, y0, z0 in F and t0′, x0′, y0′, z0′ in F′, then they can use that event as the origin, and the spacetime coordinate differences are the differences between their coordinates and this origin, e.g., Δx = x − x0, Δx′ = x′ − x0′, etc.
=== Physical implications ===
A critical requirement of the Lorentz transformations is the invariance of the speed of light, a fact used in their derivation, and contained in the transformations themselves. If in F the equation for a pulse of light along the x direction is x = ct, then in F′ the Lorentz transformations give x′ = ct′, and vice versa, for any −c < v < c.
For relative speeds much less than the speed of light, the Lorentz transformations reduce to the Galilean transformation:
t
′
≈
t
x
′
≈
x
−
v
t
{\displaystyle {\begin{aligned}t'&\approx t\\x'&\approx x-vt\end{aligned}}}
in accordance with the correspondence principle. It is sometimes said that nonrelativistic physics is a physics of "instantaneous action at a distance".
Three counterintuitive, but correct, predictions of the transformations are:
Relativity of simultaneity
Suppose two events occur along the x axis simultaneously (Δt = 0) in F, but separated by a nonzero displacement Δx. Then in F′, we find that
Δ
t
′
=
γ
−
v
Δ
x
c
2
{\displaystyle \Delta t'=\gamma {\frac {-v\,\Delta x}{c^{2}}}}
, so the events are no longer simultaneous according to a moving observer.
Time dilation
Suppose there is a clock at rest in F. If a time interval is measured at the same point in that frame, so that Δx = 0, then the transformations give this interval in F′ by Δt′ = γΔt. Conversely, suppose there is a clock at rest in F′. If an interval is measured at the same point in that frame, so that Δx′ = 0, then the transformations give this interval in F by Δt = γΔt′. Either way, each observer measures the time interval between ticks of a moving clock to be longer by a factor γ than the time interval between ticks of his own clock.
Length contraction
Suppose there is a rod at rest in F aligned along the x axis, with length Δx. In F′, the rod moves with velocity -v, so its length must be measured by taking two simultaneous (Δt′ = 0) measurements at opposite ends. Under these conditions, the inverse Lorentz transform shows that Δx = γΔx′. In F the two measurements are no longer simultaneous, but this does not matter because the rod is at rest in F. So each observer measures the distance between the end points of a moving rod to be shorter by a factor 1/γ than the end points of an identical rod at rest in his own frame. Length contraction affects any geometric quantity related to lengths, so from the perspective of a moving observer, areas and volumes will also appear to shrink along the direction of motion.
=== Vector transformations ===
The use of vectors allows positions and velocities to be expressed in arbitrary directions compactly. A single boost in any direction depends on the full relative velocity vector v with a magnitude |v| = v that cannot equal or exceed c, so that 0 ≤ v < c.
Only time and the coordinates parallel to the direction of relative motion change, while those coordinates perpendicular do not. With this in mind, split the spatial position vector r as measured in F, and r′ as measured in F′, each into components perpendicular (⊥) and parallel ( ‖ ) to v,
r
=
r
⊥
+
r
‖
,
r
′
=
r
⊥
′
+
r
‖
′
,
{\displaystyle \mathbf {r} =\mathbf {r} _{\perp }+\mathbf {r} _{\|}\,,\quad \mathbf {r} '=\mathbf {r} _{\perp }'+\mathbf {r} _{\|}'\,,}
then the transformations are
t
′
=
γ
(
t
−
r
∥
⋅
v
c
2
)
r
‖
′
=
γ
(
r
‖
−
v
t
)
r
⊥
′
=
r
⊥
{\displaystyle {\begin{aligned}t'&=\gamma \left(t-{\frac {\mathbf {r} _{\parallel }\cdot \mathbf {v} }{c^{2}}}\right)\\\mathbf {r} _{\|}'&=\gamma (\mathbf {r} _{\|}-\mathbf {v} t)\\\mathbf {r} _{\perp }'&=\mathbf {r} _{\perp }\end{aligned}}}
where · is the dot product. The Lorentz factor γ retains its definition for a boost in any direction, since it depends only on the magnitude of the relative velocity. The definition β = v/c with magnitude 0 ≤ β < 1 is also used by some authors.
Introducing a unit vector n = v/v = β/β in the direction of relative motion, the relative velocity is v = vn with magnitude v and direction n, and vector projection and rejection give respectively
r
∥
=
(
r
⋅
n
)
n
,
r
⊥
=
r
−
(
r
⋅
n
)
n
{\displaystyle \mathbf {r} _{\parallel }=(\mathbf {r} \cdot \mathbf {n} )\mathbf {n} \,,\quad \mathbf {r} _{\perp }=\mathbf {r} -(\mathbf {r} \cdot \mathbf {n} )\mathbf {n} }
Accumulating the results gives the full transformations,
The projection and rejection also applies to r′. For the inverse transformations, exchange r and r′ to switch observed coordinates, and negate the relative velocity v → −v (or simply the unit vector n → −n since the magnitude v is always positive) to obtain
The unit vector has the advantage of simplifying equations for a single boost, allows either v or β to be reinstated when convenient, and the rapidity parametrization is immediately obtained by replacing β and βγ. It is not convenient for multiple boosts.
The vectorial relation between relative velocity and rapidity is
β
=
β
n
=
n
tanh
ζ
,
{\displaystyle {\boldsymbol {\beta }}=\beta \mathbf {n} =\mathbf {n} \tanh \zeta \,,}
and the "rapidity vector" can be defined as
ζ
=
ζ
n
=
n
tanh
−
1
β
,
{\displaystyle {\boldsymbol {\zeta }}=\zeta \mathbf {n} =\mathbf {n} \tanh ^{-1}\beta \,,}
each of which serves as a useful abbreviation in some contexts. The magnitude of ζ is the absolute value of the rapidity scalar confined to 0 ≤ ζ < ∞, which agrees with the range 0 ≤ β < 1.
=== Transformation of velocities ===
Defining the coordinate velocities and Lorentz factor by
u
=
d
r
d
t
,
u
′
=
d
r
′
d
t
′
,
γ
v
=
1
1
−
v
⋅
v
c
2
{\displaystyle \mathbf {u} ={\frac {d\mathbf {r} }{dt}}\,,\quad \mathbf {u} '={\frac {d\mathbf {r} '}{dt'}}\,,\quad \gamma _{\mathbf {v} }={\frac {1}{\sqrt {1-{\dfrac {\mathbf {v} \cdot \mathbf {v} }{c^{2}}}}}}}
taking the differentials in the coordinates and time of the vector transformations, then dividing equations, leads to
u
′
=
1
1
−
v
⋅
u
c
2
[
u
γ
v
−
v
+
1
c
2
γ
v
γ
v
+
1
(
u
⋅
v
)
v
]
{\displaystyle \mathbf {u} '={\frac {1}{1-{\frac {\mathbf {v} \cdot \mathbf {u} }{c^{2}}}}}\left[{\frac {\mathbf {u} }{\gamma _{\mathbf {v} }}}-\mathbf {v} +{\frac {1}{c^{2}}}{\frac {\gamma _{\mathbf {v} }}{\gamma _{\mathbf {v} }+1}}\left(\mathbf {u} \cdot \mathbf {v} \right)\mathbf {v} \right]}
The velocities u and u′ are the velocity of some massive object. They can also be for a third inertial frame (say F′′), in which case they must be constant. Denote either entity by X. Then X moves with velocity u relative to F, or equivalently with velocity u′ relative to F′, in turn F′ moves with velocity v relative to F. The inverse transformations can be obtained in a similar way, or as with position coordinates exchange u and u′, and change v to −v.
The transformation of velocity is useful in stellar aberration, the Fizeau experiment, and the relativistic Doppler effect.
The Lorentz transformations of acceleration can be similarly obtained by taking differentials in the velocity vectors, and dividing these by the time differential.
=== Transformation of other quantities ===
In general, given four quantities A and Z = (Zx, Zy, Zz) and their Lorentz-boosted counterparts A′ and Z′ = (Z′x, Z′y, Z′z), a relation of the form
A
2
−
Z
⋅
Z
=
A
′
2
−
Z
′
⋅
Z
′
{\displaystyle A^{2}-\mathbf {Z} \cdot \mathbf {Z} ={A'}^{2}-\mathbf {Z} '\cdot \mathbf {Z} '}
implies the quantities transform under Lorentz transformations similar to the transformation of spacetime coordinates;
A
′
=
γ
(
A
−
v
n
⋅
Z
c
)
,
Z
′
=
Z
+
(
γ
−
1
)
(
Z
⋅
n
)
n
−
γ
A
v
n
c
.
{\displaystyle {\begin{aligned}A'&=\gamma \left(A-{\frac {v\mathbf {n} \cdot \mathbf {Z} }{c}}\right)\,,\\\mathbf {Z} '&=\mathbf {Z} +(\gamma -1)(\mathbf {Z} \cdot \mathbf {n} )\mathbf {n} -{\frac {\gamma Av\mathbf {n} }{c}}\,.\end{aligned}}}
The decomposition of Z (and Z′) into components perpendicular and parallel to v is exactly the same as for the position vector, as is the process of obtaining the inverse transformations (exchange (A, Z) and (A′, Z′) to switch observed quantities, and reverse the direction of relative motion by the substitution n ↦ −n).
The quantities (A, Z) collectively make up a four-vector, where A is the "timelike component", and Z the "spacelike component". Examples of A and Z are the following:
For a given object (e.g., particle, fluid, field, material), if A or Z correspond to properties specific to the object like its charge density, mass density, spin, etc., its properties can be fixed in the rest frame of that object. Then the Lorentz transformations give the corresponding properties in a frame moving relative to the object with constant velocity. This breaks some notions taken for granted in non-relativistic physics. For example, the energy E of an object is a scalar in non-relativistic mechanics, but not in relativistic mechanics because energy changes under Lorentz transformations; its value is different for various inertial frames. In the rest frame of an object, it has a rest energy and zero momentum. In a boosted frame its energy is different and it appears to have a momentum. Similarly, in non-relativistic quantum mechanics the spin of a particle is a constant vector, but in relativistic quantum mechanics spin s depends on relative motion. In the rest frame of the particle, the spin pseudovector can be fixed to be its ordinary non-relativistic spin with a zero timelike quantity st, however a boosted observer will perceive a nonzero timelike component and an altered spin.
Not all quantities are invariant in the form as shown above, for example orbital angular momentum L does not have a timelike quantity, and neither does the electric field E nor the magnetic field B. The definition of angular momentum is L = r × p, and in a boosted frame the altered angular momentum is L′ = r′ × p′. Applying this definition using the transformations of coordinates and momentum leads to the transformation of angular momentum. It turns out L transforms with another vector quantity N = (E/c2)r − tp related to boosts, see relativistic angular momentum for details. For the case of the E and B fields, the transformations cannot be obtained as directly using vector algebra. The Lorentz force is the definition of these fields, and in F it is F = q(E + v × B) while in F′ it is F′ = q(E′ + v′ × B′). A method of deriving the EM field transformations in an efficient way which also illustrates the unit of the electromagnetic field uses tensor algebra, given below.
== Mathematical formulation ==
Throughout, italic non-bold capital letters are 4 × 4 matrices, while non-italic bold letters are 3 × 3 matrices.
=== Homogeneous Lorentz group ===
Writing the coordinates in column vectors and the Minkowski metric η as a square matrix
X
′
=
[
c
t
′
x
′
y
′
z
′
]
,
η
=
[
−
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
]
,
X
=
[
c
t
x
y
z
]
{\displaystyle X'={\begin{bmatrix}c\,t'\\x'\\y'\\z'\end{bmatrix}}\,,\quad \eta ={\begin{bmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}}\,,\quad X={\begin{bmatrix}c\,t\\x\\y\\z\end{bmatrix}}}
the spacetime interval takes the form (superscript T denotes transpose)
X
⋅
X
=
X
T
η
X
=
X
′
T
η
X
′
{\displaystyle X\cdot X=X^{\mathrm {T} }\eta X={X'}^{\mathrm {T} }\eta {X'}}
and is invariant under a Lorentz transformation
X
′
=
Λ
X
{\displaystyle X'=\Lambda X}
where Λ is a square matrix which can depend on parameters.
The set of all Lorentz transformations
Λ
{\displaystyle \Lambda }
in this article is denoted
L
{\displaystyle {\mathcal {L}}}
. This set together with matrix multiplication forms a group, in this context known as the Lorentz group. Also, the above expression X·X is a quadratic form of signature (3,1) on spacetime, and the group of transformations which leaves this quadratic form invariant is the indefinite orthogonal group O(3,1), a Lie group. In other words, the Lorentz group is O(3,1). As presented in this article, any Lie groups mentioned are matrix Lie groups. In this context the operation of composition amounts to matrix multiplication.
From the invariance of the spacetime interval it follows
η
=
Λ
T
η
Λ
{\displaystyle \eta =\Lambda ^{\mathrm {T} }\eta \Lambda }
and this matrix equation contains the general conditions on the Lorentz transformation to ensure invariance of the spacetime interval. Taking the determinant of the equation using the product rule gives immediately
[
det
(
Λ
)
]
2
=
1
⇒
det
(
Λ
)
=
±
1
{\displaystyle \left[\det(\Lambda )\right]^{2}=1\quad \Rightarrow \quad \det(\Lambda )=\pm 1}
Writing the Minkowski metric as a block matrix, and the Lorentz transformation in the most general form,
η
=
[
−
1
0
0
I
]
,
Λ
=
[
Γ
−
a
T
−
b
M
]
,
{\displaystyle \eta ={\begin{bmatrix}-1&0\\0&\mathbf {I} \end{bmatrix}}\,,\quad \Lambda ={\begin{bmatrix}\Gamma &-\mathbf {a} ^{\mathrm {T} }\\-\mathbf {b} &\mathbf {M} \end{bmatrix}}\,,}
carrying out the block matrix multiplications obtains general conditions on Γ, a, b, M to ensure relativistic invariance. Not much information can be directly extracted from all the conditions, however one of the results
Γ
2
=
1
+
b
T
b
{\displaystyle \Gamma ^{2}=1+\mathbf {b} ^{\mathrm {T} }\mathbf {b} }
is useful; bTb ≥ 0 always so it follows that
Γ
2
≥
1
⇒
Γ
≤
−
1
,
Γ
≥
1
{\displaystyle \Gamma ^{2}\geq 1\quad \Rightarrow \quad \Gamma \leq -1\,,\quad \Gamma \geq 1}
The negative inequality may be unexpected, because Γ multiplies the time coordinate and this has an effect on time symmetry. If the positive equality holds, then Γ is the Lorentz factor.
The determinant and inequality provide four ways to classify Lorentz Transformations (herein LTs for brevity). Any particular LT has only one determinant sign and only one inequality. There are four sets which include every possible pair given by the intersections ("n"-shaped symbol meaning "and") of these classifying sets.
where "+" and "−" indicate the determinant sign, while "↑" for ≥ and "↓" for ≤ denote the inequalities.
The full Lorentz group splits into the union ("u"-shaped symbol meaning "or") of four disjoint sets
L
=
L
+
↑
∪
L
−
↑
∪
L
+
↓
∪
L
−
↓
{\displaystyle {\mathcal {L}}={\mathcal {L}}_{+}^{\uparrow }\cup {\mathcal {L}}_{-}^{\uparrow }\cup {\mathcal {L}}_{+}^{\downarrow }\cup {\mathcal {L}}_{-}^{\downarrow }}
A subgroup of a group must be closed under the same operation of the group (here matrix multiplication). In other words, for two Lorentz transformations Λ and L from a particular subgroup, the composite Lorentz transformations ΛL and LΛ must be in the same subgroup as Λ and L. This is not always the case: the composition of two antichronous Lorentz transformations is orthochronous, and the composition of two improper Lorentz transformations is proper. In other words, while the sets
L
+
↑
{\displaystyle {\mathcal {L}}_{+}^{\uparrow }}
,
L
+
{\displaystyle {\mathcal {L}}_{+}}
,
L
↑
{\displaystyle {\mathcal {L}}^{\uparrow }}
, and
L
0
=
L
+
↑
∪
L
−
↓
{\displaystyle {\mathcal {L}}_{0}={\mathcal {L}}_{+}^{\uparrow }\cup {\mathcal {L}}_{-}^{\downarrow }}
all form subgroups, the sets containing improper and/or antichronous transformations without enough proper orthochronous transformations (e.g.
L
+
↓
{\displaystyle {\mathcal {L}}_{+}^{\downarrow }}
,
L
−
↓
{\displaystyle {\mathcal {L}}_{-}^{\downarrow }}
,
L
−
↑
{\displaystyle {\mathcal {L}}_{-}^{\uparrow }}
) do not form subgroups.
=== Proper transformations ===
If a Lorentz covariant 4-vector is measured in one inertial frame with result
X
{\displaystyle X}
, and the same measurement made in another inertial frame (with the same orientation and origin) gives result
X
′
{\displaystyle X'}
, the two results will be related by
X
′
=
B
(
v
)
X
{\displaystyle X'=B(\mathbf {v} )X}
where the boost matrix
B
(
v
)
{\displaystyle B(\mathbf {v} )}
represents the rotation-free Lorentz transformation between the unprimed and primed frames and
v
{\displaystyle \mathbf {v} }
is the velocity of the primed frame as seen from the unprimed frame. The matrix is given by
B
(
v
)
=
[
γ
−
γ
v
x
/
c
−
γ
v
y
/
c
−
γ
v
z
/
c
−
γ
v
x
/
c
1
+
(
γ
−
1
)
v
x
2
v
2
(
γ
−
1
)
v
x
v
y
v
2
(
γ
−
1
)
v
x
v
z
v
2
−
γ
v
y
/
c
(
γ
−
1
)
v
y
v
x
v
2
1
+
(
γ
−
1
)
v
y
2
v
2
(
γ
−
1
)
v
y
v
z
v
2
−
γ
v
z
/
c
(
γ
−
1
)
v
z
v
x
v
2
(
γ
−
1
)
v
z
v
y
v
2
1
+
(
γ
−
1
)
v
z
2
v
2
]
=
[
γ
−
γ
β
→
T
−
γ
β
→
I
+
(
γ
−
1
)
β
→
β
→
T
β
2
]
,
{\displaystyle B(\mathbf {v} )={\begin{bmatrix}\gamma &-\gamma v_{x}/c&-\gamma v_{y}/c&-\gamma v_{z}/c\\-\gamma v_{x}/c&1+(\gamma -1){\dfrac {v_{x}^{2}}{v^{2}}}&(\gamma -1){\dfrac {v_{x}v_{y}}{v^{2}}}&(\gamma -1){\dfrac {v_{x}v_{z}}{v^{2}}}\\-\gamma v_{y}/c&(\gamma -1){\dfrac {v_{y}v_{x}}{v^{2}}}&1+(\gamma -1){\dfrac {v_{y}^{2}}{v^{2}}}&(\gamma -1){\dfrac {v_{y}v_{z}}{v^{2}}}\\-\gamma v_{z}/c&(\gamma -1){\dfrac {v_{z}v_{x}}{v^{2}}}&(\gamma -1){\dfrac {v_{z}v_{y}}{v^{2}}}&1+(\gamma -1){\dfrac {v_{z}^{2}}{v^{2}}}\end{bmatrix}}={\begin{bmatrix}\gamma &-\gamma {\vec {\beta }}^{T}\\-\gamma {\vec {\beta }}&I+(\gamma -1){\dfrac {{\vec {\beta }}{\vec {\beta }}^{T}}{\beta ^{2}}}\end{bmatrix}},}
where
v
=
v
x
2
+
v
y
2
+
v
z
2
{\textstyle v={\sqrt {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}}}
is the magnitude of the velocity and
γ
=
1
1
−
v
2
c
2
{\textstyle \gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}
is the Lorentz factor. This formula represents a passive transformation, as it describes how the coordinates of the measured quantity changes from the unprimed frame to the primed frame. The active transformation is given by
B
(
−
v
)
{\displaystyle B(-\mathbf {v} )}
.
If a frame F′ is boosted with velocity u relative to frame F, and another frame F′′ is boosted with velocity v relative to F′, the separate boosts are
X
″
=
B
(
v
)
X
′
,
X
′
=
B
(
u
)
X
{\displaystyle X''=B(\mathbf {v} )X'\,,\quad X'=B(\mathbf {u} )X}
and the composition of the two boosts connects the coordinates in F′′ and F,
X
″
=
B
(
v
)
B
(
u
)
X
.
{\displaystyle X''=B(\mathbf {v} )B(\mathbf {u} )X\,.}
Successive transformations act on the left. If u and v are collinear (parallel or antiparallel along the same line of relative motion), the boost matrices commute: B(v)B(u) = B(u)B(v). This composite transformation happens to be another boost, B(w), where w is collinear with u and v.
If u and v are not collinear but in different directions, the situation is considerably more complicated. Lorentz boosts along different directions do not commute: B(v)B(u) and B(u)B(v) are not equal. Although each of these compositions is not a single boost, each composition is still a Lorentz transformation as it preserves the spacetime interval. It turns out the composition of any two Lorentz boosts is equivalent to a boost followed or preceded by a rotation on the spatial coordinates, in the form of R(ρ)B(w) or B(w)R(ρ). The w and w are composite velocities, while ρ and ρ are rotation parameters (e.g. axis-angle variables, Euler angles, etc.). The rotation in block matrix form is simply
R
(
ρ
)
=
[
1
0
0
R
(
ρ
)
]
,
{\displaystyle \quad R({\boldsymbol {\rho }})={\begin{bmatrix}1&0\\0&\mathbf {R} ({\boldsymbol {\rho }})\end{bmatrix}}\,,}
where R(ρ) is a 3 × 3 rotation matrix, which rotates any 3-dimensional vector in one sense (active transformation), or equivalently the coordinate frame in the opposite sense (passive transformation). It is not simple to connect w and ρ (or w and ρ) to the original boost parameters u and v. In a composition of boosts, the R matrix is named the Wigner rotation, and gives rise to the Thomas precession. These articles give the explicit formulae for the composite transformation matrices, including expressions for w, ρ, w, ρ.
In this article the axis-angle representation is used for ρ. The rotation is about an axis in the direction of a unit vector e, through angle θ (positive anticlockwise, negative clockwise, according to the right-hand rule). The "axis-angle vector"
θ
=
θ
e
{\displaystyle {\boldsymbol {\theta }}=\theta \mathbf {e} }
will serve as a useful abbreviation.
Spatial rotations alone are also Lorentz transformations since they leave the spacetime interval invariant. Like boosts, successive rotations about different axes do not commute. Unlike boosts, the composition of any two rotations is equivalent to a single rotation. Some other similarities and differences between the boost and rotation matrices include:
inverses: B(v)−1 = B(−v) (relative motion in the opposite direction), and R(θ)−1 = R(−θ) (rotation in the opposite sense about the same axis)
identity transformation for no relative motion/rotation: B(0) = R(0) = I
unit determinant: det(B) = det(R) = +1. This property makes them proper transformations.
matrix symmetry: B is symmetric (equals transpose), while R is nonsymmetric but orthogonal (transpose equals inverse, RT = R−1).
The most general proper Lorentz transformation Λ(v, θ) includes a boost and rotation together, and is a nonsymmetric matrix. As special cases, Λ(0, θ) = R(θ) and Λ(v, 0) = B(v). An explicit form of the general Lorentz transformation is cumbersome to write down and will not be given here. Nevertheless, closed form expressions for the transformation matrices will be given below using group theoretical arguments. It will be easier to use the rapidity parametrization for boosts, in which case one writes Λ(ζ, θ) and B(ζ).
==== The Lie group SO+(3,1) ====
The set of transformations
{
B
(
ζ
)
,
R
(
θ
)
,
Λ
(
ζ
,
θ
)
}
{\displaystyle \{B({\boldsymbol {\zeta }}),R({\boldsymbol {\theta }}),\Lambda ({\boldsymbol {\zeta }},{\boldsymbol {\theta }})\}}
with matrix multiplication as the operation of composition forms a group, called the "restricted Lorentz group", and is the special indefinite orthogonal group SO+(3,1). (The plus sign indicates that it preserves the orientation of the temporal dimension).
For simplicity, look at the infinitesimal Lorentz boost in the x direction (examining a boost in any other direction, or rotation about any axis, follows an identical procedure). The infinitesimal boost is a small boost away from the identity, obtained by the Taylor expansion of the boost matrix to first order about ζ = 0,
B
x
=
I
+
ζ
∂
B
x
∂
ζ
|
ζ
=
0
+
⋯
{\displaystyle B_{x}=I+\zeta \left.{\frac {\partial B_{x}}{\partial \zeta }}\right|_{\zeta =0}+\cdots }
where the higher order terms not shown are negligible because ζ is small, and Bx is simply the boost matrix in the x direction. The derivative of the matrix is the matrix of derivatives (of the entries, with respect to the same variable), and it is understood the derivatives are found first then evaluated at ζ = 0,
∂
B
x
∂
ζ
|
ζ
=
0
=
−
K
x
.
{\displaystyle \left.{\frac {\partial B_{x}}{\partial \zeta }}\right|_{\zeta =0}=-K_{x}\,.}
For now, Kx is defined by this result (its significance will be explained shortly). In the limit of an infinite number of infinitely small steps, the finite boost transformation in the form of a matrix exponential is obtained
B
x
=
lim
N
→
∞
(
I
−
ζ
N
K
x
)
N
=
e
−
ζ
K
x
{\displaystyle B_{x}=\lim _{N\to \infty }\left(I-{\frac {\zeta }{N}}K_{x}\right)^{N}=e^{-\zeta K_{x}}}
where the limit definition of the exponential has been used (see also characterizations of the exponential function). More generally
B
(
ζ
)
=
e
−
ζ
⋅
K
,
R
(
θ
)
=
e
θ
⋅
J
.
{\displaystyle B({\boldsymbol {\zeta }})=e^{-{\boldsymbol {\zeta }}\cdot \mathbf {K} }\,,\quad R({\boldsymbol {\theta }})=e^{{\boldsymbol {\theta }}\cdot \mathbf {J} }\,.}
The axis-angle vector θ and rapidity vector ζ are altogether six continuous variables which make up the group parameters (in this particular representation), and the generators of the group are K = (Kx, Ky, Kz) and J = (Jx, Jy, Jz), each vectors of matrices with the explicit forms
K
x
=
[
0
1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
]
,
K
y
=
[
0
0
1
0
0
0
0
0
1
0
0
0
0
0
0
0
]
,
K
z
=
[
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
]
J
x
=
[
0
0
0
0
0
0
0
0
0
0
0
−
1
0
0
1
0
]
,
J
y
=
[
0
0
0
0
0
0
0
1
0
0
0
0
0
−
1
0
0
]
,
J
z
=
[
0
0
0
0
0
0
−
1
0
0
1
0
0
0
0
0
0
]
{\displaystyle {\begin{alignedat}{3}K_{x}&={\begin{bmatrix}0&1&0&0\\1&0&0&0\\0&0&0&0\\0&0&0&0\\\end{bmatrix}}\,,\quad &K_{y}&={\begin{bmatrix}0&0&1&0\\0&0&0&0\\1&0&0&0\\0&0&0&0\end{bmatrix}}\,,\quad &K_{z}&={\begin{bmatrix}0&0&0&1\\0&0&0&0\\0&0&0&0\\1&0&0&0\end{bmatrix}}\\[10mu]J_{x}&={\begin{bmatrix}0&0&0&0\\0&0&0&0\\0&0&0&-1\\0&0&1&0\\\end{bmatrix}}\,,\quad &J_{y}&={\begin{bmatrix}0&0&0&0\\0&0&0&1\\0&0&0&0\\0&-1&0&0\end{bmatrix}}\,,\quad &J_{z}&={\begin{bmatrix}0&0&0&0\\0&0&-1&0\\0&1&0&0\\0&0&0&0\end{bmatrix}}\end{alignedat}}}
These are all defined in an analogous way to Kx above, although the minus signs in the boost generators are conventional. Physically, the generators of the Lorentz group correspond to important symmetries in spacetime: J are the rotation generators which correspond to angular momentum, and K are the boost generators which correspond to the motion of the system in spacetime. The derivative of any smooth curve C(t) with C(0) = I in the group depending on some group parameter t with respect to that group parameter, evaluated at t = 0, serves as a definition of a corresponding group generator G, and this reflects an infinitesimal transformation away from the identity. The smooth curve can always be taken as an exponential as the exponential will always map G smoothly back into the group via t → exp(tG) for all t; this curve will yield G again when differentiated at t = 0.
Expanding the exponentials in their Taylor series obtains
B
(
ζ
)
=
I
−
sinh
ζ
(
n
⋅
K
)
+
(
cosh
ζ
−
1
)
(
n
⋅
K
)
2
{\displaystyle B({\boldsymbol {\zeta }})=I-\sinh \zeta (\mathbf {n} \cdot \mathbf {K} )+(\cosh \zeta -1)(\mathbf {n} \cdot \mathbf {K} )^{2}}
R
(
θ
)
=
I
+
sin
θ
(
e
⋅
J
)
+
(
1
−
cos
θ
)
(
e
⋅
J
)
2
.
{\displaystyle R({\boldsymbol {\theta }})=I+\sin \theta (\mathbf {e} \cdot \mathbf {J} )+(1-\cos \theta )(\mathbf {e} \cdot \mathbf {J} )^{2}\,.}
which compactly reproduce the boost and rotation matrices as given in the previous section.
It has been stated that the general proper Lorentz transformation is a product of a boost and rotation. At the infinitesimal level the product
Λ
=
(
I
−
ζ
⋅
K
+
⋯
)
(
I
+
θ
⋅
J
+
⋯
)
=
(
I
+
θ
⋅
J
+
⋯
)
(
I
−
ζ
⋅
K
+
⋯
)
=
I
−
ζ
⋅
K
+
θ
⋅
J
+
⋯
{\displaystyle {\begin{aligned}\Lambda &=(I-{\boldsymbol {\zeta }}\cdot \mathbf {K} +\cdots )(I+{\boldsymbol {\theta }}\cdot \mathbf {J} +\cdots )\\&=(I+{\boldsymbol {\theta }}\cdot \mathbf {J} +\cdots )(I-{\boldsymbol {\zeta }}\cdot \mathbf {K} +\cdots )\\&=I-{\boldsymbol {\zeta }}\cdot \mathbf {K} +{\boldsymbol {\theta }}\cdot \mathbf {J} +\cdots \end{aligned}}}
is commutative because only linear terms are required (products like (θ·J)(ζ·K) and (ζ·K)(θ·J) count as higher order terms and are negligible). Taking the limit as before leads to the finite transformation in the form of an exponential
Λ
(
ζ
,
θ
)
=
e
−
ζ
⋅
K
+
θ
⋅
J
.
{\displaystyle \Lambda ({\boldsymbol {\zeta }},{\boldsymbol {\theta }})=e^{-{\boldsymbol {\zeta }}\cdot \mathbf {K} +{\boldsymbol {\theta }}\cdot \mathbf {J} }.}
The converse is also true, but the decomposition of a finite general Lorentz transformation into such factors is nontrivial. In particular,
e
−
ζ
⋅
K
+
θ
⋅
J
≠
e
−
ζ
⋅
K
e
θ
⋅
J
,
{\displaystyle e^{-{\boldsymbol {\zeta }}\cdot \mathbf {K} +{\boldsymbol {\theta }}\cdot \mathbf {J} }\neq e^{-{\boldsymbol {\zeta }}\cdot \mathbf {K} }e^{{\boldsymbol {\theta }}\cdot \mathbf {J} },}
because the generators do not commute. For a description of how to find the factors of a general Lorentz transformation in terms of a boost and a rotation in principle (this usually does not yield an intelligible expression in terms of generators J and K), see Wigner rotation. If, on the other hand, the decomposition is given in terms of the generators, and one wants to find the product in terms of the generators, then the Baker–Campbell–Hausdorff formula applies.
==== The Lie algebra so(3,1) ====
Lorentz generators can be added together, or multiplied by real numbers, to obtain more Lorentz generators. In other words, the set of all Lorentz generators
V
=
{
ζ
⋅
K
+
θ
⋅
J
}
{\displaystyle V=\{{\boldsymbol {\zeta }}\cdot \mathbf {K} +{\boldsymbol {\theta }}\cdot \mathbf {J} \}}
together with the operations of ordinary matrix addition and multiplication of a matrix by a number, forms a vector space over the real numbers. The generators Jx, Jy, Jz, Kx, Ky, Kz form a basis set of V, and the components of the axis-angle and rapidity vectors, θx, θy, θz, ζx, ζy, ζz, are the coordinates of a Lorentz generator with respect to this basis.
Three of the commutation relations of the Lorentz generators are
[
J
x
,
J
y
]
=
J
z
,
[
K
x
,
K
y
]
=
−
J
z
,
[
J
x
,
K
y
]
=
K
z
,
{\displaystyle [J_{x},J_{y}]=J_{z}\,,\quad [K_{x},K_{y}]=-J_{z}\,,\quad [J_{x},K_{y}]=K_{z}\,,}
where the bracket [A, B] = AB − BA is known as the commutator, and the other relations can be found by taking cyclic permutations of x, y, z components (i.e. change x to y, y to z, and z to x, repeat).
These commutation relations, and the vector space of generators, fulfill the definition of the Lie algebra
s
o
(
3
,
1
)
{\displaystyle {\mathfrak {so}}(3,1)}
. In summary, a Lie algebra is defined as a vector space V over a field of numbers, and with a binary operation [ , ] (called a Lie bracket in this context) on the elements of the vector space, satisfying the axioms of bilinearity, alternatization, and the Jacobi identity. Here the operation [ , ] is the commutator which satisfies all of these axioms, the vector space is the set of Lorentz generators V as given previously, and the field is the set of real numbers.
Linking terminology used in mathematics and physics: A group generator is any element of the Lie algebra. A group parameter is a component of a coordinate vector representing an arbitrary element of the Lie algebra with respect to some basis. A basis, then, is a set of generators being a basis of the Lie algebra in the usual vector space sense.
The exponential map from the Lie algebra to the Lie group,
exp
:
s
o
(
3
,
1
)
→
S
O
(
3
,
1
)
,
{\displaystyle \exp \,:\,{\mathfrak {so}}(3,1)\to \mathrm {SO} (3,1),}
provides a one-to-one correspondence between small enough neighborhoods of the origin of the Lie algebra and neighborhoods of the identity element of the Lie group. In the case of the Lorentz group, the exponential map is just the matrix exponential. Globally, the exponential map is not one-to-one, but in the case of the Lorentz group, it is surjective (onto). Hence any group element in the connected component of the identity can be expressed as an exponential of an element of the Lie algebra.
=== Improper transformations ===
Lorentz transformations also include parity inversion
P
=
[
1
0
0
−
I
]
{\displaystyle P={\begin{bmatrix}1&0\\0&-\mathbf {I} \end{bmatrix}}}
which negates all the spatial coordinates only, and time reversal
T
=
[
−
1
0
0
I
]
{\displaystyle T={\begin{bmatrix}-1&0\\0&\mathbf {I} \end{bmatrix}}}
which negates the time coordinate only, because these transformations leave the spacetime interval invariant. Here I is the 3 × 3 identity matrix. These are both symmetric, they are their own inverses (see involution (mathematics)), and each have determinant −1. This latter property makes them improper transformations.
If Λ is a proper orthochronous Lorentz transformation, then TΛ is improper antichronous, PΛ is improper orthochronous, and TPΛ = PTΛ is proper antichronous.
=== Inhomogeneous Lorentz group ===
Two other spacetime symmetries have not been accounted for. In order for the spacetime interval to be invariant, it can be shown that it is necessary and sufficient for the coordinate transformation to be of the form
X
′
=
Λ
X
+
C
{\displaystyle X'=\Lambda X+C}
where C is a constant column containing translations in time and space. If C ≠ 0, this is an inhomogeneous Lorentz transformation or Poincaré transformation. If C = 0, this is a homogeneous Lorentz transformation. Poincaré transformations are not dealt further in this article.
== Tensor formulation ==
=== Contravariant vectors ===
Writing the general matrix transformation of coordinates as the matrix equation
[
x
′
0
x
′
1
x
′
2
x
′
3
]
=
[
Λ
0
0
Λ
0
1
Λ
0
2
Λ
0
3
x
′
0
Λ
1
0
Λ
1
1
Λ
1
2
Λ
1
3
x
′
0
Λ
2
0
Λ
2
1
Λ
2
2
Λ
2
3
x
′
0
Λ
3
0
Λ
3
1
Λ
3
2
Λ
3
3
x
′
0
]
[
x
0
x
′
0
x
1
x
′
0
x
2
x
′
0
x
3
x
′
0
]
{\displaystyle {\begin{bmatrix}{x'}^{0}\\{x'}^{1}\\{x'}^{2}\\{x'}^{3}\end{bmatrix}}={\begin{bmatrix}{\Lambda ^{0}}_{0}&{\Lambda ^{0}}_{1}&{\Lambda ^{0}}_{2}&{\Lambda ^{0}}_{3}{\vphantom {{x'}^{0}}}\\{\Lambda ^{1}}_{0}&{\Lambda ^{1}}_{1}&{\Lambda ^{1}}_{2}&{\Lambda ^{1}}_{3}{\vphantom {{x'}^{0}}}\\{\Lambda ^{2}}_{0}&{\Lambda ^{2}}_{1}&{\Lambda ^{2}}_{2}&{\Lambda ^{2}}_{3}{\vphantom {{x'}^{0}}}\\{\Lambda ^{3}}_{0}&{\Lambda ^{3}}_{1}&{\Lambda ^{3}}_{2}&{\Lambda ^{3}}_{3}{\vphantom {{x'}^{0}}}\\\end{bmatrix}}{\begin{bmatrix}x^{0}{\vphantom {{x'}^{0}}}\\x^{1}{\vphantom {{x'}^{0}}}\\x^{2}{\vphantom {{x'}^{0}}}\\x^{3}{\vphantom {{x'}^{0}}}\end{bmatrix}}}
allows the transformation of other physical quantities that cannot be expressed as four-vectors; e.g., tensors or spinors of any order in 4-dimensional spacetime, to be defined. In the corresponding tensor index notation, the above matrix expression is
x
′
ν
=
Λ
ν
μ
x
μ
,
{\displaystyle {x'}^{\nu }={\Lambda ^{\nu }}_{\mu }x^{\mu },}
where lower and upper indices label covariant and contravariant components respectively, and the summation convention is applied. It is a standard convention to use Greek indices that take the value 0 for time components, and 1, 2, 3 for space components, while Latin indices simply take the values 1, 2, 3, for spatial components (the opposite for Landau and Lifshitz). Note that the first index (reading left to right) corresponds in the matrix notation to a row index. The second index corresponds to the column index.
The transformation matrix is universal for all four-vectors, not just 4-dimensional spacetime coordinates. If A is any four-vector, then in tensor index notation
A
′
ν
=
Λ
ν
μ
A
μ
.
{\displaystyle {A'}^{\nu }={\Lambda ^{\nu }}_{\mu }A^{\mu }\,.}
Alternatively, one writes
A
ν
′
=
Λ
ν
′
μ
A
μ
.
{\displaystyle A^{\nu '}={\Lambda ^{\nu '}}_{\mu }A^{\mu }\,.}
in which the primed indices denote the indices of A in the primed frame. For a general n-component object one may write
X
′
α
=
Π
(
Λ
)
α
β
X
β
,
{\displaystyle {X'}^{\alpha }={\Pi (\Lambda )^{\alpha }}_{\beta }X^{\beta }\,,}
where Π is the appropriate representation of the Lorentz group, an n × n matrix for every Λ. In this case, the indices should not be thought of as spacetime indices (sometimes called Lorentz indices), and they run from 1 to n. E.g., if X is a bispinor, then the indices are called Dirac indices.
=== Covariant vectors ===
There are also vector quantities with covariant indices. They are generally obtained from their corresponding objects with contravariant indices by the operation of lowering an index; e.g.,
x
ν
=
η
μ
ν
x
μ
,
{\displaystyle x_{\nu }=\eta _{\mu \nu }x^{\mu },}
where η is the metric tensor. (The linked article also provides more information about what the operation of raising and lowering indices really is mathematically.) The inverse of this transformation is given by
x
μ
=
η
μ
ν
x
ν
,
{\displaystyle x^{\mu }=\eta ^{\mu \nu }x_{\nu },}
where, when viewed as matrices, ημν is the inverse of ημν. As it happens, ημν = ημν. This is referred to as raising an index. To transform a covariant vector Aμ, first raise its index, then transform it according to the same rule as for contravariant 4-vectors, then finally lower the index;
A
′
ν
=
η
ρ
ν
Λ
ρ
σ
η
μ
σ
A
μ
.
{\displaystyle {A'}_{\nu }=\eta _{\rho \nu }{\Lambda ^{\rho }}_{\sigma }\eta ^{\mu \sigma }A_{\mu }.}
But
η
ρ
ν
Λ
ρ
σ
η
μ
σ
=
(
Λ
−
1
)
μ
ν
,
{\displaystyle \eta _{\rho \nu }{\Lambda ^{\rho }}_{\sigma }\eta ^{\mu \sigma }={\left(\Lambda ^{-1}\right)^{\mu }}_{\nu },}
That is, it is the (μ, ν)-component of the inverse Lorentz transformation. One defines (as a matter of notation),
Λ
ν
μ
≡
(
Λ
−
1
)
μ
ν
,
{\displaystyle {\Lambda _{\nu }}^{\mu }\equiv {\left(\Lambda ^{-1}\right)^{\mu }}_{\nu },}
and may in this notation write
A
′
ν
=
Λ
ν
μ
A
μ
.
{\displaystyle {A'}_{\nu }={\Lambda _{\nu }}^{\mu }A_{\mu }.}
Now for a subtlety. The implied summation on the right hand side of
A
′
ν
=
Λ
ν
μ
A
μ
=
(
Λ
−
1
)
μ
ν
A
μ
{\displaystyle {A'}_{\nu }={\Lambda _{\nu }}^{\mu }A_{\mu }={\left(\Lambda ^{-1}\right)^{\mu }}_{\nu }A_{\mu }}
is running over a row index of the matrix representing Λ−1. Thus, in terms of matrices, this transformation should be thought of as the inverse transpose of Λ acting on the column vector Aμ. That is, in pure matrix notation,
A
′
=
(
Λ
−
1
)
T
A
.
{\displaystyle A'=\left(\Lambda ^{-1}\right)^{\mathrm {T} }A.}
This means exactly that covariant vectors (thought of as column matrices) transform according to the dual representation of the standard representation of the Lorentz group. This notion generalizes to general representations, simply replace Λ with Π(Λ).
=== Tensors ===
If A and B are linear operators on vector spaces U and V, then a linear operator A ⊗ B may be defined on the tensor product of U and V, denoted U ⊗ V according to
From this it is immediately clear that if u and v are a four-vectors in V, then u ⊗ v ∈ T2V ≡ V ⊗ V transforms as
The second step uses the bilinearity of the tensor product and the last step defines a 2-tensor on component form, or rather, it just renames the tensor u ⊗ v.
These observations generalize in an obvious way to more factors, and using the fact that a general tensor on a vector space V can be written as a sum of a coefficient (component!) times tensor products of basis vectors and basis covectors, one arrives at the transformation law for any tensor quantity T. It is given by
where Λχ′ψ is defined above. This form can generally be reduced to the form for general n-component objects given above with a single matrix (Π(Λ)) operating on column vectors. This latter form is sometimes preferred; e.g., for the electromagnetic field tensor.
==== Transformation of the electromagnetic field ====
Lorentz transformations can also be used to illustrate that the magnetic field B and electric field E are simply different aspects of the same force — the electromagnetic force, as a consequence of relative motion between electric charges and observers. The fact that the electromagnetic field shows relativistic effects becomes clear by carrying out a simple thought experiment.
An observer measures a charge at rest in frame F. The observer will detect a static electric field. As the charge is stationary in this frame, there is no electric current, so the observer does not observe any magnetic field.
The other observer in frame F′ moves at velocity v relative to F and the charge. This observer sees a different electric field because the charge moves at velocity −v in their rest frame. The motion of the charge corresponds to an electric current, and thus the observer in frame F′ also sees a magnetic field.
The electric and magnetic fields transform differently from space and time, but exactly the same way as relativistic angular momentum and the boost vector.
The electromagnetic field strength tensor is given by
F
μ
ν
=
[
0
−
1
c
E
x
−
1
c
E
y
−
1
c
E
z
1
c
E
x
0
−
B
z
B
y
1
c
E
y
B
z
0
−
B
x
1
c
E
z
−
B
y
B
x
0
]
(SI units, signature
(
+
,
−
,
−
,
−
)
)
.
{\displaystyle F^{\mu \nu }={\begin{bmatrix}0&-{\frac {1}{c}}E_{x}&-{\frac {1}{c}}E_{y}&-{\frac {1}{c}}E_{z}\\{\frac {1}{c}}E_{x}&0&-B_{z}&B_{y}\\{\frac {1}{c}}E_{y}&B_{z}&0&-B_{x}\\{\frac {1}{c}}E_{z}&-B_{y}&B_{x}&0\end{bmatrix}}{\text{(SI units, signature }}(+,-,-,-){\text{)}}.}
in SI units. In relativity, the Gaussian system of units is often preferred over SI units, even in texts whose main choice of units is SI units, because in it the electric field E and the magnetic induction B have the same units making the appearance of the electromagnetic field tensor more natural. Consider a Lorentz boost in the x-direction. It is given by
Λ
μ
ν
=
[
γ
−
γ
β
0
0
−
γ
β
γ
0
0
0
0
1
0
0
0
0
1
]
,
F
μ
ν
=
[
0
E
x
E
y
E
z
−
E
x
0
B
z
−
B
y
−
E
y
−
B
z
0
B
x
−
E
z
B
y
−
B
x
0
]
(Gaussian units, signature
(
−
,
+
,
+
,
+
)
)
,
{\displaystyle {\Lambda ^{\mu }}_{\nu }={\begin{bmatrix}\gamma &-\gamma \beta &0&0\\-\gamma \beta &\gamma &0&0\\0&0&1&0\\0&0&0&1\\\end{bmatrix}},\qquad F^{\mu \nu }={\begin{bmatrix}0&E_{x}&E_{y}&E_{z}\\-E_{x}&0&B_{z}&-B_{y}\\-E_{y}&-B_{z}&0&B_{x}\\-E_{z}&B_{y}&-B_{x}&0\end{bmatrix}}{\text{(Gaussian units, signature }}(-,+,+,+){\text{)}},}
where the field tensor is displayed side by side for easiest possible reference in the manipulations below.
The general transformation law (T3) becomes
F
μ
′
ν
′
=
Λ
μ
′
μ
Λ
ν
′
ν
F
μ
ν
.
{\displaystyle F^{\mu '\nu '}={\Lambda ^{\mu '}}_{\mu }{\Lambda ^{\nu '}}_{\nu }F^{\mu \nu }.}
For the magnetic field one obtains
B
x
′
=
F
2
′
3
′
=
Λ
2
μ
Λ
3
ν
F
μ
ν
=
Λ
2
2
Λ
3
3
F
23
=
1
×
1
×
B
x
=
B
x
,
B
y
′
=
F
3
′
1
′
=
Λ
3
μ
Λ
1
ν
F
μ
ν
=
Λ
3
3
Λ
1
ν
F
3
ν
=
Λ
3
3
Λ
1
0
F
30
+
Λ
3
3
Λ
1
1
F
31
=
1
×
(
−
β
γ
)
(
−
E
z
)
+
1
×
γ
B
y
=
γ
B
y
+
β
γ
E
z
=
γ
(
B
−
β
×
E
)
y
B
z
′
=
F
1
′
2
′
=
Λ
1
μ
Λ
2
ν
F
μ
ν
=
Λ
1
μ
Λ
2
2
F
μ
2
=
Λ
1
0
Λ
2
2
F
02
+
Λ
1
1
Λ
2
2
F
12
=
(
−
γ
β
)
×
1
×
E
y
+
γ
×
1
×
B
z
=
γ
B
z
−
β
γ
E
y
=
γ
(
B
−
β
×
E
)
z
{\displaystyle {\begin{aligned}B_{x'}&=F^{2'3'}={\Lambda ^{2}}_{\mu }{\Lambda ^{3}}_{\nu }F^{\mu \nu }={\Lambda ^{2}}_{2}{\Lambda ^{3}}_{3}F^{23}=1\times 1\times B_{x}\\&=B_{x},\\B_{y'}&=F^{3'1'}={\Lambda ^{3}}_{\mu }{\Lambda ^{1}}_{\nu }F^{\mu \nu }={\Lambda ^{3}}_{3}{\Lambda ^{1}}_{\nu }F^{3\nu }={\Lambda ^{3}}_{3}{\Lambda ^{1}}_{0}F^{30}+{\Lambda ^{3}}_{3}{\Lambda ^{1}}_{1}F^{31}\\&=1\times (-\beta \gamma )(-E_{z})+1\times \gamma B_{y}=\gamma B_{y}+\beta \gamma E_{z}\\&=\gamma \left(\mathbf {B} -{\boldsymbol {\beta }}\times \mathbf {E} \right)_{y}\\B_{z'}&=F^{1'2'}={\Lambda ^{1}}_{\mu }{\Lambda ^{2}}_{\nu }F^{\mu \nu }={\Lambda ^{1}}_{\mu }{\Lambda ^{2}}_{2}F^{\mu 2}={\Lambda ^{1}}_{0}{\Lambda ^{2}}_{2}F^{02}+{\Lambda ^{1}}_{1}{\Lambda ^{2}}_{2}F^{12}\\&=(-\gamma \beta )\times 1\times E_{y}+\gamma \times 1\times B_{z}=\gamma B_{z}-\beta \gamma E_{y}\\&=\gamma \left(\mathbf {B} -{\boldsymbol {\beta }}\times \mathbf {E} \right)_{z}\end{aligned}}}
For the electric field results
E
x
′
=
F
0
′
1
′
=
Λ
0
μ
Λ
1
ν
F
μ
ν
=
Λ
0
1
Λ
1
0
F
10
+
Λ
0
0
Λ
1
1
F
01
=
(
−
γ
β
)
(
−
γ
β
)
(
−
E
x
)
+
γ
γ
E
x
=
−
γ
2
β
2
(
E
x
)
+
γ
2
E
x
=
E
x
(
1
−
β
2
)
γ
2
=
E
x
,
E
y
′
=
F
0
′
2
′
=
Λ
0
μ
Λ
2
ν
F
μ
ν
=
Λ
0
μ
Λ
2
2
F
μ
2
=
Λ
0
0
Λ
2
2
F
02
+
Λ
0
1
Λ
2
2
F
12
=
γ
×
1
×
E
y
+
(
−
β
γ
)
×
1
×
B
z
=
γ
E
y
−
β
γ
B
z
=
γ
(
E
+
β
×
B
)
y
E
z
′
=
F
0
′
3
′
=
Λ
0
μ
Λ
3
ν
F
μ
ν
=
Λ
0
μ
Λ
3
3
F
μ
3
=
Λ
0
0
Λ
3
3
F
03
+
Λ
0
1
Λ
3
3
F
13
=
γ
×
1
×
E
z
−
β
γ
×
1
×
(
−
B
y
)
=
γ
E
z
+
β
γ
B
y
=
γ
(
E
+
β
×
B
)
z
.
{\displaystyle {\begin{aligned}E_{x'}&=F^{0'1'}={\Lambda ^{0}}_{\mu }{\Lambda ^{1}}_{\nu }F^{\mu \nu }={\Lambda ^{0}}_{1}{\Lambda ^{1}}_{0}F^{10}+{\Lambda ^{0}}_{0}{\Lambda ^{1}}_{1}F^{01}\\&=(-\gamma \beta )(-\gamma \beta )(-E_{x})+\gamma \gamma E_{x}=-\gamma ^{2}\beta ^{2}(E_{x})+\gamma ^{2}E_{x}=E_{x}(1-\beta ^{2})\gamma ^{2}\\&=E_{x},\\E_{y'}&=F^{0'2'}={\Lambda ^{0}}_{\mu }{\Lambda ^{2}}_{\nu }F^{\mu \nu }={\Lambda ^{0}}_{\mu }{\Lambda ^{2}}_{2}F^{\mu 2}={\Lambda ^{0}}_{0}{\Lambda ^{2}}_{2}F^{02}+{\Lambda ^{0}}_{1}{\Lambda ^{2}}_{2}F^{12}\\&=\gamma \times 1\times E_{y}+(-\beta \gamma )\times 1\times B_{z}=\gamma E_{y}-\beta \gamma B_{z}\\&=\gamma \left(\mathbf {E} +{\boldsymbol {\beta }}\times \mathbf {B} \right)_{y}\\E_{z'}&=F^{0'3'}={\Lambda ^{0}}_{\mu }{\Lambda ^{3}}_{\nu }F^{\mu \nu }={\Lambda ^{0}}_{\mu }{\Lambda ^{3}}_{3}F^{\mu 3}={\Lambda ^{0}}_{0}{\Lambda ^{3}}_{3}F^{03}+{\Lambda ^{0}}_{1}{\Lambda ^{3}}_{3}F^{13}\\&=\gamma \times 1\times E_{z}-\beta \gamma \times 1\times (-B_{y})=\gamma E_{z}+\beta \gamma B_{y}\\&=\gamma \left(\mathbf {E} +{\boldsymbol {\beta }}\times \mathbf {B} \right)_{z}.\end{aligned}}}
Here, β = (β, 0, 0) is used. These results can be summarized by
E
∥
′
=
E
∥
B
∥
′
=
B
∥
E
⊥
′
=
γ
(
E
⊥
+
β
×
B
⊥
)
=
γ
(
E
+
β
×
B
)
⊥
,
B
⊥
′
=
γ
(
B
⊥
−
β
×
E
⊥
)
=
γ
(
B
−
β
×
E
)
⊥
,
{\displaystyle {\begin{aligned}\mathbf {E} _{\parallel '}&=\mathbf {E} _{\parallel }\\\mathbf {B} _{\parallel '}&=\mathbf {B} _{\parallel }\\\mathbf {E} _{\bot '}&=\gamma \left(\mathbf {E} _{\bot }+{\boldsymbol {\beta }}\times \mathbf {B} _{\bot }\right)=\gamma \left(\mathbf {E} +{\boldsymbol {\beta }}\times \mathbf {B} \right)_{\bot },\\\mathbf {B} _{\bot '}&=\gamma \left(\mathbf {B} _{\bot }-{\boldsymbol {\beta }}\times \mathbf {E} _{\bot }\right)=\gamma \left(\mathbf {B} -{\boldsymbol {\beta }}\times \mathbf {E} \right)_{\bot },\end{aligned}}}
and are independent of the metric signature. For SI units, substitute E → E⁄c. Misner, Thorne & Wheeler (1973) refer to this last form as the 3 + 1 view as opposed to the geometric view represented by the tensor expression
F
μ
′
ν
′
=
Λ
μ
′
μ
Λ
ν
′
ν
F
μ
ν
,
{\displaystyle F^{\mu '\nu '}={\Lambda ^{\mu '}}_{\mu }{\Lambda ^{\nu '}}_{\nu }F^{\mu \nu },}
and make a strong point of the ease with which results that are difficult to achieve using the 3 + 1 view can be obtained and understood. Only objects that have well defined Lorentz transformation properties (in fact under any smooth coordinate transformation) are geometric objects. In the geometric view, the electromagnetic field is a six-dimensional geometric object in spacetime as opposed to two interdependent, but separate, 3-vector fields in space and time. The fields E (alone) and B (alone) do not have well defined Lorentz transformation properties. The mathematical underpinnings are equations (T1) and (T2) that immediately yield (T3). One should note that the primed and unprimed tensors refer to the same event in spacetime. Thus the complete equation with spacetime dependence is
F
μ
′
ν
′
(
x
′
)
=
Λ
μ
′
μ
Λ
ν
′
ν
F
μ
ν
(
Λ
−
1
x
′
)
=
Λ
μ
′
μ
Λ
ν
′
ν
F
μ
ν
(
x
)
.
{\displaystyle F^{\mu '\nu '}\left(x'\right)={\Lambda ^{\mu '}}_{\mu }{\Lambda ^{\nu '}}_{\nu }F^{\mu \nu }\left(\Lambda ^{-1}x'\right)={\Lambda ^{\mu '}}_{\mu }{\Lambda ^{\nu '}}_{\nu }F^{\mu \nu }(x).}
Length contraction has an effect on charge density ρ and current density J, and time dilation has an effect on the rate of flow of charge (current), so charge and current distributions must transform in a related way under a boost. It turns out they transform exactly like the space-time and energy-momentum four-vectors,
j
′
=
j
−
γ
ρ
v
n
+
(
γ
−
1
)
(
j
⋅
n
)
n
ρ
′
=
γ
(
ρ
−
j
⋅
v
n
c
2
)
,
{\displaystyle {\begin{aligned}\mathbf {j} '&=\mathbf {j} -\gamma \rho v\mathbf {n} +\left(\gamma -1\right)(\mathbf {j} \cdot \mathbf {n} )\mathbf {n} \\\rho '&=\gamma \left(\rho -\mathbf {j} \cdot {\frac {v\mathbf {n} }{c^{2}}}\right),\end{aligned}}}
or, in the simpler geometric view,
j
μ
′
=
Λ
μ
′
μ
j
μ
.
{\displaystyle j^{\mu '}={\Lambda ^{\mu '}}_{\mu }j^{\mu }.}
Charge density transforms as the time component of a four-vector. It is a rotational scalar. The current density is a 3-vector.
The Maxwell equations are invariant under Lorentz transformations.
=== Spinors ===
Equation (T1) hold unmodified for any representation of the Lorentz group, including the bispinor representation. In (T2) one simply replaces all occurrences of Λ by the bispinor representation Π(Λ),
The above equation could, for instance, be the transformation of a state in Fock space describing two free electrons.
==== Transformation of general fields ====
A general noninteracting multi-particle state (Fock space state) in quantum field theory transforms according to the rule
where W(Λ, p) is the Wigner's little group and D(j) is the (2j + 1)-dimensional representation of SO(3).
== See also ==
== Footnotes ==
== Notes ==
== References ==
=== Websites ===
O'Connor, John J.; Robertson, Edmund F. (1996), A History of Special Relativity
Brown, Harvey R. (2003), Michelson, FitzGerald and Lorentz: the Origins of Relativity Revisited
=== Papers ===
=== Books ===
== Further reading ==
Ernst, A.; Hsu, J.-P. (2001), "First proposal of the universal speed of light by Voigt 1887" (PDF), Chinese Journal of Physics, 39 (3): 211–230, Bibcode:2001ChJPh..39..211E, archived from the original (PDF) on 2011-07-16
Thornton, Stephen T.; Marion, Jerry B. (2004), Classical dynamics of particles and systems (5th ed.), Belmont, [CA.]: Brooks/Cole, pp. 546–579, ISBN 978-0-534-40896-1
Voigt, Woldemar (1887), "Über das Doppler'sche princip", Nachrichten von der Königlicher Gesellschaft den Wissenschaft zu Göttingen, 2: 41–51
== External links ==
Derivation of the Lorentz transformations. This web page contains a more detailed derivation of the Lorentz transformation with special emphasis on group properties.
The Paradox of Special Relativity. This webpage poses a problem, the solution of which is the Lorentz transformation, which is presented graphically in its next page.
Relativity Archived 2011-08-29 at the Wayback Machine – a chapter from an online textbook
Warp Special Relativity Simulator. A computer program demonstrating the Lorentz transformations on everyday objects.
Animation clip on YouTube visualizing the Lorentz transformation.
MinutePhysics video on YouTube explaining and visualizing the Lorentz transformation with a mechanical Minkowski diagram
Interactive graph on Desmos (graphing) showing Lorentz transformations with a virtual Minkowski diagram
Interactive graph on Desmos showing Lorentz transformations with points and hyperbolas
Lorentz Frames Animated from John de Pillis. Online Flash animations of Galilean and Lorentz frames, various paradoxes, EM wave phenomena, etc. | Wikipedia/Lorentz_Transformation |
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of thermodynamics, which convey a quantitative description using measurable macroscopic physical quantities but may be explained in terms of microscopic constituents by statistical mechanics. Thermodynamics applies to various topics in science and engineering, especially physical chemistry, biochemistry, chemical engineering, and mechanical engineering, as well as other complex fields such as meteorology.
Historically, thermodynamics developed out of a desire to increase the efficiency of early steam engines, particularly through the work of French physicist Sadi Carnot (1824) who believed that engine efficiency was the key that could help France win the Napoleonic Wars. Scots-Irish physicist Lord Kelvin was the first to formulate a concise definition of thermodynamics in 1854 which stated, "Thermo-dynamics is the subject of the relation of heat to forces acting between contiguous parts of bodies, and the relation of heat to electrical agency." German physicist and mathematician Rudolf Clausius restated Carnot's principle known as the Carnot cycle and gave the theory of heat a truer and sounder basis. His most important paper, "On the Moving Force of Heat", published in 1850, first stated the second law of thermodynamics. In 1865 he introduced the concept of entropy. In 1870 he introduced the virial theorem, which applied to heat.
The initial application of thermodynamics to mechanical heat engines was quickly extended to the study of chemical compounds and chemical reactions. Chemical thermodynamics studies the nature of the role of entropy in the process of chemical reactions and has provided the bulk of expansion and knowledge of the field. Other formulations of thermodynamics emerged. Statistical thermodynamics, or statistical mechanics, concerns itself with statistical predictions of the collective motion of particles from their microscopic behavior. In 1909, Constantin Carathéodory presented a purely mathematical approach in an axiomatic formulation, a description often referred to as geometrical thermodynamics.
== Introduction ==
A description of any thermodynamic system employs the four laws of thermodynamics that form an axiomatic basis. The first law specifies that energy can be transferred between physical systems as heat, as work, and with transfer of matter. The second law defines the existence of a quantity called entropy, that describes the direction, thermodynamically, that a system can evolve and quantifies the state of order of a system and that can be used to quantify the useful work that can be extracted from the system.
In thermodynamics, interactions between large ensembles of objects are studied and categorized. Central to this are the concepts of the thermodynamic system and its surroundings. A system is composed of particles, whose average motions define its properties, and those properties are in turn related to one another through equations of state. Properties can be combined to express internal energy and thermodynamic potentials, which are useful for determining conditions for equilibrium and spontaneous processes.
With these tools, thermodynamics can be used to describe how systems respond to changes in their environment. This can be applied to a wide variety of topics in science and engineering, such as engines, phase transitions, chemical reactions, transport phenomena, and even black holes. The results of thermodynamics are essential for other fields of physics and for chemistry, chemical engineering, corrosion engineering, aerospace engineering, mechanical engineering, cell biology, biomedical engineering, materials science, and economics, to name a few.
This article is focused mainly on classical thermodynamics which primarily studies systems in thermodynamic equilibrium. Non-equilibrium thermodynamics is often treated as an extension of the classical treatment, but statistical mechanics has brought many advances to that field.
== History ==
The history of thermodynamics as a scientific discipline generally begins with Otto von Guericke who, in 1650, built and designed the world's first vacuum pump and demonstrated a vacuum using his Magdeburg hemispheres. Guericke was driven to make a vacuum in order to disprove Aristotle's long-held supposition that 'nature abhors a vacuum'. Shortly after Guericke, the Anglo-Irish physicist and chemist Robert Boyle had learned of Guericke's designs and, in 1656, in coordination with English scientist Robert Hooke, built an air pump. Using this pump, Boyle and Hooke noticed a correlation between pressure, temperature, and volume. In time, Boyle's Law was formulated, which states that pressure and volume are inversely proportional. Then, in 1679, based on these concepts, an associate of Boyle's named Denis Papin built a steam digester, which was a closed vessel with a tightly fitting lid that confined steam until a high pressure was generated.
Later designs implemented a steam release valve that kept the machine from exploding. By watching the valve rhythmically move up and down, Papin conceived of the idea of a piston and a cylinder engine. He did not, however, follow through with his design. Nevertheless, in 1697, based on Papin's designs, engineer Thomas Savery built the first engine, followed by Thomas Newcomen in 1712. Although these early engines were crude and inefficient, they attracted the attention of the leading scientists of the time.
The fundamental concepts of heat capacity and latent heat, which were necessary for the development of thermodynamics, were developed by Professor Joseph Black at the University of Glasgow, where James Watt was employed as an instrument maker. Black and Watt performed experiments together, but it was Watt who conceived the idea of the external condenser which resulted in a large increase in steam engine efficiency. Drawing on all the previous work led Sadi Carnot, the "father of thermodynamics", to publish Reflections on the Motive Power of Fire (1824), a discourse on heat, power, energy and engine efficiency. The book outlined the basic energetic relations between the Carnot engine, the Carnot cycle, and motive power. It marked the start of thermodynamics as a modern science.
The first thermodynamic textbook was written in 1859 by William Rankine, originally trained as a physicist and a civil and mechanical engineering professor at the University of Glasgow. The first and second laws of thermodynamics emerged simultaneously in the 1850s, primarily out of the works of William Rankine, Rudolf Clausius, and William Thomson (Lord Kelvin).
The foundations of statistical thermodynamics were set out by physicists such as James Clerk Maxwell, Ludwig Boltzmann, Max Planck, Rudolf Clausius and J. Willard Gibbs.
Clausius, who first stated the basic ideas of the second law in his paper "On the Moving Force of Heat", published in 1850, and is called "one of the founding fathers of thermodynamics", introduced the concept of entropy in 1865.
During the years 1873–76 the American mathematical physicist Josiah Willard Gibbs published a series of three papers, the most famous being On the Equilibrium of Heterogeneous Substances, in which he showed how thermodynamic processes, including chemical reactions, could be graphically analyzed, by studying the energy, entropy, volume, temperature and pressure of the thermodynamic system in such a manner, one can determine if a process would occur spontaneously. Also Pierre Duhem in the 19th century wrote about chemical thermodynamics. During the early 20th century, chemists such as Gilbert N. Lewis, Merle Randall, and E. A. Guggenheim applied the mathematical methods of Gibbs to the analysis of chemical processes.
== Etymology ==
Thermodynamics has an intricate etymology.
By a surface-level analysis, the word consists of two parts that can be traced back to Ancient Greek. Firstly, thermo- ("of heat"; used in words such as thermometer) can be traced back to the root θέρμη therme, meaning "heat". Secondly, the word dynamics ("science of force [or power]") can be traced back to the root δύναμις dynamis, meaning "power".
In 1849, the adjective thermo-dynamic is used by William Thomson.
In 1854, the noun thermo-dynamics is used by Thomson and William Rankine to represent the science of generalized heat engines.
Pierre Perrot claims that the term thermodynamics was coined by James Joule in 1858 to designate the science of relations between heat and power, however, Joule never used that term, but used instead the term perfect thermo-dynamic engine in reference to Thomson's 1849 phraseology.
== Branches of thermodynamics ==
The study of thermodynamical systems has developed into several related branches, each using a different fundamental model as a theoretical or experimental basis, or applying the principles to varying types of systems.
=== Classical thermodynamics ===
Classical thermodynamics is the description of the states of thermodynamic systems at near-equilibrium, that uses macroscopic, measurable properties. It is used to model exchanges of energy, work and heat based on the laws of thermodynamics. The qualifier classical reflects the fact that it represents the first level of understanding of the subject as it developed in the 19th century and describes the changes of a system in terms of macroscopic empirical (large scale, and measurable) parameters. A microscopic interpretation of these concepts was later provided by the development of statistical mechanics.
=== Statistical mechanics ===
Statistical mechanics, also known as statistical thermodynamics, emerged with the development of atomic and molecular theories in the late 19th century and early 20th century, and supplemented classical thermodynamics with an interpretation of the microscopic interactions between individual particles or quantum-mechanical states. This field relates the microscopic properties of individual atoms and molecules to the macroscopic, bulk properties of materials that can be observed on the human scale, thereby explaining classical thermodynamics as a natural result of statistics, classical mechanics, and quantum theory at the microscopic level.
=== Chemical thermodynamics ===
Chemical thermodynamics is the study of the interrelation of energy with chemical reactions or with a physical change of state within the confines of the laws of thermodynamics. The primary objective of chemical thermodynamics is determining the spontaneity of a given transformation.
=== Equilibrium thermodynamics ===
Equilibrium thermodynamics is the study of transfers of matter and energy in systems or bodies that, by agencies in their surroundings, can be driven from one state of thermodynamic equilibrium to another. The term 'thermodynamic equilibrium' indicates a state of balance, in which all macroscopic flows are zero; in the case of the simplest systems or bodies, their intensive properties are homogeneous, and their pressures are perpendicular to their boundaries. In an equilibrium state there are no unbalanced potentials, or driving forces, between macroscopically distinct parts of the system. A central aim in equilibrium thermodynamics is: given a system in a well-defined initial equilibrium state, and given its surroundings, and given its constitutive walls, to calculate what will be the final equilibrium state of the system after a specified thermodynamic operation has changed its walls or surroundings.
=== Non-equilibrium thermodynamics ===
Non-equilibrium thermodynamics is a branch of thermodynamics that deals with systems that are not in thermodynamic equilibrium. Most systems found in nature are not in thermodynamic equilibrium because they are not in stationary states, and are continuously and discontinuously subject to flux of matter and energy to and from other systems. The thermodynamic study of non-equilibrium systems requires more general concepts than are dealt with by equilibrium thermodynamics. Many natural systems still today remain beyond the scope of currently known macroscopic thermodynamic methods.
== Laws of thermodynamics ==
Thermodynamics is principally based on a set of four laws which are universally valid when applied to systems that fall within the constraints implied by each. In the various theoretical descriptions of thermodynamics these laws may be expressed in seemingly differing forms, but the most prominent formulations are the following.
=== Zeroth law ===
The zeroth law of thermodynamics states: If two systems are each in thermal equilibrium with a third, they are also in thermal equilibrium with each other.
This statement implies that thermal equilibrium is an equivalence relation on the set of thermodynamic systems under consideration. Systems are said to be in equilibrium if the small, random exchanges between them (e.g. Brownian motion) do not lead to a net change in energy. This law is tacitly assumed in every measurement of temperature. Thus, if one seeks to decide whether two bodies are at the same temperature, it is not necessary to bring them into contact and measure any changes of their observable properties in time. The law provides an empirical definition of temperature, and justification for the construction of practical thermometers.
The zeroth law was not initially recognized as a separate law of thermodynamics, as its basis in thermodynamical equilibrium was implied in the other laws. The first, second, and third laws had been explicitly stated already, and found common acceptance in the physics community before the importance of the zeroth law for the definition of temperature was realized. As it was impractical to renumber the other laws, it was named the zeroth law.
=== First law ===
The first law of thermodynamics states: In a process without transfer of matter, the change in internal energy,
Δ
U
{\displaystyle \Delta U}
, of a thermodynamic system is equal to the energy gained as heat,
Q
{\displaystyle Q}
, less the thermodynamic work,
W
{\displaystyle W}
, done by the system on its surroundings.
Δ
U
=
Q
−
W
{\displaystyle \Delta U=Q-W}
.
where
Δ
U
{\displaystyle \Delta U}
denotes the change in the internal energy of a closed system (for which heat or work through the system boundary are possible, but matter transfer is not possible),
Q
{\displaystyle Q}
denotes the quantity of energy supplied to the system as heat, and
W
{\displaystyle W}
denotes the amount of thermodynamic work done by the system on its surroundings. An equivalent statement is that perpetual motion machines of the first kind are impossible; work
W
{\displaystyle W}
done by a system on its surrounding requires that the system's internal energy
U
{\displaystyle U}
decrease or be consumed, so that the amount of internal energy lost by that work must be resupplied as heat
Q
{\displaystyle Q}
by an external energy source or as work by an external machine acting on the system (so that
U
{\displaystyle U}
is recovered) to make the system work continuously.
For processes that include transfer of matter, a further statement is needed: With due account of the respective fiducial reference states of the systems, when two systems, which may be of different chemical compositions, initially separated only by an impermeable wall, and otherwise isolated, are combined into a new system by the thermodynamic operation of removal of the wall, then
U
0
=
U
1
+
U
2
{\displaystyle U_{0}=U_{1}+U_{2}}
,
where U0 denotes the internal energy of the combined system, and U1 and U2 denote the internal energies of the respective separated systems.
Adapted for thermodynamics, this law is an expression of the principle of conservation of energy, which states that energy can be transformed (changed from one form to another), but cannot be created or destroyed.
Internal energy is a principal property of the thermodynamic state, while heat and work are modes of energy transfer by which a process may change this state. A change of internal energy of a system may be achieved by any combination of heat added or removed and work performed on or by the system. As a function of state, the internal energy does not depend on the manner, or on the path through intermediate steps, by which the system arrived at its state.
=== Second law ===
A traditional version of the second law of thermodynamics states: Heat does not spontaneously flow from a colder body to a hotter body.
The second law refers to a system of matter and radiation, initially with inhomogeneities in temperature, pressure, chemical potential, and other intensive properties, that are due to internal 'constraints', or impermeable rigid walls, within it, or to externally imposed forces. The law observes that, when the system is isolated from the outside world and from those forces, there is a definite thermodynamic quantity, its entropy, that increases as the constraints are removed, eventually reaching a maximum value at thermodynamic equilibrium, when the inhomogeneities practically vanish. For systems that are initially far from thermodynamic equilibrium, though several have been proposed, there is known no general physical principle that determines the rates of approach to thermodynamic equilibrium, and thermodynamics does not deal with such rates. The many versions of the second law all express the general irreversibility of the transitions involved in systems approaching thermodynamic equilibrium.
In macroscopic thermodynamics, the second law is a basic observation applicable to any actual thermodynamic process; in statistical thermodynamics, the second law is postulated to be a consequence of molecular chaos.
=== Third law ===
The third law of thermodynamics states: As the temperature of a system approaches absolute zero, all processes cease and the entropy of the system approaches a minimum value.
This law of thermodynamics is a statistical law of nature regarding entropy and the impossibility of reaching absolute zero of temperature. This law provides an absolute reference point for the determination of entropy. The entropy determined relative to this point is the absolute entropy. Alternate definitions include "the entropy of all systems and of all states of a system is smallest at absolute zero," or equivalently "it is impossible to reach the absolute zero of temperature by any finite number of processes".
Absolute zero, at which all activity would stop if it were possible to achieve, is −273.15 °C (degrees Celsius), or −459.67 °F (degrees Fahrenheit), or 0 K (kelvin), or 0° R (degrees Rankine).
== System models ==
An important concept in thermodynamics is the thermodynamic system, which is a precisely defined region of the universe under study. Everything in the universe except the system is called the surroundings. A system is separated from the remainder of the universe by a boundary which may be a physical or notional, but serve to confine the system to a finite volume. Segments of the boundary are often described as walls; they have respective defined 'permeabilities'. Transfers of energy as work, or as heat, or of matter, between the system and the surroundings, take place through the walls, according to their respective permeabilities.
Matter or energy that pass across the boundary so as to effect a change in the internal energy of the system need to be accounted for in the energy balance equation. The volume contained by the walls can be the region surrounding a single atom resonating energy, such as Max Planck defined in 1900; it can be a body of steam or air in a steam engine, such as Sadi Carnot defined in 1824. The system could also be just one nuclide (i.e. a system of quarks) as hypothesized in quantum thermodynamics. When a looser viewpoint is adopted, and the requirement of thermodynamic equilibrium is dropped, the system can be the body of a tropical cyclone, such as Kerry Emanuel theorized in 1986 in the field of atmospheric thermodynamics, or the event horizon of a black hole.
Boundaries are of four types: fixed, movable, real, and imaginary. For example, in an engine, a fixed boundary means the piston is locked at its position, within which a constant volume process might occur. If the piston is allowed to move that boundary is movable while the cylinder and cylinder head boundaries are fixed. For closed systems, boundaries are real while for open systems boundaries are often imaginary. In the case of a jet engine, a fixed imaginary boundary might be assumed at the intake of the engine, fixed boundaries along the surface of the case and a second fixed imaginary boundary across the exhaust nozzle.
Generally, thermodynamics distinguishes three classes of systems, defined in terms of what is allowed to cross their boundaries:
As time passes in an isolated system, internal differences of pressures, densities, and temperatures tend to even out. A system in which all equalizing processes have gone to completion is said to be in a state of thermodynamic equilibrium.
Once in thermodynamic equilibrium, a system's properties are, by definition, unchanging in time. Systems in equilibrium are much simpler and easier to understand than are systems which are not in equilibrium. Often, when analysing a dynamic thermodynamic process, the simplifying assumption is made that each intermediate state in the process is at equilibrium, producing thermodynamic processes which develop so slowly as to allow each intermediate step to be an equilibrium state and are said to be reversible processes.
== States and processes ==
When a system is at equilibrium under a given set of conditions, it is said to be in a definite thermodynamic state. The state of the system can be described by a number of state quantities that do not depend on the process by which the system arrived at its state. They are called intensive variables or extensive variables according to how they change when the size of the system changes. The properties of the system can be described by an equation of state which specifies the relationship between these variables. State may be thought of as the instantaneous quantitative description of a system with a set number of variables held constant.
A thermodynamic process may be defined as the energetic evolution of a thermodynamic system proceeding from an initial state to a final state. It can be described by process quantities. Typically, each thermodynamic process is distinguished from other processes in energetic character according to what parameters, such as temperature, pressure, or volume, etc., are held fixed; Furthermore, it is useful to group these processes into pairs, in which each variable held constant is one member of a conjugate pair.
Several commonly studied thermodynamic processes are:
Adiabatic process: occurs without loss or gain of energy by heat
Isenthalpic process: occurs at a constant enthalpy
Isentropic process: a reversible adiabatic process, occurs at a constant entropy
Isobaric process: occurs at constant pressure
Isochoric process: occurs at constant volume (also called isometric/isovolumetric)
Isothermal process: occurs at a constant temperature
Steady state process: occurs without a change in the internal energy
== Instrumentation ==
There are two types of thermodynamic instruments, the meter and the reservoir. A thermodynamic meter is any device which measures any parameter of a thermodynamic system. In some cases, the thermodynamic parameter is actually defined in terms of an idealized measuring instrument. For example, the zeroth law states that if two bodies are in thermal equilibrium with a third body, they are also in thermal equilibrium with each other. This principle, as noted by James Maxwell in 1872, asserts that it is possible to measure temperature. An idealized thermometer is a sample of an ideal gas at constant pressure. From the ideal gas law pV=nRT, the volume of such a sample can be used as an indicator of temperature; in this manner it defines temperature. Although pressure is defined mechanically, a pressure-measuring device, called a barometer may also be constructed from a sample of an ideal gas held at a constant temperature. A calorimeter is a device which is used to measure and define the internal energy of a system.
A thermodynamic reservoir is a system which is so large that its state parameters are not appreciably altered when it is brought into contact with the system of interest. When the reservoir is brought into contact with the system, the system is brought into equilibrium with the reservoir. For example, a pressure reservoir is a system at a particular pressure, which imposes that pressure upon the system to which it is mechanically connected. The Earth's atmosphere is often used as a pressure reservoir. The ocean can act as temperature reservoir when used to cool power plants.
== Conjugate variables ==
The central concept of thermodynamics is that of energy, the ability to do work. By the First Law, the total energy of a system and its surroundings is conserved. Energy may be transferred into a system by heating, compression, or addition of matter, and extracted from a system by cooling, expansion, or extraction of matter. In mechanics, for example, energy transfer equals the product of the force applied to a body and the resulting displacement.
Conjugate variables are pairs of thermodynamic concepts, with the first being akin to a "force" applied to some thermodynamic system, the second being akin to the resulting "displacement", and the product of the two equaling the amount of energy transferred. The common conjugate variables are:
Pressure-volume (the mechanical parameters);
Temperature-entropy (thermal parameters);
Chemical potential-particle number (material parameters).
== Potentials ==
Thermodynamic potentials are different quantitative measures of the stored energy in a system. Potentials are used to measure the energy changes in systems as they evolve from an initial state to a final state. The potential used depends on the constraints of the system, such as constant temperature or pressure. For example, the Helmholtz and Gibbs energies are the energies available in a system to do useful work when the temperature and volume or the pressure and temperature are fixed, respectively. Thermodynamic potentials cannot be measured in laboratories, but can be computed using molecular thermodynamics.
The five most well known potentials are:
where
T
{\displaystyle T}
is the temperature,
S
{\displaystyle S}
the entropy,
p
{\displaystyle p}
the pressure,
V
{\displaystyle V}
the volume,
μ
{\displaystyle \mu }
the chemical potential,
N
{\displaystyle N}
the number of particles in the system, and
i
{\displaystyle i}
is the count of particles types in the system.
Thermodynamic potentials can be derived from the energy balance equation applied to a thermodynamic system. Other thermodynamic potentials can also be obtained through Legendre transformation.
== Axiomatic thermodynamics ==
Axiomatic thermodynamics is a mathematical discipline that aims to describe thermodynamics in terms of rigorous axioms, for example by finding a mathematically rigorous way to express the familiar laws of thermodynamics.
The first attempt at an axiomatic theory of thermodynamics was Constantin Carathéodory's 1909 work Investigations on the Foundations of Thermodynamics, which made use of Pfaffian systems and the concept of adiabatic accessibility, a notion that was introduced by Carathéodory himself. In this formulation, thermodynamic concepts such as heat, entropy, and temperature are derived from quantities that are more directly measurable. Theories that came after, differed in the sense that they made assumptions regarding thermodynamic processes with arbitrary initial and final states, as opposed to considering only neighboring states.
== Applied fields ==
== See also ==
Thermodynamic process path
=== Lists and timelines ===
List of important publications in thermodynamics
List of textbooks on thermodynamics and statistical mechanics
List of thermal conductivities
List of thermodynamic properties
Table of thermodynamic equations
Timeline of thermodynamics
Thermodynamic equations
== Notes ==
== References ==
== Further reading ==
Goldstein, Martin & Inge F. (1993). The Refrigerator and the Universe. Harvard University Press. ISBN 978-0-674-75325-9. OCLC 32826343. A nontechnical introduction, good on historical and interpretive matters.
Kazakov, Andrei; Muzny, Chris D.; Chirico, Robert D.; Diky, Vladimir V.; Frenkel, Michael (2008). "Web Thermo Tables – an On-Line Version of the TRC Thermodynamic Tables". Journal of Research of the National Institute of Standards and Technology. 113 (4): 209–220. doi:10.6028/jres.113.016. ISSN 1044-677X. PMC 4651616. PMID 27096122.
Gibbs J.W. (1928). The Collected Works of J. Willard Gibbs Thermodynamics. New York: Longmans, Green and Co. Vol. 1, pp. 55–349.
Guggenheim E.A. (1933). Modern thermodynamics by the methods of Willard Gibbs. London: Methuen & co. ltd.
Denbigh K. (1981). The Principles of Chemical Equilibrium: With Applications in Chemistry and Chemical Engineering. London: Cambridge University Press.
Stull, D.R., Westrum Jr., E.F. and Sinke, G.C. (1969). The Chemical Thermodynamics of Organic Compounds. London: John Wiley and Sons, Inc.{{cite book}}: CS1 maint: multiple names: authors list (link)
Bazarov I.P. (2010). Thermodynamics: Textbook. St. Petersburg: Lan publishing house. p. 384. ISBN 978-5-8114-1003-3. 5th ed. (in Russian)
Bawendi Moungi G., Alberty Robert A. and Silbey Robert J. (2004). Physical Chemistry. J. Wiley & Sons, Incorporated.
Alberty Robert A. (2003). Thermodynamics of Biochemical Reactions. Wiley-Interscience.
Alberty Robert A. (2006). Biochemical Thermodynamics: Applications of Mathematica. Vol. 48. John Wiley & Sons, Inc. pp. 1–458. ISBN 978-0-471-75798-6. PMID 16878778. {{cite book}}: |journal= ignored (help)
Dill Ken A., Bromberg Sarina (2011). Molecular Driving Forces: Statistical Thermodynamics in Biology, Chemistry, Physics, and Nanoscience. Garland Science. ISBN 978-0-8153-4430-8.
M. Scott Shell (2015). Thermodynamics and Statistical Mechanics: An Integrated Approach. Cambridge University Press. ISBN 978-1107656789.
Douglas E. Barrick (2018). Biomolecular Thermodynamics: From Theory to Applications. CRC Press. ISBN 978-1-4398-0019-5.
The following titles are more technical:
Bejan, Adrian (2016). Advanced Engineering Thermodynamics (4 ed.). Wiley. ISBN 978-1-119-05209-8.
Cengel, Yunus A., & Boles, Michael A. (2002). Thermodynamics – an Engineering Approach. McGraw Hill. ISBN 978-0-07-238332-4. OCLC 45791449.{{cite book}}: CS1 maint: multiple names: authors list (link)
Dunning-Davies, Jeremy (1997). Concise Thermodynamics: Principles and Applications. Horwood Publishing. ISBN 978-1-8985-6315-0. OCLC 36025958.
Kroemer, Herbert & Kittel, Charles (1980). Thermal Physics. W.H. Freeman Company. ISBN 978-0-7167-1088-2. OCLC 32932988.
== External links ==
Media related to Thermodynamics at Wikimedia Commons
Callendar, Hugh Longbourne (1911). "Thermodynamics" . Encyclopædia Britannica. Vol. 26 (11th ed.). pp. 808–814.
Thermodynamics Data & Property Calculation Websites
Thermodynamics Educational Websites
Biochemistry Thermodynamics
Thermodynamics and Statistical Mechanics
Engineering Thermodynamics – A Graphical Approach
Thermodynamics and Statistical Mechanics by Richard Fitzpatrick | Wikipedia/thermodynamics |
Science is a systematic discipline that builds and organises knowledge in the form of testable hypotheses and predictions about the universe. Modern science is typically divided into two or three major branches: the natural sciences (e.g., physics, chemistry, and biology), which study the physical world; and the social sciences (e.g., economics, psychology, and sociology), which study individuals and societies. Applied sciences are disciplines that use scientific knowledge for practical purposes, such as engineering and medicine. While sometimes referred to as the formal sciences, the study of logic, mathematics, and theoretical computer science (which study formal systems governed by axioms and rules) are typically regarded as separate because they rely on deductive reasoning instead of the scientific method or empirical evidence as their main methodology.
The history of science spans the majority of the historical record, with the earliest identifiable predecessors to modern science dating to the Bronze Age in Egypt and Mesopotamia (c. 3000–1200 BCE). Their contributions to mathematics, astronomy, and medicine entered and shaped the Greek natural philosophy of classical antiquity, whereby formal attempts were made to provide explanations of events in the physical world based on natural causes, while further advancements, including the introduction of the Hindu–Arabic numeral system, were made during the Golden Age of India.: 12 Scientific research deteriorated in these regions after the fall of the Western Roman Empire during the Early Middle Ages (400–1000 CE), but in the Medieval renaissances (Carolingian Renaissance, Ottonian Renaissance and the Renaissance of the 12th century) scholarship flourished again. Some Greek manuscripts lost in Western Europe were preserved and expanded upon in the Middle East during the Islamic Golden Age, Later, Byzantine Greek scholars contributed to their transmission by bringing Greek manuscripts from the declining Byzantine Empire to Western Europe at the beginning of the Renaissance.
The recovery and assimilation of Greek works and Islamic inquiries into Western Europe from the 10th to 13th centuries revived natural philosophy, which was later transformed by the Scientific Revolution that began in the 16th century as new ideas and discoveries departed from previous Greek conceptions and traditions. The scientific method soon played a greater role in knowledge creation and in the 19th century many of the institutional and professional features of science began to take shape, along with the changing of "natural philosophy" to "natural science".
New knowledge in science is advanced by research from scientists who are motivated by curiosity about the world and a desire to solve problems. Contemporary scientific research is highly collaborative and is usually done by teams in academic and research institutions, government agencies, and companies. The practical impact of their work has led to the emergence of science policies that seek to influence the scientific enterprise by prioritising the ethical and moral development of commercial products, armaments, health care, public infrastructure, and environmental protection.
== Etymology ==
The word science has been used in Middle English since the 14th century in the sense of "the state of knowing". The word was borrowed from the Anglo-Norman language as the suffix -cience, which was borrowed from the Latin word scientia, meaning "knowledge, awareness, understanding", a noun derivative of sciens meaning "knowing", itself the present active participle of sciō, "to know".
There are many hypotheses for science's ultimate word origin. According to Michiel de Vaan, Dutch linguist and Indo-Europeanist, sciō may have its origin in the Proto-Italic language as *skije- or *skijo- meaning "to know", which may originate from Proto-Indo-European language as *skh1-ie, *skh1-io, meaning "to incise". The Lexikon der indogermanischen Verben proposed sciō is a back-formation of nescīre, meaning "to not know, be unfamiliar with", which may derive from Proto-Indo-European *sekH- in Latin secāre, or *skh2-, from *sḱʰeh2(i)- meaning "to cut".
In the past, science was a synonym for "knowledge" or "study", in keeping with its Latin origin. A person who conducted scientific research was called a "natural philosopher" or "man of science". In 1834, William Whewell introduced the term scientist in a review of Mary Somerville's book On the Connexion of the Physical Sciences, crediting it to "some ingenious gentleman" (possibly himself).
== History ==
=== Early history ===
Science has no single origin. Rather, scientific thinking emerged gradually over the course of tens of thousands of years, taking different forms around the world, and few details are known about the very earliest developments. Women likely played a central role in prehistoric science, as did religious rituals. Some scholars use the term "protoscience" to label activities in the past that resemble modern science in some but not all features; however, this label has also been criticised as denigrating, or too suggestive of presentism, thinking about those activities only in relation to modern categories.
Direct evidence for scientific processes becomes clearer with the advent of writing systems in the Bronze Age civilisations of Ancient Egypt and Mesopotamia (c. 3000–1200 BCE), creating the earliest written records in the history of science.: 12–15 Although the words and concepts of "science" and "nature" were not part of the conceptual landscape at the time, the ancient Egyptians and Mesopotamians made contributions that would later find a place in Greek and medieval science: mathematics, astronomy, and medicine.: 12 From the 3rd millennium BCE, the ancient Egyptians developed a non-positional decimal numbering system, solved practical problems using geometry, and developed a calendar. Their healing therapies involved drug treatments and the supernatural, such as prayers, incantations, and rituals.: 9
The ancient Mesopotamians used knowledge about the properties of various natural chemicals for manufacturing pottery, faience, glass, soap, metals, lime plaster, and waterproofing. They studied animal physiology, anatomy, behaviour, and astrology for divinatory purposes. The Mesopotamians had an intense interest in medicine and the earliest medical prescriptions appeared in Sumerian during the Third Dynasty of Ur. They seem to have studied scientific subjects which had practical or religious applications and had little interest in satisfying curiosity.
=== Classical antiquity ===
In classical antiquity, there is no real ancient analogue of a modern scientist. Instead, well-educated, usually upper-class, and almost universally male individuals performed various investigations into nature whenever they could afford the time. Before the invention or discovery of the concept of phusis or nature by the pre-Socratic philosophers, the same words tend to be used to describe the natural "way" in which a plant grows, and the "way" in which, for example, one tribe worships a particular god. For this reason, it is claimed that these men were the first philosophers in the strict sense and the first to clearly distinguish "nature" and "convention".
The early Greek philosophers of the Milesian school, which was founded by Thales of Miletus and later continued by his successors Anaximander and Anaximenes, were the first to attempt to explain natural phenomena without relying on the supernatural. The Pythagoreans developed a complex number philosophy: 467–468 and contributed significantly to the development of mathematical science.: 465 The theory of atoms was developed by the Greek philosopher Leucippus and his student Democritus. Later, Epicurus would develop a full natural cosmology based on atomism, and would adopt a "canon" (ruler, standard) which established physical criteria or standards of scientific truth. The Greek doctor Hippocrates established the tradition of systematic medical science and is known as "The Father of Medicine".
A turning point in the history of early philosophical science was Socrates' example of applying philosophy to the study of human matters, including human nature, the nature of political communities, and human knowledge itself. The Socratic method as documented by Plato's dialogues is a dialectic method of hypothesis elimination: better hypotheses are found by steadily identifying and eliminating those that lead to contradictions. The Socratic method searches for general commonly-held truths that shape beliefs and scrutinises them for consistency. Socrates criticised the older type of study of physics as too purely speculative and lacking in self-criticism.
In the 4th century BCE, Aristotle created a systematic programme of teleological philosophy. In the 3rd century BCE, Greek astronomer Aristarchus of Samos was the first to propose a heliocentric model of the universe, with the Sun at the centre and all the planets orbiting it. Aristarchus's model was widely rejected because it was believed to violate the laws of physics, while Ptolemy's Almagest, which contains a geocentric description of the Solar System, was accepted through the early Renaissance instead. The inventor and mathematician Archimedes of Syracuse made major contributions to the beginnings of calculus. Pliny the Elder was a Roman writer and polymath, who wrote the seminal encyclopaedia Natural History.
Positional notation for representing numbers likely emerged between the 3rd and 5th centuries CE along Indian trade routes. This numeral system made efficient arithmetic operations more accessible and would eventually become standard for mathematics worldwide.
=== Middle Ages ===
Due to the collapse of the Western Roman Empire, the 5th century saw an intellectual decline, with knowledge of classical Greek conceptions of the world deteriorating in Western Europe.: 194 Latin encyclopaedists of the period such as Isidore of Seville preserved the majority of general ancient knowledge. In contrast, because the Byzantine Empire resisted attacks from invaders, they were able to preserve and improve prior learning.: 159 John Philoponus, a Byzantine scholar in the 6th century, started to question Aristotle's teaching of physics, introducing the theory of impetus.: 307, 311, 363, 402 His criticism served as an inspiration to medieval scholars and Galileo Galilei, who extensively cited his works ten centuries later.: 307–308
During late antiquity and the Early Middle Ages, natural phenomena were mainly examined via the Aristotelian approach. The approach includes Aristotle's four causes: material, formal, moving, and final cause. Many Greek classical texts were preserved by the Byzantine Empire and Arabic translations were made by Christians, mainly Nestorians and Miaphysites. Under the Abbasids, these Arabic translations were later improved and developed by Arabic scientists. By the 6th and 7th centuries, the neighbouring Sasanian Empire established the medical Academy of Gondishapur, which was considered by Greek, Syriac, and Persian physicians as the most important medical hub of the ancient world.
Islamic study of Aristotelianism flourished in the House of Wisdom established in the Abbasid capital of Baghdad, Iraq and the flourished until the Mongol invasions in the 13th century. Ibn al-Haytham, better known as Alhazen, used controlled experiments in his optical study. Avicenna's compilation of The Canon of Medicine, a medical encyclopaedia, is considered to be one of the most important publications in medicine and was used until the 18th century.
By the 11th century most of Europe had become Christian,: 204 and in 1088, the University of Bologna emerged as the first university in Europe. As such, demand for Latin translation of ancient and scientific texts grew,: 204 a major contributor to the Renaissance of the 12th century. Renaissance scholasticism in western Europe flourished, with experiments done by observing, describing, and classifying subjects in nature. In the 13th century, medical teachers and students at Bologna began opening human bodies, leading to the first anatomy textbook based on human dissection by Mondino de Luzzi.
=== Renaissance ===
New developments in optics played a role in the inception of the Renaissance, both by challenging long-held metaphysical ideas on perception, as well as by contributing to the improvement and development of technology such as the camera obscura and the telescope. At the start of the Renaissance, Roger Bacon, Vitello, and John Peckham each built up a scholastic ontology upon a causal chain beginning with sensation, perception, and finally apperception of the individual and universal forms of Aristotle.: Book I A model of vision later known as perspectivism was exploited and studied by the artists of the Renaissance. This theory uses only three of Aristotle's four causes: formal, material, and final.
In the 16th century, Nicolaus Copernicus formulated a heliocentric model of the Solar System, stating that the planets revolve around the Sun, instead of the geocentric model where the planets and the Sun revolve around the Earth. This was based on a theorem that the orbital periods of the planets are longer as their orbs are farther from the centre of motion, which he found not to agree with Ptolemy's model.
Johannes Kepler and others challenged the notion that the only function of the eye is perception, and shifted the main focus in optics from the eye to the propagation of light. Kepler is best known, however, for improving Copernicus' heliocentric model through the discovery of Kepler's laws of planetary motion. Kepler did not reject Aristotelian metaphysics and described his work as a search for the Harmony of the Spheres. Galileo had made significant contributions to astronomy, physics and engineering. However, he became persecuted after Pope Urban VIII sentenced him for writing about the heliocentric model.
The printing press was widely used to publish scholarly arguments, including some that disagreed widely with contemporary ideas of nature. Francis Bacon and René Descartes published philosophical arguments in favour of a new type of non-Aristotelian science. Bacon emphasised the importance of experiment over contemplation, questioned the Aristotelian concepts of formal and final cause, promoted the idea that science should study the laws of nature and the improvement of all human life. Descartes emphasised individual thought and argued that mathematics rather than geometry should be used to study nature.
=== Age of Enlightenment ===
At the start of the Age of Enlightenment, Isaac Newton formed the foundation of classical mechanics by his Philosophiæ Naturalis Principia Mathematica, greatly influencing future physicists. Gottfried Wilhelm Leibniz incorporated terms from Aristotelian physics, now used in a new non-teleological way. This implied a shift in the view of objects: objects were now considered as having no innate goals. Leibniz assumed that different types of things all work according to the same general laws of nature, with no special formal or final causes.
During this time the declared purpose and value of science became producing wealth and inventions that would improve human lives, in the materialistic sense of having more food, clothing, and other things. In Bacon's words, "the real and legitimate goal of sciences is the endowment of human life with new inventions and riches", and he discouraged scientists from pursuing intangible philosophical or spiritual ideas, which he believed contributed little to human happiness beyond "the fume of subtle, sublime or pleasing [speculation]".
Science during the Enlightenment was dominated by scientific societies and academies, which had largely replaced universities as centres of scientific research and development. Societies and academies were the backbones of the maturation of the scientific profession. Another important development was the popularisation of science among an increasingly literate population. Enlightenment philosophers turned to a few of their scientific predecessors – Galileo, Kepler, Boyle, and Newton principally – as the guides to every physical and social field of the day.
The 18th century saw significant advancements in the practice of medicine and physics; the development of biological taxonomy by Carl Linnaeus; a new understanding of magnetism and electricity; and the maturation of chemistry as a discipline. Ideas on human nature, society, and economics evolved during the Enlightenment. Hume and other Scottish Enlightenment thinkers developed A Treatise of Human Nature, which was expressed historically in works by authors including James Burnett, Adam Ferguson, John Millar and William Robertson, all of whom merged a scientific study of how humans behaved in ancient and primitive cultures with a strong awareness of the determining forces of modernity. Modern sociology largely originated from this movement. In 1776, Adam Smith published The Wealth of Nations, which is often considered the first work on modern economics.
=== 19th century ===
During the 19th century, many distinguishing characteristics of contemporary modern science began to take shape. These included the transformation of the life and physical sciences; the frequent use of precision instruments; the emergence of terms such as "biologist", "physicist", and "scientist"; an increased professionalisation of those studying nature; scientists gaining cultural authority over many dimensions of society; the industrialisation of numerous countries; the thriving of popular science writings; and the emergence of science journals. During the late 19th century, psychology emerged as a separate discipline from philosophy when Wilhelm Wundt founded the first laboratory for psychological research in 1879.
During the mid-19th century Charles Darwin and Alfred Russel Wallace independently proposed the theory of evolution by natural selection in 1858, which explained how different plants and animals originated and evolved. Their theory was set out in detail in Darwin's book On the Origin of Species, published in 1859. Separately, Gregor Mendel presented his paper, "Experiments on Plant Hybridisation" in 1865, which outlined the principles of biological inheritance, serving as the basis for modern genetics.
Early in the 19th century John Dalton suggested the modern atomic theory, based on Democritus's original idea of indivisible particles called atoms. The laws of conservation of energy, conservation of momentum and conservation of mass suggested a highly stable universe where there could be little loss of resources. However, with the advent of the steam engine and the Industrial Revolution there was an increased understanding that not all forms of energy have the same energy qualities, the ease of conversion to useful work or to another form of energy. This realisation led to the development of the laws of thermodynamics, in which the free energy of the universe is seen as constantly declining: the entropy of a closed universe increases over time.
The electromagnetic theory was established in the 19th century by the works of Hans Christian Ørsted, André-Marie Ampère, Michael Faraday, James Clerk Maxwell, Oliver Heaviside, and Heinrich Hertz. The new theory raised questions that could not easily be answered using Newton's framework. The discovery of X-rays inspired the discovery of radioactivity by Henri Becquerel and Marie Curie in 1896, Marie Curie then became the first person to win two Nobel Prizes. In the next year came the discovery of the first subatomic particle, the electron.
=== 20th century ===
In the first half of the century the development of antibiotics and artificial fertilisers improved human living standards globally. Harmful environmental issues such as ozone depletion, ocean acidification, eutrophication, and climate change came to the public's attention and caused the onset of environmental studies.
During this period scientific experimentation became increasingly larger in scale and funding. The extensive technological innovation stimulated by World War I, World War II, and the Cold War led to competitions between global powers, such as the Space Race and nuclear arms race. Substantial international collaborations were also made, despite armed conflicts.
In the late 20th century active recruitment of women and elimination of sex discrimination greatly increased the number of women scientists, but large gender disparities remained in some fields. The discovery of the cosmic microwave background in 1964 led to a rejection of the steady-state model of the universe in favour of the Big Bang theory of Georges Lemaître.
The century saw fundamental changes within science disciplines. Evolution became a unified theory in the early 20th-century when the modern synthesis reconciled Darwinian evolution with classical genetics. Albert Einstein's theory of relativity and the development of quantum mechanics complement classical mechanics to describe physics in extreme length, time and gravity. Widespread use of integrated circuits in the last quarter of the 20th century combined with communications satellites led to a revolution in information technology and the rise of the global internet and mobile computing, including smartphones. The need for mass systematisation of long, intertwined causal chains and large amounts of data led to the rise of the fields of systems theory and computer-assisted scientific modelling.
=== 21st century ===
The Human Genome Project was completed in 2003 by identifying and mapping all of the genes of the human genome. The first induced pluripotent human stem cells were made in 2006, allowing adult cells to be transformed into stem cells and turn into any cell type found in the body. With the affirmation of the Higgs boson discovery in 2013, the last particle predicted by the Standard Model of particle physics was found. In 2015, gravitational waves, predicted by general relativity a century before, were first observed. In 2019, the international collaboration Event Horizon Telescope presented the first direct image of a black hole's accretion disc.
== Branches ==
Modern science is commonly divided into three major branches: natural science, social science, and formal science. Each of these branches comprises various specialised yet overlapping scientific disciplines that often possess their own nomenclature and expertise. Both natural and social sciences are empirical sciences, as their knowledge is based on empirical observations and is capable of being tested for its validity by other researchers working under the same conditions.
=== Natural ===
Natural science is the study of the physical world. It can be divided into two main branches: life science and physical science. These two branches may be further divided into more specialised disciplines. For example, physical science can be subdivided into physics, chemistry, astronomy, and earth science. Modern natural science is the successor to the natural philosophy that began in Ancient Greece. Galileo, Descartes, Bacon, and Newton debated the benefits of using approaches that were more mathematical and more experimental in a methodical way. Still, philosophical perspectives, conjectures, and presuppositions, often overlooked, remain necessary in natural science. Systematic data collection, including discovery science, succeeded natural history, which emerged in the 16th century by describing and classifying plants, animals, minerals, and other biotic beings. Today, "natural history" suggests observational descriptions aimed at popular audiences.
=== Social ===
Social science is the study of human behaviour and the functioning of societies. It has many disciplines that include, but are not limited to anthropology, economics, history, human geography, political science, psychology, and sociology. In the social sciences, there are many competing theoretical perspectives, many of which are extended through competing research programmes such as the functionalists, conflict theorists, and interactionists in sociology. Due to the limitations of conducting controlled experiments involving large groups of individuals or complex situations, social scientists may adopt other research methods such as the historical method, case studies, and cross-cultural studies. Moreover, if quantitative information is available, social scientists may rely on statistical approaches to better understand social relationships and processes.
=== Formal ===
Formal science is an area of study that generates knowledge using formal systems. A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. It includes mathematics, systems theory, and theoretical computer science. The formal sciences share similarities with the other two branches by relying on objective, careful, and systematic study of an area of knowledge. They are, however, different from the empirical sciences as they rely exclusively on deductive reasoning, without the need for empirical evidence, to verify their abstract concepts. The formal sciences are therefore a priori disciplines and because of this, there is disagreement on whether they constitute a science. Nevertheless, the formal sciences play an important role in the empirical sciences. Calculus, for example, was initially invented to understand motion in physics. Natural and social sciences that rely heavily on mathematical applications include mathematical physics, chemistry, biology, finance, and economics.
=== Applied ===
Applied science is the use of the scientific method and knowledge to attain practical goals and includes a broad range of disciplines such as engineering and medicine. Engineering is the use of scientific principles to invent, design and build machines, structures and technologies. Science may contribute to the development of new technologies. Medicine is the practice of caring for patients by maintaining and restoring health through the prevention, diagnosis, and treatment of injury or disease.
=== Basic ===
The applied sciences are often contrasted with the basic sciences, which are focused on advancing scientific theories and laws that explain and predict events in the natural world.
=== Blue skies ===
=== Computational ===
Computational science applies computer simulations to science, enabling a better understanding of scientific problems than formal mathematics alone can achieve. The use of machine learning and artificial intelligence is becoming a central feature of computational contributions to science, for example in agent-based computational economics, random forests, topic modeling and various forms of prediction. However, machines alone rarely advance knowledge as they require human guidance and capacity to reason; and they can introduce bias against certain social groups or sometimes underperform against humans.
=== Interdisciplinary ===
Interdisciplinary science involves the combination of two or more disciplines into one, such as bioinformatics, a combination of biology and computer science or cognitive sciences. The concept has existed since the ancient Greek period and it became popular again in the 20th century.
== Research ==
Scientific research can be labelled as either basic or applied research. Basic research is the search for knowledge and applied research is the search for solutions to practical problems using this knowledge. Most understanding comes from basic research, though sometimes applied research targets specific practical problems. This leads to technological advances that were not previously imaginable.
=== Scientific method ===
Scientific research involves using the scientific method, which seeks to objectively explain the events of nature in a reproducible way. Scientists usually take for granted a set of basic assumptions that are needed to justify the scientific method: there is an objective reality shared by all rational observers; this objective reality is governed by natural laws; these laws were discovered by means of systematic observation and experimentation. Mathematics is essential in the formation of hypotheses, theories, and laws, because it is used extensively in quantitative modelling, observing, and collecting measurements. Statistics is used to summarise and analyse data, which allows scientists to assess the reliability of experimental results.
In the scientific method an explanatory thought experiment or hypothesis is put forward as an explanation using parsimony principles and is expected to seek consilience – fitting with other accepted facts related to an observation or scientific question. This tentative explanation is used to make falsifiable predictions, which are typically posted before being tested by experimentation. Disproof of a prediction is evidence of progress.: 4–5 Experimentation is especially important in science to help establish causal relationships to avoid the correlation fallacy, though in some sciences such as astronomy or geology, a predicted observation might be more appropriate.
When a hypothesis proves unsatisfactory it is modified or discarded. If the hypothesis survives testing, it may become adopted into the framework of a scientific theory, a validly reasoned, self-consistent model or framework for describing the behaviour of certain natural events. A theory typically describes the behaviour of much broader sets of observations than a hypothesis; commonly, a large number of hypotheses can be logically bound together by a single theory. Thus, a theory is a hypothesis explaining various other hypotheses. In that vein, theories are formulated according to most of the same scientific principles as hypotheses. Scientists may generate a model, an attempt to describe or depict an observation in terms of a logical, physical or mathematical representation, and to generate new hypotheses that can be tested by experimentation.
While performing experiments to test hypotheses, scientists may have a preference for one outcome over another. Eliminating the bias can be achieved through transparency, careful experimental design, and a thorough peer review process of the experimental results and conclusions. After the results of an experiment are announced or published, it is normal practice for independent researchers to double-check how the research was performed, and to follow up by performing similar experiments to determine how dependable the results might be. Taken in its entirety, the scientific method allows for highly creative problem solving while minimising the effects of subjective and confirmation bias. Intersubjective verifiability, the ability to reach a consensus and reproduce results, is fundamental to the creation of all scientific knowledge.
=== Literature ===
Scientific research is published in a range of literature. Scientific journals communicate and document the results of research carried out in universities and various other research institutions, serving as an archival record of science. The first scientific journals, Journal des sçavans followed by Philosophical Transactions, began publication in 1665. Since that time the total number of active periodicals has steadily increased. In 1981, one estimate for the number of scientific and technical journals in publication was 11,500.
Most scientific journals cover a single scientific field and publish the research within that field; the research is normally expressed in the form of a scientific paper. Science has become so pervasive in modern societies that it is considered necessary to communicate the achievements, news, and ambitions of scientists to a wider population.
=== Challenges ===
The replication crisis is an ongoing methodological crisis that affects parts of the social and life sciences. In subsequent investigations, the results of many scientific studies have been proven to be unrepeatable. The crisis has long-standing roots; the phrase was coined in the early 2010s as part of a growing awareness of the problem. The replication crisis represents an important body of research in metascience, which aims to improve the quality of all scientific research while reducing waste.
An area of study or speculation that masquerades as science in an attempt to claim legitimacy that it would not otherwise be able to achieve is sometimes referred to as pseudoscience, fringe science, or junk science. Physicist Richard Feynman coined the term "cargo cult science" for cases in which researchers believe, and at a glance, look like they are doing science but lack the honesty to allow their results to be rigorously evaluated. Various types of commercial advertising, ranging from hype to fraud, may fall into these categories. Science has been described as "the most important tool" for separating valid claims from invalid ones.
There can also be an element of political bias or ideological bias on all sides of scientific debates. Sometimes, research may be characterised as "bad science", research that may be well-intended but is incorrect, obsolete, incomplete, or over-simplified expositions of scientific ideas. The term scientific misconduct refers to situations such as where researchers have intentionally misrepresented their published data or have purposely given credit for a discovery to the wrong person.
== Philosophy ==
There are different schools of thought in the philosophy of science. The most popular position is empiricism, which holds that knowledge is created by a process involving observation; scientific theories generalise observations. Empiricism generally encompasses inductivism, a position that explains how general theories can be made from the finite amount of empirical evidence available. Many versions of empiricism exist, with the predominant ones being Bayesianism and the hypothetico-deductive method.
Empiricism has stood in contrast to rationalism, the position originally associated with Descartes, which holds that knowledge is created by the human intellect, not by observation. Critical rationalism is a contrasting 20th-century approach to science, first defined by Austrian-British philosopher Karl Popper. Popper rejected the way that empiricism describes the connection between theory and observation. He claimed that theories are not generated by observation, but that observation is made in the light of theories, and that the only way theory A can be affected by observation is after theory A were to conflict with observation, but theory B were to survive the observation.
Popper proposed replacing verifiability with falsifiability as the landmark of scientific theories, replacing induction with falsification as the empirical method. Popper further claimed that there is actually only one universal method, not specific to science: the negative method of criticism, trial and error, covering all products of the human mind, including science, mathematics, philosophy, and art.
Another approach, instrumentalism, emphasises the utility of theories as instruments for explaining and predicting phenomena. It views scientific theories as black boxes, with only their input (initial conditions) and output (predictions) being relevant. Consequences, theoretical entities, and logical structure are claimed to be things that should be ignored. Close to instrumentalism is constructive empiricism, according to which the main criterion for the success of a scientific theory is whether what it says about observable entities is true.
Thomas Kuhn argued that the process of observation and evaluation takes place within a paradigm, a logically consistent "portrait" of the world that is consistent with observations made from its framing. He characterised normal science as the process of observation and "puzzle solving", which takes place within a paradigm, whereas revolutionary science occurs when one paradigm overtakes another in a paradigm shift. Each paradigm has its own distinct questions, aims, and interpretations. The choice between paradigms involves setting two or more "portraits" against the world and deciding which likeness is most promising. A paradigm shift occurs when a significant number of observational anomalies arise in the old paradigm and a new paradigm makes sense of them. That is, the choice of a new paradigm is based on observations, even though those observations are made against the background of the old paradigm. For Kuhn, acceptance or rejection of a paradigm is a social process as much as a logical process. Kuhn's position, however, is not one of relativism.
Another approach often cited in debates of scientific scepticism against controversial movements like "creation science" is methodological naturalism. Naturalists maintain that a difference should be made between natural and supernatural, and science should be restricted to natural explanations. Methodological naturalism maintains that science requires strict adherence to empirical study and independent verification.
== Community ==
The scientific community is a network of interacting scientists who conduct scientific research. The community consists of smaller groups working in scientific fields. By having peer review, through discussion and debate within journals and conferences, scientists maintain the quality of research methodology and objectivity when interpreting results.
=== Scientists ===
Scientists are individuals who conduct scientific research to advance knowledge in an area of interest. Scientists may exhibit a strong curiosity about reality and a desire to apply scientific knowledge for the benefit of public health, nations, the environment, or industries; other motivations include recognition by peers and prestige. In modern times, many scientists study within specific areas of science in academic institutions, often obtaining advanced degrees in the process. Many scientists pursue careers in various fields such as academia, industry, government, and nonprofit organisations.
Science has historically been a male-dominated field, with notable exceptions. Women have faced considerable discrimination in science, much as they have in other areas of male-dominated societies. For example, women were frequently passed over for job opportunities and denied credit for their work. The achievements of women in science have been attributed to the defiance of their traditional role as labourers within the domestic sphere.
=== Learned societies ===
Learned societies for the communication and promotion of scientific thought and experimentation have existed since the Renaissance. Many scientists belong to a learned society that promotes their respective scientific discipline, profession, or group of related disciplines. Membership may either be open to all, require possession of scientific credentials, or conferred by election. Most scientific societies are nonprofit organisations, and many are professional associations. Their activities typically include holding regular conferences for the presentation and discussion of new research results and publishing or sponsoring academic journals in their discipline. Some societies act as professional bodies, regulating the activities of their members in the public interest, or the collective interest of the membership.
The professionalisation of science, begun in the 19th century, was partly enabled by the creation of national distinguished academies of sciences such as the Italian Accademia dei Lincei in 1603, the British Royal Society in 1660, the French Academy of Sciences in 1666, the American National Academy of Sciences in 1863, the German Kaiser Wilhelm Society in 1911, and the Chinese Academy of Sciences in 1949. International scientific organisations, such as the International Science Council, are devoted to international cooperation for science advancement.
=== Awards ===
Science awards are usually given to individuals or organisations that have made significant contributions to a discipline. They are often given by prestigious institutions; thus, it is considered a great honour for a scientist receiving them. Since the early Renaissance, scientists have often been awarded medals, money, and titles. The Nobel Prize, a widely regarded prestigious award, is awarded annually to those who have achieved scientific advances in the fields of medicine, physics, and chemistry.
== Society ==
=== Funding and policies ===
Funding of science is often through a competitive process in which potential research projects are evaluated and only the most promising receive funding. Such processes, which are run by government, corporations, or foundations, allocate scarce funds. Total research funding in most developed countries is between 1.5% and 3% of GDP. In the OECD, around two-thirds of research and development in scientific and technical fields is carried out by industry, and 20% and 10%, respectively, by universities and government. The government funding proportion in certain fields is higher, and it dominates research in social science and the humanities. In less developed nations, the government provides the bulk of the funds for their basic scientific research.
Many governments have dedicated agencies to support scientific research, such as the National Science Foundation in the United States, the National Scientific and Technical Research Council in Argentina, Commonwealth Scientific and Industrial Research Organisation in Australia, National Centre for Scientific Research in France, the Max Planck Society in Germany, and National Research Council in Spain. In commercial research and development, all but the most research-orientated corporations focus more heavily on near-term commercialisation possibilities than research driven by curiosity.
Science policy is concerned with policies that affect the conduct of the scientific enterprise, including research funding, often in pursuance of other national policy goals such as technological innovation to promote commercial product development, weapons development, health care, and environmental monitoring. Science policy sometimes refers to the act of applying scientific knowledge and consensus to the development of public policies. In accordance with public policy being concerned about the well-being of its citizens, science policy's goal is to consider how science and technology can best serve the public. Public policy can directly affect the funding of capital equipment and intellectual infrastructure for industrial research by providing tax incentives to those organisations that fund research.
=== Education and awareness ===
Science education for the general public is embedded in the school curriculum, and is supplemented by online pedagogical content (for example, YouTube and Khan Academy), museums, and science magazines and blogs. Major organisations of scientists such as the American Association for the Advancement of Science (AAAS) consider the sciences to be a part of the liberal arts traditions of learning, along with philosophy and history. Scientific literacy is chiefly concerned with an understanding of the scientific method, units and methods of measurement, empiricism, a basic understanding of statistics (correlations, qualitative versus quantitative observations, aggregate statistics), and a basic understanding of core scientific fields such as physics, chemistry, biology, ecology, geology, and computation. As a student advances into higher stages of formal education, the curriculum becomes more in depth. Traditional subjects usually included in the curriculum are natural and formal sciences, although recent movements include social and applied science as well.
The mass media face pressures that can prevent them from accurately depicting competing scientific claims in terms of their credibility within the scientific community as a whole. Determining how much weight to give different sides in a scientific debate may require considerable expertise regarding the matter. Few journalists have real scientific knowledge, and even beat reporters who are knowledgeable about certain scientific issues may be ignorant about other scientific issues that they are suddenly asked to cover.
Science magazines such as New Scientist, Science & Vie, and Scientific American cater to the needs of a much wider readership and provide a non-technical summary of popular areas of research, including notable discoveries and advances in certain fields of research. The science fiction genre, primarily speculative fiction, can transmit the ideas and methods of science to the general public. Recent efforts to intensify or develop links between science and non-scientific disciplines, such as literature or poetry, include the Creative Writing Science resource developed through the Royal Literary Fund.
=== Anti-science attitudes ===
While the scientific method is broadly accepted in the scientific community, some fractions of society reject certain scientific positions or are sceptical about science. Examples are the common notion that COVID-19 is not a major health threat to the US (held by 39% of Americans in August 2021) or the belief that climate change is not a major threat to the US (also held by 40% of Americans, in late 2019 and early 2020). Psychologists have pointed to four factors driving rejection of scientific results:
Scientific authorities are sometimes seen as inexpert, untrustworthy, or biased.
Some marginalised social groups hold anti-science attitudes, in part because these groups have often been exploited in unethical experiments.
Messages from scientists may contradict deeply held existing beliefs or morals.
The delivery of a scientific message may not be appropriately targeted to a recipient's learning style.
Anti-science attitudes often seem to be caused by fear of rejection in social groups. For instance, climate change is perceived as a threat by only 22% of Americans on the right side of the political spectrum, but by 85% on the left. That is, if someone on the left would not consider climate change as a threat, this person may face contempt and be rejected in that social group. In fact, people may rather deny a scientifically accepted fact than lose or jeopardise their social status.
=== Politics ===
Attitudes towards science are often determined by political opinions and goals. Government, business and advocacy groups have been known to use legal and economic pressure to influence scientific researchers. Many factors can act as facets of the politicisation of science such as anti-intellectualism, perceived threats to religious beliefs, and fear for business interests. Politicisation of science is usually accomplished when scientific information is presented in a way that emphasises the uncertainty associated with the scientific evidence. Tactics such as shifting conversation, failing to acknowledge facts, and capitalising on doubt of scientific consensus have been used to gain more attention for views that have been undermined by scientific evidence. Examples of issues that have involved the politicisation of science include the global warming controversy, health effects of pesticides, and health effects of tobacco.
== See also ==
List of scientific occupations
List of years in science
Logology (science)
Science (Wikiversity)
Scientific integrity
== Notes ==
== References ==
== External links == | Wikipedia/Sciences |
Residual entropy is the difference in entropy between a non-equilibrium state and crystal state of a substance close to absolute zero. This term is used in condensed matter physics to describe the entropy at zero kelvin of a glass or plastic crystal referred to the crystal state, whose entropy is zero according to the third law of thermodynamics. It occurs if a material can exist in many different states when cooled. The most common non-equilibrium state is vitreous state, glass.
A common example is the case of carbon monoxide, which has a very small dipole moment. As the carbon monoxide crystal is cooled to absolute zero, few of the carbon monoxide molecules have enough time to align themselves into a perfect crystal (with all of the carbon monoxide molecules oriented in the same direction). Because of this, the crystal is locked into a state with
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different corresponding microstates, giving a residual entropy of
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, rather than zero.
Another example is any amorphous solid (glass). These have residual entropy, because the atom-by-atom microscopic structure can be arranged in a huge number of different ways across a macroscopic system.
The residual entropy has a somewhat special significance compared to other residual properties, in that it has a role in the framework of residual entropy scaling, which is used to compute transport coefficients (coefficients governing non-equilibrium phenomena) directly from the equilibrium property residual entropy, which can be computed directly from any equation of state.
== History ==
One of the first examples of residual entropy was pointed out by Pauling to describe water ice. In water, each oxygen atom is bonded to two hydrogen atoms. However, when water freezes it forms a tetragonal structure where each oxygen atom has four hydrogen neighbors (due to neighboring water molecules). The hydrogen atoms sitting between the oxygen atoms have some degree of freedom as long as each oxygen atom has two hydrogen atoms that are 'nearby', thus forming the traditional H2O water molecule. However, it turns out that for a large number of water molecules in this configuration, the hydrogen atoms have a large number of possible configurations that meet the 2-in 2-out rule (each oxygen atom must have two 'near' (or 'in') hydrogen atoms, and two far (or 'out') hydrogen atoms). This freedom exists down to absolute zero, which was previously seen as an absolute one-of-a-kind configuration. The existence of these multiple configurations (choices for each H of orientation along O--O axis) that meet the rules of absolute zero (2-in 2-out for each O) amounts to randomness, or in other words, entropy. Thus systems that can take multiple configurations at or near absolute zero are said to have residual entropy.
Although water ice was the first material for which residual entropy was proposed, it is generally very difficult to prepare pure defect-free crystals of water ice for studying. A great deal of research has thus been undertaken into finding other systems that exhibit residual entropy. Geometrically frustrated systems in particular often exhibit residual entropy. An important example is spin ice, which is a geometrically frustrated magnetic material where the magnetic moments of the magnetic atoms have Ising-like magnetic spins and lie on the corners of network of corner-sharing tetrahedra. This material is thus analogous to water ice, with the exception that the spins on the corners of the tetrahedra can point into or out of the tetrahedra, thereby producing the same 2-in, 2-out rule as in water ice, and therefore the same residual entropy. One of the interesting properties of geometrically frustrated magnetic materials such as spin ice is that the level of residual entropy can be controlled by the application of an external magnetic field. This property can be used to create one-shot refrigeration systems.
== See also ==
Proton disorder in ice
Ice rules
Geometrical frustration
== Notes == | Wikipedia/Residual_entropy |
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applications include many problems in a wide variety of fields such as biology, neuroscience, computer science, information theory and sociology. Its main purpose is to clarify the properties of matter in aggregate, in terms of physical laws governing atomic motion.
Statistical mechanics arose out of the development of classical thermodynamics, a field for which it was successful in explaining macroscopic physical properties—such as temperature, pressure, and heat capacity—in terms of microscopic parameters that fluctuate about average values and are characterized by probability distributions.: 1–4
While classical thermodynamics is primarily concerned with thermodynamic equilibrium, statistical mechanics has been applied in non-equilibrium statistical mechanics to the issues of microscopically modeling the speed of irreversible processes that are driven by imbalances.: 3 Examples of such processes include chemical reactions and flows of particles and heat. The fluctuation–dissipation theorem is the basic knowledge obtained from applying non-equilibrium statistical mechanics to study the simplest non-equilibrium situation of a steady state current flow in a system of many particles.: 572–573
== History ==
In 1738, Swiss physicist and mathematician Daniel Bernoulli published Hydrodynamica which laid the basis for the kinetic theory of gases. In this work, Bernoulli posited the argument, still used to this day, that gases consist of great numbers of molecules moving in all directions, that their impact on a surface causes the gas pressure that we feel, and that what we experience as heat is simply the kinetic energy of their motion.
The founding of the field of statistical mechanics is generally credited to three physicists:
Ludwig Boltzmann, who developed the fundamental interpretation of entropy in terms of a collection of microstates
James Clerk Maxwell, who developed models of probability distribution of such states
Josiah Willard Gibbs, who coined the name of the field in 1884
In 1859, after reading a paper on the diffusion of molecules by Rudolf Clausius, Scottish physicist James Clerk Maxwell formulated the Maxwell distribution of molecular velocities, which gave the proportion of molecules having a certain velocity in a specific range. This was the first-ever statistical law in physics. Maxwell also gave the first mechanical argument that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium. Five years later, in 1864, Ludwig Boltzmann, a young student in Vienna, came across Maxwell's paper and spent much of his life developing the subject further.
Statistical mechanics was initiated in the 1870s with the work of Boltzmann, much of which was collectively published in his 1896 Lectures on Gas Theory. Boltzmann's original papers on the statistical interpretation of thermodynamics, the H-theorem, transport theory, thermal equilibrium, the equation of state of gases, and similar subjects, occupy about 2,000 pages in the proceedings of the Vienna Academy and other societies. Boltzmann introduced the concept of an equilibrium statistical ensemble and also investigated for the first time non-equilibrium statistical mechanics, with his H-theorem.
The term "statistical mechanics" was coined by the American mathematical physicist J. Willard Gibbs in 1884. According to Gibbs, the term "statistical", in the context of mechanics, i.e. statistical mechanics, was first used by the Scottish physicist James Clerk Maxwell in 1871:
"In dealing with masses of matter, while we do not perceive the individual molecules, we are compelled to adopt what I have described as the statistical method of calculation, and to abandon the strict dynamical method, in which we follow every motion by the calculus."
"Probabilistic mechanics" might today seem a more appropriate term, but "statistical mechanics" is firmly entrenched. Shortly before his death, Gibbs published in 1902 Elementary Principles in Statistical Mechanics, a book which formalized statistical mechanics as a fully general approach to address all mechanical systems—macroscopic or microscopic, gaseous or non-gaseous. Gibbs' methods were initially derived in the framework classical mechanics, however they were of such generality that they were found to adapt easily to the later quantum mechanics, and still form the foundation of statistical mechanics to this day.
== Principles: mechanics and ensembles ==
In physics, two types of mechanics are usually examined: classical mechanics and quantum mechanics. For both types of mechanics, the standard mathematical approach is to consider two concepts:
The complete state of the mechanical system at a given time, mathematically encoded as a phase point (classical mechanics) or a pure quantum state vector (quantum mechanics).
An equation of motion which carries the state forward in time: Hamilton's equations (classical mechanics) or the Schrödinger equation (quantum mechanics)
Using these two concepts, the state at any other time, past or future, can in principle be calculated.
There is however a disconnect between these laws and everyday life experiences, as we do not find it necessary (nor even theoretically possible) to know exactly at a microscopic level the simultaneous positions and velocities of each molecule while carrying out processes at the human scale (for example, when performing a chemical reaction). Statistical mechanics fills this disconnection between the laws of mechanics and the practical experience of incomplete knowledge, by adding some uncertainty about which state the system is in.
Whereas ordinary mechanics only considers the behaviour of a single state, statistical mechanics introduces the statistical ensemble, which is a large collection of virtual, independent copies of the system in various states. The statistical ensemble is a probability distribution over all possible states of the system. In classical statistical mechanics, the ensemble is a probability distribution over phase points (as opposed to a single phase point in ordinary mechanics), usually represented as a distribution in a phase space with canonical coordinate axes. In quantum statistical mechanics, the ensemble is a probability distribution over pure states and can be compactly summarized as a density matrix.
As is usual for probabilities, the ensemble can be interpreted in different ways:
an ensemble can be taken to represent the various possible states that a single system could be in (epistemic probability, a form of knowledge), or
the members of the ensemble can be understood as the states of the systems in experiments repeated on independent systems which have been prepared in a similar but imperfectly controlled manner (empirical probability), in the limit of an infinite number of trials.
These two meanings are equivalent for many purposes, and will be used interchangeably in this article.
However the probability is interpreted, each state in the ensemble evolves over time according to the equation of motion. Thus, the ensemble itself (the probability distribution over states) also evolves, as the virtual systems in the ensemble continually leave one state and enter another. The ensemble evolution is given by the Liouville equation (classical mechanics) or the von Neumann equation (quantum mechanics). These equations are simply derived by the application of the mechanical equation of motion separately to each virtual system contained in the ensemble, with the probability of the virtual system being conserved over time as it evolves from state to state.
One special class of ensemble is those ensembles that do not evolve over time. These ensembles are known as equilibrium ensembles and their condition is known as statistical equilibrium. Statistical equilibrium occurs if, for each state in the ensemble, the ensemble also contains all of its future and past states with probabilities equal to the probability of being in that state. (By contrast, mechanical equilibrium is a state with a balance of forces that has ceased to evolve.) The study of equilibrium ensembles of isolated systems is the focus of statistical thermodynamics. Non-equilibrium statistical mechanics addresses the more general case of ensembles that change over time, and/or ensembles of non-isolated systems.
== Statistical thermodynamics ==
The primary goal of statistical thermodynamics (also known as equilibrium statistical mechanics) is to derive the classical thermodynamics of materials in terms of the properties of their constituent particles and the interactions between them. In other words, statistical thermodynamics provides a connection between the macroscopic properties of materials in thermodynamic equilibrium, and the microscopic behaviours and motions occurring inside the material.
Whereas statistical mechanics proper involves dynamics, here the attention is focused on statistical equilibrium (steady state). Statistical equilibrium does not mean that the particles have stopped moving (mechanical equilibrium), rather, only that the ensemble is not evolving.
=== Fundamental postulate ===
A sufficient (but not necessary) condition for statistical equilibrium with an isolated system is that the probability distribution is a function only of conserved properties (total energy, total particle numbers, etc.).
There are many different equilibrium ensembles that can be considered, and only some of them correspond to thermodynamics. Additional postulates are necessary to motivate why the ensemble for a given system should have one form or another.
A common approach found in many textbooks is to take the equal a priori probability postulate. This postulate states that
For an isolated system with an exactly known energy and exactly known composition, the system can be found with equal probability in any microstate consistent with that knowledge.
The equal a priori probability postulate therefore provides a motivation for the microcanonical ensemble described below. There are various arguments in favour of the equal a priori probability postulate:
Ergodic hypothesis: An ergodic system is one that evolves over time to explore "all accessible" states: all those with the same energy and composition. In an ergodic system, the microcanonical ensemble is the only possible equilibrium ensemble with fixed energy. This approach has limited applicability, since most systems are not ergodic.
Principle of indifference: In the absence of any further information, we can only assign equal probabilities to each compatible situation.
Maximum information entropy: A more elaborate version of the principle of indifference states that the correct ensemble is the ensemble that is compatible with the known information and that has the largest Gibbs entropy (information entropy).
Other fundamental postulates for statistical mechanics have also been proposed. For example, recent studies shows that the theory of statistical mechanics can be built without the equal a priori probability postulate. One such formalism is based on the fundamental thermodynamic relation together with the following set of postulates:
where the third postulate can be replaced by the following:
=== Three thermodynamic ensembles ===
There are three equilibrium ensembles with a simple form that can be defined for any isolated system bounded inside a finite volume. These are the most often discussed ensembles in statistical thermodynamics. In the macroscopic limit (defined below) they all correspond to classical thermodynamics.
Microcanonical ensemble
describes a system with a precisely given energy and fixed composition (precise number of particles). The microcanonical ensemble contains with equal probability each possible state that is consistent with that energy and composition.
Canonical ensemble
describes a system of fixed composition that is in thermal equilibrium with a heat bath of a precise temperature. The canonical ensemble contains states of varying energy but identical composition; the different states in the ensemble are accorded different probabilities depending on their total energy.
Grand canonical ensemble
describes a system with non-fixed composition (uncertain particle numbers) that is in thermal and chemical equilibrium with a thermodynamic reservoir. The reservoir has a precise temperature, and precise chemical potentials for various types of particle. The grand canonical ensemble contains states of varying energy and varying numbers of particles; the different states in the ensemble are accorded different probabilities depending on their total energy and total particle numbers.
For systems containing many particles (the thermodynamic limit), all three of the ensembles listed above tend to give identical behaviour. It is then simply a matter of mathematical convenience which ensemble is used.: 227 The Gibbs theorem about equivalence of ensembles was developed into the theory of concentration of measure phenomenon, which has applications in many areas of science, from functional analysis to methods of artificial intelligence and big data technology.
Important cases where the thermodynamic ensembles do not give identical results include:
Microscopic systems.
Large systems at a phase transition.
Large systems with long-range interactions.
In these cases the correct thermodynamic ensemble must be chosen as there are observable differences between these ensembles not just in the size of fluctuations, but also in average quantities such as the distribution of particles. The correct ensemble is that which corresponds to the way the system has been prepared and characterized—in other words, the ensemble that reflects the knowledge about that system.
=== Calculation methods ===
Once the characteristic state function for an ensemble has been calculated for a given system, that system is 'solved' (macroscopic observables can be extracted from the characteristic state function). Calculating the characteristic state function of a thermodynamic ensemble is not necessarily a simple task, however, since it involves considering every possible state of the system. While some hypothetical systems have been exactly solved, the most general (and realistic) case is too complex for an exact solution. Various approaches exist to approximate the true ensemble and allow calculation of average quantities.
==== Exact ====
There are some cases which allow exact solutions.
For very small microscopic systems, the ensembles can be directly computed by simply enumerating over all possible states of the system (using exact diagonalization in quantum mechanics, or integral over all phase space in classical mechanics).
Some large systems consist of many separable microscopic systems, and each of the subsystems can be analysed independently. Notably, idealized gases of non-interacting particles have this property, allowing exact derivations of Maxwell–Boltzmann statistics, Fermi–Dirac statistics, and Bose–Einstein statistics.
A few large systems with interaction have been solved. By the use of subtle mathematical techniques, exact solutions have been found for a few toy models. Some examples include the Bethe ansatz, square-lattice Ising model in zero field, hard hexagon model.
==== Monte Carlo ====
Although some problems in statistical physics can be solved analytically using approximations and expansions, most current research utilizes the large processing power of modern computers to simulate or approximate solutions. A common approach to statistical problems is to use a Monte Carlo simulation to yield insight into the properties of a complex system. Monte Carlo methods are important in computational physics, physical chemistry, and related fields, and have diverse applications including medical physics, where they are used to model radiation transport for radiation dosimetry calculations.
The Monte Carlo method examines just a few of the possible states of the system, with the states chosen randomly (with a fair weight). As long as these states form a representative sample of the whole set of states of the system, the approximate characteristic function is obtained. As more and more random samples are included, the errors are reduced to an arbitrarily low level.
The Metropolis–Hastings algorithm is a classic Monte Carlo method which was initially used to sample the canonical ensemble.
Path integral Monte Carlo, also used to sample the canonical ensemble.
==== Other ====
For rarefied non-ideal gases, approaches such as the cluster expansion use perturbation theory to include the effect of weak interactions, leading to a virial expansion.
For dense fluids, another approximate approach is based on reduced distribution functions, in particular the radial distribution function.
Molecular dynamics computer simulations can be used to calculate microcanonical ensemble averages, in ergodic systems. With the inclusion of a connection to a stochastic heat bath, they can also model canonical and grand canonical conditions.
Mixed methods involving non-equilibrium statistical mechanical results (see below) may be useful.
== Non-equilibrium statistical mechanics ==
Many physical phenomena involve quasi-thermodynamic processes out of equilibrium, for example:
heat transport by the internal motions in a material, driven by a temperature imbalance,
electric currents carried by the motion of charges in a conductor, driven by a voltage imbalance,
spontaneous chemical reactions driven by a decrease in free energy,
friction, dissipation, quantum decoherence,
systems being pumped by external forces (optical pumping, etc.),
and irreversible processes in general.
All of these processes occur over time with characteristic rates. These rates are important in engineering. The field of non-equilibrium statistical mechanics is concerned with understanding these non-equilibrium processes at the microscopic level. (Statistical thermodynamics can only be used to calculate the final result, after the external imbalances have been removed and the ensemble has settled back down to equilibrium.)
In principle, non-equilibrium statistical mechanics could be mathematically exact: ensembles for an isolated system evolve over time according to deterministic equations such as Liouville's equation or its quantum equivalent, the von Neumann equation. These equations are the result of applying the mechanical equations of motion independently to each state in the ensemble. These ensemble evolution equations inherit much of the complexity of the underlying mechanical motion, and so exact solutions are very difficult to obtain. Moreover, the ensemble evolution equations are fully reversible and do not destroy information (the ensemble's Gibbs entropy is preserved). In order to make headway in modelling irreversible processes, it is necessary to consider additional factors besides probability and reversible mechanics.
Non-equilibrium mechanics is therefore an active area of theoretical research as the range of validity of these additional assumptions continues to be explored. A few approaches are described in the following subsections.
=== Stochastic methods ===
One approach to non-equilibrium statistical mechanics is to incorporate stochastic (random) behaviour into the system. Stochastic behaviour destroys information contained in the ensemble. While this is technically inaccurate (aside from hypothetical situations involving black holes, a system cannot in itself cause loss of information), the randomness is added to reflect that information of interest becomes converted over time into subtle correlations within the system, or to correlations between the system and environment. These correlations appear as chaotic or pseudorandom influences on the variables of interest. By replacing these correlations with randomness proper, the calculations can be made much easier.
=== Near-equilibrium methods ===
Another important class of non-equilibrium statistical mechanical models deals with systems that are only very slightly perturbed from equilibrium. With very small perturbations, the response can be analysed in linear response theory. A remarkable result, as formalized by the fluctuation–dissipation theorem, is that the response of a system when near equilibrium is precisely related to the fluctuations that occur when the system is in total equilibrium. Essentially, a system that is slightly away from equilibrium—whether put there by external forces or by fluctuations—relaxes towards equilibrium in the same way, since the system cannot tell the difference or "know" how it came to be away from equilibrium.: 664
This provides an indirect avenue for obtaining numbers such as ohmic conductivity and thermal conductivity by extracting results from equilibrium statistical mechanics. Since equilibrium statistical mechanics is mathematically well defined and (in some cases) more amenable for calculations, the fluctuation–dissipation connection can be a convenient shortcut for calculations in near-equilibrium statistical mechanics.
A few of the theoretical tools used to make this connection include:
Fluctuation–dissipation theorem
Onsager reciprocal relations
Green–Kubo relations
Landauer–Büttiker formalism
Mori–Zwanzig formalism
GENERIC formalism
=== Hybrid methods ===
An advanced approach uses a combination of stochastic methods and linear response theory. As an example, one approach to compute quantum coherence effects (weak localization, conductance fluctuations) in the conductance of an electronic system is the use of the Green–Kubo relations, with the inclusion of stochastic dephasing by interactions between various electrons by use of the Keldysh method.
== Applications ==
The ensemble formalism can be used to analyze general mechanical systems with uncertainty in knowledge about the state of a system. Ensembles are also used in:
propagation of uncertainty over time,
regression analysis of gravitational orbits,
ensemble forecasting of weather,
dynamics of neural networks,
bounded-rational potential games in game theory and non-equilibrium economics.
Statistical physics explains and quantitatively describes superconductivity, superfluidity, turbulence, collective phenomena in solids and plasma, and the structural features of liquid. It underlies the modern astrophysics and virial theorem. In solid state physics, statistical physics aids the study of liquid crystals, phase transitions, and critical phenomena. Many experimental studies of matter are entirely based on the statistical description of a system. These include the scattering of cold neutrons, X-ray, visible light, and more. Statistical physics also plays a role in materials science, nuclear physics, astrophysics, chemistry, biology and medicine (e.g. study of the spread of infectious diseases).
Analytical and computational techniques derived from statistical physics of disordered systems, can be extended to large-scale problems, including machine learning, e.g., to analyze the weight space of deep neural networks. Statistical physics is thus finding applications in the area of medical diagnostics.
=== Quantum statistical mechanics ===
Quantum statistical mechanics is statistical mechanics applied to quantum mechanical systems. In quantum mechanics, a statistical ensemble (probability distribution over possible quantum states) is described by a density operator S, which is a non-negative, self-adjoint, trace-class operator of trace 1 on the Hilbert space H describing the quantum system. This can be shown under various mathematical formalisms for quantum mechanics. One such formalism is provided by quantum logic.
== Index of statistical mechanics topics ==
=== Physics ===
Probability amplitude
Statistical physics
Boltzmann factor
Feynman–Kac formula
Fluctuation theorem
Information entropy
Vacuum expectation value
Cosmic variance
Negative probability
Gibbs state
Master equation
Partition function (mathematics)
Quantum probability
=== Percolation theory ===
Percolation theory
Schramm–Loewner evolution
== See also ==
List of textbooks in thermodynamics and statistical mechanics
Laplace transform § Statistical mechanics
== References ==
== Further reading ==
Reif, F. (2009). Fundamentals of Statistical and Thermal Physics. Waveland Press. ISBN 978-1-4786-1005-2.
Müller-Kirsten, Harald J W. (2013). Basics of Statistical Physics (PDF). doi:10.1142/8709. ISBN 978-981-4449-53-3.
Kadanoff, Leo P. "Statistical Physics and other resources". Archived from the original on August 12, 2021. Retrieved June 18, 2023.
Kadanoff, Leo P. (2000). Statistical Physics: Statics, Dynamics and Renormalization. World Scientific. ISBN 978-981-02-3764-6.
Flamm, Dieter (1998). "History and outlook of statistical physics". arXiv:physics/9803005.
== External links ==
Philosophy of Statistical Mechanics article by Lawrence Sklar for the Stanford Encyclopedia of Philosophy.
Sklogwiki - Thermodynamics, statistical mechanics, and the computer simulation of materials. SklogWiki is particularly orientated towards liquids and soft condensed matter.
Thermodynamics and Statistical Mechanics by Richard Fitzpatrick
Cohen, Doron (2011). "Lecture Notes in Statistical Mechanics and Mesoscopics". arXiv:1107.0568 [quant-ph].
Videos of lecture series in statistical mechanics on YouTube taught by Leonard Susskind.
Vu-Quoc, L., Configuration integral (statistical mechanics), 2008. this wiki site is down; see this article in the web archive on 2012 April 28. | Wikipedia/Fundamental_postulate_of_statistical_mechanics |
Thermal energy storage (TES) is the storage of thermal energy for later reuse. Employing widely different technologies, it allows surplus thermal energy to be stored for hours, days, or months. Scale both of storage and use vary from small to large – from individual processes to district, town, or region. Usage examples are the balancing of energy demand between daytime and nighttime, storing summer heat for winter heating, or winter cold for summer cooling (Seasonal thermal energy storage). Storage media include water or ice-slush tanks, masses of native earth or bedrock accessed with heat exchangers by means of boreholes, deep aquifers contained between impermeable strata; shallow, lined pits filled with gravel and water and insulated at the top, as well as eutectic solutions and phase-change materials.
Other sources of thermal energy for storage include heat or cold produced with heat pumps from off-peak, lower cost electric power, a practice called peak shaving; heat from combined heat and power (CHP) power plants; heat produced by renewable electrical energy that exceeds grid demand and waste heat from industrial processes. Heat storage, both seasonal and short term, is considered an important means for cheaply balancing high shares of variable renewable electricity production and integration of electricity and heating sectors in energy systems almost or completely fed by renewable energy.
== Categories ==
The kinds of thermal energy storage can be divided into three separate categories: sensible heat, latent heat, and thermo-chemical heat storage. Each of these has different advantages and disadvantages that determine their applications.
=== Sensible heat storage ===
Sensible heat storage (SHS) is the most straightforward method. It simply means the temperature of some medium is either increased or decreased. This type of storage is the most commercially available out of the three; other techniques are less developed.
The materials are generally inexpensive and safe. One of the cheapest, most commonly used options is a water tank, but materials such as molten salts or metals can be heated to higher temperatures and therefore offer a higher storage capacity. Energy can also be stored underground (UTES), either in an underground tank or in some kind of heat-transfer fluid (HTF) flowing through a system of pipes, either placed vertically in U-shapes (boreholes) or horizontally in trenches. Yet another system is known as a packed-bed (or pebble-bed) storage unit, in which some fluid, usually air, flows through a bed of loosely packed material (usually rock, pebbles or ceramic brick) to add or extract heat.
A disadvantage of SHS is its dependence on the properties of the storage medium. Storage capacities are limited by the specific heat capacity of the storage material, and the system needs to be properly designed to ensure energy extraction at a constant temperature.
==== Molten salt technology ====
The sensible heat of molten salt is also used for storing solar energy at a high temperature, termed molten-salt technology or molten salt energy storage (MSES). Molten salts can be employed as a thermal energy storage method to retain thermal energy. Presently, this is a commercially used technology to store the heat collected by concentrated solar power (e.g., from a solar tower or solar trough). The heat can later be converted into superheated steam to power conventional steam turbines and generate electricity at a later time. It was demonstrated in the Solar Two project from 1995 to 1999. Estimates in 2006 predicted an annual efficiency of 99%, a reference to the energy retained by storing heat before turning it into electricity, versus converting heat directly into electricity. Various eutectic mixtures of different salts are used (e.g., sodium nitrate, potassium nitrate and calcium nitrate). Experience with such systems exists in non-solar applications in the chemical and metals industries as a heat-transport fluid.
The salt melts at 131 °C (268 °F). It is kept liquid at 288 °C (550 °F) in an insulated "cold" storage tank. The liquid salt is pumped through panels in a solar collector where the focused sun heats it to 566 °C (1,051 °F). It is then sent to a hot storage tank. With proper insulation of the tank the thermal energy can be usefully stored for up to a week. When electricity is needed, the hot molten salt is pumped to a conventional steam-generator to produce superheated steam for driving a conventional turbine/generator set as used in any coal, oil, or nuclear power plant. A 100-megawatt turbine would need a tank of about 9.1 metres (30 ft) tall and 24 metres (79 ft) in diameter to drive it for four hours by this design.A single tank with a divider plate to separate cold and hot molten salt is under development. It is more economical by achieving 100% more heat storage per unit volume over the dual tanks system as the molten-salt storage tank is costly due to its complicated construction. Phase Change Material (PCMs) are also used in molten-salt energy storage, while research on obtaining shape-stabilized PCMs using high porosity matrices is ongoing.
Most solar thermal power plants use this thermal energy storage concept. The Solana Generating Station in the U.S. can store 6 hours worth of generating capacity in molten salt. During the summer of 2013 the Gemasolar Thermosolar solar power-tower/molten-salt plant in Spain achieved a first by continuously producing electricity 24 hours per day for 36 days. The Cerro Dominador Solar Thermal Plant, inaugurated in June 2021, has 17.5 hours of heat storage.
==== Heat storage in tanks, ponds or rock caverns ====
A steam accumulator consists of an insulated steel pressure tank containing hot water and steam under pressure. As a heat storage device, it is used to mediate heat production by a variable or steady source from a variable demand for heat. Steam accumulators may take on a significance for energy storage in solar thermal energy projects.
Large stores, mostly hot water storage tanks, are widely used in Nordic countries to store heat for several days, to decouple heat and power production and to help meet peak demands. Some towns use insulated ponds heated by solar power as a heat source for district heating pumps. Intersessional storage in caverns has been investigated and appears to be economical and plays a significant role in heating in Finland. Energy producer Helen Oy estimates an 11.6 GWh capacity and 120 MW thermal output for its 260,000 m3 water cistern under Mustikkamaa (fully charged or discharged in 4 days at capacity), operating from 2021 to offset days of peak production/demand; while the 300,000 m3 rock caverns 50 m under sea level in Kruunuvuorenranta (near Laajasalo) were designated in 2018 to store heat in summer from warm seawater and release it in winter for district heating. In 2024, it was announced that the municipal energy supplier of Vantaa had commissioned an underground heat storage facility of over 1,100,000 cubic metres (39,000,000 cu ft) in size and 90GWh in capacity to be built, expected to be operational in 2028.
==== Hot silicon technology ====
Solid or molten silicon offers much higher storage temperatures than salts with consequent greater capacity and efficiency. It is being researched as a possible more energy efficient storage technology. Silicon is able to store more than 1 MWh of energy per cubic meter at 1400 °C. An additional advantage is the relative abundance of silicon when compared to the salts used for the same purpose.
==== Molten aluminum ====
Another medium that can store thermal energy is molten (recycled) aluminum. This technology was developed by the Swedish company Azelio. The material is heated to 600 °C. When needed, the energy is transported to a Stirling engine using a heat-transfer fluid.
==== Heat storage using oils ====
Using oils as sensible heat storage materials is an effective approach for storing thermal energy, particularly in medium- to high-temperature applications. Different types of oils are used based on the temperature range and the specific requirements of the thermal energy storage system: mineral oils, synthetic oils are more recently, vegetable oils are gaining interest because they are renewable and biodegradable. Numerious criteria are used to select an oil for a particular application: high energy storage capacity and specific heat capacity, high thermal conductivity, high chemical and physical stability, low coefficient of expansion, low cost, availability, low corrosion and compatibility with compounds materials, limited environmental issues, etc. Regarding the selection of a low-cost or cost-effective thermal oil, it is important to consider not only the acquisition or purchase cost, but also the operating and replacement costs or even final disposal costs. An oil that is initially more expensive may prove to be more cost-effective in the long run if it offers higher thermal stability, thereby reducing the frequency of replacement.
==== Heat storage in hot rocks or concrete ====
Water has one of the highest thermal capacities at 4.2 kJ/(kg⋅K) whereas concrete has about one third of that. On the other hand, concrete can be heated to much higher temperatures (1200 °C) by for example electrical heating and therefore has a much higher overall volumetric capacity. Thus in the example below, an insulated cube of about 2.8 m3 would appear to provide sufficient storage for a single house to meet 50% of heating demand. This could, in principle, be used to store surplus wind or solar heat due to the ability of electrical heating to reach high temperatures. At the neighborhood level, the Wiggenhausen-Süd solar development at Friedrichshafen in southern Germany has received international attention. This features a 12,000 m3 (420,000 cu ft) reinforced concrete thermal store linked to 4,300 m2 (46,000 sq ft) of solar collectors, which will supply the 570 houses with around 50% of their heating and hot water. Siemens-Gamesa built a 130 MWh thermal storage near Hamburg with 750 °C in basalt and 1.5 MW electric output. A similar system is scheduled for Sorø, Denmark, with 41–58% of the stored 18 MWh heat returned for the town's district heating, and 30–41% returned as electricity.
“Brick toaster” is a recently (August 2022) announced innovative heat reservoir operating at up to 1,500 °C (2,732 °F) that its maker, Titan Cement/Rondo claims should be able cut global CO2 output by 15% over 15 years.
Research into using sintered bauxite proppants as the thermal store, heating them up to 1000 °C. This material was tested against plasma-sprayed alumina and mullite, alumina fiber reinforced/alumina matrix and mullite fiber reinforced/mullite ceramic matrix composites. These four materials were considered because of their usefulness as solar receivers, transport tubes and storage tanks.
=== Latent heat storage ===
Because latent heat storage (LHS) is associated with a phase transition, the general term for the associated media is Phase-Change Material (PCM). During these transitions, heat can be added or extracted without affecting the material's temperature, giving it an advantage over SHS-technologies. Storage capacities are often higher as well.
There are a multitude of PCMs available, including but not limited to salts, polymers, gels, paraffin waxes, metal alloys and semiconductor-metal alloys, each with different properties. This allows for a more target-oriented system design. As the process is isothermal at the PCM's melting point, the material can be picked to have the desired temperature range. Desirable qualities include high latent heat and thermal conductivity. Furthermore, the storage unit can be more compact if volume changes during the phase transition are small.
PCMs are further subdivided into organic, inorganic and eutectic materials. Compared to organic PCMs, inorganic materials are less flammable, cheaper and more widely available. They also have higher storage capacity and thermal conductivity. Organic PCMs, on the other hand, are less corrosive and not as prone to phase-separation. Eutectic materials, as they are mixtures, are more easily adjusted to obtain specific properties, but have low latent and specific heat capacities.
Another important factor in LHS is the encapsulation of the PCM. Some materials are more prone to erosion and leakage than others. The system must be carefully designed in order to avoid unnecessary loss of heat.
==== Miscibility gap alloy technology ====
Miscibility gap alloys rely on the phase change of a metallic material (see: latent heat) to store thermal energy.
Rather than pumping the liquid metal between tanks as in a molten-salt system, the metal is encapsulated in another metallic material that it cannot alloy with (immiscible). Depending on the two materials selected (the phase changing material and the encapsulating material) storage densities can be between 0.2 and 2 MJ/L.
A working fluid, typically water or steam, is used to transfer the heat into and out of the system. Thermal conductivity of miscibility gap alloys is often higher (up to 400 W/(m⋅K)) than competing technologies which means quicker "charge" and "discharge" of the thermal storage is possible. The technology has not yet been implemented on a large scale.
==== Ice-based technology ====
Several applications are being developed where ice is produced during off-peak periods and used for cooling at a later time. For example, air conditioning can be provided more economically by using low-cost electricity at night to freeze water into ice, then using the cooling capacity of ice in the afternoon to reduce the electricity needed to handle air conditioning demands. Thermal energy storage using ice makes use of the large heat of fusion of water. Historically, ice was transported from mountains to cities for use as a coolant. One metric ton of water (= one cubic meter) can store 334 million joules (MJ) or 317,000 BTUs (93 kWh). A relatively small storage facility can hold enough ice to cool a large building for a day or a week.
In addition to using ice in direct cooling applications, it is also being used in heat pump-based heating systems. In these applications, the phase change energy provides a very significant layer of thermal capacity that is near the bottom range of temperature that water source heat pumps can operate in. This allows the system to ride out the heaviest heating load conditions and extends the timeframe by which the source energy elements can contribute heat back into the system.
==== Cryogenic energy storage ====
Cryogenic energy storage uses liquification of air or nitrogen as an energy store.
A pilot cryogenic energy system that uses liquid air as the energy store, and low-grade waste heat to drive the thermal re-expansion of the air, operated at a power station in Slough, UK in 2010.
=== Thermo-chemical heat storage ===
Thermo-chemical heat storage (TCS) involves some kind of reversible exotherm/endotherm chemical reaction with thermo-chemical materials (TCM) . Depending on the reactants, this method can allow for an even higher storage capacity than LHS.
In one type of TCS, heat is applied to decompose certain molecules. The reaction products are then separated, and mixed again when required, resulting in a release of energy. Some examples are the decomposition of potassium oxide (over a range of 300–800 °C, with a heat decomposition of 2.1 MJ/kg), lead oxide (300–350 °C, 0.26 MJ/kg) and calcium hydroxide (above 450 °C, where the reaction rates can be increased by adding zinc or aluminum). The photochemical decomposition of nitrosyl chloride can also be used and, since it needs photons to occur, works especially well when paired with solar energy.
==== Adsorption (or Sorption) solar heating and storage ====
Adsorption processes also fall into this category. It can be used to not only store thermal energy, but also control air humidity. Zeolites (microporous crystalline alumina-silicates) and silica gels are well suited for this purpose. In hot, humid environments, this technology is often used in combination with lithium chloride to cool water.
The low cost ($200/ton) and high cycle rate (2,000×) of synthetic zeolites such as Linde 13X with water adsorbate has garnered much academic and commercial interest recently for use for thermal energy storage (TES), specifically of low-grade solar and waste heat. Several pilot projects have been funded in the EU from 2000 to the present (2020). The basic concept is to store solar thermal energy as chemical latent energy in the zeolite. Typically, hot dry air from flat plate solar collectors is made to flow through a bed of zeolite such that any water adsorbate present is driven off. Storage can be diurnal, weekly, monthly, or even seasonal depending on the volume of the zeolite and the area of the solar thermal panels. When heat is called for during the night, or sunless hours, or winter, humidified air flows through the zeolite. As the humidity is adsorbed by the zeolite, heat is released to the air and subsequently to the building space. This form of TES, with specific use of zeolites, was first taught by Guerra in 1978. Advantages over molten salts and other high temperature TES include that (1) the temperature required is only the stagnation temperature typical of a solar flat plate thermal collector, and (2) as long as the zeolite is kept dry, the energy is stored indefinitely. Because of the low temperature, and because the energy is stored as latent heat of adsorption, thus eliminating the insulation requirements of a molten salt storage system, costs are significantly lower.
==== Salt hydrate technology ====
One example of an experimental storage system based on chemical reaction energy is the salt hydrate technology. The system uses the reaction energy created when salts are hydrated or dehydrated. It works by storing heat in a container containing 50% sodium hydroxide (NaOH) solution. Heat (e.g. from using a solar collector) is stored by evaporating the water in an endothermic reaction. When water is added again, heat is released in an exothermic reaction at 50 °C (120 °F). Current systems operate at 60% efficiency. The system is especially advantageous for seasonal thermal energy storage, because the dried salt can be stored at room temperature for prolonged times, without energy loss. The containers with the dehydrated salt can even be transported to a different location. The system has a higher energy density than heat stored in water and the capacity of the system can be designed to store energy from a few months to years.
In 2013 the Dutch technology developer TNO presented the results of the MERITS project to store heat in a salt container. The heat, which can be derived from a solar collector on a rooftop, expels the water contained in the salt. When the water is added again, the heat is released, with almost no energy losses. A container with a few cubic meters of salt could store enough of this thermochemical energy to heat a house throughout the winter. In a temperate climate like that of the Netherlands, an average low-energy household requires about 6.7 GJ/winter. To store this energy in water (at a temperature difference of 70 °C), 23 m3 insulated water storage would be needed, exceeding the storage abilities of most households. Using salt hydrate technology with a storage density of about 1 GJ/m3, 4–8 m3 could be sufficient.
As of 2016, researchers in several countries are conducting experiments to determine the best type of salt, or salt mixture. Low pressure within the container seems favorable for the energy transport. Especially promising are organic salts, so called ionic liquids. Compared to lithium halide-based sorbents they are less problematic in terms of limited global resources and compared to most other halides and sodium hydroxide (NaOH) they are less corrosive and not negatively affected by CO2 contaminations.
However, a recent meta-analysis on studies of thermochemical heat storage suggests that salt hydrates offer very low potential for thermochemical heat storage, that absorption processes have prohibitive performance for long-term heat storage, and that thermochemical storage may not be suitable for long-term solar heat storage in buildings.
==== Molecular bonds ====
Storing energy in molecular bonds is being investigated. Energy densities equivalent to lithium-ion batteries have been achieved. This has been done by a DSPEC (dys-sensitized photoelectrosythesis cell). This is a cell that can store energy that has been acquired by solar panels during the day for night-time (or even later) use. It is designed by taking an indication from, well known, natural photosynthesis.
The DSPEC generates hydrogen fuel by making use of the acquired solar energy to split water molecules into its elements. As the result of this split, the hydrogen is isolated and the oxygen is released into the air. This sounds easier than it actually is. Four electrons of the water molecules need to be separated and transported elsewhere. Another difficult part is the process of merging the two separate hydrogen molecules.
The DSPEC consists of two components: a molecule and a nanoparticle. The molecule is called a chromophore-catalyst assembly which absorbs sunlight and kick starts the catalyst. This catalyst separates the electrons and the water molecules. The nanoparticles are assembled into a thin layer and a single nanoparticle has many chromophore-catalyst on it. The function of this thin layer of nanoparticles is to transfer away the electrons which are separated from the water. This thin layer of nanoparticles is coated by a layer of titanium dioxide. With this coating, the electrons that come free can be transferred more quickly so that hydrogen could be made. This coating is, again, coated with a protective coating that strengthens the connection between the chromophore-catalyst and the nanoparticle.
Using this method, the solar energy acquired from the solar panels is converted into fuel (hydrogen) without releasing the so-called greenhouse gasses. This fuel can be stored into a fuel cell and, at a later time, used to generate electricity.
==== Molecular Solar Thermal System (MOST) ====
Another promising way to store solar energy for electricity and heat production is a so-called molecular solar thermal system (MOST). With this approach a molecule is converted by photoisomerization into a higher-energy isomer. Photoisomerization is a process in which one (cis trans) isomer is converted into another by light (solar energy). This isomer is capable of storing the solar energy until the energy is released by a heat trigger or catalyst (then, the isomer is converted into its original isomer). A promising candidate for such a MOST is Norbornadiene (NBD). This is because there is a high energy difference between the NBD and the quadricyclane (QC) photoisomer. This energy difference is approximately 96 kJ/mol. It is also known that for such systems, the donor-acceptor substitutions provide an effective means for red shifting the longest-wavelength absorption. This improves the solar spectrum match.
A crucial challenge for a useful MOST system is to acquire a satisfactory high energy storage density (if possible, higher than 300 kJ/kg). Another challenge of a MOST system is that light can be harvested in the visible region. The functionalization of the NBD with the donor and acceptor units is used to adjust this absorption maxima. However, this positive effect on the solar absorption is compensated by a higher molecular weight. This implies a lower energy density. This positive effect on the solar absorption has another downside. Namely, that the energy storage time is lowered when the absorption is redshifted. A possible solution to overcome this anti-correlation between the energy density and the red shifting is to couple one chromophore unit to several photo switches. In this case, it is advantageous to form so called dimers or trimers. The NBD share a common donor and/or acceptor.
Kasper Moth-Poulsen and his team tried to engineer the stability of the high energy photo isomer by having two electronically coupled photo switches with separate barriers for thermal conversion. By doing so, a blue shift occurred after the first isomerization (NBD-NBD to QC-NBD). This led to a higher energy of isomerization of the second switching event (QC-NBD to QC-QC). Another advantage of this system, by sharing a donor, is that the molecular weight per norbornadiene unit is reduced. This leads to an increase of the energy density.
Eventually, this system could reach a quantum yield of photoconversion up 94% per NBD unit. A quantum yield is a measure of the efficiency of photon emission. With this system the measured energy densities reached up to 559 kJ/kg (exceeding the target of 300 kJ/kg). So, the potential of the molecular photo switches is enormous—not only for solar thermal energy storage but for other applications as well.
In 2022, researchers reported combining the MOST with a chip-sized thermoelectric generator to generate electricity from it. The system can reportedly store solar energy for up to 18 years and may be an option for renewable energy storage.
== Thermal Battery ==
A thermal energy battery is a physical structure used for the purpose of storing and releasing thermal energy. Such a thermal battery (a.k.a. TBat) allows energy available at one time to be temporarily stored and then released at another time. The basic principles involved in a thermal battery occur at the atomic level of matter, with energy being added to or taken from either a solid mass or a liquid volume which causes the substance's temperature to change. Some thermal batteries also involve causing a substance to transition thermally through a phase transition which causes even more energy to be stored and released due to the delta enthalpy of fusion or delta enthalpy of vaporization.
Thermal batteries are very common, and include such familiar items as a hot water bottle. Early examples of thermal batteries include stone and mud cook stoves, rocks placed in fires, and kilns. While stoves and kilns are ovens, they are also thermal storage systems that depend on heat being retained for an extended period of time. Thermal energy storage systems can also be installed in domestic situations with heat batteries and thermal stores being amongst the most common types of energy storage systems installed at homes in the UK.
=== Types of thermal batteries ===
Thermal batteries generally fall into 4 categories with different forms and applications, although fundamentally all are for the storage and retrieval of thermal energy. They also differ in method and density of heat storage.
==== Phase change thermal battery ====
Phase change materials used for thermal storage are capable of storing and releasing significant thermal capacity at the temperature that they change phase. These materials are chosen based on specific applications because there is a wide range of temperatures that may be useful in different applications and a wide range of materials that change phase at different temperatures. These materials include salts and waxes that are specifically engineered for the applications they serve. In addition to manufactured materials, water is a phase change material. The latent heat of water is 334 joules/gram. The phase change of water occurs at 0 °C (32 °F).
Some applications use the thermal capacity of water or ice as cold storage; others use it as heat storage. It can serve either application; ice can be melted to store heat then refrozen to warm an environment. The advantage of using a phase change in this way is that a given mass of material can absorb a large quantity of energy without its temperature changing. Hence a thermal battery that uses a phase change can be made lighter, or more energy can be put into it without raising the internal temperature unacceptably.
==== Encapsulated thermal battery ====
An encapsulated thermal battery is physically similar to a phase change thermal battery in that it is a confined amount of physical material which is thermally heated or cooled to store or extract energy. However, in a non-phase change encapsulated thermal battery, the temperature of the substance is changed without inducing a phase change. Since a phase change is not needed many more materials are available for use in an encapsulated thermal battery. One of the key properties of an encapsulated thermal battery is its volumetric heat capacity (VHC), also termed volume-specific heat capacity. Several substances are used for these thermal batteries, for example water, concrete, and wet or dry sand.
An example of an encapsulated thermal battery is a residential water heater with a storage tank. This thermal battery is usually slowly charged over a period of about 30–60 minutes for rapid use when needed (e.g., 10–15 minutes). Many utilities, understanding the "thermal battery" nature of water heaters, have begun using them to absorb excess renewable energy power when available for later use by the homeowner. According to the above-cited article, "net savings to the electricity system as a whole could be $200 per year per heater — some of which may be passed on to its owner".
Research into using sand as a heat storage medium has been performed in Finland, where a prototype 8 MWh sand battery was built in 2022 to store renewable solar and wind power as heat, for later use as district heating, and possible later power generation. In Canada, single building thermal storage also stores renewable solar and wind power as heat, for later use as space or water heating for the building in which it's installed. It differs from the system in Finland by being compact, using low pressure pumped fluids, and can only heat one building rather than several. It can take in waste heat from alternate sources such as computer server rooms or compost heaps and store it for later distribution.
==== Ground heat exchange thermal battery ====
A ground heat exchanger (GHEX) is an area of the earth that is utilized as a seasonal/annual cycle thermal battery. These thermal batteries are areas of the earth into which pipes have been placed in order to transfer thermal energy. Energy is added to the GHEX by running a higher temperature fluid through the pipes and thus raising the temperature of the local earth. Energy can also be taken from the GHEX by running a lower-temperature fluid through those same pipes.
GHEX are usually implemented in two forms. The picture above depicts what is known as a "horizontal" GHEX where trenching is used to place an amount of pipe in a closed loop in the ground. They are also formed by drilling boreholes into the ground, either vertically or horizontally, and then the pipes are inserted in the form of a closed-loop with a "u-bend" fitting on the far end of the loop.
Heat energy can be added to or removed from a GHEX at any point in time. However, they are most often used as a Seasonal thermal energy storage operating on an annual cycle where energy is extracted from a building during the summer season to cool a building and added to the GHEX. Then that same energy is later extracted from the GHEX in the winter season to heat the building. This annual cycle of energy addition and subtraction is highly predictable based on energy modelling of the building served. A thermal battery used in this mode is a renewable energy source as the energy extracted in the winter will be restored to the GHEX the next summer in a continually repeating cycle. This type is solar powered because it is the heat from the sun in the summer that is removed from a building and stored in the ground for use in the next winter season for heating. There are two main methods of Thermal Response Testing that are used to characterize the thermal conductivity and Thermal Capacity/Diffusivity of GHEX Thermal Batteries—Log-Time 1-Dimensional Curve Fit and newly released Advanced Thermal Response Testing.
A good example of the Annual Cycle nature of a GHEX Thermal Battery can be seen in the ASHRAE Building study. As seen there in the 'Ground Loop and Ambient Air temperatures by date' graphic (Figure 2–7), one can easily see the annual cycle sinusoidal shape of the ground temperature as heat is seasonally extracted from the ground in winter and rejected to the ground in summer, creating a ground "thermal charge" in one season that is not uncharged and driven the other direction from neutral until a later season. Other more advanced examples of Ground-based Thermal Batteries utilizing intentional well-bore thermal patterns are currently in research and early use.
==== Other thermal batteries ====
In the defense industry primary molten-salt batteries are termed "thermal batteries". They are non-rechargeable electrical batteries using a low-melting eutectic mixture of ionic metal salts (sodium, potassium and lithium chlorides, bromides, etc.) as the electrolyte, manufactured with the salts in solid form. As long as the salts remain solid, the battery has a long shelf life of up to 50 years. Once activated (usually by a pyrotechnic heat source) and the electrolyte melts, it is very reliable with a high energy and power density. They are extensively used for military applications such as small to large guided missiles, and nuclear weapons.
There are other items that have historically been termed "thermal batteries", such as energy-storage heat packs that skiers use for keeping hands and feet warm (see hand warmer). These contain iron powder moist with oxygen-free salt water which rapidly corrodes over a period of hours, releasing heat, when exposed to air. Instant cold packs absorb heat by a non-chemical phase-change such as by absorbing the endothermic heat of solution of certain compounds.
The one common principle of these other thermal batteries is that the reaction involved is not reversible. Thus, these batteries are not used for storing and retrieving heat energy.
== Electric thermal storage ==
Storage heaters are commonplace in European homes with time-of-use metering (traditionally using cheaper electricity at nighttime). They consist of high-density ceramic bricks or feolite blocks heated to a high temperature with electricity and may or may not have good insulation and controls to release heat over a number of hours. Some advice not to use them in areas with young children or where there is an increased risk of fires due to poor housekeeping, both due to the high temperatures involved.
With the rise of wind and solar power (and other renewable energies) providing an ever increasing share of energy input into the electricity grids in some countries, the use of larger scale electric energy storage is being explored by several commercial companies. Ideally, the utilisation of surplus renewable energy is transformed into high temperature high grade heat in highly insulated heat stores, for release later when needed. An emerging technology is the use of vacuum super insulated (VSI) heat stores. The use of electricity to generate heat, and not say direct heat from solar thermal collectors, means that very high temperatures can be realised, potentially allowing for inter seasonal heat transfer—storing high grade heat in summer from surplus photovoltaics generation into heat stored for the following winter with relatively minimal standing losses.
== Solar energy storage ==
Solar energy is an application of thermal energy storage. Most practical solar thermal storage systems provide storage from a few hours to a day's worth of energy. However, a growing number of facilities use seasonal thermal energy storage (STES), enabling solar energy to be stored in summer to heat space during winter. In 2017 Drake Landing Solar Community in Alberta, Canada, achieved a year-round 97% solar heating fraction, a world record made possible by incorporating STES.
The combined use of latent heat and sensible heat are possible with high temperature solar thermal input. Various eutectic metal mixtures, such as aluminum and silicon (AlSi12) offer a high melting point suited to efficient steam generation, while high alumina cement-based materials offer good storage capabilities.
== Pumped-heat electricity storage ==
In pumped-heat electricity storage (PHES), a reversible heat-pump system is used to store energy as a temperature difference between two heat stores.
=== Isentropic ===
Isentropic systems involve two insulated containers filled, for example, with crushed rock or gravel: a hot vessel storing thermal energy at high temperature/pressure, and a cold vessel storing thermal energy at low temperature/pressure. The vessels are connected at top and bottom by pipes and the whole system is filled with an inert gas such as argon.
While charging, the system can use off-peak electricity to work as a heat pump. One prototype used argon at ambient temperature and pressure from the top of the cold store is compressed adiabatically, to a pressure of, for example, 12 bar, heating it to around 500 °C (900 °F). The compressed gas is transferred to the top of the hot vessel where it percolates down through the gravel, transferring heat to the rock and cooling to ambient temperature. The cooled, but still pressurized, gas emerging at the bottom of the vessel is then adiabatically expanded to 1 bar, which lowers its temperature to −150 °C. The cold gas is then passed up through the cold vessel where it cools the rock while warming to its initial condition.
The energy is recovered as electricity by reversing the cycle. The hot gas from the hot vessel is expanded to drive a generator and then supplied to the cold store. The cooled gas retrieved from the bottom of the cold store is compressed which heats the gas to ambient temperature. The gas is then transferred to the bottom of the hot vessel to be reheated.
The compression and expansion processes are provided by a specially designed reciprocating machine using sliding valves. Surplus heat generated by inefficiencies in the process is shed to the environment through heat exchangers during the discharging cycle.
The developer claimed that a round trip efficiency of 72–80% was achievable. This compares to >80% achievable with pumped hydro energy storage.
Another proposed system uses turbomachinery and is capable of operating at much higher power levels. Use of phase change material as heat storage material could enhance performance.
== See also ==
Renewable energy portal
== References ==
== External links ==
ASHRAE white paper on the economies of load shifting
MSN article on Ice Storage Air Conditioning at archive.today (archived 19 January 2013)
ICE TES Thermal Energy Storage — IDE-Tech
Laramie, Wyoming
"Prepared for the Thermal Energy-Storage Systems Collaborative of the California Energy Commission" Report titled "Source Energy and Environmental Impacts of Thermal Energy Storage." Tabors Caramanis & Assoc energy.ca.gov Archived 23 August 2014 at the Wayback Machine
Competence Center Thermal Energy Storage at Lucerne School of Engineering and Architecture
== Further reading == | Wikipedia/Thermal_energy_storage |
The Clausius–Clapeyron relation, in chemical thermodynamics, specifies the temperature dependence of pressure, most importantly vapor pressure, at a discontinuous phase transition between two phases of matter of a single constituent. It is named after Rudolf Clausius and Benoît Paul Émile Clapeyron. However, this relation was in fact originally derived by Sadi Carnot in his Reflections on the Motive Power of Fire, which was published in 1824 but largely ignored until it was rediscovered by Clausius, Clapeyron, and Lord Kelvin decades later. Kelvin said of Carnot's argument that "nothing in the whole range of Natural Philosophy is more remarkable than the establishment of general laws by such a process of reasoning."
Kelvin and his brother James Thomson confirmed the relation experimentally in 1849–50, and it was historically important as a very early successful application of theoretical thermodynamics. Its relevance to meteorology and climatology is the increase of the water-holding capacity of the atmosphere by about 7% for every 1 °C (1.8 °F) rise in temperature.
== Definition ==
=== Exact Clapeyron equation ===
On a pressure–temperature (P–T) diagram, for any phase change the line separating the two phases is known as the coexistence curve. The Clapeyron relation gives the slope of the tangents to this curve. Mathematically,
d
P
d
T
=
L
T
Δ
v
=
Δ
s
Δ
v
,
{\displaystyle {\frac {\mathrm {d} P}{\mathrm {d} T}}={\frac {L}{T\,\Delta v}}={\frac {\Delta s}{\Delta v}},}
where
d
P
/
d
T
{\displaystyle \mathrm {d} P/\mathrm {d} T}
is the slope of the tangent to the coexistence curve at any point,
L
{\displaystyle L}
is the molar change in enthalpy (latent heat, the amount of energy absorbed in the transformation),
T
{\displaystyle T}
is the temperature,
Δ
v
{\displaystyle \Delta v}
is the molar volume change of the phase transition, and
Δ
s
{\displaystyle \Delta s}
is the molar entropy change of the phase transition. Alternatively, the specific values may be used instead of the molar ones.
=== Clausius–Clapeyron equation ===
The Clausius–Clapeyron equation: 509 applies to vaporization of liquids where vapor follows ideal gas law using the ideal gas constant
R
{\displaystyle R}
and liquid volume is neglected as being much smaller than vapor volume V. It is often used to calculate vapor pressure of a liquid.
d
P
d
T
=
P
L
T
2
R
,
{\displaystyle {\frac {\mathrm {d} P}{\mathrm {d} T}}={\frac {PL}{T^{2}R}},}
v
=
V
n
=
R
T
P
.
{\displaystyle v={\frac {V}{n}}={\frac {RT}{P}}.}
The equation expresses this in a more convenient form just in terms of the latent heat, for moderate temperatures and pressures.
== Derivations ==
=== Derivation from state postulate ===
Using the state postulate, take the molar entropy
s
{\displaystyle s}
for a homogeneous substance to be a function of molar volume
v
{\displaystyle v}
and temperature
T
{\displaystyle T}
.: 508
d
s
=
(
∂
s
∂
v
)
T
d
v
+
(
∂
s
∂
T
)
v
d
T
.
{\displaystyle \mathrm {d} s=\left({\frac {\partial s}{\partial v}}\right)_{T}\,\mathrm {d} v+\left({\frac {\partial s}{\partial T}}\right)_{v}\,\mathrm {d} T.}
The Clausius–Clapeyron relation describes a Phase transition in a closed system composed of two contiguous phases, condensed matter and ideal gas, of a single substance, in mutual thermodynamic equilibrium, at constant temperature and pressure. Therefore,: 508
d
s
=
(
∂
s
∂
v
)
T
d
v
.
{\displaystyle \mathrm {d} s=\left({\frac {\partial s}{\partial v}}\right)_{T}\,\mathrm {d} v.}
Using the appropriate Maxwell relation gives: 508
d
s
=
(
∂
P
∂
T
)
v
d
v
,
{\displaystyle \mathrm {d} s=\left({\frac {\partial P}{\partial T}}\right)_{v}\,\mathrm {d} v,}
where
P
{\displaystyle P}
is the pressure. Since pressure and temperature are constant, the derivative of pressure with respect to temperature does not change.: 57, 62, 671 Therefore, the partial derivative of molar entropy may be changed into a total derivative
d
s
=
d
P
d
T
d
v
,
{\displaystyle \mathrm {d} s={\frac {\mathrm {d} P}{\mathrm {d} T}}\,\mathrm {d} v,}
and the total derivative of pressure with respect to temperature may be factored out when integrating from an initial phase
α
{\displaystyle \alpha }
to a final phase
β
{\displaystyle \beta }
,: 508 to obtain
d
P
d
T
=
Δ
s
Δ
v
,
{\displaystyle {\frac {\mathrm {d} P}{\mathrm {d} T}}={\frac {\Delta s}{\Delta v}},}
where
Δ
s
≡
s
β
−
s
α
{\displaystyle \Delta s\equiv s_{\beta }-s_{\alpha }}
and
Δ
v
≡
v
β
−
v
α
{\displaystyle \Delta v\equiv v_{\beta }-v_{\alpha }}
are respectively the change in molar entropy and molar volume. Given that a phase change is an internally reversible process, and that our system is closed, the first law of thermodynamics holds:
d
u
=
δ
q
+
δ
w
=
T
d
s
−
P
d
v
,
{\displaystyle \mathrm {d} u=\delta q+\delta w=T\,\mathrm {d} s-P\,\mathrm {d} v,}
where
u
{\displaystyle u}
is the internal energy of the system. Given constant pressure and temperature (during a phase change) and the definition of molar enthalpy
h
{\displaystyle h}
, we obtain
d
h
=
T
d
s
+
v
d
P
,
{\displaystyle \mathrm {d} h=T\,\mathrm {d} s+v\,\mathrm {d} P,}
d
h
=
T
d
s
,
{\displaystyle \mathrm {d} h=T\,\mathrm {d} s,}
d
s
=
d
h
T
.
{\displaystyle \mathrm {d} s={\frac {\mathrm {d} h}{T}}.}
Given constant pressure and temperature (during a phase change), we obtain: 508
Δ
s
=
Δ
h
T
.
{\displaystyle \Delta s={\frac {\Delta h}{T}}.}
Substituting the definition of molar latent heat
L
=
Δ
h
{\displaystyle L=\Delta h}
gives
Δ
s
=
L
T
.
{\displaystyle \Delta s={\frac {L}{T}}.}
Substituting this result into the pressure derivative given above (
d
P
/
d
T
=
Δ
s
/
Δ
v
{\displaystyle \mathrm {d} P/\mathrm {d} T=\Delta s/\Delta v}
), we obtain: 508
d
P
d
T
=
L
T
Δ
v
.
{\displaystyle {\frac {\mathrm {d} P}{\mathrm {d} T}}={\frac {L}{T\,\Delta v}}.}
This result (also known as the Clapeyron equation) equates the slope
d
P
/
d
T
{\displaystyle \mathrm {d} P/\mathrm {d} T}
of the coexistence curve
P
(
T
)
{\displaystyle P(T)}
to the function
L
/
(
T
Δ
v
)
{\displaystyle L/(T\,\Delta v)}
of the molar latent heat
L
{\displaystyle L}
, the temperature
T
{\displaystyle T}
, and the change in molar volume
Δ
v
{\displaystyle \Delta v}
. Instead of the molar values, corresponding specific values may also be used.
=== Derivation from Gibbs–Duhem relation ===
Suppose two phases,
α
{\displaystyle \alpha }
and
β
{\displaystyle \beta }
, are in contact and at equilibrium with each other. Their chemical potentials are related by
μ
α
=
μ
β
.
{\displaystyle \mu _{\alpha }=\mu _{\beta }.}
Furthermore, along the coexistence curve,
d
μ
α
=
d
μ
β
.
{\displaystyle \mathrm {d} \mu _{\alpha }=\mathrm {d} \mu _{\beta }.}
One may therefore use the Gibbs–Duhem relation
d
μ
=
M
(
−
s
d
T
+
v
d
P
)
{\displaystyle \mathrm {d} \mu =M(-s\,\mathrm {d} T+v\,\mathrm {d} P)}
(where
s
{\displaystyle s}
is the specific entropy,
v
{\displaystyle v}
is the specific volume, and
M
{\displaystyle M}
is the molar mass) to obtain
−
(
s
β
−
s
α
)
d
T
+
(
v
β
−
v
α
)
d
P
=
0.
{\displaystyle -(s_{\beta }-s_{\alpha })\,\mathrm {d} T+(v_{\beta }-v_{\alpha })\,\mathrm {d} P=0.}
Rearrangement gives
d
P
d
T
=
s
β
−
s
α
v
β
−
v
α
=
Δ
s
Δ
v
,
{\displaystyle {\frac {\mathrm {d} P}{\mathrm {d} T}}={\frac {s_{\beta }-s_{\alpha }}{v_{\beta }-v_{\alpha }}}={\frac {\Delta s}{\Delta v}},}
from which the derivation of the Clapeyron equation continues as in the previous section.
=== Ideal gas approximation at low temperatures ===
When the phase transition of a substance is between a gas phase and a condensed phase (liquid or solid), and occurs at temperatures much lower than the critical temperature of that substance, the specific volume of the gas phase
v
g
{\displaystyle v_{\text{g}}}
greatly exceeds that of the condensed phase
v
c
{\displaystyle v_{\text{c}}}
. Therefore, one may approximate
Δ
v
=
v
g
(
1
−
v
c
v
g
)
≈
v
g
{\displaystyle \Delta v=v_{\text{g}}\left(1-{\frac {v_{\text{c}}}{v_{\text{g}}}}\right)\approx v_{\text{g}}}
at low temperatures. If pressure is also low, the gas may be approximated by the ideal gas law, so that
v
g
=
R
T
P
,
{\displaystyle v_{\text{g}}={\frac {RT}{P}},}
where
P
{\displaystyle P}
is the pressure,
R
{\displaystyle R}
is the specific gas constant, and
T
{\displaystyle T}
is the temperature. Substituting into the Clapeyron equation
d
P
d
T
=
L
T
Δ
v
,
{\displaystyle {\frac {\mathrm {d} P}{\mathrm {d} T}}={\frac {L}{T\,\Delta v}},}
we can obtain the Clausius–Clapeyron equation: 509
d
P
d
T
=
P
L
T
2
R
{\displaystyle {\frac {\mathrm {d} P}{\mathrm {d} T}}={\frac {PL}{T^{2}R}}}
for low temperatures and pressures,: 509 where
L
{\displaystyle L}
is the specific latent heat of the substance. Instead of the specific, corresponding molar values (i.e.
L
{\displaystyle L}
in kJ/mol and R = 8.31 J/(mol⋅K)) may also be used.
Let
(
P
1
,
T
1
)
{\displaystyle (P_{1},T_{1})}
and
(
P
2
,
T
2
)
{\displaystyle (P_{2},T_{2})}
be any two points along the coexistence curve between two phases
α
{\displaystyle \alpha }
and
β
{\displaystyle \beta }
. In general,
L
{\displaystyle L}
varies between any two such points, as a function of temperature. But if
L
{\displaystyle L}
is approximated as constant,
d
P
P
≅
L
R
d
T
T
2
,
{\displaystyle {\frac {\mathrm {d} P}{P}}\cong {\frac {L}{R}}{\frac {\mathrm {d} T}{T^{2}}},}
∫
P
1
P
2
d
P
P
≅
L
R
∫
T
1
T
2
d
T
T
2
,
{\displaystyle \int _{P_{1}}^{P_{2}}{\frac {\mathrm {d} P}{P}}\cong {\frac {L}{R}}\int _{T_{1}}^{T_{2}}{\frac {\mathrm {d} T}{T^{2}}},}
ln
P
|
P
=
P
1
P
2
≅
−
L
R
⋅
1
T
|
T
=
T
1
T
2
,
{\displaystyle \ln P{\Big |}_{P=P_{1}}^{P_{2}}\cong -{\frac {L}{R}}\cdot \left.{\frac {1}{T}}\right|_{T=T_{1}}^{T_{2}},}
or: 672
ln
P
2
P
1
≅
−
L
R
(
1
T
2
−
1
T
1
)
.
{\displaystyle \ln {\frac {P_{2}}{P_{1}}}\cong -{\frac {L}{R}}\left({\frac {1}{T_{2}}}-{\frac {1}{T_{1}}}\right).}
These last equations are useful because they relate equilibrium or saturation vapor pressure and temperature to the latent heat of the phase change without requiring specific-volume data. For instance, for water near its normal boiling point, with a molar enthalpy of vaporization of 40.7 kJ/mol and R = 8.31 J/(mol⋅K),
P
vap
(
T
)
≅
1
bar
⋅
exp
[
−
40
700
K
8.31
(
1
T
−
1
373
K
)
]
.
{\displaystyle P_{\text{vap}}(T)\cong 1~{\text{bar}}\cdot \exp \left[-{\frac {40\,700~{\text{K}}}{8.31}}\left({\frac {1}{T}}-{\frac {1}{373~{\text{K}}}}\right)\right].}
=== Clapeyron's derivation ===
In the original work by Clapeyron, the following argument is advanced.
Clapeyron considered a Carnot process of saturated water vapor with horizontal isobars. As the pressure is a function of temperature alone, the isobars are also isotherms. If the process involves an infinitesimal amount of water,
d
x
{\displaystyle \mathrm {d} x}
, and an infinitesimal difference in temperature
d
T
{\displaystyle \mathrm {d} T}
, the heat absorbed is
Q
=
L
d
x
,
{\displaystyle Q=L\,\mathrm {d} x,}
and the corresponding work is
W
=
d
p
d
T
d
T
(
V
″
−
V
′
)
,
{\displaystyle W={\frac {\mathrm {d} p}{\mathrm {d} T}}\,\mathrm {d} T(V''-V'),}
where
V
″
−
V
′
{\displaystyle V''-V'}
is the difference between the volumes of
d
x
{\displaystyle \mathrm {d} x}
in the liquid phase and vapor phases.
The ratio
W
/
Q
{\displaystyle W/Q}
is the efficiency of the Carnot engine,
d
T
/
T
{\displaystyle \mathrm {d} T/T}
. Substituting and rearranging gives
d
p
d
T
=
L
T
(
v
″
−
v
′
)
,
{\displaystyle {\frac {\mathrm {d} p}{\mathrm {d} T}}={\frac {L}{T(v''-v')}},}
where lowercase
v
″
−
v
′
{\displaystyle v''-v'}
denotes the change in specific volume during the transition.
== Applications ==
=== Chemistry and chemical engineering ===
For transitions between a gas and a condensed phase with the approximations described above, the expression may be rewritten as
ln
(
P
1
P
0
)
=
L
R
(
1
T
0
−
1
T
1
)
{\displaystyle \ln \left({\frac {P_{1}}{P_{0}}}\right)={\frac {L}{R}}\left({\frac {1}{T_{0}}}-{\frac {1}{T_{1}}}\right)}
where
P
0
,
P
1
{\displaystyle P_{0},P_{1}}
are the pressures at temperatures
T
0
,
T
1
{\displaystyle T_{0},T_{1}}
respectively and
R
{\displaystyle R}
is the ideal gas constant. For a liquid–gas transition,
L
{\displaystyle L}
is the molar latent heat (or molar enthalpy) of vaporization; for a solid–gas transition,
L
{\displaystyle L}
is the molar latent heat of sublimation. If the latent heat is known, then knowledge of one point on the coexistence curve, for instance (1 bar, 373 K) for water, determines the rest of the curve. Conversely, the relationship between
ln
P
{\displaystyle \ln P}
and
1
/
T
{\displaystyle 1/T}
is linear, and so linear regression is used to estimate the latent heat.
=== Meteorology and climatology ===
Atmospheric water vapor drives many important meteorologic phenomena (notably, precipitation), motivating interest in its dynamics. The Clausius–Clapeyron equation for water vapor under typical atmospheric conditions (near standard temperature and pressure) is
d
e
s
d
T
=
L
v
(
T
)
e
s
R
v
T
2
,
{\displaystyle {\frac {\mathrm {d} e_{s}}{\mathrm {d} T}}={\frac {L_{v}(T)e_{s}}{R_{v}T^{2}}},}
where
The temperature dependence of the latent heat
L
v
(
T
)
{\displaystyle L_{v}(T)}
can be neglected in this application. The August–Roche–Magnus formula provides a solution under that approximation:
e
s
(
T
)
=
6.1094
exp
(
17.625
T
T
+
243.04
)
,
{\displaystyle e_{s}(T)=6.1094\exp \left({\frac {17.625T}{T+243.04}}\right),}
where
e
s
{\displaystyle e_{s}}
is in hPa, and
T
{\displaystyle T}
is in degrees Celsius (whereas everywhere else on this page,
T
{\displaystyle T}
is an absolute temperature, e.g. in kelvins).
This is also sometimes called the Magnus or Magnus–Tetens approximation, though this attribution is historically inaccurate. But see also the discussion of the accuracy of different approximating formulae for saturation vapour pressure of water.
Under typical atmospheric conditions, the denominator of the exponent depends weakly on
T
{\displaystyle T}
(for which the unit is degree Celsius). Therefore, the August–Roche–Magnus equation implies that saturation water vapor pressure changes approximately exponentially with temperature under typical atmospheric conditions, and hence the water-holding capacity of the atmosphere increases by about 7% for every 1 °C rise in temperature.
== Example ==
One of the uses of this equation is to determine if a phase transition will occur in a given situation. Consider the question of how much pressure is needed to melt ice at a temperature
Δ
T
{\displaystyle {\Delta T}}
below 0 °C. Note that water is unusual in that its change in volume upon melting is negative. We can assume
Δ
P
=
L
T
Δ
v
Δ
T
,
{\displaystyle \Delta P={\frac {L}{T\,\Delta v}}\,\Delta T,}
and substituting in
we obtain
Δ
P
Δ
T
=
−
13.5
MPa
/
K
.
{\displaystyle {\frac {\Delta P}{\Delta T}}=-13.5~{\text{MPa}}/{\text{K}}.}
To provide a rough example of how much pressure this is, to melt ice at −7 °C (the temperature many ice skating rinks are set at) would require balancing a small car (mass ~ 1000 kg) on a thimble (area ~ 1 cm2). This shows that ice skating cannot be simply explained by pressure-caused melting point depression, and in fact the mechanism is quite complex.
== Second derivative ==
While the Clausius–Clapeyron relation gives the slope of the coexistence curve, it does not provide any information about its curvature or second derivative. The second derivative of the coexistence curve of phases 1 and 2 is given by
d
2
P
d
T
2
=
1
v
2
−
v
1
[
c
p
2
−
c
p
1
T
−
2
(
v
2
α
2
−
v
1
α
1
)
d
P
d
T
]
+
1
v
2
−
v
1
[
(
v
2
κ
T
2
−
v
1
κ
T
1
)
(
d
P
d
T
)
2
]
,
{\displaystyle {\begin{aligned}{\frac {\mathrm {d} ^{2}P}{\mathrm {d} T^{2}}}&={\frac {1}{v_{2}-v_{1}}}\left[{\frac {c_{p2}-c_{p1}}{T}}-2(v_{2}\alpha _{2}-v_{1}\alpha _{1}){\frac {\mathrm {d} P}{\mathrm {d} T}}\right]\\{}&+{\frac {1}{v_{2}-v_{1}}}\left[(v_{2}\kappa _{T2}-v_{1}\kappa _{T1})\left({\frac {\mathrm {d} P}{\mathrm {d} T}}\right)^{2}\right],\end{aligned}}}
where subscripts 1 and 2 denote the different phases,
c
p
{\displaystyle c_{p}}
is the specific heat capacity at constant pressure,
α
=
(
1
/
v
)
(
d
v
/
d
T
)
P
{\displaystyle \alpha =(1/v)(\mathrm {d} v/\mathrm {d} T)_{P}}
is the thermal expansion coefficient, and
κ
T
=
−
(
1
/
v
)
(
d
v
/
d
P
)
T
{\displaystyle \kappa _{T}=-(1/v)(\mathrm {d} v/\mathrm {d} P)_{T}}
is the isothermal compressibility.
== See also ==
Van 't Hoff equation
Antoine equation
Lee–Kesler method
== References ==
== Bibliography ==
== Notes == | Wikipedia/Clausius–Clapeyron_equation |
The renewable-energy industry is the part of the energy industry focusing on new and appropriate renewable energy technologies. Investors worldwide are increasingly paying greater attention to this emerging industry. In many cases, this has translated into rapid renewable energy commercialization and considerable industry expansion. The wind power, solar power and hydroelectric power industries provide good examples of this.
In 2020, the global renewable energy market was valued at $881.7 billion and consumption grew 2.9 EJ. China was the largest contributor to renewable growth, accounting an increment of 1.0 EJ in consumption, followed by the US, Japan, the United Kingdom, India, and Germany. In Europe, renewable consumption incremented 0.7 EJ.
== Overview ==
Net-zero and 100% renewable energy global goals create market opportunities for renewable industries such as solar and wind energy and lithium-ion batteries. By 2050, it’s estimated that the renewable market will reach a value of one trillion dollars, the same size as the current oil market.
In 1997, world leaders adopted the Kyoto Protocol as a step in mitigating the climate crisis. This Protocol was the precursor to the 2016 Paris Climate Accord, and helped push the renewable energy industry forward. This protocol places more responsibility on developed nations in an act of recognition that developed countries are responsible for emitting more greenhouse gases. That responsibility is reflected in the investment into the renewable energy industry. According to IRENA, China and the United States are the leading countries in renewable energy. Overall in 2022, China generated 2,673,556GW of energy through renewable sources, and the U.S. produced 981,697GW of electricity through the renewable energy industry. Brazil, commonly recognized as a developing country, has routinely come in third for overall electricity generated through renewable energy, and is recognized for its use of hydropower.
In 2020, renewable sources incorporated into energy consumption at its fastest rate in two decades.
During 2006/2007, several renewable energy companies went through high profile initial public offerings (IPOs), resulting in market capitalization near or above $1 billion. These corporations included the solar PV companies First Solar (USA), Trina Solar (USA), Centrosolar (Germany), and Renesola (U.K.), wind power company Iberdrola (Spain), and U.S. biofuels producers VeraSun Energy, Aventine, and Pacific Ethanol.
Renewable energy industries expanded during most of 2008, with large increases in manufacturing capacity, diversification of manufacturing locations, and shifts in leadership. By August 2008, there were at least 160 publicly traded renewable energy companies with a market capitalization greater than $100 million. The number of companies in this category has expanded from around 60 in 2005. : 15
Some $150 billion was invested in renewable energy globally in 2009, including new capacity (asset finance and projects) and biofuels refineries. This is more than double the 2006 investment figure of $63 billion. Almost all of the increase was due to greater investment in wind power, solar PV, and biofuels.: 27
In 2000, venture capital (VC) investment in renewable energy was about 1% of total VC investment. In 2007 that figure was closer to 10%, with solar power alone making up about 3% of the entire Venture Capital asset class of ~$33B. More than 60 start-ups have been funded by
VCs in the last three years. Venture capital and private equity investments in renewable energy companies increased by 167 percent in 2006, according to investment analysts at New Energy Finance Limited.
New investment into the sector jumped US$148 billion in 2007, up 60 per cent over 2006, noted a report by the Sustainable Energy Finance Initiative (SEFI). Wind energy attracted one-third of the new capital and solar one-fifth. But interest in solar is growing rapidly on the back of major technological advances which saw solar investment increase 254 per cent. The IEA predicts US$20 trillion will be invested into alternative energy projects over the next 22 years. As of 2025, BloombergNEF reports that China was the leading country in investing in the transition to renewable energy. The report also finds that well-developed technologies, like solar and wind energies, are seeing a growth in investments, while newer renewable energy technologies are being invested in less.
In 2012, world leaders met in Rio for the United Nations Conference on Climate Change (UNFCCC), where they established 17 Sustainable Development Goals meant to unite leaders in mitigating challenges like climate change. Among these goals are: affordable and clean energy, responsible consumption and production, and climate action. 11 years later, the 2023 United Nations Climate Change Conference, also referred to as COP28 saw the first ever Global Stocktake, where countries are evaluated on their progress towards the sustainable goals they previously set. Also referred to as the UAE Consensus, this evaluation concluded that countries are not on target to reach their sustainable development goals. Subsequently, there has been a call to “tripling renewables and doubling energy efficiency by 2030.”
== Wind power ==
In 2020, wind power accounted for more than six percent of global electricity with 743 GW of global capacity. In the same year, 93 GW capacity was installed. For reach a 'net zero' emission status, the world needs to install at least 180 GW of new wind energy capacity by year.
=== Companies ===
Vestas was the largest wind turbine manufacturer in the world with and 16% market share in 2020. The company operates plants in Denmark, Germany, India, Italy, Britain, Spain, Sweden, Norway, Australia and China, and employs more than 20,000 people globally. After a sales slump in 2005, Vestas recovered and was voted Top Green Company of 2006.
In 2020, Siemens Gamesa was the world's second largest wind turbine manufacturer in 2020 thank to its position in the offshore sector of India. The company lead the offshore wind market.
Other major wind power companies include GE Power, Suzlon, Sinovel and Goldwind.
=== Wind potential ===
Africa's onshore wind energy potential is calculated of almost 180,000 Terawatt hours (TWh) per annum, which is able to satisfy the electricity demands of the continent 250 times over. In 2009, a technical study by the Wind Energy Technologies Office estimated that the onshore wind energy potential for the United States is 10,500 gigawatt (GW) capacity at 80 meters.
== Photovoltaics ==
=== Trends ===
Solar production has been increasing by an average of some 20 percent each year since 2002, making it the world’s fastest-growing energy technology. At the end of 2009, the cumulative global PV installations surpassed 21,000 megawatts.: 12
According to the China Greentech Report 2009, jointly issued by the PricewaterhouseCoopers and American Chamber of Commerce in Shanghai and released on 10 Sept in Dalian, China, the estimated size of China's green technology market could be between US$500 billion and US$1 trillion annually, or as much as 15 percent of China's forecasted GDP, in 2013. With the positive drivers from the Chinese government’s policies to develop green technology solution, China has already played a more important role in green technology market development. Following the announcements of the Chinese government in 2009 about the new subsidy scheme of “Golden Sun” to support solar industry development in China, some of the worldwide industry players have announced their development plans in this region, such as the agreement signed by LDK Solar regarding a solar project in Jiangsu province with a total capacity of 500MW, manufacturing facilities of polysilicon ingots and wafers, PV cells and PV modules to be built by Yingli Green Energy in Hainan Province, and the new thin film manufacturing plants of Tianwei Baoding and Anwell Technologies. In 2022, solar power market is expected to reach a value of $422 billion.
=== Companies ===
In 2017, main manufacturers of photovoltaics cells are based in Asia. Nine out of twelve major companies are based in China. The manufacturer Jinko Solar was the leader company in the sector, with 9.86% of the market share, followed by Trina Solar, JA Solar, Canadian Solar and Hanwha Q-Cells.
Tengger Desert Solar Park is the largest solar park in the world, with a capacity of 1,547MW. The park is located in Zhongwei, Ningxia, and it's called the Great Wall of Solar.
== Biofuels ==
Brazil continued its ethanol expansion plans which began in the 70's and now has the largest ethanol distribution and the largest fleet of cars run by any mix of ethanol and gasoline.: 19
In the ethanol fuel industry, the United States dominated, with 130 operating ethanol plants in 2007, and production capacity of 26 billion liters/year (6.87 billion gallons/year), a 60 percent increase over 2005. Another 84 plants were under construction or undergoing expansion, and this will result in a doubled production capacity.
The biodiesel industry opened many new production facilities during 2006/2007 and continued expansion plans in several countries. New biodiesel capacity appeared throughout Europe, including in
Belgium, Czech Republic, France, Germany, Italy, Poland, Portugal, Spain, Sweden, and the United Kingdom.: 19
Commercial investment in second-generation biofuels began in 2006/2007, and much of this investment went beyond pilot-scale plants. The world’s first commercial wood-to-ethanol plant began operation in Japan in 2007, with a capacity of 1.4 million liters/year. The first wood-to-ethanol plant in the United States is planned for 2008 with an initial output of 75 million liters/year.: 19
== Hydropower ==
see also: Hydropower
China and Brazil are home to the largest dams in the world. As the largest dam in the world, China’s Three Gorges Dam is the world’s leading source of hydropower. The Itaipu Dam, bordering Paraguay and Brazil, is the second largest dam. The Xiluodu and Belo Monte dams, located in China and Brazil, respectively, follow behind.
=== Companies ===
China Yangtze Power Co. is the world’s biggest hydroelectric power company. They oversee the Three Gorges Dam and the Xiluodu dam, and have plans to build two new hydroelectric power plants- the Baihetan Dam and the Wudongde Dam.
Centrais Eletricas Brasileiras in Brazil is the second largest company in the world for hydroelectric power, and is in charge of the Belo Monte dam located by the Xingu River.
== Employment ==
Renewable energy use tends to be more labor-intensive than fossil fuels, and so a transition toward renewables promises employment gains. In 2019, 11.5 million people work either directly in renewables or indirectly in supplier industries. The wind power industry employs some 1.7 million people, the photovoltaic sector accounts for an estimated 3.7 million jobs, and the solar thermal industry accounts for about 820,000. More than 3.58 million jobs are located in the biomass and biofuels sector.
== Challenges in the Renewable Energy Industry ==
While renewable energy sources such as solar, wind, and hydropower are crucial for reducing carbon emissions and building a sustainable energy development, their adoption is affect by factors such as high capital costs, infrastructural inadequacies, and market inefficiencies. Additionally, the transition to renewable energy has raised concerns regarding equitable access, particularly for underrepresent communities that lack the financial and infrastructural resources to benefit from cleaner energy sources.
Renewable energy has gained much attention over the past two decades due to technological developments, government support measures, and the need to fight the growing threat of climate change. Renewable energy sources include solar, wind, hydro, geothermal and biomass energy sources. These sources are an efficient substitute for fossil energy and a significant weapon in the fight against the world’s energy issues. Renewable energy sources hold a significant market share on the total power capacity indicating the high acceptance rate of the technology around the world.
The most important potential driver for renewable energy adoption is the need to address the issue of climate change. It helps reduce carbon emissions, mitigate climate change, and enhance environmental sustainability, which makes them crucial in serving the battle against climate change. Governments across the globe are setting ambitious policies regarding renewable energy targets, like the European Union’s target of 42.5% of energy from renewable sources by 2030. Likewise, the United States and China, which are the largest consumers of energy globally, have also invested massively in clean energy infrastructure, especially in capacities for solar and wind energy.
Renewable energy has been revolutionary in developing countries. Some developing countries use decentralized renewable energy systems like solar microgrids, and off-grid wind turbines in inaccessible places to generate power. For instance, India and Kenya have led large-scale installation of solar projects to replace costly and environmentally unfriendly diesel generators. The African Development Bank’s “Desert to Power” plan envisions the deployment of at least 10 GW of solar in the Sahel region to light up millions of homes.
However, the renewable energy sector has its challenges. Wind and solar power production are as unpredictable as the weather and this means there must be ways of storing energy, perhaps through batteries or hydrogen fuel cells. However, the integration of renewable energy in existing power systems calls for massive investment in both infrastructure and smart grid systems. Challenges are mainly related to financing as many countries, especially the developing ones, cannot afford the start-up costs.
Some of those challenges are, however, being overcome by the developments in technology in the recent past. The advancement in perovskite solar cells has increased the efficiency and affordability of the panels. Large blade lengths and the use of floating structures are increasing the wind energy capabilities. At the same time, renewable energy sources like wind, photovoltaic solar energy, and so on have become more reliable due to energy storage options like large batteries, and green hydrogen.
== Technological and Infrastructure Challenges ==
One of the key challenges in the renewable energy sector is the technological and infrastructural limitations associated with integrating renewable sources into existing energy grids. Many developing countries, particularly in South and Southeast Asia, struggle with outdated power grids that are not equipped to handle variable energy sources such as wind and solar power. The lack of efficient storage technologies further raise this concern, as energy generated from renewable sources must be consumed in real-time or stored in costly battery systems.
== Financial and Economic Barriers ==
The high initial capital costs associated with renewable energy infrastructure pose a significant barrier to its widespread adoption. Unlike fossil fuels, which benefit from well-established systems, renewable energy projects often require substantial upfront investments. In ASEAN countries, for example, financial constraints have limited the adaptation of large-scale renewable energy initiatives. Green bonds and other financial instruments have been proposed as solutions to bridge this gap, but their impact remains limited due to underdeveloped capital markets in many regions.A key example is Indonesia's green sukuk bond, which raised $1.25 billion to support renewable energy projects, particularly in underdeveloped regions.
== Policy and Regulatory Challenges ==
Inconsistent policies and regulatory frameworks present another significant challenge to the expansion of renewable energy. Many countries lack a unified approach to renewable energy investment, leading to fragmented and uncoordinated efforts. For instance, while ASEAN nations have set ambitious renewable energy targets, the implementation of these policies varies widely, with some governments prioritizing fossil fuels due to economic and political considerations. The absence of centralized monitoring systems further complicates policy enforcement and evaluation.
== Equity and Access to Renewable Energy ==
The transition to renewable energy has also highlighted disparities in access to clean energy, particularly among underprivileged communities. Rural populations in many developing countries still rely on traditional energy sources such as firewood, which are inefficient and pose health risks. Large-scale hydropower projects, while providing renewable energy, have led to environmental degradation in countries like Cambodia and Laos. Decentralized renewable solutions, such as solar mini-grids, have been proposed as a means to address energy poverty, but their implementation remains uneven due to financial constraints.
== See also ==
== References ==
== Bibliography ==
"WEO 2017 : Key Findings". International Energy Agency. 2017-11-14. Retrieved 2018-01-05.
"Summary and Conclusions". World Energy Outlook 2006 (PDF). International Energy Agency/OECD. 2006. pp. 27–47.
Renewables in global energy supply: An IEA facts sheet (PDF). International Energy Agency/OECD. 2007.
Renewables 2007 Global Status Report (PDF). REN21. 2008.
Renewables Global Status Report: 2009 Update (PDF). REN21. 2009.
Renewables Global Status Report: 2010 (PDF). REN21. 2010.
Renewables Global Status Report: 2017. REN21. 2017.
Changing climates: The Role of Renewable Energy in a Carbon-constrained World (PDF). United Nations Environment Programme. 2006.
Global Trends in Sustainable Energy Investment 2007: Analysis of Trends and Issues in the Financing of Renewable Energy and Energy Efficiency in OECD and Developing Countries. United Nations Environment Programme and New Energy Finance Ltd. 2007. ISBN 978-92-807-2859-0.
American energy: The renewable path to energy security. Worldwatch Institute and Center for American Progress. 2006.
== External links ==
"The Sunny Side of the Street: Investing in Solar". SocialFunds.com. 2008-06-23. Retrieved 2018-01-05.
"Solar continues to shine despite the dark future". www.businessgreen.com. 2008-10-10. Retrieved 2018-01-05.
"Optimism Abounds Throughout Renewable Energy Industry". Renewable Energy World. 16 March 2009. Retrieved 2018-01-05.
Reuters Editorial (2009-03-16). "GE Energy turbine sales to grow fivefold in Europe". U.S. Retrieved 2018-01-05. {{cite web}}: |author= has generic name (help)
"Renewable Energy Database". Renewable Energy Database. 2012-02-08. Retrieved 2018-01-05.
"Renewable Energy". Financial Times. 2017-09-07. Retrieved 2018-01-05.
"2018 Renewable Energy Industry Outlook". Deloitte United States. 2017-12-19. Retrieved 2018-01-05.
"Four Renewable Energy Trends to Follow in 2018". Renewable Energy World. 27 December 2017. Retrieved 2018-01-05.
This article incorporates public domain material from the United States government | Wikipedia/Renewable_energy_industry |
Corporate sustainability is an approach aiming to create long-term stakeholder value through the implementation of a business strategy that focuses on the ethical, social, environmental, cultural, and economic dimensions of doing business. The strategies created are intended to foster longevity, transparency, and proper employee development within business organizations. Firms will often express their commitment to corporate sustainability through a statement of Corporate Sustainability Standards (CSS), which are usually policies and measures that aim to meet, or exceed, minimum regulatory requirements.
Corporate sustainability is often confused with corporate social responsibility (CSR), though the two are not the same. Bansal and DesJardine (2014) state that the notion of 'time' discriminates sustainability from CSR and other similar concepts. Whereas ethics, morality, and norms permeate CSR, sustainability only obliges businesses to make intertemporal trade-offs to safeguard intergenerational equity. Short-termism is the bane of sustainability.
== Origin ==
The phrase is derived from the concept of "sustainable development" and Elkington's (1997) "triple bottom line." The Brundtland Commission's Report, Our Common Future, defined sustainable development as "development that meets the needs of the present without compromising the ability of future generations to meet their own needs. It contains within it two key concepts:
"the concept of 'needs', in particular the essential needs of the world's poor, to which overriding priority should be given; and
"the idea of limitations imposed by the state of technology and social organization on the environment's ability to meet present and future needs."
The idea of meeting present economic needs without reducing the ability of future generations to meet their own economic needs became a popular approach in the business world's implementation of sustainable development, referred to as "corporate sustainable development."
"Triple bottom line" proposes that business goals were inseparable from the societies and environments within which they operate. While short-term economic gains could be pursued, failure to account the social and environmental impacts of these pursuits is believed to make those business practices unsustainable.
Therefore, in the literature, corporate sustainability is often referred to as a three-dimensional construct integrating social, environmental, and economic factors. Of these three dimensions, the social dimension is the least represented in current research, and further work on conceptualization is needed. Research primarily approaches the topic from a country and/or industry perspective.
Whether corporate sustainability can be measured remains contested. There are composite measures that include measures of environmental, social, corporate governance, and economic performance, such as the Complex Performance Indicator (CPI). And there are many different definitions of sustainability applied to and used by companies. It remains difficult to say whether a company or other actor is operating sustainably or not because "there is no generally accepted set of indicators that could clearly delineate a status of sustainability from one of unsustainability. Therefore, the global status of sustainability, as well as the exact status of different actors, such as countries, companies, or individuals, is almost impossible to measure."
== Scope ==
The most broadly accepted criterion for corporate sustainability constitutes a firm's efficient use of natural capital. Natural capital not only includes the responsible consumption of renewable and non-renewable resources, but also the preservation of vital ecosystem services such as climate regulation and water purification, which have no viable substitutes. Without balancing industrial inputs and outputs with nature’s regenerative capacities, firms risk contributing to long-term ecological unsustainability.This eco-efficiency is usually calculated as the economic value added by a firm in relation to its aggregated ecological impact.
Similar to the eco-efficiency concept but so far less explored is the second criterion for corporate sustainability. Socio-efficiency describes the relation between a firm's value added and its social impact. Whereas, it can be assumed that most corporate impacts on the environment are negative (apart from rare exceptions such as the planting of trees) this is not true for social impacts. These can be either positive (e.g. corporate giving, creation of employment) or negative (e.g. work accidents, human rights abuses).
Both eco-efficiency and socio-efficiency are concerned primarily with increasing economic sustainability. In this process they instrumentalise both natural and social capital aiming to benefit from win-win situations. Some point towards eco-effectiveness, socio-effectiveness, sufficiency, and eco-equity as four criteria that need to be met if sustainable development is to be reached.
Theorists agree that respect for issues other than economics is an important matter. The Business Case for Sustainability (BCS) has had many different approaches for ways to approve or disapprove the economic rationale for corporate sustainability management.
== Principles for corporate sustainability ==
Transparency
proposes that by having an engaging environment within a company and within the community it operates will improve performance and increase profits. This can be attained through open communications with stakeholders characterized by high levels of information disclosure, clarity, and accuracy.
Stakeholder engagement
is attained when a company educates its employees and outside stakeholders (customers, suppliers, and the entire community) and move them to act on matters such as waste reduction or energy efficiency.
Thinking ahead
Envisioning the future enables companies to generate fresh ideas for implementation. These ideas can either reduce productions costs, increase profits, or provide a better image for the organization.
Diversity, Equity and Inclusion
A a 2012 study by the University of California Berkeley's Haas School of Business found that companies with a high number of female board members were more likely to reduce their environmental impact and improve energy efficiency.
== Pillars of corporate sustainability ==
As previously mentioned, corporate sustainability can be seen through the lens of guiding principles: transparency, stakeholder engagement, thinking ahead, and DEI. However, another way to categorize these core principles is to separate between three key aspects of maintaining a sustainable business:
Environmental
Social Responsibility
Economic and Regulatory
While corporate sustainability is typically associated with solely environmental concerns, such as a carbon footprint reduction or a transition to clean energy sources, it is equally important that companies maintain these other two components. Specifically, social responsibility entails a company's treatment of employees, consideration for consumers, and broader community impacts, and economic sustainability encompasses a corporation's compliance to government regulations, transparency in accounting and reporting, and overall integrity with respect to larger policies and the markets they impact. Considering this separation, "transparency" would fall under economic and governmental, "stakeholder engagement" and "diversity, equity, and inclusion" would fall under social responsibility, and "thinking ahead" would be a general approach conducive to the execution and balancing of all three aspects.
== See also ==
Sustainable business
Sustainable finance
Project finance
EthicalQuote (CEQ)
Environmental, social and corporate governance
Index of sustainability articles
== References ==
== Sources ==
Atkinson, Giles (March 2000). "Measuring Corporate Sustainability". Journal of Environmental Planning and Management. 43 (2): 235–252. Bibcode:2000JEPM...43..235A. doi:10.1080/09640560010694. S2CID 154282221.
Dyllick, Thomas; Hockerts, Kai (March 2002). "Beyond the business case for corporate sustainability". Business Strategy and the Environment. 11 (2): 130–141. doi:10.1002/bse.323.
van Marrewijk, Marcel (1 May 2003). "Concepts and Definitions of CSR and Corporate Sustainability: Between Agency and Communion". Journal of Business Ethics. 44 (2): 95–105. doi:10.1023/A:1023331212247. S2CID 189900614.
Dunphy, Dexter Colboyd; Griffiths, Andrew; Benn, Suzanne (2003). Organizational Change for Corporate Sustainability: A Guide for Leaders and Change Agents of the Future. Psychology Press. ISBN 978-0-415-28740-1.
Werbach, Adam. Strategy for Sustainability: a Business Manifesto. Boston, Mass.: Harvard Business, 2009. Print.
"Strengthen Your Business by Developing Your Employees", by Leslie Levine—Business Resources, Advice and Forms for Large and Small Businesses, 2010
"Walmart Sustainability Report 2010 – Environment – Waste." Walmartstores.com. Web. 1 July 2010.
"Handbook on Corporate Social Responsibility in India", Sachin Shukla et al – PwC India, 2013.
== External links ==
International Society of Sustainability Professionals – Non-profit association supporting sustainability professionals | Wikipedia/Corporate_sustainability |
Unexploded ordnance (UXO, sometimes abbreviated as UO) and unexploded bombs (UXBs) are explosive weapons (bombs, shells, grenades, land mines, naval mines, cluster munition, and other munitions) that did not explode when they were deployed and remain at risk for detonation, sometimes many decades after they were used or discarded. When unwanted munitions are found, they are sometimes destroyed in controlled explosions, but accidental detonation of even very old explosives might also occur, sometimes with fatal consequences.
For example, UXO from World War I continues to be a hazard, with poisonous gas filled munitions still a problem. UXO does not always originate from conflict; areas such as military training bases can also hold significant numbers, even after the area has been abandoned.
Seventy-eight countries are contaminated by land mines, which kill or maim 15,000–20,000 people every year. Approximately 80% of casualties are civilian, with children being the most affected age group. An average estimate of 50% of deaths occur within hours of the blast. In recent years, mines have been used increasingly as weapons of terror; especially against local populations, such as in the Syrian civil war.
In addition to the obvious danger of explosion, buried UXO can cause environmental contamination. In some heavily used military training areas, munitions-related chemicals such as explosives and perchlorate (a component of pyrotechnics and rocket fuel) may enter soil and groundwater, thereby contaminating the water supply, likewise with preventing agrarian uses such as farming and food distribution.
== Risks and problems ==
Unexploded ordnance, no matter how old, may explode. It might seem that the dangers of UXO decrease over time, but this is not always the case. Corrosion and damages sustained on impact pose significant difficulties to defuse UXO safely and also make the consequences of defusion harder to predict. Mixed explosive agents might separate or migrate over time and leave contact explosives like nitroglycerine at random places in the shell. Sometimes components of the original explosives, in the presence of moisture, can form new explosive compounds with the metals in the shells like picrates that can leave a shell highly explosive, even when it is defused and the detonator destroyed or removed.
Even if unexploded ordnance does not explode, environmental pollutants are released as it degrades. The toxic compounds and heavy metals can contaminate water and soil over time. Recovery, particularly of deeply-buried projectiles, is difficult and hazardous—jarring may detonate the charge. Once uncovered, explosives can often be transported safely to a site where they can be destroyed; if this is not possible, they must be detonated on site which might require evacuation of the surrounding area.
Unexploded ordnance dating as far back as the mid-19th century still poses a hazard worldwide, both in current and former combat areas and at military firing ranges. A major problem with unexploded ordnance is that over the years, the detonator and main charge deteriorate to such an extent that they frequently become more sensitive to disturbance and therefore more dangerous to handle. Construction works may disturb unsuspected unexploded bombs, which may then explode. Forest fires may be aggravated if buried ordnance explodes. Heat waves, causing the water level to drop severely, may increase the danger of immersed ordnance. There are countless examples of people tampering with unexploded ordnance that is many years old, often with fatal results. For this reason, it is universally recommended that unexploded ordnance should not be touched or handled by unqualified persons. Instead, the location should be reported to the local police so that bomb disposal or Explosive Ordnance Disposal (EOD) professionals can render it safe.
Although professional EOD personnel have expert knowledge, skills and equipment, they are not immune to misfortune because of the inherent dangers: in June 2010, construction workers in Göttingen, Germany discovered an Allied 500-kilogram (1,100 lb) bomb dating from World War II buried approximately 7 metres (23 ft) below the ground. German EOD experts were notified and attended the scene. Whilst residents living nearby were being evacuated and the EOD personnel were preparing to disarm the bomb, it detonated, killing three of them and severely injuring six others. The dead and injured each had over 20 years of hands-on experience, and had previously rendered safe between 600 and 700 unexploded bombs. The bomb which killed and injured the EOD personnel was of a particularly dangerous type because it was fitted with a delayed-action chemical fuze (with an integral anti-handling device) which had not operated as designed, but had become highly unstable after over 65 years underground. The type of delayed-action fuze in the Göttingen bomb was commonly used: a glass vial containing acetone was smashed after the bomb was released; the acetone was intended, as it dripped downwards, to disintegrate celluloid discs holding back a spring-loaded trigger that would strike a detonator when the discs degraded sufficiently after some minutes or hours. These bombs, when striking soft earth at an angle, often end their trajectory not pointing downwards, so that the acetone did not drip onto and weaken the celluloid; but over many years the discs degraded until the trigger was released and the bomb detonated spontaneously, or when weakened by being jarred.
In November 2013, four US Marines were killed by an explosion whilst clearing unexploded ordnance from a firing range at Camp Pendleton. The exact cause is not known, although the Marines had been handing grenades they were collecting to each other, a practice permitted but discouraged. It is thought that a grenade may have exploded after being kicked or bumped, setting off hundreds of other grenades and shells.
A dramatic example of munitions and explosives of concern (MEC) threat is the wreck of the SS Richard Montgomery, which was sunk in shallow water about 2.4 kilometres (1.5 miles) from the town of Sheerness and 8.0 kilometres (5 miles) from Southend. The wreckage still contains 1,400 tons of explosives. In comparison with the World War II wreck of the SS Kielce which rests at a higher depth, with a smaller load of explosives, it still exploded after a salvaging operation in 1967 and produced a tremor measuring 4.5 on the Richter scale.
== Around the world ==
=== Africa ===
==== Effects of the North African campaign of World War II ====
During the fighting in North Africa between the Axis and Allied forces, much of North Africa was heavily mined to prevent military advances. During the conflict, in addition to the millions of mines that were placed, some of the millions of shells which were fired did not explode, and remain deadly to this day. Algeria, Egypt, Libya and Tunisia are all affected by this issue, with civilians being injured and killed every year. UXO also slows progress, with areas having to be demined before being developed.
==== Algeria ====
Algeria has been contaminated with large numbers of mines and UXO throughout several wars, starting from World War II. During the Algerian war for independence, French forces laid up to 10 million mines on the Morice and Challe lines, on the eastern and western sides of the country. In 2007, France officially handed over maps to Algerian authorities showing the locations of minefields. The lack of these maps had previously severely hampered Algerian demining efforts.
Further mines were laid in the Algerian civil war by both warring parties, requiring further demining efforts. However, these mining operations were not on nearly as large a scale as French operations.
By July 2016, Algeria reported that it had cleared all major minefields it had identified to clear. Thereafter, Algeria called on French authorities to provide compensation to the families of the 4000 people who are estimated to have been killed by mines, and thousands who have been left disabled from French ordnance.
==== Chad ====
Chad has been dealing with contamination issues stemming from its numerous conflicts between the 1960s and the 1980s. A significant portion of this contamination comes from the presence of anti-personnel mines, many of which are believed to have originated from Libyan sources during that period. As of 2020, estimates provided by the Mine Action Review indicated that approximately 10 square kilometers (or 3.9 square miles) of Chadian territory remained contaminated with these dangerous antipersonnel mines. Additionally, a smaller portion of unexploded ordnance (UXO) related to cluster munitions continues to affect some regions in the northern part of the country.
In recent years, the ongoing jihadist insurgency led by Boko Haram has further complicated the situation. According to the Chadian government, Boko Haram and similar insurgent groups are likely responsible for laying additional mines. These groups are also known for scavenging explosives from pre-existing UXO in order to manufacture improvised explosive devices (IEDs), making the clearance of these remnants of war even more critical for national security. Effective mine clearance and UXO removal are essential not only to reduce the threat of accidental detonation, but also to limit the availability of materials that insurgents might use for their attacks.
==== Egypt ====
Egypt is the most heavily mined country in the world (by number) with as many as 22.7 million mines as of 2024. It is estimated that 22% of Egypt's territory is mined. These mines are from both World War II and wars that Egypt has fought with Israel. Mines contaminate large amounts of agricultural land, slowing development efforts. De-mining is a priority in the country to open up more land for agriculture purposes, oil drilling and mining. Nevertheless, Egypt stresses its need to deploy mines in order to protect its borders.
==== Ethiopia ====
Ethiopia was heavily mined in World War II, the Eritrean War of Independence, the Eritrean-Ethiopian War, and the Tigray War. The most heavily affected regions are Afar, Somali, and Tigray regions which have seen repeated conflict. A study in 2004 found that landmines and UXO affected an estimated 1.5 million people. Between 2000 and 2004, they caused 588 fatalities and 1,300 injuries.
==== Libya ====
Libya was first contaminated with UXO in World War II, in areas such as Tobruk, where heavy fighting took place. The contamination from World War II is largely unexploded ordnance and anti vehicle mines.
Libya was contaminated during its wars with Egypt and Chad, and it is also believed that the border with Tunisia is contaminated. While Muammar Gaddafi was in power in Libya, mines were placed around military facilities and other key infrastructure.
In the first Libyan civil war that began in 2011, both government and opposition forces used mines. According to the Libyan mine action centre, 30–35,000 mines were laid; however, these were largely cleared after the downfall of the Gaddafi regime by ex-soldiers. With the downfall of the Gaddafi regime, in March 2011 large ammunition depots were left unatetended, and easily accessible by the civilian population, as well as soldiers and paramilitary forces. The government did not regain control of these depots, and munitions from the same depots were spread across the country. Several of the stores also exploded, spreading dangerous ordnance over a wide area. Many military vehicles were also destroyed in fighting all across the country, and these often contained ordnance in an unstable condition.
With hostilities breaking out again in 2014, there were reports of both landmines and IEDs being laid by opposition groups, particularly in urban areas. This complicated clearance operations as these areas are often densely populated.
In 2019, clashes between the Libyan National Army (LNA) and government forces around Tripoli escalated, with the LNA surrounding Tripoli in January 2020 and launching constant rocket and artillery attacks. Both sides were also reported to be using weapons indiscriminately against international law and endangering civilian lives. Weapons such as drones from Turkey and China were used, violating the UN arms embargo placed on Libya. When the LNA forces withdrew from the east of Tripoli in June 2021, they left behind an unspecified amount of IEDs. It was reported by the UN mine action service that booby traps were left in civilian homes with their only purpose being to cause civilian casualties. In January 2020, the UN estimated that Libya was contaminated by up to 20 million mines and pieces of UXO.
The Russian paramilitary organisation Wagner which was operating in the area, also reportedly left munitions and mines in southern Tripoli. Human Rights Watch said that the Wagner Group and other militias left behind "enormous" amounts of ordnance. In August 2021, the BBC reported receiving an electronic tablet containing information on Wagner operators' role in laying mines. Deminers in Tripoli reported finding documents in Russian in rooms that they were demining. On 24 May 2022, the Human Rights Watch wrote to the Russian foreign minister, asking to review their findings connecting with the Wagner group operations in laying mines in Tripoli, and clarify on the group's role in the conflict. The Russian authorities did not respond.
==== Mali ====
Major contamination of Mali with UXO stems from the resurgence of conflict in 2012 Mali. Mines and IEDs were laid more heavily in the north of the country. The situation deteriorated in 2019; however, the extent of the contamination is unknown, as there has been no clear mapping of the country's minefields.
==== Mauritania ====
Mine and UXO contamination stems from Mauritania's 1976–1978 war in the Western Sahara, while fighting against the Polisario front over the region. UXO is largely concentrated in the north of the country, around urban centres, where heavy fighting took place.
Following the urbanisation of 70% of the country's nomadic population, urban expansion has strayed into mine belts. As many of these nomads still follow pastoral practises, valuable livestock and people can stray into contact with mines. Despite this, people are unwilling to move due to the fact that Northern Mauritania is known as the best place to raise camels. It is also difficult to precisely mark mines, due to the fact that dunes can rapidly change their location.
Although the country was declared mine free in 2018, Mauritania reported the discovery of previously unknown mined areas. As of 2023, an estimated 11.52 square kilometres (4.45 sq mi) of Mauritania was contaminated with mines.
==== Morocco ====
The contamination of Moroccan territory is a consequence of the conflict between the Royal Moroccan Army and the Polisario Front over the Western Sahara. The majority of the contamination is confined to the area around the Moroccan Western Sahara wall. All along the length of the wall (on the Eastern side) runs a minefield, sometimes claimed to be the world's longest continual minefield. During the 1975–1991 conflict, the Moroccan army used cluster munitions, and unexploded bomblets still kill and maim uneducated citizens to this day.
Prior to the resumption of hostilities in November 2020, both the UN and the Moroccan army claimed to have destroyed tens of thousands of land mines, and cleared hundreds of square kilometres of land.
==== Niger ====
In 2018 Niger reported a known contaminated area near Madama military base, totalling just over 0.2 square kilometres (0.077 sq mi). Clearance of approximately 18,000 square metres (190,000 sq ft) took place up to March 2020, however no clearance is thought to have taken place since then. In 2023, Niger reported that there were just under 0.2 km2 of contaminated areas near the Madama military base.
The spread of conflicts in the Lake Chad and Liptako-Gourma regions has contributed new UXO to the regions, with some insurgencies spreading to Niger. IEDs have seen increased use, some victim activated and some indiscriminate. Many of the mines used by insurgencies such as Boko Haram are used to target military convoys and vehicles, however inevitably there are civilian casualties. Between 2016 and the end of 2022, the National Commission for the Collection and Control of Illicit weapons reported 183 explosive ordnance incidents, killing 203 and wounding 204. 80% of the incidents occurred in the Tillabéri and Diffa regions.
==== Sudan ====
Sudan's mine contamination largely stems from its civil war and other wars since the country's independence from Britain. In 2005, a peace agreement between the rebel forces (mainly the Sudan People's Liberation Movement) and the government brought an end to fighting, and along with it mine laying. In 2009, the UN Mine Action Service (UNMAS) reported that across 16 Sudanese states, contamination totalled 107 square kilometres (41 sq mi). Despite conflict breaking out in 2011, by early 2023 it was reported that only just over 13 km2 (5.0 sq mi) of Sudanese land was contaminated with mines, and slightly more contaminated with UXO.
In April 2023, heavy fighting broke out between the Sudanese Armed Forces (SAF), and the Rapid Support Forces, (RSF), a paramilitary organisation. The SAF alleges that the RSF has laid mines, but as of April 2024 no evidence has emerged to support that claim.
=== Americas ===
==== Canada ====
After World War II, much unused ordnance in Canada was dumped along the country's eastern and western coasts at sites selected by the Canadian military. Other UXO in Canada is found on sites used by the Canadian military for operations, training and weapons tests. These sites are labeled under the "legacy sites" program created in 2005 to identify areas and quantify risk due to UXO. As of 2019, the Department of National Defence has confirmed 62 locations as legacy sites, with a further 774 sites in assessment. There has been controversy because some lands appropriated by the military during World War II were owned by First Nations, such as 8 square kilometres (2,000 acres) that make up Camp Ipperwash in Ontario, which was given with the understanding that the land would be given back at the end of the war. These lands have required and still need extensive clean-up efforts due to the possible presence of UXO.
==== Colombia ====
During the long Colombian conflict that began around 1964, a large number of landmines were deployed in rural areas across Colombia. The landmines are homemade and were placed primarily during the last 25 years of the conflict, hindering rural development significantly. The rebel groups of Revolutionary Armed Forces of Colombia (FARC) and the smaller ELN are usually blamed for having placed the mines. All departments of Colombia are affected, but Antioquia, where the city of Medellín is located, holds the largest amounts. After Afghanistan, Colombia has the second-highest number of landmine casualties, with more than 11,500 people killed or injured by landmines since 1990, according to Colombian government figures.
In September 2012, the Colombian peace process began officially in Havana and in August 2016, the US and Norway initiated an international five-year demining program, now supported by another 24 countries and the European Union. Both the Colombian military and FARC are taking part in the demining efforts. The program intends to rid Colombia of landmines and other UXO by 2021 and it has been funded with nearly US$112 million, including US$33 million from the US (as part of the larger US foreign policy Plan Colombia) and US$20 million from Norway. Experts however, have estimated that it will take at least a decade due to the difficult terrain.
==== United States ====
Unlike many countries in Europe and Asia, the United States has not been subjected to significant aerial bombardment. Nevertheless, according to the Department of Defense, "millions of acres" of US territory may contain UXO, discarded military munitions (DMM) and munitions constituents (e.g., explosive compounds).
According to United States Environmental Protection Agency documents released in late 2002, UXO at 16,000 domestic inactive military ranges within the United States pose an "imminent and substantial" public health risk and could require the largest environmental cleanup ever, at a cost of at least US$14 billion. Some individual ranges cover 1,300 square kilometres (500 sq mi), and, taken together, the ranges comprise an area the size of Florida.
On Joint Base Cape Cod (JBCC) on Cape Cod, Massachusetts, decades of artillery training have contaminated the only drinking water for thousands of surrounding residents. A costly UXO recovery effort is under way.
UXO on US military bases has caused problems for transferring and restoring Base Realignment and Closure (BRAC) land. The Environmental Protection Agency's efforts to commercialize former munitions testing grounds are complicated by UXO, making investments and development risky.
The area around Fort St. Philip, Louisiana is also covered in UXO from the naval bombardment, and caution would be taken when visiting the ruins.
UXO cleanup in the US involves over 40,000 square kilometres (10 million acres) of land and 1,400 different sites. Estimated cleanup costs are tens of billions of dollars. It costs roughly $1,000 to demolish a UXO on site. Other costs include surveying and mapping, removing vegetation from the site, transportation, and personnel to manually detect UXOs with metal detectors. Searching for UXOs is tedious work and often 100 holes are dug to every 1 UXO found. Other methods of finding UXOs include digital geophysics detection with land and airborne systems.
===== Examples =====
In December 2007, UXO was discovered in new development areas outside Orlando, Florida, and construction had to be halted.
In 1917, in response to other nations' extensive use of chemical weapons in World War I, the US Army Chemical Warfare Service (CWS) opened a weapons research laboratory and production facility at American University in Washington, D.C. CWS troops at the station routinely fired incendiary and chemical projectiles into a nearby undeveloped area that became known as "No Man's Land". When the station was deactivated after the war in 1919, UXO in No Man's Land was abandoned there, and unused projectiles and toxic chemicals were buried in deep, poorly mapped pits. Collegiate athletic fields, businesses and homes were subsequently built in the area. Chemical UXO continues to be periodically found on and near campus, and in 2001, the USACE began cleanup efforts after arsenic was found in soil at the athletic fields. In 2017, the USACE was cautiously excavating a university-owned property in an adjacent neighborhood where investigators believed that a large unmapped cache of mustard gas projectiles was buried.
Although comparatively rare, unexploded ordnance from the American Civil War is still occasionally found and is still deadly over 150 years later. Union and Confederate troops fired an estimated 1.5 million artillery shells at each other from 1861 to 1865. As many as one in five did not explode. In 1973, during the restoration of Weston Manor, an 18th-century plantation house in Hopewell, Virginia, that was shelled by Union gunboats during the Civil War, a live shell was found embedded in the dining room ceiling. The ball was disarmed and is shown to visitors to the plantation. In late March 2008, a 20-kilogram (44 lb), 20-centimetre (8 in) mortar shell was uncovered at the Petersburg National Battlefield, the site of a 292-day siege. The shell was taken to the city landfill where it was safely detonated by ordnance disposal experts. Also in 2008, a Civil War enthusiast was killed in the explosion of a 23-centimetre (9 in), 34-kilogram (75 lb) naval shell he was attempting to disarm in the driveway of his home near Richmond, Virginia. The explosion sent a chunk of shrapnel crashing into a house four hundred metres (1⁄4 mi) away.
According to Alaska State Troopers, an unexploded aerial bomb, found at a home off Warner Road, was safely detonated by Fort Wainwright soldiers on September 19, 2019.
=== Asia ===
==== Japan ====
Thousands of tons of UXO remains buried across Japan, particularly in Okinawa, where over 200,000 tons of ordnance were dropped during the final year of World War II. From 1945 until the end of the U.S. occupation of the island in 1972, the Japan Self-Defense Forces (JSDF) and the US military disposed of 5,500 tons of UXO. Over 30,000 UXO disposal operations have been conducted on Okinawa by the JSDF since 1972, and it is estimated it could take close to a century to dispose of the remaining unexploded munitions on the islands. No injuries or deaths have been reported as a result of UXO disposal, however. Tokyo and other major cities, including Kobe, Yokohama and Fukuoka, were targeted by several massive air raids during World War II, which left behind large amounts of UXO. Shells from Imperial Army and Imperial Navy guns also continue to be discovered.
On 29 October 2012, an unexploded 250-kilogram (550 lb) US bomb with a functioning detonator was discovered near a runway at Sendai Airport during reconstruction following the 2011 Tōhoku earthquake and tsunami, resulting in the airport being closed and all flights cancelled. The airport reopened the next day after the bomb was safely contained, but closed again on 14 November while the bomb was defused and safely removed.
In March 2013, an unexploded Imperial Army anti-aircraft shell measuring 40 centimetres (16 in) long was discovered at a construction site in Tokyo's Kita Ward, close to the Kaminakazato Station on the JR Keihin Tohoku Line. The shell was detonated in place by a JGSDF UXO disposal squad in June, causing 150 scheduled rail and Shinkansen services to be halted for three hours and affecting 90,000 commuters. In July, an unexploded 1,000-kilogram (2,200 lb) US bomb from an air raid was discovered near the Akabane Station in the Kita Ward and defused on site by the JGSDF in November, resulting in the evacuation of 3,000 households nearby and causing several trains to be halted for an hour while the bomb was being defused.
On 13 April 2014, the JGSDF defused and removed an unexploded 250-kilogram (550 lb) US oil incendiary bomb discovered at a construction site in Kurume, Fukuoka Prefecture, which required the evacuation of 740 people living nearby.
On 16 March 2015, a 2,000-pound (910 kg) bomb was found in central Osaka.
In December 2019, 100 buildings were evacuated to remove a 500-pound (230 kg) World War II bomb found on Okinawa's Camp Kinser.
On 2 October 2024, more than 80 flights were cancelled at Miyazaki Airport after a previously unknown 500-pound (230 kg) World War II bomb detonated under a taxiway, leaving a substantial crater. No aircraft were nearby and no injuries were reported. Officials launched an investigation into what caused the bomb to suddenly explode.
==== Indian Administered Kashmir ====
Tosa Maidan, a scenic meadow in the Budgam district of Indian-administered Kashmir, was used as a military firing range by the Indian Army and Indian Air Force from 1964 to 2014. Decades of artillery exercises left the area littered with UXO, resulting in civilian casualties. Official records attribute at least 63 deaths and over 150 injuries to UXO explosions, though local reports suggest higher figures. In 2014, after significant public protests, the government declined to renew the military’s lease. The Indian Army subsequently initiated "Operation Falah" to clear the area of unexploded ordnance. Despite these efforts, sporadic explosions continue to pose risks, leading to ongoing demands for thorough demining and compensation for affected families.
==== South Asia ====
===== Afghanistan =====
According to The Guardian, since 2001, the coalition forces dropped about 20,000 tonnes of ammunition over Afghanistan with an estimated 10% of munitions not detonated according to some experts. Many valleys, fields and dry riverbeds in Macca have been used by foreign soldiers as firing ranges, leaving them peppered with undetonated ammunition. Despite the removal of 16.5 million items since mine-clearing programmes were established in 1989 after the Soviet withdrawal, Macca and its predecessors have recorded 22,000 casualties in the same period.
===== Sri Lanka =====
According to The HALO Trust, following the Sri Lankan Civil War in 2009, over 1,600,000 landmines were left in the country. Since 2009 over 270,000 landmines have been safely destroyed and 280,000 people have been able to return to their homes. Following the signing of the Ottawa Treaty, Sri Lanka has committed to clearing all known landmines by 2028.
==== Southeast Asia ====
===== Cambodia =====
===== Laos =====
Laos is considered the world's most heavily bombed nation per capita. During the period of the Laotian Civil War and Vietnam War, over half a million American bombing missions dropped more than 2 million tons of ordnance on Laos, most of it anti-personnel cluster bombs. Each cluster bomb shell contained hundreds of individual bomblets, "bombies", about the size of a tennis ball. An estimated 30% of these munitions did not detonate. Some 288 million cluster munitions and about 75 million unexploded bombs were left across Laos after the war ended. Estimates are that present rate of defining will require nearly 100 years to clear. Around 30% of Laos is considered heavily contaminated with UXOs and ten of the eighteen Laotian provinces have been described as "severely contaminated" with artillery and mortar shells, mines, rockets, grenades, and other devices from various countries of origin. These munitions pose a continuing obstacle to agriculture and a special threat to children, who are attracted by the toylike devices.
From 1996 to 2009, more than 1 million items of UXO were destroyed, freeing up 23,000 hectares of land. Between 1999 and 2008, there were 2,184 casualties (including 834 deaths) from UXO incidents. Since the end of the conflict in 1975, unexploded ordnance, mostly from US bombing, has killed or injured over 25,000 people, half of them being children.
UXOs continue to be a contentious issue as it has impeded infrastructure development and railway construction within the nation, including the Boten–Vientiane railway which required clearing thousands of hectares for UXO and shrapnel.
===== Vietnam =====
In Vietnam, 800,000 tons of landmines and unexploded ordnance is buried in the land and mountains. From 1975 to 2015, up to 100,000 people have been injured or killed by bombs left over from the second Indochina war. Nearly one-fifth of the land is contaminated by UXOs.
At present, all 63 provinces and cities are contaminated with UXO and landmines. However, it is possible to prioritize demining for the Northern border provinces of Lang Son, Ha Giang and the six Central provinces of Nghe An, Ha Tinh, Quang Binh, Quang Tri, Thua Thien and Quang Ngai. Particularly in these 6 central provinces, up to 2010, there were 22,760 victims of landmines and UXO, of which 10,529 died and 12,231 were injured.
One of the most heavily contaminated province, Quảng Trị, has seen at least 3500 deaths since the end of the war and ongoing efforts will require over a decade to clear.
"The National Action Plan for the Prevention and Fighting of Unexploded Ordnance and Mines from 2010 to 2025" has been prepared and promulgated by the Vietnamese Government in April 2010.
=== Middle East ===
==== Iraq ====
Iraq is widely contaminated with unexploded remnants of war from the Iran–Iraq War (1980–1988), the Gulf War (1990–1991), the Iraq War (2003–2011) and the Iraqi Civil War (2014–2017). The UXO in Iraq poses a particularly serious threat to civilians as millions of cluster bomb munitions were dropped in towns and densely populated areas by Coalition forces, mostly in the first few weeks of the 2003 invasion of Iraq. An estimated 30% of the munitions failed to detonate on impact and small unexploded bombs are regularly found in and around homes in Iraq, frequently maiming or killing civilians and restricting land use. From 1991 to 2009, an estimated 8,000 people were killed or maimed by cluster bomblets alone, 2,000 of which were children. Land mines are another part of the UXO problem in Iraq as they litter large areas of farmland and many oil fields, severely affecting economic recovery and development.
Reporting and monitoring is lacking in Iraq and no completely reliable survey and overview of the local threat levels exists. Useful statistics on injuries and deaths caused by UXO are also missing; only singular local reports exist. UNDP and UNICEF however, issued a partial survey report in 2009, concluding that the entire country is contaminated and more than 1.6 million Iraqis are affected by UXO. More than 1,730 km2 (670 square miles) in total are saturated with unexploded ordnance (including land mines). The south-east region and Baghdad are the most heavily contaminated areas and UNDP has designated around 4,000 communities as "hazard areas".
==== Kuwait ====
The government of Kuwait has launched the Kuwait Environmental Remediation Program, a set of deals of the scale of US$3 billion to promote, among other initiatives, the clearance of unexploded ordnance remaining from the First Gulf War.
Kuwait has the largest amount of landmines per square mile in the world. Following the start of UXO removal, an estimated 1,486 casualties have occurred.
There are numerous mines, bombs and other explosives left from the Persian Gulf war, which makes a simple U-turn on a dirt road a life-threatening maneuver, unless performed entirely in an area covered by fresh tire tracks. Risking walking or driving in unknown areas puts oneself in danger of detonating those forgotten explosives.
In Kuwait City, there are some signs that warn people to keep distance from the broad and gleaming beaches, for example. Although, even the experts still have trouble. According to a New York Times article: Several Saudi soldiers involved in mine clearing have been killed or wounded. Two were hurt while demonstrating mine clearing for reporters.
Weeks right after the Gulf, hospitals in Kuwait reported that mines did not appear to be a major cause of injury. Six weeks after the Iraqi retreat, at Ahmadi Hospital, in an area thick with cluster bombs and Iraqi mines, the only injury was a hospital employee who had picked up an anti-personnel bomb as a souvenir.
==== Lebanon ====
Lebanon was initially contaminated by mines during its civil war, with both sides laying mines in the conflict. During several Israeli invasions of South Lebanon, up to 400,000 anti-personnel and anti-tank mines were laid along the Blue line, the 75 mile long demarcation line drawn up by the UN to mark the withdrawal of Israeli forces.
In 2014, fighting from the Syrian civil war spilled over into Lebanon when members of the Al-Nusra Front militant group attacked the town of Arsal, after one of their leaders was arrested. Fighting ensued for several days, and improvised explosive devices (IEDs) were left behind when the militants retreated. In 2015, the al-Nusra front attacked and seized some Israeli territory, and it took until 2017 for the LBF to fully dislodge them. They left behind IEDs to harm civilians, but these were fully cleared by 2023.
During the 2006 war between Israel and Lebanon, the Israel Defense Forces used large amounts of cluster weapons. For the majority of the war, they were used to target Hezbollah rocket launch points after they were detected by radar. Civilian casualties were reasonably low at this time, as many civilians had fled or were sheltering in basement. During the conflict, four million subminitions are estimated to have been dropped on South Lebanon.
However, during the final 72 hours of this war, before the ceasefire, both Hezbollah and Israeli rates of fire greatly increased. It is estimated that 90% cluster bombs used during the war were used in this time. Large areas were affected. It is thought that the Israeli bomblets have a failure rate of about 40%, which is much higher compared to other weapons. For this reason, hundreds of thousands of bomblets still litter the Israeli countryside, killing and maiming people every year. This is also the case for the borderland in South Lebanon as Khayyat argues, where the areas in which south Lebanese farmers work and herd their sheep are filled with ordinances and mines left from both the Israeli occupation of Southern Lebanon and the 2006 Lebanon War. This leaves the farmers to need to adapt to the bomb-filled environment as post-war efforts to remove unexploded ordinances and mines by international humanitarian organisations has arguably faltered out with time.
==== Yemen ====
Since the start of the Yemeni Civil War, the country has been plagued with unexploded munitions. In 2022 alone, the United Nations Development Programme (UNDP), Yemen Executive Mine Action Centre (YEMAC), and Yemen Mine Action Co-ordination Centre (Y-MACC) destroyed or removed 81,000 explosive devices, including 9,054 anti-vehicle landmines, 861 anti-personnel landmines, and 3,149 improvised explosive devices (IEDs), which in turn significantly reduced the risk of death or injury from IEDs over 6,500,000 square miles (17,000,000 km2).
=== Europe ===
Despite massive demining efforts, Europe is still affected by UXO from mainly World War I and World War II, some countries more than others. However, more recent military conflicts have also affected some areas severely, in particular Ukraine and the western Balkans. After WWII, large quantities of unexploded ordnance were disposed of primarily in the Baltic Sea and North Sea, as well as other lakes and rivers to a smaller extent. These submerged munitions still represent a major threat to fishers and marine wildlife.
==== Austria ====
Unexploded ordnance from World War II in Austria is blown up twice a year in the military training area near Allentsteig. Moreover, explosives are still being recovered from lakes, rivers and mountains dating back to World War I on the Italian Front between Austria-Hungary and Italy.
==== Balkans ====
As a result of the Yugoslav Wars (1991–2001), the countries of Albania, Bosnia-Herzegovina, Croatia and Kosovo have all been affected by UXOs, mostly land mines in regions where intense fighting took place. Due to the lack of awareness of these post-war landmines, civilian casualties have risen since the end of the wars. As many as 2,000 people have been killed by these landmines alone, with countless others dying due to different unexploded munitions. Many efforts made by peacekeeping forces in Bosnia such as IFOR, SFOR (and its successor EUFOR ALTHEA), and in Kosovo with KFOR in order to contain these landmines have been met with some difficulty. Landslides caused by heavy rainfall and flooding have led to migration of landmines, further complicating efforts.
The Federal Civil Protection Administration (FUCZ) team deactivated and destroyed four World War II bombs found at a construction site in the centre of Sarajevo in September 2019.
In November 2023, a US-funded project cleared over 395 acres of mined land in Mostar, Bosnia and Herzegovina's sixth-largest city, and declared the area mine-free. As of September 2023, the Bosnia and Herzegovina Mine Action Center estimates that over 200,000 acres in the country are still hazardous in contrast to the over 1 million acres considered unsafe in 1996. The US is also supporting the government in an effort to clear Brčko District by the end of 2024.
==== France and Belgium ====
In the Ardennes region of France, large-scale citizen evacuations were necessary during MEC removal operations in 2001. In the forests of Verdun, French government démineurs working for the Département du Déminage still hunt for poisonous, volatile, and/or explosive munitions and recover about 900 tons every year. The most feared are corroded artillery shells containing chemical warfare agents such as mustard gas. French and Flemish farmers still find many UXOs when ploughing their fields, the so-called "iron harvest".
In Belgium, Dovo, the country's bomb disposal unit, recovers between 150 and 200 tons of unexploded bombs each year. Over 20 members of the unit have been killed since it was formed in 1919.
In February 2019, a 450 kg (1,000 lb) bomb was found at a construction site at Porte de la Chapelle, near the Gare du Nord in Paris. The bomb, which led to a temporary cancellation of Eurostar trains to Paris and evacuation of 2,000 people, was probably dropped by the RAF in April 1944, targeting the Nazi-occupied Paris before the D-Day landings in Normandy.
==== Germany ====
In Germany, the responsibility for UXO disposal falls to the states, each of which operates a bomb disposal unit. These are known as the Kampfmittelbeseitigungsdienst (KMBD) or Kampfmittelräumdienst (KRD) ("Explosive Ordnance Disposal Service") and are commonly part of the state police or report directly to a mid-level administrative district. Germany's bomb squads are considered some of the busiest worldwide, deactivating a bomb every two weeks.
The presence of UXO is an ongoing task. Areas that have been subjected to aircraft bombs and artillery shells or were known battle grounds are mapped. The reconnaissance photos of the allies taken after airstrikes may show UXO and are still used to this day for location. In mapped areas New road projects, demolition, new land developments require clearing with metal detectors by the authorities to get the permits.
An estimated 5,500 UXOs from World War II are still uncovered each year in Germany, an average of 15 per day. Concentration is especially high in Berlin, where many artillery shells and smaller munitions from the Battle of Berlin are uncovered each year. One of the largest individual pieces ever found was an unexploded 'Tallboy' bomb uncovered in the Sorpe Dam in 1958.
===== 2010s =====
In 2011, a 1,800 kg (4,000 lb) RAF bomb from World War II was uncovered in Koblenz on the bottom of the Rhine River after a prolonged drought. It caused the evacuation of 45,000 people from the city. While most cases only make local news, one of the more spectacular finds in was an American 230-kilogram (500 lb) aerial bomb discovered in Munich on 28 August 2012. As it was deemed too unsafe for transport, it had to be exploded on site, shattering windows over a wide area of Schwabing and causing structural damage to several homes despite precautions to minimize damage. In February 2015, a British unexploded bomb was discovered near Signal Iduna Park in Dortmund. In May 2015, some 20,000 people had to leave their homes in Cologne in order to be safe while a 1,000 kg (2,200 lb) bomb was defused.
On December 20, 2016, another 1,800 kg RAF bomb was found in the city centre of Augsburg and prompted the evacuation of 54,000 people on December 25, which was considered the biggest bomb-related evacuation in Germany's post-war history at the time. In May 2017, 50,000 people in Hanover had to be evacuated in order to defuse three British unexploded bombs.
On 29 August 2017, a British HC 4000 bomb was discovered during construction work near the Goethe University in Frankfurt, requiring the evacuation of approximately 70,000 people within a radius of 1.5 km (0.9 mi). This was the largest evacuation in Germany since World War II. Later, it was successfully defused on 3 September. In the meantime, 21,000 residents in Koblenz were evacuated due to an unexploded 500 kg (1,100 lb) bomb dropped by the United States.
On 8 April 2018, a 1,800 kg bomb was defused in Paderborn, which caused the evacuation of more than 26,000 people. On 24 May 2018, a 250 kg (550 lb) bomb was defused in Dresden after the initial attempts of deactivation failed, and caused a small explosion. On 3 July 2018, a 250 kg bomb was disabled in Potsdam which caused 10,000 people to be evacuated from the region. In August 2018, 18,500 people in the city of Ludwigshafen had to be evacuated, in order to detonate a 500 kg (1,100 lb) bomb dropped by American forces.
In Summer 2018, high temperatures caused a decrease in the water level of the Elbe River in which grenades, mines and other explosives founded in the eastern German states of Saxony-Anhalt and Saxony were dumped. In October 2018, a World War II bomb was found during construction work in Europaviertel, Frankfurt, 16,000 people were affected within a radius of 700 m (2,300 ft). In November 2018, 10,000 people had to be evacuated, in order to defuse an American unexploded bomb found in Cologne. In December 2018, a 250 kg (550 lb) World War II bomb was discovered in Mönchengladbach.
On 31 January 2019, a World War II bomb was detonated in Lingen, Lower Saxony, which caused property damage of shattering windows and the evacuation of 9,000 people. In February 2019, an American unexploded bomb was found in Essen, which led to the evacuation of 4,000 residents within a radius of 250 to 500 metres (800 to 1,600 ft) of defusing work. A few weeks later, a 250 kg (550 lb) bomb led to the evacuation of 8,000 people in Nuremberg. In March 2019, another 250 kg bomb was found in Rostock. In April 2019, a World War II bomb was found near the U.S. military facilities in Wiesbaden.
On 14 April 2019, 600 people were evacuated when a bomb was discovered in Frankfurt's River Main. Divers with the city's fire service were participating in a routine training exercise when they found the 250 kg device. Later in April, thousands were evacuated in both Regensburg and Cologne, upon the discovery of unexploded ordnance.
On 23 June 2019, a World War II aerial bomb that was buried 4 metres (13 ft) underground in a field in Limburg self-detonated and left a crater that measured 10 metres (33 ft) wide and 4 metres (13 ft) deep. Though no one was injured, the explosion was powerful enough to register a minor tremor of 1.7 on the Richter scale. In June 2019, a World War II bomb, weighing 500 kilograms (1,100 lb), was found near the European Central Bank in Frankfurt am Main. More than 16,000 people were told to evacuate the location before the bomb was defused by the ordnance authorities on July 7, 2019. On September 2, 2019, over 15,000 people were evacuated in Hanover, after a World War II aerial bomb, weighing 230 kilograms (500 lb), was found at a construction site.
===== 2020s =====
In January 2020, 14,000 residents in Dortmund were ordered to leave their homes, during the disposal of two 250 kg (550 lb) bombs dropped by American and British forces. On August 2, 2021, 3,000 residents had to evacuate a 300-metre (980 ft) radius of the discovery site of a 250 kg (550 lb) unexploded bomb in Borsigplatz area of Dortmund.
On October 29, 2021, a five-year-old boy discovered a British hand grenade from World War II on the playground of his kindergarten "An der Beverbäke" in Oldenburg. He took it home in his backpack. The kindergarten is located on a former barracks site used by the Bundeswehr until 2007, which was converted into a residential area. On December 1, 2021, an old aircraft bomb exploded in the city of Munich during construction near Donnersbergerbruecke station.
On October 11, 2023, authorities ordered residents in Huckarde, Dortmund to leave their homes, with a 250 m (820 ft) radius from the discovery site of a 250-kilogram (550 lb) unexploded ordnance. A month later, on November 10, a 500-metre (1,600 ft) security perimeter was established in Nordhausen, following the discovery of a 450-kilogram (990 lb) unexploded bomb. On April 26, 2024, authorities defused a 500-kilogram (1,100 lb) unexploded American bomb that had been discovered two days earlier at a university expansion site in Mainz. The discovery prompted the evacuation of residents within a radius of 500 to 1,000 metres (1,600 to 3,300 ft), affecting approximately 3,500 people.
In August and October 2024, four bombs were found and safely defused in Cologne, including a 1-ton U.S. WWII bomb which was discovered during construction work in Merheim. Authorities initially tried to defuse the bomb but could only remove one of its two fuses, leading to a controlled detonation on October 11, 2024. The operation, described as the most complex since 1945, required evacuating 6,400 residents and clearing three nearby hospitals. On 4 June 2025, three WWII-era U.S. bombs were defused in Cologne's Deutz district after being uncovered during construction work. The devices—two weighing approximately 1,000 kilograms (2,200 lb) and one around 500 kilograms (1,100 lb)—were equipped with impact fuses and triggered the evacuation of roughly 20,000 people.
==== Malta ====
Malta, then a British colony, was heavily bombarded by Italian and German aircraft during World War II. During the war the Royal Engineers had a Bomb Disposal Section which cleared about 7,300 unexploded bombs between 1940 and 1942. UXO is still being found intermittently in Malta as of the early 21st century, and the Explosive Ordnance Disposal unit of the Armed Forces of Malta (AFM) is responsible for removing such ordnance. In July 2021, a Hedgehog anti-submarine mortar which likely fell off a British warship during the war was discovered on a beach in Marsaxlokk and it was successfully removed by the AFM.
==== Poland ====
In October 2020, Polish Navy divers discovered a six-ton "Tallboy" British bomb. During the attempt to remotely neutralise the bomb, it exploded in a shipping canal off the Polish port city of Świnoujscie. The Polish Navy considered it a success because the divers were able to ultimately destroy the munition with zero casualties reported. The government reportedly took all necessary measures before they started to defuse the bomb, which included evacuating 750 residents from the site.
==== Spain ====
Since the 1980s, more than 750,000 pieces of UXO from the Spanish Civil War (1936–1939) has been recovered and destroyed by the Guardia Civil in Spain. In the 2010s, around 1,000 bombs, artillery shells and grenades have been defused every year.
==== Ukraine ====
Ukraine is contaminated with UXO from World War II, former Soviet military training and the current Russo-Ukrainian War. Most of the UXO from the World Wars has presumably been removed by demining efforts in the mid-1970s, but sporadic remnants may remain in unknown locations. The UXO from the recent military conflicts includes both landmines and cluster bomblets dropped and set by both Ukrainian, anti-government and Russian forces. Reports of booby traps harming civilians also exist. Ukraine reports that Donetsk and Luhansk Oblast are the regions mostly affected by unexploded submunitions. Proper, reliable statistics are currently unavailable, and information from the involved combatants are possibly politically biased and partly speculative. However, 600 deaths and 2,000 injured due to UXO in 2014 and 2015 alone have been accounted for.
Since the beginning of the 2022 Russian invasion of Ukraine, both Russia and Ukraine have extensively used mines. As of the 22 July 2023, it is estimated that an area of 174,000 km2 (67,000 sq mi) of Ukraine are mined. The World Bank estimates that it will take $37.4 billion to clear the currently mined areas of Ukraine over a period of ten years. As of September 10, 2023, the estimated number of civilians killed by mines and unexploded ordinance is 989, and this number will increase as the conflict continues and well after the conflict has ended.
The Ukraine Mine Action Conference (UMAC2024) hosted by Switzerland and Ukraine aims to clear 10 million hectares (12.3 million acres) of land from land mines and UXO, this equates to roughly 10% of Ukraine's arable land. Before the invasion of Ukraine, agriculture made up some 11% of Ukraine's GDP, at the end of 2023 this figure had fallen to 7.4%. According to data presented in a Tony Blair Institute report, land mines are "suppressing Ukraine’s GDP by $11.2 billion (€10.27 billion) each year — equivalent to roughly 5.6% of GDP in 2021".
==== United Kingdom ====
UXO is standard terminology in the United Kingdom, although in artillery, especially on practice ranges, an unexploded shell is referred to as a blind, and during the Blitz in World War II an unexploded bomb was referred to as a UXB.
Most current UXO risk is limited to areas in cities, mainly London, Sheffield and Portsmouth, that were heavily bombed during the Blitz, and to land used by the military to store ammunition and for training. According to the Construction Industry Research and Information Association (CIRIA), from 2006 to 2009 over 15,000 items of ordnance were found in construction sites in the UK. It is not uncommon for many homes to be evacuated temporarily when a bomb is found. In April 2007, 1,000 residents were evacuated in Plymouth when a World War II bomb was discovered, and in June 2008 a 1,000-kilogram (2,200 lb) bomb was found in Bow in East London. In 2009 CIRIA published Unexploded Ordnance (UXO) – a guide for the construction industry to provide advice on assessing the risk posed by UXO.
The burden of Explosive Ordnance Disposal in the UK is split between Royal Engineers Bomb Disposal Officers, Royal Logistic Corps Ammunition Technicians in the Army, Clearance Divers of the Royal Navy and the Armourers of the Royal Air Force. The Metropolitan Police of London is the only force not to rely on the Ministry of Defence, although they generally focus on contemporary terrorist devices rather than unexploded ordnance and will often call military teams in to deal with larger and historical bombs.
In May 2016, a 230 kg (500 lb) bomb was found at the former Royal High Junior School in Bath which led to 1,000 houses being evacuated. In September 2016, a 500 kg (1,100 lb) bomb was discovered on the seabed in Portsmouth Harbour. In March 2017, a 230 kg (500 lb) bomb was found in Brondesbury Park, London. In May 2017, a 250 kg (550 lb) device was detonated in Birmingham. In February 2018, a 500 kg (1,100 lb) bomb was discovered in the Thames which forced London City Airport to cancel all the scheduled flights. In February 2019, a 76 mm (3 in) explosive device was located and destroyed in Dovercourt, near Harwich, Essex.
On September 26, 2019, Invicta Valley Primary School in Kings Hill was reportedly evacuated after an unexploded World War II bomb was discovered in its vicinity.
In February 2021, thousands of residents of Exeter were evacuated from their homes prior to the detonation of a 1,000 kg (2,200 lb) World War II bomb; the ensuing blast blew out windows and caused structural damage to nearby homes, leaving some uninhabitable.
On 20 February 2024, a 500 kg (1,100 lb) bomb from World War II was found in the garden of a residential property in Keyham, Plymouth. This prompted one of the largest evacuations in the UK since World War II, with more than 10,000 people evacuated. On 24 February, the bomb was taken out to sea and detonated, and the cordon in the area lifted.
=== Pacific ===
Buried and abandoned aerial and mortar bombs, artillery shells, and other unexploded ordnance from World War II have threatened communities across the islands of the South Pacific. As of 2014 the Office of Weapons Removal and Abatement in the U.S. Department of State's Bureau of Political-Military Affairs invested more than $5.6 million in support of conventional weapons destruction programs in the Pacific Islands.
On the battlefield of Peleliu Island in the Republic of Palau UXO removal made the island safe for tourism. At Hell's Point Guadalcanal Province in the Solomon Islands an explosive ordnance disposal training program was established which safely disposed of hundreds of items of UXO. It trained police personnel to respond to EOD call-outs in the island's highly populated areas. On Mili Atoll and Maloelap Atoll in the Marshall Islands removal of UXO has allowed for population expansion into formerly inaccessible areas.
In the Marianas, World War II-era unexploded ordnance is still often found and detonated under controlled conditions.
In September 2020, two Norwegian People's Aid employees were killed in an explosion in a residential area of Honiara, Solomon Islands, while clearing unexploded ordnance left over from the Pacific War of World War II.
== In international law ==
Protocol V of the Convention on Certain Conventional Weapons requires that when active hostilities have ended the parties must clear the areas under their control from "explosive remnants of war". Land mines are covered similarly by Protocol II. In addition to clearance obligations, Protocol V of the CCW requires parties to record information on the use and location of explosive ordnance and to provide this data to facilitate post-conflict clearance. It also encourages cooperation and assistance, allowing affected states to request international help with resources or expertise for ERW (Explosive Remnants of War) removal. Protocol V aims to reduce the long-term dangers posed by unexploded munitions to civilians and supports safer post-conflict recovery.
The Ottawa Treaty, signed in 1997 by 122 countries and effective in 1999, sought to eliminate anti-personnel landmines. It prohibits use, stockpiling, production, and transfer of Anti-personnel mines and mandates affected countries to clear mined areas within 10 years. While the treaty does not cover UXO (unexploded ordinance) directly, its principles indirectly influence UXO management, with clearance, victim assistance, and transparency obligations encouraging similar actions for UXO. Over 160 countries are now parties, with major non-signatories including the United States, China, and Russia.
The Geneva Conventions and International humanitarian law address UXO indirectly through principles focused on civilian protection. Under Protocol I (1977), parties to a conflict are required to take precautions to minimize harm to civilians, which includes managing the risks posed by UXO with an emphasis on preventing long-term civilian casualties.
== Detection technology ==
Many weapons, including aerial bombs in particular, are discovered during construction work, after lying undetected for decades. Having failed to explode while resting undiscovered is no guarantee that a bomb will not explode when disturbed. Such discoveries are common in heavily bombed cities, without a serious enough threat to warrant systematic searching.
Where there is known to be much unexploded ordnance, in cases of unexploded subsoil ordnance a remote investigation is done by visual interpretation of available historical aerial photographs. Modern techniques can combine geophysical and survey methods with modern electromagnetic and magnetic detectors. This provides digital mapping of UXO contamination with the aim to better target subsequent excavations, reducing the cost of digging on every metallic contact and speeding the clearance process. Magnetometer probes can detect UXO and provide geotechnical data before drilling or piling is carried out.
In the U.S., the Strategic Environmental Research and Development Program (SERDP) and Environmental Security Technology Certification Program (ESTCP) Department of Defense programs fund research into the detection and discrimination of UXO from scrap metal. Much of the cost of UXO removal comes from removing non-explosive items that the metal detectors have identified, so improved discrimination is critical. New techniques such as shape reconstruction from magnetic data and better de-noising techniques will reduce cleanup costs and enhance recovery.
The Interstate Technology & Regulatory Council published a Geophysical Classification for Munitions Response guidance document in August 2015.
UXO or UXBs (as they are called in some countries – unexploded bombs) are broadly classified into buried and unburied. The disposal team carries out reconnaissance of the area and determines the location of the ordnance. If is not buried it may be dug up carefully and disposed of. But if the bomb is buried it becomes a huge task. A team is formed to find the location of the bomb using metal detectors and then the earth is dug carefully.
== Effects post-conflict ==
There are a variety of effects unexploded ordnance contamination has on post-conflict societies other than physical harm from detonation. Segments of society which are also negatively affected include foreign direct investment, education, aid distribution, industrialization, and the environment.
=== Industrialisation ===
UXO presence reduces farming communities’ ability to use industrial machinery due to higher likelihood of triggering a buried munition. As well as this, large scale infrastructure projects such as road, rail, dam, or bridge building which require heavy machinery are prevented due to the risk of setting off UXO. These two factors in turn reduce road building and therefore prevent other more remote communities from industrializing themselves.
=== Aid distribution ===
Contaminated areas experience more difficulties in providing humanitarian aid to rural or remote communities. Infrastructure for transportation is either impossible to develop, or preexisting infrastructure is difficult to demine.
=== Environmental effects ===
Demining procedures destroy topsoil. This causes increased erosion and can reduce the fertility of arable land.
Munitions which are left over a long period of time degrade and eventually poison the soil or groundwater around them.
=== Education ===
The inhibition of necessary resources correlates with decreases in education. Injuries experienced by older members of the community take children away from classrooms to support a family's subsistence agriculture techniques.
=== Foreign direct investment ===
Foreign direct investment from more developed nations is discouraged due to difficulty in clearing contaminated areas.
== See also ==
Ammunition dump
Bombhunters, a 2006 documentary film about the effects of unexploded ordnance on Cambodian people
Danger UXB, a 1979 British ITV television series set during the World War II
Land of Mine, a 2015 movie about post-WWII demining in Denmark
Delay-action bomb
Dud
Mines Advisory Group
Ordnance
Red Zone
ZEUS-HLONS (HMMWV Laser Ordnance Neutralization System)
== References ==
== Further reading ==
Owen, James (2010). Danger UXB. Little, Brown. ISBN 978-1-4087-0255-0.
Webster, Donovan (1996). Aftermath: The Remnants of War. Pantheon. ISBN 0-679-43195-0.
== External links ==
Mines Advisory Group
US Department of Defense UXO Awareness web site | Wikipedia/Unexploded_ordnance |
A materials recovery facility, materials reclamation facility, materials recycling facility or multi re-use facility (MRF, pronounced "murf") is a specialized waste sorting and recycling system that receives, separates and prepares recyclable materials for marketing to end-user manufacturers. Generally, the main recyclable materials include ferrous metal, non-ferrous metal, plastics, paper, glass. Organic food waste is used to assist anaerobic digestion or composting. Inorganic inert waste is used to make building materials. Non-recyclable high calorific value waste is used to making refuse-derived fuel (RDF) and solid recovered fuel (SRF).
== Industry and locations ==
In the United States, there are over 300 materials recovery facilities. The total market size is estimated at $6.6B as of 2019.
As of 2016, the top 75 were headed by Sims Municipal Recycling out of Brooklyn, New York. Waste Management operated 95 MRF facilities total, with 26 in the top 75. ReCommunity operated 6 in the top 75. Republic Services operated 6 in the top 75. Waste Connections operated 4 in the top 75.
== Business economics ==
In 2018, a survey in the Northeast United States found that the processing cost per ton was $82, versus a value of around $45 per ton. Composition of the ton included 28% mixed paper and 24% old corrugated containers (OCC).
Prices for OCC declined into 2019. Three paper mill companies have announced initiatives to use more recycled fiber.
Glass recycling is expensive for these facilities, but a study estimated that costs could be cut significantly by investments in improved glass processing. In Texas, Austin and Houston have facilities which have invested glass recycling, built and operated by Balcones Recycling and FCC Environment, respectively.
Robots have spread across the industry, helping with sorting.
== Process ==
Waste enters a MRF when it is dumped onto the tipping floor by the collection trucks. The materials are then scooped up and placed onto conveyor belts, which transports it to the pre-sorting area. Here, human workers remove some items that are not recyclable, which will either be sent to a landfill or an incinerator. Between 5 and 45% of "dirty" MRF material is recovered. Potential hazards are also removed, such as lithium batteries, propane tanks, and aerosol cans, which can create fires. Materials like plastic bags and hoses, which can entangle the recycling equipment, are also removed. From there, materials are transported via another conveyer belt to the disk screen, which separates wide and flat materials like flattened cardboard boxes from items like cans, jars, paper, and bottles. Flattened boxes ride across the disk screen to the other side, while all other materials fall below, where paper is separated from the waste stream with a blower. The stream of cardboard and paper is overseen by more human workers, who ensure no plastic, metal, or glass is present. Newer MRFs or retrofitted ones may use industrial robots instead of humans for pre-sorting and for quality control. However, complete removal of human labor from the sortation process is unlikely for the foreseeable future, as one needs to replicate the dexterity of the human hand and nervous system for removing every type of contaminant within a material stream. The technical limitations of this involve advanced concepts in mechatronics and computer science, where a robot hand would need to be designed, and a highly flexible algorithm that creates another precise movement algorithm within the time constraints of the system (say, the highly approximate estimate of 30,000 lines of code to do this on a modern processor would trigger too long of a delay to be effective on a sortation line). In other words, one would need to search an encyclopedia of said robotic hand motions for every configuration of waste for every pick, and this may be computationally insurmountable, even with quantum computing, as every conditional would need to be checked every iteration.
Metal is separated from plastics and glass first with electromagnets, which removes ferrous metals. Non-ferrous metals like aluminum are then removed with eddy current separators.
The glass and plastic streams are separated by further disk screens. The glass is crushed into cullet for ease of transportation. The plastics are then separated by polymer type, often using infrared technology (optical sorting). Infrared light reflects differently off different polymer types; once identified, a jet of air shoots the plastic into the appropriate bin. MRFs might only collect and recycle a few polymers of plastic, sending the rest to landfills or incinerators. The separated materials are baled and sent to the shipping dock of the facility.
== Types ==
=== Clean ===
A clean MRF accepts recyclable materials that have already been separated at the source from municipal solid waste generated by either residential or commercial sources. There are a variety of clean MRFs. The most common are single stream where all recyclable material is mixed, or dual stream MRFs, where source-separated recyclables are delivered in a mixed container stream (typically glass, ferrous metal, aluminum and other non-ferrous metals, PET [No.1] and HDPE [No.2] plastics) and a mixed paper stream including corrugated cardboard boxes, newspapers, magazines, office paper and junk mail. Material is sorted to specifications, then baled, shredded, crushed, compacted, or otherwise prepared for shipment to market.
=== Mixed-waste processing facility (MWPF) / Dirty MRF ===
A mixed-waste processing system, sometimes referred to as a dirty MRF, accepts a mixed solid waste stream and then proceeds to separate out designated recyclable materials through a combination of manual and mechanical sorting. The sorted recyclable materials may undergo further processing required to meet technical specifications established by end-markets while the balance of the mixed waste stream is sent to a disposal facility such as a landfill. Today, MWPFs are attracting renewed interest as a way to address low participation rates for source-separated recycling collection systems and prepare fuel products and/or feedstocks for conversion technologies. MWPFs can give communities the opportunity to recycle at much higher rates than has been demonstrated by curbside or other waste collection systems. Advances in technology make today’s MWPF different and, in many respects better, than older versions.
=== Wet MRF ===
Around 2004, new mechanical biological treatment technologies were beginning to utilise wet MRFs. These combine a dirty MRF with water, which acts to densify, separate and clean the output streams. It also hydrocrushes and dissolves biodegradable organics in solution to make them suitable for anaerobic digestion.
== History ==
In the United States, modern MRFs began in the 1970s. Peter Karter established Resource Recovery Systems, Inc. in Branford, Connecticut, the "first materials recovery facility (MRF)" in the US.
== See also ==
Cradle-to-cradle design
Curbside collection
List of waste treatment technologies
List of waste types
Mechanical biological treatment
Resource recovery
Transfer station (waste management)
Waste characterization
Waste sorting
== References ==
== External links ==
"Coming soon! van der Linde's amazing recycling machine"
"Materials Recovery Facility Solutions"
The Role of MRFS in Modern Day Waste Management | Wikipedia/Materials_recovery_facility |
Corporate environmental responsibility (CER) refers to a company's duties to abstain from damaging natural environments. The term derives from corporate social responsibility (CSR).
== Background ==
The environmental aspect of corporate social responsibility has been debated over the past few decades, as stakeholders increasingly require organizations to become more environmentally aware and socially responsible. In the traditional business model, environmental protection was considered only in relation to the "public interest". Hitherto, governments had maintained principal responsibility for ensuring environmental management and conservation.
The public sector has been focused on the development of regulations and the imposition of sanctions as a means to facilitating environmental protection. Recently, the private sector has adopted the approach of co-responsibility towards the prevention and alleviation of environmental damage. The sectors and their roles have been changing, with the private sector becoming more active in the protection of the environment. Many governments, corporations, and big companies are now providing strategies for environmental protection and economic growth.
The World Commission on Environment published the Brundtland Report in 1987 to address sustainable development. Since then, managers, scholars, and business owners have tried to determine why and how big corporations should incorporate environmental aspects into their own policies. In recent years, an increasing number of companies have pledged to protect natural environments.
== Relations to corporate social responsibility ==
There are different perceptions of corporate social responsibility between government, the private sector, non-governmental organizations (NGOs) and society in general, and thus, the concept has no single definition.
Corporate social responsibility may cover:
A company running its business responsibly in relation to internal stakeholders (shareholders, employees, customers and suppliers)
The role of business in relation to the state (locally and nationally) as well as to inter-state institutions or standards
Business performance as a responsible member of the society in which it operates and the global community."
The European Union defines corporate social responsibility as "...the concept that an enterprise is accountable for its impact on all relevant stakeholders. It is the continuing commitment by a business to behave fairly and responsibly and contribute to economic development while
improving the quality of life of the workforce and their families as well as of the local community and society at large." According to this definition, a CSR strategy is more focused on social aspects, particularly the interests of stakeholders.
Corporate environmental responsibility (CER) is, in many ways, connected to CSR, as both of them influence environmental protection. CER, however, is strictly about the consideration of environmental implications and protection within corporate strategy. The understanding of CER cannot be separated from CSR—both are interconnected and based on environmental protection. There are three major areas related to these two concepts—economic, environmental and social. CER is focused more on economic and environmental while CSR relates to social and environmental aspects. Economy, society, and environment all play significant roles in the development of an efficient and effective company strategy.
== Main elements ==
These cover the environmental implications of a company's operations:
Eliminate waste and emissions
Maximize the efficient use of resources and productivity
Minimize activities that might impair the enjoyment of resources by future generations.
== Drivers and challenges ==
Among the main drivers for CER are government policies and regulations. Many states provide their own legislation, regulations and policies, which are important in creating a positive environmental attitude within companies. Subsidies, tariffs and taxes play a vital role in the implementation of these policies. Another significant factor is the competitive environment among companies generated by media, public, shareholder and NGO awareness, which are also major drivers of CER. Another significant driver of corporate responsibility is that the private sector is largely responsible for the development of green technology and renewable energy sources meaning they are contributing towards climate change mitigation while still operating as a business.
Challenges include the cost of regulation and difficulties in predicting economic gains, which could become problematic for a company's management. Additionally, new technologies are frequently too expensive for a lot of companies. Another challenge is the lack of harmonization of regulations among different states—often there is a mosaic of propositions, leading to unclear strategies for environmental behavior, especially in multinational corporations. Further challenges of CES are whether corporations have a responsibility to go further than the current governmental legislation and Corporation a firstly responsible to produce profit for shareholders and producing goods for customers. Furthermore Companies work within the framework of the society and country that they operate in meaning that corporations cannot be held solely responsible for lack of legislation on pollution and emissions.
Corporations emissions are also fractured between different sectors such as supply and outsourcing which can make it unclear what emissions the corporation is responsible. Further challenges is the argument of whether corporations should be held responsible for past emissions when the negative impacts were not known.
== Worldwide perspectives on corporate environmental responsibility ==
The majority of international CSR studies focus on business practices and its aspects, such as business economics and the legality of environmental law. Most companies are noticing the importance of taking into account one of its most important stakeholders: employees and customers and their commitment to sustainability. Studies have demonstrated that once companies place sustainability practices they can be directly linked to financial success and customer satisfaction, which in turn can be used as a marketing tool. An additional study highlighted that these practices are in effect at larger firms with more resources to fund environmental responsibility. Although every country has a different culture, and each country determines their own scale of environmental responsibility, research has shown that there is a standard global human values that drive customer needs and wants. Companies have taken initiatives to take sustainability and align it with each company's economic goals. Part of this initiative has included publishing sustainability reports, offering more transparency of company operations to customers. Managers and other people at the top, play the key role in decision-making and implementing the firm's sustainability practices.
== Benefits of corporate environmental responsibility ==
Corporate social responsibility can prove to be more profitable for companies and to extend it survivability in markets because greater awareness on this topic, in both social and business markets, has been in higher demand. Customers have responded with overall satisfaction and loyalty when companies have a better CSR, especially in countries like Spain and Brazil. Culture has an impact on the CSR ratings and studies, as well as human values across different nations.
This topic can also be found under sustainable development. This area is concerned with not only protecting the environment but maintaining economical growth. There were several agreements internationally to help adopt new business practices that held these standards, but they were considered individual and there was no law-abiding body to regulate nor implement them.
One of the other factors that is considered an integral part of sustainable development are human beings, and specific groups and their habitat. Counties and companies that more developed would lead, and other small countries and business would slowly make gains. It is important to recognize that just because corporate environmental responsibility is being recognized that consumption is something that is not discouraged.
The idea of corporate environmental responsibility is for humans to be more aware of the environmental impact and counteract their pollution/carbon footprint on the natural resources. One of the main factors is to reduce carbon footprint and carbon emissions. Many of the studies focus on trying to find a balance between economic growth and reducing waste and cleaner environments.
Furthermore, many firms are discovering that there is an advantage to advocating for environmental regulations and preparing for them to be implemented before they become law. In a recent study, the researcher found that firms support climate change legislation as a means of gaining power over their competitors. Essentially, even if a new regulation hurts a firm in the short term, the firm may embrace it because they know that it will hurt their competitors even more. This allows them to come out on top in the long run.
== Summary ==
The environmental aspects of security have increasingly become a major issue being considered by states.
Globalization also plays a key role in the adoption of new environmental strategies as a multi-faceted process influencing modern societies, and creating interconnected and multidimensional environments.
Corporate environmental responsibility is used by multinational corporations as well as small, local organizations. It is highlighted and more institutionalized because of stakeholders' awareness of the huge impacts of business activities on the environment. To understand CER, its relations with CSR strategies need to be recognized. CER and CSR are the main strategies that help in the creation of efficient and environmentally sustainable businesses.
== See also ==
Triple bottom line
== References == | Wikipedia/Corporate_environmental_responsibility |
Energy flow is the flow of energy through living things within an ecosystem. All living organisms can be organized into producers and consumers, and those producers and consumers can further be organized into a food chain. Each of the levels within the food chain is a trophic level. In order to more efficiently show the quantity of organisms at each trophic level, these food chains are then organized into trophic pyramids. The arrows in the food chain show that the energy flow is unidirectional, with the head of an arrow indicating the direction of energy flow; energy is lost as heat at each step along the way.
The unidirectional flow of energy and the successive loss of energy as it travels up the food web are patterns in energy flow that are governed by thermodynamics, which is the theory of energy exchange between systems. Trophic dynamics relates to thermodynamics because it deals with the transfer and transformation of energy (originating externally from the sun via solar radiation) to and among organisms.
== Energetics and the carbon cycle ==
The first step in energetics is photosynthesis, where in water and carbon dioxide from the air are taken in with energy from the sun, and are converted into oxygen and glucose. Cellular respiration is the reverse reaction, wherein oxygen and sugar are taken in and release energy as they are converted back into carbon dioxide and water. The carbon dioxide and water produced by respiration can be recycled back into plants.
Energy loss can be measured either by efficiency (how much energy makes it to the next level), or by biomass (how much living material exists at those levels at one point in time, measured by standing crop). Of all the net primary productivity at the producer trophic level, in general only 10% goes to the next level, the primary consumers, then only 10% of that 10% goes on to the next trophic level, and so on up the food pyramid. Ecological efficiency may be anywhere from 5% to 20% depending on how efficient or inefficient that ecosystem is. This decrease in efficiency occurs because organisms need to perform cellular respiration to survive, and energy is lost as heat when cellular respiration is performed. That is also why there are fewer tertiary consumers than there are producers.
== Primary production ==
A producer is any organism that performs photosynthesis. Producers are important because they convert energy from the sun into a storable and usable chemical form of energy, glucose, as well as oxygen. The producers themselves can use the energy stored in glucose to perform cellular respiration. Or, if the producer is consumed by herbivores in the next trophic level, some of the energy is passed on up the pyramid. The glucose stored within producers serves as food for consumers, and so it is only through producers that consumers are able to access the sun’s energy. Some examples of primary producers are algae, mosses, and other plants such as grasses, trees, and shrubs.
Chemosynthetic bacteria perform a process similar to photosynthesis, but instead of energy from the sun they use energy stored in chemicals like hydrogen sulfide. This process, referred to as chemosynthesis, usually occurs deep in the ocean at hydrothermal vents that produce heat and chemicals such as hydrogen, hydrogen sulfide and methane. Chemosynthetic bacteria can use the energy in the bonds of the hydrogen sulfide and oxygen to convert carbon dioxide to glucose, releasing water and sulfur in the process. Organisms that consume the chemosynthetic bacteria can take in the glucose and use oxygen to perform cellular respiration, similar to herbivores consuming producers.
One of the factors that controls primary production is the amount of energy that enters the producer(s), which can be measured using productivity. Only one percent of solar energy enters the producer, the rest bounces off or moves through. Gross primary productivity is the amount of energy the producer actually gets. Generally, 60% of the energy that enters the producer goes to the producer’s own respiration. The net primary productivity is the amount that the plant retains after the amount that it used for cellular respiration is subtracted. Another factor controlling primary production is organic/inorganic nutrient levels in the water or soil that the producer is living in. An example of the nutrients that can impact the efficiency of primary plant production are nitrogen (N) and phosphorus (P).
=== Carnivorous plants ===
When it comes to dealing with environments that have low nutrient availability, some plants have developed unique ways to adapt to be able to perform photosynthesis. In order to do so, these plants have evolved to be able to obtain important nutrients such as nitrogen from other organisms, just as heterotrophs would giving them the unique title of carnivorous plants. With methods such as the pitfall trap (pitcher plant), the flypaper trap (Drosera capensis), or the snap trap (venus flytrap) these plants have learned to lure insects in and digest them.
Pitcher plants lure insects in using a variety of attractive cues such as scent and color. Once an insect or small organism falls into the bulb shaped body of the plant, a variety of enzymes are secreted beginning the digestion process of the organism and preventing it from escaping. Flypaper trap plants, the most common of carnivorous plants, secret a special liquid that allow an insect to land on its leaves but then prevents the insect from escaping. The snap trap plant, use similar methods to the pitcher plant in order to attract various insects. However, these carnivorous plants are able to detect when an insect is touching its leaves thus triggering the "mouth" of the plant to close and encase the insect. In developing this method of nutrient acquisition, carnivorous plants are able to survive in almost any environment around the world, excluding Antarctica and the Arctic Circle.
== Secondary production ==
Secondary production is the use of energy stored in plants converted by consumers to their own biomass. Different ecosystems have different levels of consumers, all end with one top consumer. Most energy is stored in organic matter of plants, and as the consumers eat these plants they take up this energy. This energy in the herbivores and omnivores is then consumed by carnivores. There is also a large amount of energy that is in primary production and ends up being waste or litter, referred to as detritus. The detrital food chain includes a large amount of microbes, macroinvertebrates, meiofauna, fungi, and bacteria. These organisms are consumed by omnivores and carnivores and account for a large amount of secondary production. Secondary consumers can vary widely in how efficient they are in consuming. The efficiency of energy being passed on to consumers is estimated to be around 10%. Energy flow through consumers differs in aquatic and terrestrial environments.
=== In aquatic environments ===
Heterotrophs contribute to secondary production and it is dependent on primary productivity and the net primary products. Secondary production is the energy that herbivores and decomposers use and thus depends on primary productivity. Primarily herbivores and decomposers consume all the carbon from two main organic sources in aquatic ecosystems, autochthonous and allochthonous. Autochthonous carbon comes from within the ecosystem and includes aquatic plants, algae and phytoplankton. Allochthonous carbon from outside the ecosystem is mostly dead organic matter from the terrestrial ecosystem entering the water. In stream ecosystems, approximately 66% of annual energy input can be washed downstream. The remaining amount is consumed and lost as heat.
=== In terrestrial environments ===
Secondary production is often described in terms of trophic levels, and while this can be useful in explaining relationships it overemphasizes the rarer interactions. Consumers often feed at multiple trophic levels. Energy transferred above the third trophic level is relatively unimportant. The assimilation efficiency can be expressed by the amount of food the consumer has eaten, how much the consumer assimilates and what is expelled as feces or urine. While a portion of the energy is used for respiration, another portion of the energy goes towards biomass in the consumer. There are two major food chains: The primary food chain is the energy coming from autotrophs and passed on to the consumers; and the second major food chain is when carnivores eat the herbivores or decomposers that consume the autotrophic energy. Consumers are broken down into primary consumers, secondary consumers and tertiary consumers. Carnivores have a much higher assimilation of energy, about 80% and herbivores have a much lower efficiency of approximately 20 to 50%. Energy in a system can be affected by animal emigration/immigration. The movements of organisms are significant in terrestrial ecosystems. Energetic consumption by herbivores in terrestrial ecosystems has a low range of ~3-7%. The flow of energy is similar in many terrestrial environments. The fluctuation in the amount of net primary product consumed by herbivores is generally low. This is in large contrast to aquatic environments of lakes and ponds where grazers have a much higher consumption of around ~33%. Ectotherms and endotherms have very different assimilation efficiencies.
== Detritivores ==
Detritivores consume organic material that is decomposing and are in turn consumed by carnivores. Predator productivity is correlated with prey productivity. This confirms that the primary productivity in ecosystems affects all productivity following.
Detritus is a large portion of organic material in ecosystems. Organic material in temperate forests is mostly made up of dead plants, approximately 62%.
In an aquatic ecosystem, leaf matter that falls into streams gets wet and begins to leech organic material. This happens rather quickly and will attract microbes and invertebrates. The leaves can be broken down into large pieces called coarse particulate organic matter (CPOM). The CPOM is rapidly colonized by microbes. Meiofauna is extremely important to secondary production in stream ecosystems. Microbes breaking down and colonizing this leaf matter are very important to the detritovores. The detritovores make the leaf matter more edible by releasing compounds from the tissues; it ultimately helps soften them. As leaves decay nitrogen will decrease since cellulose and lignin in the leaves is difficult to break down. Thus the colonizing microbes bring in nitrogen in order to aid in the decomposition. Leaf breakdown can depend on initial nitrogen content, season, and species of trees. The species of trees can have variation when their leaves fall. Thus the breakdown of leaves is happening at different times, which is called a mosaic of microbial populations.
Species effect and diversity in an ecosystem can be analyzed through their performance and efficiency. In addition, secondary production in streams can be influenced heavily by detritus that falls into the streams; production of benthic fauna biomass and abundance decreased an additional 47–50% during a study of litter removal and exclusion.
== Energy flow across ecosystems ==
Research has demonstrated that primary producers fix carbon at similar rates across ecosystems. Once carbon has been introduced into a system as a viable source of energy, the mechanisms that govern the flow of energy to higher trophic levels vary across ecosystems. Among aquatic and terrestrial ecosystems, patterns have been identified that can account for this variation and have been divided into two main pathways of control: top-down and bottom-up. The acting mechanisms within each pathway ultimately regulate community and trophic level structure within an ecosystem to varying degrees. Bottom-up controls involve mechanisms that are based on resource quality and availability, which control primary productivity and the subsequent flow of energy and biomass to higher trophic levels. Top-down controls involve mechanisms that are based on consumption by consumers. These mechanisms control the rate of energy transfer from one trophic level to another as herbivores or predators feed on lower trophic levels.
=== Aquatic vs terrestrial ecosystems ===
Much variation in the flow of energy is found within each type of ecosystem, creating a challenge in identifying variation between ecosystem types. In a general sense, the flow of energy is a function of primary productivity with temperature, water availability, and light availability. For example, among aquatic ecosystems, higher rates of production are usually found in large rivers and shallow lakes than in deep lakes and clear headwater streams. Among terrestrial ecosystems, marshes, swamps, and tropical rainforests have the highest primary production rates, whereas tundra and alpine ecosystems have the lowest. The relationships between primary production and environmental conditions have helped account for variation within ecosystem types, allowing ecologists to demonstrate that energy flows more efficiently through aquatic ecosystems than terrestrial ecosystems due to the various bottom-up and top-down controls in play.
==== Bottom-up ====
The strength of bottom-up controls on energy flow are determined by the nutritional quality, size, and growth rates of primary producers in an ecosystem. Photosynthetic material is typically rich in nitrogen (N) and phosphorus (P) and supplements the high herbivore demand for N and P across all ecosystems. Aquatic primary production is dominated by small, single-celled phytoplankton that are mostly composed of photosynthetic material, providing an efficient source of these nutrients for herbivores. In contrast, multi-cellular terrestrial plants contain many large supporting cellulose structures of high carbon but low nutrient value. Because of this structural difference, aquatic primary producers have less biomass per photosynthetic tissue stored within the aquatic ecosystem than in the forests and grasslands of terrestrial ecosystems. This low biomass relative to photosynthetic material in aquatic ecosystems allows for a more efficient turnover rate compared to terrestrial ecosystems. As phytoplankton are consumed by herbivores, their enhanced growth and reproduction rates sufficiently replace lost biomass and, in conjunction with their nutrient dense quality, support greater secondary production.
Additional factors impacting primary production includes inputs of N and P, which occurs at a greater magnitude in aquatic ecosystems. These nutrients are important in stimulating plant growth and, when passed to higher trophic levels, stimulate consumer biomass and growth rate. If either of these nutrients are in short supply, they can limit overall primary production. Within lakes, P tends to be the greater limiting nutrient while both N and P limit primary production in rivers. Due to these limiting effects, nutrient inputs can potentially alleviate the limitations on net primary production of an aquatic ecosystem. Allochthonous material washed into an aquatic ecosystem introduces N and P as well as energy in the form of carbon molecules that are readily taken up by primary producers. Greater inputs and increased nutrient concentrations support greater net primary production rates, which in turn supports greater secondary production.
==== Top-down ====
Top-down mechanisms exert greater control on aquatic primary producers due to the roll of consumers within an aquatic food web. Among consumers, herbivores can mediate the impacts of trophic cascades by bridging the flow of energy from primary producers to predators in higher trophic levels. Across ecosystems, there is a consistent association between herbivore growth and producer nutritional quality. However, in aquatic ecosystems, primary producers are consumed by herbivores at a rate four times greater than in terrestrial ecosystems. Although this topic is highly debated, researchers have attributed the distinction in herbivore control to several theories, including producer to consumer size ratios and herbivore selectivity.
Modeling of top-down controls on primary producers suggests that the greatest control on the flow of energy occurs when the size ratio of consumer to primary producer is the highest. The size distribution of organisms found within a single trophic level in aquatic systems is much narrower than that of terrestrial systems. On land, the consumer size ranges from smaller than the plant it consumes, such as an insect, to significantly larger, such as an ungulate, while in aquatic systems, consumer body size within a trophic level varies much less and is strongly correlated with trophic position. As a result, the size difference between producers and consumers is consistently larger in aquatic environments than on land, resulting in stronger herbivore control over aquatic primary producers.
Herbivores can potentially control the fate of organic matter as it is cycled through the food web. Herbivores tend to select nutritious plants while avoiding plants with structural defense mechanisms. Like support structures, defense structures are composed of nutrient poor, high carbon cellulose. Access to nutritious food sources enhances herbivore metabolism and energy demands, leading to greater removal of primary producers. In aquatic ecosystems, phytoplankton are highly nutritious and generally lack defense mechanisms. This results in greater top-down control because consumed plant matter is quickly released back into the system as labile organic waste. In terrestrial ecosystems, primary producers are less nutritionally dense and are more likely to contain defense structures. Because herbivores prefer nutritionally dense plants and avoid plants or plant parts with defense structures, a greater amount of plant matter is left unconsumed within the ecosystem. Herbivore avoidance of low-quality plant matter may be why terrestrial systems exhibit weaker top-down control on the flow of energy.
== See also ==
Food web – Natural interconnection of food chains
Ecological stoichiometry
Energy – Physical quantity
== References ==
== Further reading == | Wikipedia/Energy_flow_(ecology) |
Nanomaterials can be both incidental and engineered. Engineered nanomaterials (ENMs) are nanoparticles that are made for use, are defined as materials with dimensions between 1 and 100nm, for example in cosmetics or pharmaceuticals like zinc oxide and TiO2 as well as microplastics. Incidental nanomaterials are found from sources such as cigarette smoke and building demolition. Engineered nanoparticles have become increasingly important for many applications in consumer and industrial products, which has resulted in an increased presence in the environment. This proliferation has instigated a growing body of research into the effects of nanoparticles on the environment. Natural nanoparticles include particles from natural processes like dust storms, volcanic eruptions, forest fires, and ocean water evaporation.
== Sources ==
Products containing nanoparticles such as cosmetics, coatings, paints, and catalytic additives can release nanoparticles into the environment in different ways. There are three main ways that nanoparticles enter the environment. The first is emission during the production of raw materials such as mining and refining operations. The second is emission during use, like cosmetics or sunblock getting washed into the environment. The third is emission after disposal of nanoparticle products or use during waste treatment, like nanoparticles in sewage and wastewater streams.
The first emission scenario, causing 2% of emissions, results from the production of materials. Studies of a precious metals refinery found that the mining and refining of metals releases a significant amount of nanoparticles into the air. Further analysis showed concentration levels of silver nanoparticles far higher than OSHA standards in the air despite operational ventilation. Wind speed can also cause nanoparticles generated in mining or related activities to spread further and have increased penetration power. A high wind speed can cause aerosolized particles to penetrate enclosures at a much higher rate than particles not exposed to wind.
Construction also generates nanoparticles during the manufacture and use of materials. The release of nanoscale materials can occur during the evacuation of waste from cleanout operations, losses during spray drying, filter residuals, and emissions from filters. Pump sprays and propellants on average can emit 1.1 x 10^8 and 8.6 x 10^9 particles/g.
A significant amount of nanoparticles are also released during the handling of dry powders, even when contained in fume hoods. Particles on construction sites can have prolonged exposure to the atmosphere and thus are more likely to enter the environment. Nanoparticles in concrete construction and recycling introduce a new hazard during the demolition process, which can pose even higher environmental exposure risks. Concrete modified with nanoparticles is almost impossible to separate from conventional concrete, so the release may be uncontrollable if demolished using conventional means. Even normal abrasion and deterioration of buildings can release nanoparticles into the environment on a long-term basis.
Normal weathering can release 10 to 10^5 mg/m^2 fragments containing nanomaterials.
Another emission scenario is release during use. Sunscreens can release a significant amount of Titanium dioxide (TiO2) nanoparticles into surface waters. Testing of the Old Danube Lake indicated that there were significant concentrations of nanoparticles from cosmetics in the water. Conservative estimates calculate that there were approximately 27.2 micrograms/L of TiO2, if TiO2 was distributed throughout the entire 3.5*10^6 M^3 volume of the lake.
Although TiO2 is generally considered weakly soluble, these nanoparticles undergo weathering and transformation under conditions in acidic soils with high proportions of organic and inorganic acids. There are observable differences in particle morphology between manufactured and natural TIO2 nanoparticles, though differences may attenuate over time due to weathering. However, these processes are likely to take decades.
Copper and zinc oxide nanoparticles that get into the water can additionally act as chemosensitizers in sea urchin embryos. It is predicted that for animals in aquatic systems sunscreen is probably the most important exposure route to harmful metal particles. ZnOs from sunblock and other applications like paints, optoelectronics, and pharmaceuticals are entering the environment at an increasing rate. Their effects can be genotoxic, mutagenic, and cytotoxic.
Nanoparticles can be transported through different mediums depending on their type. Emissions patterns have found that TiO2 NPs accumulate in sludge-treated soils. This means that the dominating emission pathway is through wastewater. ZnO generally collects in natural and urban soil as well as landfills. Silver nanoparticles from production and mining operations generally enter landfills and wastewater. Comparing different reservoirs by how readily nanoparticles pollute them, ~63-91% of NPs accumulate in landfills, 8-28% in soils, aquatic environments receive ~7%, and air around 1.5%.
== Exposure toxicity ==
Knowledge of the effects of industrial nanoparticles (NPs) released into the environment remains limited. Effects vary widely over aquatic and terrestrial environments as well as types of organisms. The characteristics of the nanoparticle itself plays a wide variety of roles including size, charge, composition, surface chemistry, etc.
Nanoparticles released into the environment can potentially interact with pre-existing contaminants, leading to cascading biological effects that are currently poorly understood.
Several scientific studies have indicated that nanoparticles can cause a series of adverse physiological and cellular effects on plants including root length inhibition, biomass reduction, altered transpiration rate, developmental delay, chlorophyll synthesis disruption, cell membrane damage, and chromosomal aberration. Though genetic damage induced by metal nanoparticles in plants has been documented, the mechanism of that damage, its severity, and whether the damage is reversible remain active areas of study. Studies of CeO2 nanoparticles were shown to greatly diminish nitrogen fixation in the root nodules of soybean plants, leading to stunted growth. Positive charges on nanoparticles were shown to destroy the membrane lipid bilayers in animal cells and interfere with overall cellular structure. For animals, it has been shown that nanoparticles can provoke inflammation, oxidative stress, and modification of mitochondrial distribution. These effects were dose-dependent and varied by nanoparticle type.
Present research indicates that biomagnification of nanoparticles through trophic levels is highly dependent upon the type of nanoparticles and biota in question. While some instances of bioaccumulation of nanoparticles exist, there is no general consensus.
== Difficulties in measurement ==
There is no clear consensus on potential human and ecological impacts stemming from exposure to ENMs. As a result, developing reliable methods for testing ENM toxicity assessment has been a high priority for commercial usage. However, ENMs are found in a variety of conditions making a universal testing method non-viable. Currently, both in-vitro and in-vivo assessments are used, where the effects of NPs on events such as apoptosis, or conditions like cell viability, are observed.
In measuring ENMs, addressing and accounting for uncertainties such as impurities and biological variability is crucial. In the case of ENMs, some concerns include changes that occur during testing such as agglomeration and interaction with substances in the testing media, as well as how ENMS disperse in the environment. For example, one investigation into how the presence of fullerenes impacted largemouth bass in 2004 concluded that fullerenes were responsible for neurological damage done to the fish, whereas subsequent studies revealed this was actually a result of byproducts resulting from the dispersal of fullerenes into tetrahydrofuran (THF) and minimal toxicity was observed when water was used in its place. Fortunately, greater thoroughness in the process of testing could help to resolve these issues. One method that has proven useful in avoiding artifacts is the thorough characterization of ENMS in the laboratory conducting the testing rather than just relying on the information provided by manufacturers.
In addition to problems that can arise due to testing, there is contention on how to ensure testing is done for environmentally relevant conditions, partly due to the difficulty of detecting and quantifying ENMs in complex environmental matrices. Currently, straightforward analytical methods are not available for the detection of NPs in the environment, although computer modeling is thought to be a potential pathway moving forward. A push to focus on the development of internationally agreed upon unbiased toxicological models holds promise to provide greater consensus within the field as well as enable more accurate determinations of ENMs in the environment.
== Regulation and organizations ==
The regulation of nanomaterials is present in the U.S. and many other countries globally. Policy is directed mainly at manufacturing exposure of NPs in the environment.
=== International / intergovernmental organizations ===
As of 2013, the OECD Working Party on Nanomaterials (WPN) worked on a multitude of projects with the purpose of mitigating potential threats and hazards associated with nanoparticles. The WPN conducted research on methods for testing, improvements on field assessments, exposure relief, and efforts to educate individuals and organizations on environmental sustainability with respect to NPs.
The International Organization for Standardization TC 229 focuses on standardizing manufacturing, nomenclature/terminology, instrumentation, testing and assessment methodology, and safety, health, and environmental practices.
=== North America ===
In the United States, the FDA and OSHA focus on regulations that prevent toxic harm to people from NPs, whereas the EPA takes on environmental policies to inhibit harmful effects nanomaterials may pose on the planet.
As of 2019, there were supporters and opponents of increased regulation. Supporters of regulation want NPs to be seen as a class and/or have the precautionary principle applied. Opponents believe that over-regulation could lead to harmful effects on the economy and customer and economic freedom. As of 2019, there were multiple policies up for consideration for the purpose of changing nanomaterial regulation.
The EPA is tackling regulations through two approaches under the TSCA: information gathering rule on new to old NMs and required premanufacturing notification for novice NMs. The gathering rule requires companies that produce or import NMs to provide the EPA with chemical properties, production/use amounts, manufacturing methods, and any found health, safety, and environmental impact for any nanomaterials being used. The premanufacturing notifications gives the EPA better governance over nanomaterial exposure, health testing, manufacturing/process and worker safety, and release amount which can allow the agency to take control of a NM if it poses concerning risk.
The United States National Nanotechnology Initiative involves 20 departments and independent agencies that focus on nanotechnology innovation and regulation in the United States. Projects and activities of NNI span from R&D to policy on environment and safety regulations of NMs.
NIEHS built itself from the complications that came with conducting research and assessment on nanomaterials. NIEHS realized the rapid adoption of NMs in products from a large variety of industries, and since then the organization has supported research focused on understanding the underlying threats NMs may pose on the environment and people.
The Canada-U.S. Regulatory Cooperation Council (RCC) Nanotechnology Initiative was constructed in order for the U.S. and Canada to protect and improve safety and environmental impacts of NMs without hindering growth and investment in NMs for both countries. The RCC oversees both countries and has maintained regulations, worked to create new regulations with the goal of alignment, secure transparency, and ensure that new and beneficial opportunities in the nanotechnology sector were shared with both countries.
=== Europe ===
Nanomaterials are defined consistently in both Registration, Evaluation, Authorisation and Restriction of Chemicals and Classification, Labeling, and Packaging legislations, in order to promote harmony in industry use. In January, 2020 REACH listed explicit requirements for businesses that import or manufacture NMs in Annex I, III, VI, VII-XI, and XII. Reporting of chemical characteristics/properties, safety assessments, and downstream user obligations of NMs are all required for reporting to the ECHA.
The Biocidal Products Regulation (BPR) has different regulation and reporting requirements than what is stated in REACH and CLP. Data and risk assessments are required for substance approval, specific labeling requirements are needed, and reporting on the substance which includes current use and potential risks must be done every 5 years.
=== Asia ===
The Asia Nano Forum (ANF) focuses on ensuring responsible manufacturing of nanomaterials that are environmentally, economically, and population safe. ANF supports joint projects with a focus on supporting safe development in emerging economies and technical research. Overall, the organization helps promote homogenous regulation and policy on NMs in Asia.
The Chinese National Nanotechnology Standardization Technical Committee (NSTC) reviews standards and regulation policies. The technical committee SAC/TC279 focuses on normalizing terminology, methodology, assessment methods, and material use in the field. The committee develops specific test protocols and technical standards for companies manufacturing NMs. In addition, the NSTC is constantly adding to their nano-material toxicology database in order to better standards and regulation.
== See also ==
Nanotoxicology
== References == | Wikipedia/Pollution_from_nanomaterials |
Industrial symbiosis is a subset of industrial ecology. It describes how a network of diverse organizations can foster eco-innovation and long-term culture change, create and share mutually profitable transactions—and improve business and technical processes.
Although geographic proximity is often associated with industrial symbiosis, it is neither necessary nor sufficient—nor is a singular focus on physical resource exchange. Strategic planning is required to optimize the synergies of co-location. In practice, using industrial symbiosis as an approach to commercial operations—using, recovering and redirecting resources for reuse—results in resources remaining in productive use in the economy for longer. This in turn creates business opportunities, reduces demands on the earth's resources, and provides a stepping-stone towards creating a circular economy.
Industrial symbiosis is a subset of industrial ecology, with a particular focus on material and energy exchange. Industrial ecology is a relatively new field that is based on a natural paradigm, claiming that an industrial ecosystem may behave in a similar way to the natural ecosystem wherein everything gets recycled, albeit the simplicity and applicability of this paradigm has been questioned.
== Introduction ==
Eco-industrial development is one of the ways in which industrial ecology contributes to the integration of economic growth and environmental protection. Some of the examples of eco-industrial development are:
Circular economy (single material and/or energy exchange)
Greenfield eco-industrial development (geographically confined space)
Brownfield eco-industrial development (geographically confined space)
Eco-industrial network (no strict requirement of geographical proximity)
Virtual eco-industrial network (networks spread in large areas e.g. regional network)
Networked Eco-industrial System (macro level developments with links across regions)
Industrial symbiosis engages traditionally separate industries in a collective approach to competitive advantage involving physical exchange of materials, energy, water, and/or by-products. The keys to industrial symbiosis are collaboration and the synergistic possibilities offered by geographic proximity". Notably, this definition and the stated key aspects of industrial symbiosis, i.e., the role of collaboration and geographic proximity, in its variety of forms, has been explored and empirically tested in the UK through the research and published activities of the National Industrial Symbiosis Programme.
Industrial symbiosis systems collectively optimize material and energy use at efficiencies beyond those achievable by any individual process alone. IS systems such as the web of materials and energy exchanges among companies in Kalundborg, Denmark have spontaneously evolved from a series of micro innovations over a long time scale; however, the engineered design and implementation of such systems from a macro planner's perspective, on a relatively short time scale, proves challenging.
Often, access to information on available by-products is difficult to obtain. These by-products are considered waste and typically not traded or listed on any type of exchange. Only a small group of specialized waste marketplaces addresses this particular kind of waste trading.
== Example ==
Recent work reviewed government policies necessary to construct a multi-gigaWatt photovoltaic factory and complementary policies to protect existing solar companies are outlined and the technical requirements for a symbiotic industrial system are explored to increase the manufacturing efficiency while improving the environmental impact of solar photovoltaic cells. The results of the analysis show that an eight-factory industrial symbiotic system can be viewed as a medium-term investment by any government, which will not only obtain direct financial return, but also an improved global environment.
This is because synergies have been identified for co-locating glass manufacturing and photovoltaic manufacturing.
The waste heat from glass manufacturing can be used in industrial-sized greenhouses for food production. Even within the PV plant itself a secondary chemical recycling plant can reduce environmental impact while improving economic performance for the group of manufacturing facilities.
In DCM Shriram consolidated limited (Kota unit) produces caustic soda, calcium carbide, cement and PVC resins. Chlorine and hydrogen are obtained as by-products from caustic soda production, while calcium carbide produced is partly sold and partly is treated with water to form slurry(aqueous solution of calcium hydroxide) and ethylene. The chlorine and ethylene produced are utilised to form PVC compounds, while the slurry is consumed for cement production by wet process. Hydrochloric acid is prepared by direct synthesis where the pure chlorine gas can be combined with hydrogen to produce hydrogen chloride in the presence of UV light.
== See also ==
Eco-industrial park
Industrial ecology
Industrial metabolism
Waste valorization
== References ==
== External links ==
International Group of Industrial Symbiosis Researchers & Practitioners
Marian Chertow interview on Industrial Symbiosis (audio)
Western Cape Industrial Symbiosis Programme (WISP) | Wikipedia/Industrial_symbiosis |
Design for the environment (DfE) is a design approach to reduce the overall human health and environmental impact of a product, process or service, where impacts are considered across its life cycle. Different software tools have been developed to assist designers in finding optimized products or processes/services. DfE is also the original name of a United States Environmental Protection Agency (EPA) program, created in 1992, that works to prevent pollution, and the risk pollution presents to humans and the environment. The program provides information regarding safer chemical formulations for cleaning and other products. EPA renamed its program "Safer Choice" in 2015.
== Introduction ==
Initial guidelines for a DfE approach were written in 1990 by East Meets West, a New York-based non-governmental organization founded by Anneke van Waesberghe. It became a global movement targeting design initiatives and incorporating environmental motives to improve product design in order to minimize health and environmental impacts by incorporating it from design stage all the way to the manufacturing process. The DfE strategy aims to improve technology and design tactics to expand the scope of products. By incorporating eco-efficiency into design tactics, DfE takes into consideration the entire life-cycle of the product, while still making products usable but minimizing resource use. The key focus of DfE is to minimize the environmental-economic cost to consumers while still focusing on the life-cycle framework of the product. By balancing both customer needs as well as environmental and social impacts DfE aims to "improve the product use experience both for consumers and producers, while minimally impacting the environment".
== Practices ==
Four main concepts that fall under the DfE umbrella.
Design for environmental processing and manufacturing: Raw material extraction (mining, drilling, etc.), processing (processing reusable materials, metal melting, etc.) and manufacturing are done using materials and processes which are not dangerous to the environment or the employees working on said processes. This includes the minimization of waste and hazardous by-products, air pollution, energy expenditure and other factors.
Design for environmental packaging: Materials used in packaging are environmentally responsible, which can be achieved through the reuse of shipping products, elimination of unnecessary paper and packaging products, efficient use of materials and space, use of recycled and/or recyclable materials.
Design for disposal or reuse: The end-of-life of a product is very important, because some products emit dangerous chemicals into the air, ground and water after they are disposed of in a landfill. Planning for the reuse or refurbishing of a product will change the types of materials that would be used, how they could later be disassembled and reused, and the environmental impacts such materials have.
Design for energy efficiency: The design of products to reduce overall energy consumption throughout the product's life.
Life-cycle assessment (LCA) is employed to forecast the impacts of different (production) alternatives of the product in question, thus being able to choose the most environmentally friendly. A life cycle analysis can serve as a tool when determining the environmental impact of a product or process. Proper LCAs can help a designer compare several different products according to several categories, such as energy use, toxicity, acidification, CO2 emissions, ozone depletion, resource depletion and many others. By comparing different products, designers can make decisions about which environmental hazard to focus on in order to make the product more environmentally friendly.
== Rationale ==
Modern day businesses aim to produce goods at a low cost while maintaining quality, staying competitive in the global marketplace, and meeting consumer preferences for more environment friendly products. To help businesses meet these challenges, EPA encourages businesses to incorporate environmental considerations into the design process. The benefits of incorporating DfE include: cost savings, reduced business and environmental risks, expanded business and market opportunities, and to meet environmental regulations.
== Companies and products ==
Starbucks: Starbucks is decreasing its carbon footprint by building more energy efficient stores and facilities, conserving energy and water, and purchasing renewable energy credits. Starbucks has achieved LEED certificates in 116 stores in 12 countries. Starbucks has even created a portable, LEED certified store in Denver. It is Starbucks' goal to reduce energy consumption by 25% and to cover 100% of its electricity with renewable energy by 2015.
Hewlett Packard: HP is working towards reducing energy used in manufacturing, developing materials that have less environmental impact, and designing easily recyclable equipment.
IBM: Their goal is to extend product life beyond just production, and to use reusable and recyclable products. This means that IBM is currently working on creating products that can be safely disposed of at the end of its product life. They are also reducing consumption of energy to minimize their carbon footprint.
Philips: For almost 20 years now, sustainable development has been a crucial part of Philips decision making and manufacturing process. Philips' goal is to produce products with their environmental responsibility in mind. Not only are they working on reducing energy during the manufacturing process, Phillips is also participating in a unique project, philanthropy through design. Since 2005, Philips has been working on and developing philanthropy through design. They collaborate with other organizations to use their expertise and innovation to help the more fragile parts of our society.
Besides these large brand names there are several other consumer product companies in the DfE program this including:
Atlantic Chemical & Equipment Co.
American Cleaning Solutions
BCD Supply
Beta Technology
Brighton USA
== Design process ==
A business can design for the environment by:
Evaluating the human health and environmental impacts of its processes and products.
Identifying what information is needed to make human health and environment decisions
Conducting an assessment of alternatives
Considering cross-media impacts and the benefits of substituting chemicals
Reducing the use and release of toxic chemicals through the innovation of cleaner technologies that use safer chemicals.
Implementing pollution prevention, energy efficiency, and other resource conservation measures.
Making products that can be reused and recycled
Monitoring the environmental impacts and costs associated with each product or process
Recognizing that although change can be rapid, in many cases a cycle of evaluation and continuous improvement is needed.
== Safer Choice labeling program ==
EPA's DfE labeling program was renamed "Safer Choice" in 2015.
== Current U.S. laws and regulations encouraging DfE in the electronics industry ==
=== National Ambient Air Quality Standards (NAAQS) ===
EPA promulgated the National Ambient Air Quality Standards (NAAQS) to establish basic air pollution control requirements across the U.S. The NAAQS sets standards on six main sources of pollutants, which include emissions of: ozone (0.12 ppm per 1 hour), carbon monoxide (35 ppm per 1 hour; primary standard), particulate matter (50g/m^3 at an annual arithmetic mean), sulfur dioxide (80g/m^3 at an annual arithmetic mean), nitrogen dioxide (100g/m^3 at an annual arithmetic mean), and lead emissions (1.5g/m^3 at an annual arithmetic mean).
=== Stratospheric ozone protection ===
Stratospheric ozone protection is required by section 602 of the Clean Air Act of 1990. This regulation aims to decrease emission of chlorofluorocarbons (CFCs) and other chemicals that are destroying the stratospheric ozone layer. The protection initiative categorizes ozone-depleting substances into two classes: Class I, and Class II.
Class I substances include 20 different kinds of chemicals and have all been phased-out of production processes since 2000. Class II substances consist of 33 different hydro-chlorofluorocarbons (HCFCs). The EPA has already begun plans to decrease emissions in HCFCs and plan to completely phase out the class II substances by 2030.
=== Reporting requirements for releases of toxic substances ===
A firm operating in the electronics industry in Standard Industrial Classification (SIC) Codes 20-39 that has more than 10 full-time employees and consumes more than 10,000 lbs per year of any toxic chemical lists in 40 CFR 372.65 must file a toxic release inventory.
=== Other regulations ===
National Emissions Standards for Hazardous Air Pollutants (NESHAP)
National Pollutant Discharge Elimination System (NPDES–Water pollution permit program)
Underground Injection Control Program
Hazardous waste management
Underground storage tank management
== See also ==
== References ==
== External links ==
The European Union: The European Platform on Life Cycle Assessment
Sustainable design for the environment
Department Life Cycle Engineering, University of Stuttgart (English)
Sustainable Building Alliance.org
Sustainable Residential Design.org: Using Low-Impact Materials Resource Guide | Wikipedia/Design_for_Environment |
Pollution is the introduction of contaminants into the natural environment that cause harm. Pollution can take the form of any substance (solid, liquid, or gas) or energy (such as radioactivity, heat, sound, or light). Pollutants, the components of pollution, can be either foreign substances/energies or naturally occurring contaminants.
Although environmental pollution can be caused by natural events, the word pollution generally implies that the contaminants have a human source, such as manufacturing, extractive industries, poor waste management, transportation or agriculture. Pollution is often classed as point source (coming from a highly concentrated specific site, such as a factory, mine, construction site), or nonpoint source pollution (coming from a widespread distributed sources, such as microplastics or agricultural runoff).
Many sources of pollution were unregulated parts of industrialization during the 19th and 20th centuries until the emergence of environmental regulation and pollution policy in the later half of the 20th century. Sites where historically polluting industries released persistent pollutants may have legacy pollution long after the source of the pollution is stopped. Major forms of pollution include air pollution, water pollution, litter, noise pollution, plastic pollution, soil contamination, radioactive contamination, thermal pollution, light pollution, and visual pollution.
Pollution has widespread consequences on human and environmental health, having systematic impact on social and economic systems. In 2019, pollution killed approximately nine million people worldwide (about one in six deaths that year); about three-quarters of these deaths were caused by air pollution. A 2022 literature review found that levels of anthropogenic chemical pollution have exceeded planetary boundaries and now threaten entire ecosystems around the world. Pollutants frequently have outsized impacts on vulnerable populations, such as children and the elderly, and marginalized communities, because polluting industries and toxic waste sites tend to be collocated with populations with less economic and political power. This outsized impact is a core reason for the formation of the environmental justice movement, and continues to be a core element of environmental conflicts, particularly in the Global South.
Because of the impacts of these chemicals, local and international countries' policy have increasingly sought to regulate pollutants, resulting in increasing air and water quality standards, alongside regulation of specific waste streams. Regional and national policy is typically supervised by environmental agencies or ministries, while international efforts are coordinated by the UN Environmental Program and other treaty bodies. Pollution mitigation is an important part of all of the Sustainable Development Goals.
== Definitions and types ==
Various definitions of pollution exist, which may or may not recognize certain types, such as noise pollution or greenhouse gases. The United States Environmental Protection Administration defines pollution as "Any substances in water, soil, or air that degrade the natural quality of the environment, offend the senses of sight, taste, or smell, or cause a health hazard. The usefulness of the natural resource is usually impaired by the presence of pollutants and contaminants." In contrast, the United Nations considers pollution to be the "presence of substances and heat in environmental media (air, water, land) whose nature, location, or quantity produces undesirable environmental effects."
The major forms of pollution are listed below along with the particular contaminants relevant to each of them:
Air pollution: the release of chemicals and particulates into the atmosphere. Common gaseous pollutants include carbon monoxide, sulfur dioxide, chlorofluorocarbons (CFCs) and nitrogen oxides produced by industry and motor vehicles. Photochemical ozone and smog are created as nitrogen oxides and hydrocarbons react to sunlight. Particulate matter, or fine dust is characterized by their micrometre size PM10 to PM2.5.
Electromagnetic pollution: the overabundance of electromagnetic radiation in their non-ionizing form, such as radio and television transmissions, Wi-fi etc. Although there is no demonstrable effect on humans there can be interference with radio-astronomy and effects on safety systems of aircraft and cars.
Light pollution: includes light trespass, over-illumination and astronomical interference.
Littering: the criminal throwing of inappropriate man-made objects, unremoved, onto public and private properties.
Noise pollution: which encompasses roadway noise, aircraft noise, industrial noise as well as high-intensity sonar.
Plastic pollution: involves the accumulation of plastic products and microplastics in the environment that adversely affects wildlife, wildlife habitat, or humans.
Soil contamination occurs when chemicals are released by spill or underground leakage. Among the most significant soil contaminants are hydrocarbons, heavy metals, MTBE, herbicides, pesticides and chlorinated hydrocarbons.
Radioactive contamination, resulting from 20th century activities in atomic physics, such as nuclear power generation and nuclear weapons research, manufacture and deployment. (See alpha emitters and actinides in the environment.)
Thermal pollution, is a temperature change in natural water bodies caused by human influence, such as use of water as coolant in a power plant.
Visual pollution, which can refer to the presence of overhead power lines, motorway billboards, scarred landforms (as from strip mining), open storage of trash, municipal solid waste or space debris.
Water pollution, caused by the discharge of industrial wastewater from commercial and industrial waste (intentionally or through spills) into surface waters; discharges of untreated sewage and chemical contaminants, such as chlorine, from treated sewage; and releases of waste and contaminants into surface runoff flowing to surface waters (including urban runoff and agricultural runoff, which may contain chemical fertilizers and pesticides, as well as human feces from open defecation).
== Natural causes ==
One of the most significant natural sources of pollution are volcanoes, which during eruptions release large quantities of harmful gases into the atmosphere. Volcanic gases include carbon dioxide, which can be fatal in large concentrations and contributes to climate change, hydrogen halides which can cause acid rain, sulfur dioxides, which are harmful to animals and damage the ozone layer, and hydrogen sulfides, which are capable of killing humans at concentrations of less than 1 part per thousand. Volcanic emissions also include fine and ultrafine particles which may contain toxic chemicals and substances such as arsenic, lead, and mercury.
Wildfires, which can be caused naturally by lightning strikes, are also a significant source of air pollution. Wildfire smoke contains significant quantities of both carbon dioxide and carbon monoxide, which can cause suffocation. Large quantities of fine particulates are found within wildfire smoke as well, which pose a health risk to animals.
== Human generation ==
Motor vehicle emissions are one of the leading causes of air pollution. China, United States, Russia, India, Mexico, and Japan are the world leaders in air pollution emissions. Principal stationary pollution sources include chemical plants, coal-fired power plants, oil refineries, petrochemical plants, nuclear waste disposal activity, incinerators, large livestock farms (dairy cows, pigs, poultry, etc.), PVC factories, metals production factories, plastics factories, and other heavy industry. Agricultural air pollution comes from contemporary practices which include clear felling and burning of natural vegetation as well as spraying of pesticides and herbicides.
About 400 million metric tons of hazardous wastes are generated each year. The United States alone produces about 250 million metric tons. Americans constitute less than 5% of the world's population, but produce roughly 25% of the world's CO2, and generate approximately 30% of world's waste. In 2007, China overtook the United States as the world's biggest producer of CO2, while still far behind based on per capita pollution (ranked 78th among the world's nations).
Chlorinated hydrocarbons (CFH), heavy metals (such as chromium, cadmium – found in rechargeable batteries, and lead – found in lead paint, aviation fuel, and even in certain countries, gasoline), MTBE, zinc, arsenic, and benzene are some of the most frequent soil contaminants. A series of press reports published in 2001, culminating in the publication of the book Fateful Harvest, revealed a widespread practise of recycling industrial leftovers into fertilizer, resulting in metal poisoning of the soil. Ordinary municipal landfills are the source of many chemical substances entering the soil environment (and often groundwater), emanating from the wide variety of refuse accepted, especially substances illegally discarded there, or from pre-1970 landfills that may have been subject to little control in the U.S. or EU. There have also been some unusual releases of polychlorinated dibenzodioxins, commonly called dioxins for simplicity, such as TCDD.
Pollution can also occur as a result of natural disasters. Hurricanes, for example, frequently result in sewage contamination and petrochemical spills from burst boats or automobiles. When coastal oil rigs or refineries are involved, larger-scale and environmental damage is not unusual. When accidents occur, some pollution sources, such as nuclear power stations or oil ships, can create extensive and potentially catastrophic emissions.
Plastic pollution is choking our oceans by making plastic gyres, entangling marine animals, poisoning our food and water supply, and ultimately inflicting havoc on the health and well-being of humans and wildlife globally. With the exception of a small amount that has been incinerating, virtually every piece of plastic that was ever made in the past still exists in one form or another. And since most of the plastics do not biodegrade in any meaningful sense, all that plastic waste could exist for hundreds or even thousands of years. If plastic production is not circumscribed, plastic pollution will be disastrous and will eventually outweigh fish in oceans.
Raised levels of greenhouse gases such as carbon dioxide in the atmosphere are affecting the Earth's climate. Disruption of the environment can also highlight the connection between areas of pollution that would normally be classified separately, such as those of water and air. Recent studies have investigated the potential for long-term rising levels of atmospheric carbon dioxide to cause slight but critical increases in the acidity of ocean waters, and the possible effects of this on marine ecosystems.
In February 2007, a report by the Intergovernmental Panel on Climate Change (IPCC), representing the work of 2,500 scientists, economists, and policymakers from more than 120 countries, confirmed that humans have been the primary cause of global warming since 1950. Humans have ways to cut greenhouse gas emissions and avoid the consequences of global warming, a major climate report concluded. But to change the climate, the transition from fossil fuels like coal and oil needs to occur within decades, according to the final report this year from the UN's Intergovernmental Panel on Climate Change (IPCC).
== Effects ==
=== Human health ===
Pollution affects humans in every part of the world. An October 2017 study by the Lancet Commission on Pollution and Health found that global pollution, specifically toxic air, water, soil and workplaces, kills nine million people annually, which is triple the number of deaths caused by AIDS, tuberculosis and malaria combined, and 15 times higher than deaths caused by wars and other forms of human violence. The study concluded that "pollution is one of the great existential challenges of the Anthropocene era. Pollution endangers the stability of the Earth's support systems and threatens the continuing survival of human societies."
Adverse air quality can kill many organisms, including humans. Ozone pollution can cause respiratory disease, cardiovascular disease, throat inflammation, chest pain, and congestion. A 2010 analysis estimated that 1.2 million people died prematurely each year in China alone because of air pollution. China's high smog levels can damage the human body and cause various diseases. In 2019, air pollution caused 1.67 million deaths in India (17.8% of total deaths nationally). Studies have estimated that the number of people killed annually in the United States could be over 50,000. A study published in 2022 in GeoHealth concluded that energy-related fossil fuel emissions in the United States cause 46,900–59,400 premature deaths each year and PM2.5-related illness and death costs the nation $537–$678 billion annually. In the US, deaths caused by coal pollution were highest in 1999, but decreased sharply after 2007. The number dropped by about 95% by 2020, as coal plants have been closed or have scrubbers installed.
In 2019, water pollution caused 1.4 million premature deaths. Contamination of drinking water by untreated sewage in developing countries is an issue, for example, over 732 million Indians (56% of the population) and over 92 million Ethiopians (92.9% of the population) do not have access to basic sanitation. In 2013 over 10 million people in India fell ill with waterborne illnesses in 2013, and 1,535 people died, most of them children. As of 2007, nearly 500 million Chinese lack access to safe drinking water.
Acute exposure to certain pollutants can have short and long term effects. Oil spills can cause skin irritations and rashes. Noise pollution induces hearing loss, high blood pressure, stress, and sleep disturbance. Mercury has been linked to developmental deficits in children and neurologic symptoms. Older people are significantly exposed to diseases induced by air pollution. Those with heart or lung disorders are at additional risk. Children and infants are also at serious risk. Lead and other heavy metals have been shown to cause neurological problems, intellectual disabilities and behavioural problems. Chemical and radioactive substances can cause cancer and birth defects.
==== Socio economic impacts ====
The health impacts of pollution have both direct and lasting social consequences. A 2021 study found that exposure to pollution causes an increase in violent crime. A 2019 paper linked pollution to adverse school outcomes for children. A number of studies show that pollution has an adverse effect on the productivity of both indoor and outdoor workers.
=== Environment ===
Pollution has been found to be present widely in the natural environment. A 2022 study published in Environmental Science & Technology found that levels of anthropogenic chemical pollution have exceeded planetary boundaries and now threaten entire ecosystems around the world.
There are a number of effects of this:
Biomagnification describes situations where toxins (such as heavy metals) may pass through trophic levels, becoming exponentially more concentrated in the process.
Carbon dioxide emissions cause ocean acidification, the ongoing decrease in the pH of the Earth's oceans as CO2 becomes dissolved.
The emission of greenhouse gases leads to global warming which affects ecosystems in many ways.
Invasive species can outcompete native species and reduce biodiversity. Invasive plants can contribute debris and biomolecules (allelopathy) that can alter soil and chemical compositions of an environment, often reducing native species competitiveness.
Nitrogen oxides are removed from the air by rain and fertilise land which can change the species composition of ecosystems.
Smog and haze can reduce the amount of sunlight received by plants to carry out photosynthesis and leads to the production of tropospheric ozone which damages plants.
Soil can become infertile and unsuitable for plants. This will affect other organisms in the food web.
Sulfur dioxide and nitrogen oxides can cause acid rain which lowers the pH value of soil.
Organic pollution of watercourses can deplete oxygen levels and reduce species diversity.
== Regulation and monitoring ==
== Control ==
Pollution control is a term used in environmental management. It refers to the control of emissions and effluents into air, water or soil. Without pollution control, the waste products from overconsumption, heating, agriculture, mining, manufacturing, transportation and other human activities, whether they accumulate or disperse, will degrade the environment. In the hierarchy of controls, pollution prevention and waste minimization are more desirable than pollution control. In the field of land development, low impact development is a similar technique for the prevention of urban runoff.
Policy, law and monitoring/transparency/life-cycle assessment-attached economics could be developed and enforced to control pollution. A review concluded that there is a lack of attention and action such as work on a globally supported "formal science–policy interface", e.g. to "inform intervention, influence research, and guide funding".
In September 2023 a Global Framework on Chemicals aiming to reduce pollution was agreed during an international conference in Bonn, Germany. The framework includes 28 targets, for example, to "end the use of hazardous pesticides in agriculture where the risks have not been managed" by 2035.
=== Practices ===
Recycling
Reusing
Waste minimisation
Mitigating
Pollution prevention
Compost
=== Devices ===
Air pollution control
Green wall
Smog Tower
Thermal oxidizer
Bioremediation
Dust collection systems
Baghouses
Cyclones
Electrostatic precipitators
Scrubbers
Baffle spray scrubber
Cyclonic spray scrubber
Ejector venturi scrubber
Mechanically aided scrubber
Spray tower
Wet scrubber
Sewage treatment
Sedimentation (Primary treatment)
Activated sludge biotreaters (Secondary treatment; also used for industrial wastewater)
Aerated lagoons
Constructed wetlands (also used for urban runoff)
Industrial wastewater treatment
API oil-water separators
Biofilters
Dissolved air flotation (DAF)
Powdered activated carbon treatment
Ultrafiltration
Vapor recovery systems
Phytoremediation
== Cost ==
Pollution has a cost. Manufacturing activities that cause air pollution impose health and clean-up costs on the whole of society. A manufacturing activity that causes air pollution is an example of a negative externality in production. A negative externality in production occurs "when a firm's production reduces the well-being of others who are not compensated by the firm." For example, if a laundry firm exists near a polluting steel manufacturing firm, there will be increased costs for the laundry firm because of the dirt and smoke produced by the steel manufacturing firm. If external costs exist, such as those created by pollution, the manufacturer will choose to produce more of the product than would be produced if the manufacturer were required to pay all associated environmental costs. Because responsibility or consequence for self-directed action lies partly outside the self, an element of externalization is involved. If there are external benefits, such as in public safety, less of the good may be produced than would be the case if the producer were to receive payment for the external benefits to others. Goods and services that involve negative externalities in production, such as those that produce pollution, tend to be overproduced and underpriced since the externality is not being priced into the market.
Pollution can also create costs for the firms producing the pollution. Sometimes firms choose, or are forced by regulation, to reduce the amount of pollution that they are producing. The associated costs of doing this are called abatement costs, or marginal abatement costs if measured by each additional unit. In 2005 pollution abatement capital expenditures and operating costs in the US amounted to nearly $27 billion.
== Dirtiest industries ==
The Pure Earth, an international non-for-profit organization dedicated to eliminating life-threatening pollution in the developing world, issues an annual list of some of the world's most polluting industries. Below is the list for 2016:
Lead–acid battery recycling
Mining and extractive metallurgy
Lead smelting
Tanning
Artisanal mining
Landfills
Industrial parks
Chemical industry
Manufacturing
Dyeing
A 2018 report by the Institute for Agriculture and Trade Policy and GRAIN says that the meat and dairy industries are poised to surpass the oil industry as the world's worst polluters.
=== Textile industry ===
=== Fossil fuel related industries ===
Outdoor air pollution attributable to fossil fuel use alone causes ~3.61 million deaths annually, making it one of the top contributors to human death, beyond being a major driver of climate change whereby greenhouse gases are considered per se as a form of pollution (see above).
== Socially optimal level ==
Society derives some indirect utility from pollution; otherwise, there would be no incentive to pollute. This utility may come from the consumption of goods and services that inherently create pollution (albeit the level can vary) or lower prices or lower required efforts (or inconvenience) to abandon or substitute these goods and services. Therefore, it is important that policymakers attempt to balance these indirect benefits with the costs of pollution in order to achieve an efficient outcome.
It is possible to use environmental economics to determine which level of pollution is deemed the social optimum. For economists, pollution is an "external cost and occurs only when one or more individuals suffer a loss of welfare". There is a socially optimal level of pollution at which welfare is maximized. This is because consumers derive utility from the good or service manufactured, which will outweigh the social cost of pollution until a certain point. At this point the damage of one extra unit of pollution to society, the marginal cost of pollution, is exactly equal to the marginal benefit of consuming one more unit of the good or service.
Moreover, the feasibility of pollution reduction rates could also be a factor of calculating optimal levels. While a study puts the global mean loss of life expectancy (LLE; similar to YPLL) from air pollution in 2015 at 2.9 years (substantially more than, for example, 0.3 years from all forms of direct violence), it also indicated that a significant fraction of the LLE is unavoidable in terms of current economical-technological feasibility such as aeolian dust and wildfire emission control.
In markets with pollution, or other negative externalities in production, the free market equilibrium will not account for the costs of pollution on society. If the social costs of pollution are higher than the private costs incurred by the firm, then the true supply curve will be higher. The point at which the social marginal cost and market demand intersect gives the socially optimal level of pollution. At this point, the quantity will be lower and the price will be higher in comparison to the free market equilibrium. Therefore, the free market outcome could be considered a market failure because it "does not maximize efficiency".
This model can be used as a basis to evaluate different methods of internalizing the externality, such as tariffs, a Pigouvian tax (such as a carbon tax) and cap and trade systems.
== History ==
=== Prior to 19th century ===
Air pollution has always accompanied civilizations. Pollution started from prehistoric times, when humans created the first fires. According to a 1983 article in the journal Science, "soot" found on ceilings of prehistoric caves provides ample evidence of the high levels of pollution that was associated with inadequate ventilation of open fires."
Metal forging appears to be a key turning point in the creation of significant air pollution levels outside the home. Core samples of glaciers in Greenland indicate increases in pollution associated with Greek, Roman, and Chinese metal production.
The burning of coal and wood, and the presence of many horses in concentrated areas made the cities the primary sources of pollution. King Edward I of England banned the burning of mineral coal by proclamation in London in 1306, after its smoke became a problem; the fuel was named sea-coal at the time, getting its name from the fact that it was delivered form overseas (as opposed to charcoal, which was referred to as "coal").
=== 19th century ===
The Industrial Revolution gave birth to environmental pollution as we know it today. London also recorded one of the earliest extreme cases of water quality problems with the Great Stink on the Thames of 1858, which led to the construction of the London sewerage system soon afterward. Pollution issues escalated as population growth far exceeded the ability of neighborhoods to handle their waste problem. Reformers began to demand sewer systems and clean water.
In 1870, the sanitary conditions in Berlin were among the worst in Europe. August Bebel recalled conditions before a modern sewer system was built in the late 1870s:
Waste-water from the houses collected in the gutters running alongside the curbs and emitted a truly fearsome smell. There were no public toilets in the streets or squares. Visitors, especially women, often became desperate when nature called. In the public buildings the sanitary facilities were unbelievably primitive....As a metropolis, Berlin did not emerge from a state of barbarism into civilization until after 1870.
=== 20th and 21st century ===
The primitive conditions were intolerable for a world national capital, and the Imperial German government brought in its scientists, engineers, and urban planners to solve the deficiencies and forge Berlin as the world's model city. A British expert in 1906 concluded that Berlin represented "the most complete application of science, order and method of public life," adding "it is a marvel of civic administration, the most modern and most perfectly organized city that there is."
The emergence of great factories and consumption of immense quantities of coal gave rise to unprecedented air pollution, and the large volume of industrial chemical discharges added to the growing load of untreated human waste. Chicago and Cincinnati were the first two American cities to enact laws ensuring cleaner air in 1881. Pollution became a significant issue in the United States in the early twentieth century, as progressive reformers took issue with air pollution caused by coal burning, water pollution caused by bad sanitation, and street pollution caused by the three million horses who worked in American cities in 1900, generating large quantities of urine and manure. As historian Martin Melosi notes, the generation that first saw automobiles replacing horses saw cars as "miracles of cleanliness". By the 1940s, automobile-caused smog was a significant issue in Los Angeles.
Other cities followed around the country until early in the 20th century when the short-lived Office of Air Pollution was created under the Department of the Interior. The cities of Los Angeles experienced extreme smog events and Donora, Pennsylvania, in the late 1940s, serving as another public reminder.
Air pollution would continue to be a problem in England, especially later during the Industrial Revolution, and extending into the recent past with the Great Smog of 1952. Awareness of atmospheric pollution spread widely after World War II, with fears triggered by reports of radioactive fallout from atomic warfare and testing. Then a non-nuclear event – the Great Smog of 1952 in London – killed at least 4000 people. This prompted some of the first major modern environmental legislation: the Clean Air Act of 1956.
Pollution began to draw significant public attention in the United States between the mid-1950s and early 1970s, when Congress passed the Noise Control Act, the Clean Air Act, the Clean Water Act, and the National Environmental Policy Act.
Severe incidents of pollution helped increase consciousness. PCB dumping in the Hudson River resulted in a ban by the EPA on consumption of its fish in 1974. National news stories in the late 1970s – especially the long-term dioxin contamination at Love Canal starting in 1947 and uncontrolled dumping in Valley of the Drums – led to the Superfund legislation of 1980. The pollution of industrial land gave rise to the name brownfield, a term now common in city planning.
The development of nuclear science introduced radioactive contamination, which can remain lethally radioactive for hundreds of thousands of years. Lake Karachay – named by the Worldwatch Institute as the "most polluted spot" on earth – served as a disposal site for the Soviet Union throughout the 1950s and 1960s. Chelyabinsk, Russia, is considered the "Most polluted place on the planet".
Nuclear weapons continued to be tested in the Cold War, especially in the earlier stages of their development. The toll on the worst-affected populations and the growth since then in understanding the critical threat to human health posed by radioactivity has also been a prohibitive complication associated with nuclear power. Though extreme care is practiced in that industry, the potential for disaster suggested by incidents such as those at Three Mile Island, Chernobyl, and Fukushima pose a lingering specter of public mistrust. Worldwide publicity has been intense on those disasters. Widespread support for test ban treaties has ended almost all nuclear testing in the atmosphere.
International catastrophes such as the wreck of the Amoco Cadiz oil tanker off the coast of Brittany in 1978 and the Bhopal disaster in 1984 have demonstrated the universality of such events and the scale on which efforts to address them needed to engage. The borderless nature of the atmosphere and oceans inevitably resulted in the implication of pollution on a planetary level with the issue of global warming. Most recently, the term persistent organic pollutant (POP) has come to describe a group of chemicals such as PBDEs and PFCs, among others. Though their effects remain poorly understood owing to a lack of experimental data, they have been detected in various ecological habitats far removed from industrial activity, such as the Arctic, demonstrating diffusion and bioaccumulation after only a relatively brief period of widespread use.
The Great Pacific Garbage Patch is a concentration of plastics in the North Pacific Gyre. It and other garbage patches contain debris that can transport invasive species and that can entangle and be ingested by wildlife. Organizations such as 5 Gyres and the Algalita Marine Research Foundation have researched the Great Pacific Garbage Patch and found microplastics in the water.
Pollution introduced by light at night is becoming a global problem, more severe in urban centres, but contaminating also large territories, far away from towns.
Growing evidence of local and global pollution and an increasingly informed public over time have given rise to environmentalism and the environmental movement, which generally seek to limit human impact on the environment.
== See also ==
Biological contamination
Brain health and pollution
Chemical contamination
Environmental health
Environmental racism
Hazardous Substances Data Bank
Overpopulation
Neuroplastic effects of pollution
Pollutant release and transfer register
Polluter pays principle
Pollution haven hypothesis
Regulation of greenhouse gases under the Clean Air Act
Pollution is Colonialism
Sacrifice zone
== References ==
== External links ==
OEHHA proposition 65 list
National Toxicology Program – from US National Institutes of Health. Reports and studies on how pollutants affect people
TOXNET – NIH databases and reports on toxicology
TOXMAP – Geographic Information System (GIS) that uses maps of the United States to help users visually explore data from the United States Environmental Protection Agency (EPA) Toxics Release Inventory and Superfund Basic Research Programs
EPA.gov – manages Superfund sites and the pollutants in them (CERCLA). Map the EPA Superfund
Toxic Release Inventory – tracks how much waste US companies release into the water and air. Gives permits for releasing specific quantities of these pollutants each year.
Agency for Toxic Substances and Disease Registry – Top 20 pollutants, how they affect people, what US industries use them and the products in which they are found
Chelyabinsk: The Most Contaminated Spot on the Planet Documentary Film by Slawomir Grünberg (1996)
Nieman Reports | Tracking Toxics When the Data Are Polluted | Wikipedia/Pollution_control |
Design for the environment (DfE) is a design approach to reduce the overall human health and environmental impact of a product, process or service, where impacts are considered across its life cycle. Different software tools have been developed to assist designers in finding optimized products or processes/services. DfE is also the original name of a United States Environmental Protection Agency (EPA) program, created in 1992, that works to prevent pollution, and the risk pollution presents to humans and the environment. The program provides information regarding safer chemical formulations for cleaning and other products. EPA renamed its program "Safer Choice" in 2015.
== Introduction ==
Initial guidelines for a DfE approach were written in 1990 by East Meets West, a New York-based non-governmental organization founded by Anneke van Waesberghe. It became a global movement targeting design initiatives and incorporating environmental motives to improve product design in order to minimize health and environmental impacts by incorporating it from design stage all the way to the manufacturing process. The DfE strategy aims to improve technology and design tactics to expand the scope of products. By incorporating eco-efficiency into design tactics, DfE takes into consideration the entire life-cycle of the product, while still making products usable but minimizing resource use. The key focus of DfE is to minimize the environmental-economic cost to consumers while still focusing on the life-cycle framework of the product. By balancing both customer needs as well as environmental and social impacts DfE aims to "improve the product use experience both for consumers and producers, while minimally impacting the environment".
== Practices ==
Four main concepts that fall under the DfE umbrella.
Design for environmental processing and manufacturing: Raw material extraction (mining, drilling, etc.), processing (processing reusable materials, metal melting, etc.) and manufacturing are done using materials and processes which are not dangerous to the environment or the employees working on said processes. This includes the minimization of waste and hazardous by-products, air pollution, energy expenditure and other factors.
Design for environmental packaging: Materials used in packaging are environmentally responsible, which can be achieved through the reuse of shipping products, elimination of unnecessary paper and packaging products, efficient use of materials and space, use of recycled and/or recyclable materials.
Design for disposal or reuse: The end-of-life of a product is very important, because some products emit dangerous chemicals into the air, ground and water after they are disposed of in a landfill. Planning for the reuse or refurbishing of a product will change the types of materials that would be used, how they could later be disassembled and reused, and the environmental impacts such materials have.
Design for energy efficiency: The design of products to reduce overall energy consumption throughout the product's life.
Life-cycle assessment (LCA) is employed to forecast the impacts of different (production) alternatives of the product in question, thus being able to choose the most environmentally friendly. A life cycle analysis can serve as a tool when determining the environmental impact of a product or process. Proper LCAs can help a designer compare several different products according to several categories, such as energy use, toxicity, acidification, CO2 emissions, ozone depletion, resource depletion and many others. By comparing different products, designers can make decisions about which environmental hazard to focus on in order to make the product more environmentally friendly.
== Rationale ==
Modern day businesses aim to produce goods at a low cost while maintaining quality, staying competitive in the global marketplace, and meeting consumer preferences for more environment friendly products. To help businesses meet these challenges, EPA encourages businesses to incorporate environmental considerations into the design process. The benefits of incorporating DfE include: cost savings, reduced business and environmental risks, expanded business and market opportunities, and to meet environmental regulations.
== Companies and products ==
Starbucks: Starbucks is decreasing its carbon footprint by building more energy efficient stores and facilities, conserving energy and water, and purchasing renewable energy credits. Starbucks has achieved LEED certificates in 116 stores in 12 countries. Starbucks has even created a portable, LEED certified store in Denver. It is Starbucks' goal to reduce energy consumption by 25% and to cover 100% of its electricity with renewable energy by 2015.
Hewlett Packard: HP is working towards reducing energy used in manufacturing, developing materials that have less environmental impact, and designing easily recyclable equipment.
IBM: Their goal is to extend product life beyond just production, and to use reusable and recyclable products. This means that IBM is currently working on creating products that can be safely disposed of at the end of its product life. They are also reducing consumption of energy to minimize their carbon footprint.
Philips: For almost 20 years now, sustainable development has been a crucial part of Philips decision making and manufacturing process. Philips' goal is to produce products with their environmental responsibility in mind. Not only are they working on reducing energy during the manufacturing process, Phillips is also participating in a unique project, philanthropy through design. Since 2005, Philips has been working on and developing philanthropy through design. They collaborate with other organizations to use their expertise and innovation to help the more fragile parts of our society.
Besides these large brand names there are several other consumer product companies in the DfE program this including:
Atlantic Chemical & Equipment Co.
American Cleaning Solutions
BCD Supply
Beta Technology
Brighton USA
== Design process ==
A business can design for the environment by:
Evaluating the human health and environmental impacts of its processes and products.
Identifying what information is needed to make human health and environment decisions
Conducting an assessment of alternatives
Considering cross-media impacts and the benefits of substituting chemicals
Reducing the use and release of toxic chemicals through the innovation of cleaner technologies that use safer chemicals.
Implementing pollution prevention, energy efficiency, and other resource conservation measures.
Making products that can be reused and recycled
Monitoring the environmental impacts and costs associated with each product or process
Recognizing that although change can be rapid, in many cases a cycle of evaluation and continuous improvement is needed.
== Safer Choice labeling program ==
EPA's DfE labeling program was renamed "Safer Choice" in 2015.
== Current U.S. laws and regulations encouraging DfE in the electronics industry ==
=== National Ambient Air Quality Standards (NAAQS) ===
EPA promulgated the National Ambient Air Quality Standards (NAAQS) to establish basic air pollution control requirements across the U.S. The NAAQS sets standards on six main sources of pollutants, which include emissions of: ozone (0.12 ppm per 1 hour), carbon monoxide (35 ppm per 1 hour; primary standard), particulate matter (50g/m^3 at an annual arithmetic mean), sulfur dioxide (80g/m^3 at an annual arithmetic mean), nitrogen dioxide (100g/m^3 at an annual arithmetic mean), and lead emissions (1.5g/m^3 at an annual arithmetic mean).
=== Stratospheric ozone protection ===
Stratospheric ozone protection is required by section 602 of the Clean Air Act of 1990. This regulation aims to decrease emission of chlorofluorocarbons (CFCs) and other chemicals that are destroying the stratospheric ozone layer. The protection initiative categorizes ozone-depleting substances into two classes: Class I, and Class II.
Class I substances include 20 different kinds of chemicals and have all been phased-out of production processes since 2000. Class II substances consist of 33 different hydro-chlorofluorocarbons (HCFCs). The EPA has already begun plans to decrease emissions in HCFCs and plan to completely phase out the class II substances by 2030.
=== Reporting requirements for releases of toxic substances ===
A firm operating in the electronics industry in Standard Industrial Classification (SIC) Codes 20-39 that has more than 10 full-time employees and consumes more than 10,000 lbs per year of any toxic chemical lists in 40 CFR 372.65 must file a toxic release inventory.
=== Other regulations ===
National Emissions Standards for Hazardous Air Pollutants (NESHAP)
National Pollutant Discharge Elimination System (NPDES–Water pollution permit program)
Underground Injection Control Program
Hazardous waste management
Underground storage tank management
== See also ==
== References ==
== External links ==
The European Union: The European Platform on Life Cycle Assessment
Sustainable design for the environment
Department Life Cycle Engineering, University of Stuttgart (English)
Sustainable Building Alliance.org
Sustainable Residential Design.org: Using Low-Impact Materials Resource Guide | Wikipedia/Design_for_the_Environment |
Drug recycling, also referred to as medication redispensing or medication re-use, is the idea that health care organizations or patients with unused drugs can transfer them in a safe and appropriate way to another patient in need. The purpose of such a program is reducing medication waste, thereby saving healthcare costs, enlarging medications’ availability and alleviating the environmental burden of medication.
== The debate ==
Despite the need for waste-preventive measures, the debate of drug recycling programs is ongoing. It is traditional to expect that consumers get prescription drugs from a pharmacy and that the pharmacy got their drugs from a trusted source, such as manufacturer or wholesaler. In a drug recycling program, consumers would access drugs through a less standardized supply chain. Consequently, concerns of the quality of the recycled drugs arise.
However, in a regulated process, monitored by specialized pharmacies or medical organization, these uncertainties can be overcome. For example, monitoring the storage conditions, including temperature, light, humidity and agitation of medication, can contribute to regulation of the quality of recycled drugs. For this purpose, pharmaceutical packaging could be upgraded with sensing technologies, that can also be designed to detect counterfeits. Such packaging requires an initial investment, but this can be compensated with potential cost savings obtained by a drug recycling program. Accordingly, drugs recycling seems economically viable for expensive drugs, such as HIV post-exposure prophylaxis medication.
== Donating practices ==
In some countries, drug recycling programs operate successfully by donating unused drugs to the less fortunate. In the United States drug recycling programs exist locally. As of 2010, Canada had fewer drug recycling programs than the United States. These programs occur in specific pharmacies only, since these pharmacies are prepared to address the special requirements of participating in a recycling program. Usually, drug returns happen without financial compensation. In Greece, the organization GIVMED operates in drug recycling, and saved over half a million euros by recycling almost 60k drug packages since 2016.
However, in other countries, such as Canada, implementation of drug recycling programs is limited. Other initiatives focus on donating drugs to third world countries. However, this is accompanied with ethical constraints due to uncertainties in quality, as well as practical constraints, due to making the drugs only temporarily available and not necessarily addressing local needs. The World Health Organization provided guidelines on appropriate drug donation, thereby discouraging donation practices that do not consider recipient's needs, government policies, effective coordination or quality standards.
== Towards redispensing as standard of care ==
Alternatively, drug recycling programs could be set as routine clinical practice with the aim of reducing the economic and environmental burden of medication waste. Still, for general implementation of drug recycling programs, clear professional guidelines are required. Research could provide the rationale for these guidelines. For example, research showed that a majority of patients is willing to use recycled drugs if the quality is maintained, and explored requirements for a drug recycling program perceived by stakeholders, including the general public, pharmacists. and policy-makers.
One can assume that implementing drug recycling as routine clinical practice is only attractive from an economical perspective, if the savings exceed the operational pharmacy costs. For this purpose, research should assess the feasibility of drug recycling. In the Netherlands, redispensing of unused oral anticancer drugs is currently tested in routine clinical practice to determined cost-savings of a quality-controlled process. This data could help policy-makers to prioritize drug recycling on their agenda, thereby facilitating guidelines for general implementation of drug recycling.
== References ==
== External links ==
Guidance Document: Best Management Practices for Unused Pharmaceuticals at Health Care Facilities, a 2010 publication from the United States Environmental Protection Agency | Wikipedia/Drug_recycling |
Kalundborg Eco-Industrial Park is an industrial symbiosis network located in Kalundborg, Denmark, in which companies in the region collaborate to use each other's by-products and otherwise share resources.
The Kalundborg Eco-Industrial Park is the first full realization of industrial symbiosis. The collaboration and its environmental implications arose unintentionally through private initiatives, as opposed to government planning, making it a model for private planning of eco-industrial parks. At the center of the exchange network is the Asnæs Power Station, a 1500MW coal-fired power plant, which has material and energy links with the community and several other companies. Surplus heat from this power plant is used to heat 3500 local homes in addition to a nearby fish farm, whose sludge is then sold as a fertilizer. Steam from the power plant is sold to Novo Nordisk, a pharmaceutical and enzyme manufacturer, in addition to Statoil oil refinery. This reuse of heat reduces the amount thermal pollution discharged to a nearby fjord. Additionally, a by-product from the power plant's sulfur dioxide scrubber contains gypsum, which is sold to a wallboard manufacturer. Almost all of the manufacturer's gypsum needs are met this way, which reduces the amount of open-pit mining needed. Furthermore, fly ash and clinker from the power plant is used for road building and cement production. These exchanges of waste, water and materials have greatly increased environmental and economic efficiency, as well as created other less tangible benefits for these actors, including sharing of personnel, equipment, and information.
== History ==
The Kalundborg Industrial Park was not originally planned for industrial symbiosis. Its current state of waste heat and materials sharing developed over a period of 20 years. Early sharing at Kalundborg tended to involve the sale of waste products without significant pretreatment. Each further link in the system was negotiated as an independent business deal, and was established only if it was expected to be economically beneficial.
The park began in 1959 with the start up of the Asnæs Power Station. The first episode of sharing between two entities was in 1972 when Gyproc, a plaster-board manufacturing plant, established a pipeline to supply gas from Tidewater Oil Company. In 1981 the Kalundborg municipality completed a district heating distribution network within the city of Kalundborg, which utilized waste heat from the power plant.
Since then, the facilities in Kalundborg have been expanding, and have been sharing a variety of materials and waste products, some for the purpose of industrial symbiosis and some out of necessity, for example, freshwater scarcity in the area has led to water reuse schemes. In particular, 700,000 cubic meters per year of cooling water is piped from Statoil to Asnaes.
A timeline of the creation of the industrial park:
1959 The Asnæs Power Station was started up
1961 Tidewater Oil Company constructed a pipeline from Lake Tissø to provide water for its operation
1963 Tidewater Oil Company's oil refinery is taken over by Esso
1972 Gyproc establishes plaster-board manufacturing plant. A pipeline from the refinery to the Gyproc facility is constructed to supply excess refinery gas
1973 The Asnæs Power Station is expanded. A connection is built to the Lake Tissø-Statoil pipeline
1976 Novo Nordisk starts delivering biological sludge to neighboring farms
1979 Asnæs Power Station starts supplying fly ash to cement manufacturers in northern Denmark
1981 the Kalundborg municipality completes a district heating distribution network within the city that utilizes waste heat from the power plant
1982 Novo Nordisk and the Statoil refinery complete construction of steam supply pipelines from the power plant. By purchasing process steam from the power plant, the companies are able to shut down inefficient steam boilers
1987 The Statoil refinery completes a pipeline to supply its effluent cooling water to the power plant for use as raw boiler feed water.
1989 The power plant starts using waste heat from its salt cooling water to produce trout and turbot at its local fish farm
1989 Novo Nordisk enters into agreement with Kalundborg municipality, the power plant, and the refinery to connect to the water supply grid from Lake Tissø
1990 The Statoil refinery completes construction of a sulphur recovery plant. The recovered sulphur is sold as raw material to a sulfuric acid manufacturer in Fredericia
1991 The Statoil refinery commissions the building of a pipeline to supply biologically treated refinery effluent water to the power plant for cleaning purposes, and for fly ash stabilization
1992 The Statoil refinery commissions the building of a pipeline to supply flare gas to the power plant as a supplementary fuel
1993 The power plant completes a stack flue gas desulfurization project. The resulting calcium sulphate is sold to Gyproc, where it replaces imported natural gypsum
== The Symbiosis ==
The relationships among the firms comprising the Kalundborg Eco-Industrial Park form an industrial symbiosis. Generally speaking, the actors involved in the symbiosis at Kalundborg exchange material wastes, energy, water, and information. The Kalundborg network involves a number of actors, including a power station, two big energy firms, a plaster board company, and a soil remediation company. Other actors include farmers, recycling facilities, and fish factories that use some of the material flows. Kalundborg Municipality Archived 2013-07-01 at archive.today plays an active role. Additionally, other actors, such as Novoren, a recycling and urban land field firm, are formally part of the network but do not contribute
tangibly in the exchange. A researcher studying the evolution of the Kalundborg Symbiosis concluded that a high level of trust between the actors involved represented an essential element to collaborative success.
=== Partners ===
The Kalundborg Eco-Industrial Park today includes nine private and public enterprises, some of which are some of the largest enterprises in Denmark. The enterprises are:
Novo Nordisk - Danish company and largest producer of insulin in the world
Novozymes - Danish company and largest enzyme producer in the world
Gyproc - French producer of gypsum board
Kalundborg Municipality
Ørsted A/S - owner of Asnaes Power Station, the largest power plant in Denmark
RGS 90 - Danish soil remediation and recovery company
Statoil - Norwegian company which owns Denmark's largest oil refinery
Kara/Novoren - Danish waste treatment company
Kalundborg Forsyning A/S - water and heat supplier, as well as waste disposer, for Kalundborg citizens
=== Material Exchanges ===
There are currently over thirty exchanges of materials among the actors of Kalundborg. The Asnaes Power Station is at the heart of the network. The power company gives its steam residuals to the Statoil Refinery, meeting 40% of its steam requirements, in exchange for waste gas from the refinery. The power plant creates electricity and steam from this gas. These products are sent to a fish farm and Novo Nordisk, who receive all of their required steam from Asnaes, and a heating system that supplies 3500 homes. These homeowners pay for the underground piping that supplies their heat, but receive the heat reliably and at a low price. Fly ash from Asnaes is sent to a cement company, and gypsum from its desulfurization process is sent to Gyproc for use in gypsum board. Two-thirds of Gyproc's gypsum needs are met by Asnaes. Statoil Refinery removes sulfur from its natural gas and sells it to a sulfuric acid manufacturer, Kemira. The fish farm sells sludge from its ponds as fertilizer to nearby farms, and Novo Nordisk gives away its own sludge, of which it produces 3,000 cubic meters per day. The sludge is to be refined for biogas for the power plant.
Water reuse schemes have also been developed within Kalundborg. Statoil pipes 700,000 cubic meters of cooling water per year to Asnaes, which purifies it and uses it as "boiler feed-water." Asnaes also uses approximately 200,000 cubic meters of Statoil's treated wastewater per year for cleaning. The 90 °C residual heat from the refinery is not used for district heating due to taxes. Instead, heat pumps are used with the 24 °C waste water as a heat reservoir.
== Savings and environmental impacts ==
Since its start over 25 years ago, Kalundborg has been operating successfully as an eco-industrial park. One of the main goals of industrial symbiosis is to make goods and services that use the least-cost combination of inputs. These relationships were formed on an economic and environmental basis. As mentioned above, there are over thirty exchanges occurring in Kalundborg. While Kalundborg does operate using trades between various firms in the vicinity, it itself is not self-sufficient or contained to the industrial park. There are many trades that occur with companies outside of this park region.
All of these exchanges have contributed to water savings, and savings in fuel and input chemicals. Wastes were also avoided through these interchanges. For example, in 1997, Asnaes (the power station) saved 30,000 tons of coal (~2% of throughput) by using Statoil (large oil refinery) fuel gas. And 200,000 tons of fly ash and clinker were avoided from Asnaes landfill. These resources savings and waste avoidances, documented before 1997, are illustrated in the tables to the right.
A study in 2002 showed that these exchanges also contributed to more than 95% of the total water supply to the power plant. This is up from 70% in 1990. So, the system is becoming more comprehensive in its ability to save groundwater, however, there is still room for improvement. Out of the 1.2 million m3 of wastewater discharged from Statoil (the refinery), only 9000 m3 were reused at the power plant.
More recent numbers show a vast improvement, when comparing to the numbers from 1997, in resource savings. Data from around 2004 show annual savings of 2.9 million cubic meters of ground water, and 1 million cubic meters of surface water. Gypsum savings are estimated around 170,000 tons, and sulfur dioxide waste avoidance is estimated around 53 Tn. These numbers are mostly estimations. Aspects of the eco-industrial park have changed, and there are many levels to consider when doing these calculations. All together though, these interchanges have shown annual savings of up to $15 million (US), with investments around $78.5 million (US). The total accumulated savings is estimated around $310 million (US).
== As a Model ==
Kalundborg was the first example of separate industries grouping together to gain competitive advantage by material exchange, energy exchange, information exchange, and/or product exchange. The very term industrial symbiosis (IS) was first defined by a station manager in Kalundborg as "a cooperation between different industries by which the presence of each…increases the viability of the others, and by which the demands of society for resource savings and environmental protection are considered".
Kalundborg's success helped generate interest in industrial symbiosis. Developed nations such as the United States began to formulate incentives for corporations to implement materials exchange with other corporations. Industrial and political circles began to look into the implementation of eco-industrial parks (EIPs). Specifically, the United States worked to put into service several planned EIPs. The U.S. President's Council on Sustainable Development in 1996 proposed fifteen eco-industrial parks to pursue the idea of industrial symbiosis. These parks were created by grouping diverse stakeholders with common material flows together, with added governmental incentives to encourage materials exchange. The goal of these planned EIPs was to test if the industrial symbiosis that worked so well in Kalundborg could be replicated. The Council on Sustainable Development also defined 5 major characteristics of a successful EIP to help guide EIP development. These characteristics include: (1) some form of material exchange between multiple separate entities, (2) industries in close proximity to each other, (3) cooperation between plant management of the different corporations, (4) an existing infrastructure for material sharing that does not require much retooling, and (5) "anchor" tenants (large corporation with resources to support early implementation). Devens Regional Enterprise Zone is a good example of a successful EIP in the United States.
Kalundborg became an attractive topic in academia as well because of the obvious sustainability advantages of industrial symbiosis. Research conducted on planning and implementation of eco-industrial parks revealed interesting results. Experts argued over the idea of "planned parks" versus "self organized parks". Research showed systematic failure of forced or planned EIPs. Most successful EIPs originate from industrial symbiosis that occurs naturally during industry life, much like the Kalundborg case. This conclusion served to deflate the momentum that the success of Kalundborg generated. Organizations began to recognize the difficulties associated with forcing eco-industrial parks to coalesce and abandoned the idea.
== See also ==
Eco-industrial park
EcoPark - EIP in Hong-Kong
Industrial ecology
Industrial symbiosis
== References ==
== External links ==
The Kalundborg Centre for Industrial Symbiosis
Indigo Development Eco-Industrial Park page and handbook
Existing and Developing Eco-Industrial Park Sites in the U.S.
The Kalundborg eco-industrial park with a perspective of sustainable city planning (Chinese version) Archived 2015-06-17 at the Wayback Machine | Wikipedia/Kalundborg_Eco-industrial_Park |
Green industrial policy (GIP) is strategic government policy that attempts to accelerate the development and growth of green industries to transition towards a low-carbon economy. Green industrial policy is necessary because green industries such as renewable energy and low-carbon public transportation infrastructure face high costs and many risks in terms of the market economy. Therefore, they need support from the public sector in the form of industrial policy until they become commercially viable.
Natural scientists warn that immediate action must occur to lower greenhouse gas emissions and mitigate the effects of climate change. Social scientists argue that the mitigation of climate change requires state intervention and governance reform. Thus, governments use GIP to address the economic, political, and environmental issues of climate change. GIP is conducive to sustainable economic, institutional, and technological transformation. It goes beyond the free market economic structure to address market failures and commitment problems that hinder sustainable investment. Effective GIP builds political support for carbon regulation, which is necessary to transition towards a low-carbon economy. Several governments use different types of GIP that lead to various outcomes. The Green Industry plays a pivotal role in creating a sustainable and environmentally responsible future; By prioritizing resource efficiency, renewable energy, and eco-friendly practices, this industry significantly benefits society and the planet at large.
GIP and industrial policy are similar, although GIP has unique challenges and goals. GIP faces the particular challenge of reconciling economic and environmental issues. It deals with a high degree of uncertainty about green investment profitability. Furthermore, it addresses the reluctance of industry to invest in green development, and it helps current governments influence future climate policy.
GIP offers opportunities for energy transition to renewables and a low-carbon economy. A large challenge for climate policy is a lack of industry and public support, but GIP creates benefits that attract support for sustainability. It can create strategic niche management and generate a "green spiral," or a process of feedback that combines industrial interests with climate policy. GIP can protect employees in emerging and declining industries, which increases political support for other climate policy. Carbon pricing, sustainable energy transitions, and decreases in greenhouse gas emissions have higher chances of success as political support increases. GID is closely associated with the green recovery, a set of policy directives to address the economic effects of COVID-19 and the environmental effects of climate change by encouraging renewable energy expansion and green job growth. However, GIP faces many risks. Some risks include poor government choices about which industries to support; political capture of economic policy; wasted resources; ineffective action to combat climate change; poor policy design that lacks policy objectives and exit strategies; trade disputes; and coordination failure. Strategic steps can be taken to manage the risks of GIP. Some include public and private sector communication, transparency, and accountability; policy with clear objectives, evaluation techniques, and exit strategies; policy learning and policy experimentation; green rent management; strong institutions; and a free press.
Governments in various countries, states, provinces, territories, and cities use different types of green industrial policy. Distinct policy instruments lead to several outcomes. Examples include sunrise and sunset policies, subsidies, research and development, local content requirements, feed-in tariffs, tax credits, export restrictions, consumer mandates, green public procurement rules, and renewable portfolio standards.
== Versus industrial policy ==
GIP and industrial policy (IP) have similarities. Both seek to promote the development of industries and the creation of new technology. Each approach also involves government intervention in the economy to address economic issues and market failures. Both use similar policy approaches, like research and development subsidies and tax credits. Further, they face comparable risks, such as implementation failure that occurs when the government fails to monitor the policy adequately. Additionally, the two are related because policymakers can use information from past IP when they design and implement GIP. Policymakers can apply policy learning and lesson drawing from the failures and successes of IP to GIP to lower its risks. For example, an important lesson from IP is that what works for one region will not necessarily work for another, so policymakers cannot directly adopt policy from a different region because it must address an area's local context to ensure success. Overall, the two approaches have many things in common.
However, GIP differs significantly from IP because it addresses environmental concerns, whereas IP does not. The current economy focuses on private benefits, such as immediate profitability, rather than social benefits, like reducing pollution. Since green investment has less private benefits than social benefits, GIP deals with the unique commitment problem that green investment profitability is highly uncertain, so firms are reluctant to invest. As a result, governments use GIP to promote green investments. Future environmental policy success, like carbon taxation policy, hinges on the future availability of renewable energy. Current investment is the only way to ensure future availability, and GIP addresses this fact. Efficient and accessible green technology will also make it politically easier to adopt future low-carbon policies. Thus, a transition towards a low-carbon economy depends on current investment, and as such, it depends on GIP.
== Energy transitions ==
The persistence of a carbon-based economy has led to environmentally destructive path dependency, and energy transitions are vital to divert from the reliance. Strategic niche management (SNM) offers an opportunity for energy transitions. New, sustainable technologies cannot immediately compete on the market with existing, unsustainable technologies due to path dependency. Green innovations that are not immediately profitable are vital for inducing sustainable development and achieving societal goals of mitigating climate change. Thus, governments must create technological niches and use forms of GIP to subsidize and nurture technological niches to ensure that green innovations develop. Technical niches provide protected space for innovative sustainable development that co-evolves with user practices, regulatory structures, and technology. Co-evolutionary dynamics are necessary for successful niche innovation -- multiple actors from multiple layers must work together for sustainable transitions. Social networks are essential for this niche development because numerous stakeholders lead to many points of view, more commitment and resources, and more innovation.
Sustainable urbanization models in cities are examples of SNM. In these instances, municipal governments and social networks help create small-scale testing spaces that allow for technological and social innovation, such as developing electric car technology and encouraging car-sharing. Overall, electric cars have not become a norm in the automobile industry. However, if a technological niche successfully emerges in the market, it can transform into a market niche and solidify its place in the industry and the socio-technical regime. In turn, the regime, or industry, influences the landscape, which can change the economic climate and induce sustainable energy transitions. Therefore, SNM and GIP can break path dependency and solidify the place of green technologies in markets and society.
Green industrial policy can induce a green spiral and can also break path dependency. Economists view carbon pricing as the most compelling approach to the mitigation of climate change, but their opinion ignores the political cost of the radical adoption of carbon pricing and its lack of political feasibility. Consequently, the immediate adoption of carbon pricing often fails, and carbon pricing schemes often adapt to the demands of the polluters, which makes them ineffective. GIP addresses the issue of a lack of political feasibility through green spiral.
Green spiral means that GIP and carbon pricing approaches are most effective when policymakers produce them in a sequence to increase climate policy support over time and encourage positive feedback. GIP encourages increases in policy support as it contributes to the growth of a political landscape of coalitions and interests, such as renewable energy firms and investors, that benefit from energy transformation. Those alliances and interests generate political support for GIP, even when unsustainable industries may oppose it. They also become political allies during the development of stricter climate policy that negatively affects polluters. Thus, GIP creates positive feedback. Early GIP helps green industries expand, and the more they expand, the more support increases for decarbonized energy systems, and the easier it becomes to apply stricter climate policy. A green spiral makes sustainability feasible, attractive, and profitable for industries, which encourages the adoption of sustainable business techniques. For example, feed-in tariffs create direct incentives for the growth of green industry groups and can push sustainable shifts in investment and revenues. These shifts then create support for policy and technology experimentation, and they induce progress towards system-wide transformation. A green spiral can create energy transitions to renewables and lower the political costs of transitions.
=== Environmental benefits ===
GIP does not immediately create a radical transformation to a green economy, but it represents practical steps towards it, and energy transitions are one of its primary goals. Without government intervention in the economy, it is unlikely that the current market will transition towards a low-carbon economy. GIP also increases political support for further climate policy. Therefore, GIP has the potential for environmental benefits. Green technologies emit fewer greenhouse gases (GHG) and use fewer resources or economize on renewable resources. A majority of natural scientists agree that an enormous reduction in GHGs is essential to mitigate the effects of climate change, such as a rise in global temperatures, droughts, floods, extreme weather events, diseases, food shortages, and species extinction. Since GIP can reduce GHG emissions, it can protect the environment, and in turn, it can preserve the health, safety, and security of humans and other species. Not all green industrial policies are successful in achieving a reduction in emissions, but some form of failure is inevitable within the policy and economic realms, and governments learn from failures to improve future policy. Immediate action is necessary to address climate change and protect the environment, and GIP offers the tools to do so.
=== Worker benefits ===
GIP creates sunrise policies and sunset policies that produce benefits for employees. Sunrise policies aim to set up and develop new technologies or grow green sectors, and they create new employment opportunities in green industries. For instance, GIP investment in research and development helped develop the renewable energy sector in Germany. GIP led to a booming German renewable energy industry that employs over 371,000 people, which is double the number of jobs that were available in 2004. Investment in innovation can also increase economic growth, which can create further benefits, such as job availability, job stability, and increased salaries. In contrast, sunset policies support declining industries to allow for a smooth economic transition away from energy-intensive industries towards sustainable ones. Sunset policies are expensive, but they are often a requirement for the political acceptability of energy transitions. Examples include retraining schemes for workers in declining industries, funding to adjust production technologies to make them more sustainable, and social safety nets, including unemployment insurance. To conclude, GIP is beneficial for both the environment and workers, which creates political support for climate policy and makes energy transitions just and feasible.
== Risks ==
Proponents and skeptics of GIP acknowledge that it involves numerous risks. Arguments against GIP state that governments cannot make practical choices about which firms or industries to support, and subsequently, they will make mistakes and waste valuable resources. Additionally, GIP raises concerns about rent-seeking and regulatory capture. Government intervention in markets can create rent-seeking behaviour - or the manipulation of policy to increase profits - so GIP may become driven by political concerns rather than economic ones. Subsidies are particularly prone to rent-seeking as special interests may lobby intensely to maintain subsidies, even when they are no longer needed, while taxpayers who may want to abolish subsidies have fewer resources for lobbying. Political capture of economic policy leads to a reluctance to abandon a failing or expensive policy, and if rent-seeking occurs, a policy is bound to be ineffective, which will waste resources.
Inadequate policy design can also lead to the failure of GIP. Failure is likely if GIP does not have clear objectives, benchmarks to measure success, close monitoring, and exit strategies. For instance, the U.S. government partially funded Solyndra, an energy efficiency firm in California, United States. The funding came from poorly planned policy, and it experienced political capture, which led to its failure.
GIP is also not an immediate solution, so skeptics argue that it constitutes ineffective action to address climate change.
Trade disputes are another risk because GIP created a new strand of trade and environment conflicts within the World Trade Organization (WTO). For example, policies with local content requirements have induced several trade disputes.
Finally, coordination failure is a significant risk, as green innovation requires inter-agency, inter-sectoral, and public-private coordination, which can be difficult to produce, and requires strong institutions. Thus, there are several potential issues of GIP, but there are several approaches to address the risks.
=== Addressing risks ===
While proponents of GIP discuss several ways to mitigate risks, it is important to note that some instances of targeting the wrong firms or industries are inevitable because some degree of failure is inherent in GIP effort. Profit cannot measure success, but rather, success occurs with the creation of environmental and technological externalities. Governments can take several steps to lower risks and ensure success. For example, they can make sufficient choices about which industries or companies to support to avoid failure. Governments can also avoid using the wrong policy instruments if they experiment in select parts of the country before applying policy country-wide. Policy learning and lesson drawing from industrial policy and GIP can also foster the adoption of correct policy instruments. Further, rent-seeking can be an issue, but the creation of rent attracts investors into risky green technology fields. Rent management can avoid the problem by dictating the correct amount of profit, appropriately offering profit incentives, and withdrawing them when markets can function on their own. Governments must also work with the private sector, and the two should have a mutual interest and understanding of the issues each seeks to address, although governments must avoid capture by the private sector. Independent monitoring of policy progress, strong institutions, consumer protection agencies, and a free press can deal with the risk of political capture. Furthermore, clear objectives, consistent monitoring, evaluation techniques, and exit strategies can strengthen policies. Policies can avoid trade disputes through the process of policy learning and by adhering to WTO rules. Policymakers can also evade ineffective GIP through the creation of a transparent and accountable political coalition of actors, which includes public-private partnerships, business alliances, and civil society. A strong coalition also addresses coordination failures. The extra risks of GIP options could avoid future costs by increasing progress toward more ambitious cuts in emissions. As a result, GIP that is politically optimal may be economically optimal in the long-run, even if it experiences immediate inefficiencies.
== Examples ==
The following section includes examples of GIP.
=== Subsidies ===
Subsidies help offset the private costs of green investments. Subsidies for a targeted sector are the most common form of GIP. The WTO defines three types of government subsidies. The first is governments transfers or private transfers mandated by the government that create budgetary outlays. The second is programs that provide goods or services below cost, and the third is regulatory policies that create transfers from one person or group to another. The International Energy Agency predicts that subsidies for green energy will expand to almost $250 billion in 2035, compared to $39 billion in 2007. Subsidies directly contributed to the growth of renewable energy industries, and the positive benefits spread globally as the cost of renewables steadily declined. The WTO has rules that constrain subsidies to avoid rent-seeking.
=== Research and development ===
Research and development (R&D) is an essential GIP instrument because it generates green technologies. An example of green R&D is the scientific agency United States Geological Survey (USGS), which is a part of the United States government. It receives government funding for the USGS Climate R&D Program, which seeks to mitigate the complex issues of climate change. Another example is the Program of Energy Research and Development, which is run by the Canadian federal government. It provides R&D funding for federal departments and agencies, such as Agriculture and Agri-Food Canada and Transport Canada. The federal government encourages the departments and agencies to collaborate with the private sector, international organizations, universities, and provincial and municipal governments. Similar to the American program, the objective of the Canadian program is to create a sustainable energy future.
=== Local content requirements ===
Local content requirements (LCRs) mean that in the production process, producers must obtain a certain minimum percentage of goods, labour, or services from local sources. Ontario, Canada passed legislation with local content requirements in 2009 called the Green Energy and Green Economy Act. Its objectives were to expand renewable energy production and use, promote the conservation of energy, and create new green employment. The Act required Ontario-made content in renewable electricity generators, such as wind and solar farms, for the generators to be eligible for government subsidies. It created many jobs, lowered GHG emissions, and vastly expanded the renewable energy industry in Ontario. Japan and the European Union disputed the requirements, and the WTO ruled that Ontario must remove LCRs from the Act. The trade dispute and its WTO decision had adverse effects in Ontario, as support for green innovation declined, and worldwide, as many countries that used LCRs in successful GIP learned that LCRs violate WTO regulations.
=== Feed-in tariffs ===
Feed-in tariffs (FITs) are a series of policies that create long-term financial encouragement for renewable energy generation. There are different versions of FITs. One version provides a fixed price for renewables, and the price is usually higher than the market rate for non-renewable energy. The fixed price guarantee counteracts the increased costs that renewable energy producers experience, and the elimination of a cost disadvantage encourages investment and innovation. Germany's FIT approach has received worldwide acclaim as it transformed Germany into a renewable energy leader.
=== Tax credits and incentives ===
There are several green tax credits available for individuals and businesses to create financial incentives for eco-conscious actions. Several countries have tax credits for electric vehicles, including Canada, the United States, Australia, and countries in Europe. In the United States, Internal Revenue Code Section 30D provides a tax credit for plug-in electric vehicles, and the total amount of credit available is $7,500. In Belgium, the registration fee for vehicles does not apply to electric cars and plug-in hybrids. Additionally, corporations with zero-emissions automobiles have a deductibility rate of 120 percent. Several other European countries have exemptions from car-related taxes, including Austria, Bulgaria, Czech Republic, Denmark, Finland, France, Germany, Greece, Hungary, Ireland, Italy, Luxembourg, the Netherlands, Portugal, Romania, Slovakia, Spain, Sweden, and the United Kingdom.
=== Export restrictions ===
Export restrictions involve inhibiting exports of a resource with the objective of increasing competitiveness of a domestic industry that relies on the resource. The limits use taxes or quotas, or a combination of them. China restricted the export of minerals and rare earth elements and argued that restrictions constrain production, which decreases environmental harm. The limitations are for China's economic benefit, but extracting and refining the resources indeed causes environmental damage, so the policy does protect the environment. However, export restrictions can distort the trade market and negatively affect foreign consumers, which can lead to WTO challenges.
=== Mandates ===
Renewable energy mandates require that companies or consumers produce or sell a certain amount of energy from renewables. Australia's Small-scale Renewable Energy Scheme is an incentive for individual citizens and small-scale businesses to install renewable energy systems, such as rooftop solar systems. Its Large-scale Renewable Energy Target requires an increase in annual renewable electricity generation. Of the power that electricity retailers provide, 12.75 percent of it must be renewable to be eligible for subsidies. Australian electricity consumers pay for the subsidies that support the scheme.
=== Green public procurement ===
Green public procurement (GPP) occurs when governments obtain goods, works, and services that are sustainable and environmentally friendly. Rules encourage the public sector to purchase green products and supplies, such as energy efficient computers, recycled paper, green cleaning services, electric vehicles, and renewable energy. These rules can drive green innovation and produce financial savings. Also, GPP can create economic growth and increase the sales of eco-industries. An example of GPP is A Plan for Public Procurement: food and catering in the United Kingdom, which encourages sustainable food procurement for the public sector and its suppliers, and it sets out a vision for specific targets and outcomes. The policy addresses issues such as energy use, water and waste, seasonality, animal welfare, and fair trade.
=== Renewable portfolio standards ===
Renewable portfolio standards (RPS) are regulatory mandates that support increased production of renewables. Standards set a minimum amount for annual production of renewable energy. In Michigan, the United States, the 2016 Clean, Renewable and Efficient Energy Act requires that electric providers increase their supply of renewables from 10 percent in 2015 to 15 percent in 2021, with an interim requirement of 12.5 percent in 2019 and 2020. In the United States, state-level RPS have driven the development of renewable energy. RPS-motivated development accounted for 60 percent of American new renewable development in 2012.
== See also ==
Climate change
Climate change mitigation
Energy transition
Green New Deal
Greenhouse gas
Industrial policy
Low-carbon economy
The Lucas Plan
Path dependence
Positive feedback
Public-private partnership
== References == | Wikipedia/Green_industrial_policy |
An organigraph is a graphical representation of a company's structure or processes. It is used as an alternative to a traditional organizational chart as it does not imply the same degree of linear hierarchy that an organizational chart does.
Organigraphs are used to expose critical associations and competitive opportunities as opposed to viewing all parties, departments, and business units as separate entities. They also can reveal relationships between departments, products, supply chains, and more within an organization that might not otherwise be apparent. Business strategists, consultants, and academics use organigraphs.
Around the year 2000, Henry Mintzberg and Ludo Van der Heyden conceived the organigraph.
Organigraphs can be created as diagrams or as images which represent the nature of the firm. For example, a computer company's organigraph could be in the form of a computer. The hard drive could represent employees, the power supply could relate to its financing, and the web browser could indicate the firm's strategy.
== See also ==
Ecosystem
Industrial ecology
== References == | Wikipedia/Organigraph |
Human population planning is the practice of managing the growth rate of a human population. The practice, traditionally referred to as population control, had historically been implemented mainly with the goal of increasing population growth, though from the 1950s to the 1980s, concerns about overpopulation and its effects on poverty, the environment and political stability led to efforts to reduce population growth rates in many countries. More recently, however, several countries such as China, Japan, South Korea, Russia, Iran, Italy, Spain, Finland, Hungary and Estonia have begun efforts to boost birth rates once again, generally as a response to looming demographic crises.
While population planning can involve measures that improve people's lives by giving them greater control of their reproduction, a few programs, such as the Chinese government's "one-child policy and two-child policy", have employed coercive measures.
== Types ==
Three types of population planning policies pursued by governments can be identified:
Increasing or decreasing the overall population growth rate.
Increasing or decreasing the relative population growth of a subgroup of people, such as those of high or low intelligence or those with special abilities or disabilities. Policies that aim to boost relative growth rates are known as positive eugenics; those that aim to reduce relative growth rates are known as negative eugenics.
Attempts to ensure that all population groups of a certain type (e.g. all social classes within a society) have the same average rate of population growth.
== History ==
=== Ancient times through Middle Ages ===
A number of ancient writers have reflected on the issue of population. At about 300 BC, the Indian political philosopher Chanakya (c. 350-283 BC) considered population a source of political, economic, and military strength. Though a given region can house too many or too few people, he considered the latter possibility to be the greater evil. Chanakya favored the remarriage of widows (which at the time was forbidden in India), opposed taxes encouraging emigration, and believed in restricting asceticism to the aged.
In ancient Greece, Plato (427-347 BC) and Aristotle (384-322 BC) discussed the best population size for Greek city-states such as Sparta, and concluded that cities should be small enough for efficient administration and direct citizen participation in public affairs, but at the same time needed to be large enough to defend themselves against hostile neighbors. In order to maintain a desired population size, the philosophers advised that procreation, and if necessary, immigration, should be encouraged if the population size was too small. Emigration to colonies would be encouraged should the population become too large. Aristotle concluded that a large increase in population would bring, "certain poverty on the citizenry and poverty is the cause of sedition and evil." To halt rapid population increase, Aristotle advocated the use of abortion and the exposure of newborns (that is, infanticide).
Confucius (551-478 BC) and other Chinese writers cautioned that, "excessive growth may reduce output per worker, repress levels of living for the masses and engender strife." Some Chinese writers may also have observed that "mortality increases when food supply is insufficient; that premature marriage makes for high infantile mortality rates, that war checks population growth." It is particularly noteworthy that Han Fei (281-233 BC), long before Malthus, had already noted the conflict between a population growing at the exponential rate and a food supply growing at the arithmetic rate. Not only did he conclude that overpopulation was the root cause of the intensification of political and social conflict, but he also reduced traditional morality to an evolutionary product of material surplus rather than having any objective value. Nevertheless, during the Han Dynasty, the emperors enacted a large number of laws to encourage early marriage and childbirth.
Ancient Rome, especially in the time of Augustus (63 BC-AD 14), needed manpower to acquire and administer the vast Roman Empire. A series of laws were instituted to encourage early marriage and frequent childbirth. Lex Julia (18 BC) and the Lex Papia Poppaea (AD 9) are two well-known examples of such laws, which among others, provided tax breaks and preferential treatment when applying for public office for those who complied with the laws. Severe limitations were imposed on those who did not. For example, the surviving spouse of a childless couple could only inherit one-tenth of the deceased fortune, while the rest was taken by the state. These laws encountered resistance from the population which led to the disregard of their provisions and to their eventual abolition.
Tertullian, an early Christian author (ca. AD 160-220), was one of the first to describe famine and war as factors that can prevent overpopulation. He wrote: "The strongest witness is the vast population of the earth to which we are a burden and she scarcely can provide for our needs; as our demands grow greater, our complaints against Nature's inadequacy are heard by all. The scourges of pestilence, famine, wars, and earthquakes have come to be regarded as a blessing to overcrowded nations since they serve to prune away the luxuriant growth of the human race."
Ibn Khaldun, a North African polymath (1332–1406), considered population changes to be connected to economic development, linking high birth rates and low death rates to times of economic upswing, and low birth rates and high death rates to economic downswing. Khaldoun concluded that high population density rather than high absolute population numbers were desirable to achieve more efficient division of labour and cheap administration.
During the Middle Ages in Christian Europe, population issues were rarely discussed in isolation. Attitudes were generally pro-natalist in line with the Biblical command, "Be ye fruitful and multiply."
When Russian explorer Otto von Kotzebue visited the Marshall Islands in Micronesia in 1817, he noted that Marshallese families practiced infanticide after the birth of a third child as a form of population planning due to frequent famines.
=== 16th and 17th centuries ===
European cities grew more rapidly than before, and throughout the 16th century and early 17th century discussions on the advantages and disadvantages of population growth were frequent. Niccolò Machiavelli, an Italian Renaissance political philosopher, wrote, "When every province of the world so teems with inhabitants that they can neither subsist where they are nor remove themselves elsewhere... the world will purge itself in one or another of these three ways," listing floods, plague and famine. Martin Luther concluded, "God makes children. He is also going to feed them."
Jean Bodin, a French jurist and political philosopher (1530–1596), argued that larger populations meant more production and more exports, increasing the wealth of a country. Giovanni Botero, an Italian priest and diplomat (1540–1617), emphasized that, "the greatness of a city rests on the multitude of its inhabitants and their power," but pointed out that a population cannot increase beyond its food supply. If this limit was approached, late marriage, emigration, and the war would serve to restore the balance.
Richard Hakluyt, an English writer (1527–1616), observed that, "Through our longe peace and seldom sickness... we are grown more populous than ever heretofore;... many thousands of idle persons are within this realme, which, having no way to be sett on work, be either mutinous and seek alteration in the state, or at least very burdensome to the commonwealth." Hakluyt believed that this led to crime and full jails and in A Discourse on Western Planting (1584), Hakluyt advocated for the emigration of the surplus population. With the onset of the Thirty Years' War (1618–48), characterized by widespread devastation and deaths brought on by hunger and disease in Europe, concerns about depopulation returned.
== Population planning movement ==
In the 20th century, population planning proponents have drawn from the insights of Thomas Malthus, a British clergyman and economist who published An Essay on the Principle of Population in 1798. Malthus argued that, "Population, when unchecked, increases in a geometrical ratio. Subsistence only increases in an arithmetical ratio." He also outlined the idea of "positive checks" and "preventative checks." "Positive checks", such as diseases, wars, disasters, famines, and genocides are factors which Malthus believed could increase the death rate.
"Preventative checks" were factors which Malthus believed could affect the birth rate such as moral restraint, abstinence and birth control. He predicted that "positive checks" on exponential population growth would ultimately save humanity from itself and he also believed that human misery was an "absolute necessary consequence". Malthus went on to explain why he believed that this misery affected the poor in a disproportionate manner.
There is a constant effort towards an increase in population which tends to subject the lower classes of society to distress and to prevent any great permanent amelioration of their condition…. The way in which these effects are produced seems to be this. We will suppose the means of subsistence in any country just equal to the easy support of its inhabitants. The constant effort towards population... increases the number of people before the means of subsistence are increased. The food, therefore which before supplied seven million must now be divided among seven million and a half or eight million. The poor consequently must live much worse, and many of them are reduced to severe distress.
Finally, Malthus advocated for the education of the lower class about the use of "moral restraint" or voluntary abstinence, which he believed would slow the growth rate.
Paul R. Ehrlich, a US biologist and environmentalist, published The Population Bomb in 1968, advocating stringent population planning policies. His central argument on population is as follows:
A cancer is an uncontrolled multiplication of cells; the population explosion is an uncontrolled multiplication of people. Treating only the symptoms of cancer may make the victim more comfortable at first, but eventually, he dies - often horribly. A similar fate awaits a world with a population explosion if only the symptoms are treated. We must shift our efforts from the treatment of the symptoms to the cutting out of cancer. The operation will demand many apparently brutal and heartless decisions. The pain may be intense. But the disease is so far advanced that only with radical surgery does the patient have a chance to survive.
In his concluding chapter, Ehrlich offered a partial solution to the "population problem",
"[We need] compulsory birth regulation... [through] the addition of temporary sterilants to water supplies or staple food. Doses of the antidote would be carefully rationed by the government to produce the desired family size".
Ehrlich's views came to be accepted by many population planning advocates in the United States and Europe in the 1960s and 1970s. Since Ehrlich introduced his idea of the "population bomb", overpopulation has been blamed for a variety of issues, including increasing poverty, high unemployment rates, environmental degradation, famine and genocide. In a 2004 interview, Ehrlich reviewed the predictions in his book and found that while the specific dates within his predictions may have been wrong, his predictions about climate change and disease were valid. Ehrlich continued to advocate for population planning and co-authored the book The Population Explosion, released in 1990 with his wife Anne Ehrlich.
However, it is controversial as to whether human population stabilization will avert environmental risks. A 2014 study published in the Proceedings of the National Academy of Sciences of the United States of America found that given the "inexorable demographic momentum of the global human population", even mass mortality events and draconian one-child policies implemented on a global scale would still likely result in a population of 5 to 10 billion by 2100. Therefore, while reduced fertility rates are positive for society and the environment, the short term focus should be on mitigating the human impact on the environment through technological and social innovations, along with reducing overconsumption, with population planning being a long-term goal. A letter in response, published in the same journal, argued that a reduction in population by 1 billion people in 2100 could help reduce the risk of catastrophic climate disruption. A 2021 article published in Sustainability Science said that sensible population policies could advance social justice (such as by abolishing child marriage, expanding family planning services and reforms that improve education for women and girls) and avoid the abusive and coercive population control schemes of the past while at the same time mitigating the human impact on the climate, biodiversity and ecosystems by slowing fertility rates.
Paige Whaley Eager argues that the shift in perception that occurred in the 1960s must be understood in the context of the demographic changes that took place at the time. It was only in the first decade of the 19th century that the world's population reached one billion. The second billion was added in the 1930s, and the next billion in the 1960s. 90 percent of this net increase occurred in developing countries. Eager also argues that, at the time, the United States recognised that these demographic changes could significantly affect global geopolitics. Large increases occurred in China, Mexico and Nigeria, and demographers warned of a "population explosion", particularly in developing countries from the mid-1950s onwards.
In the 1980s, tension grew between population planning advocates and women's health activists who advanced women's reproductive rights as part of a human rights-based approach. Growing opposition to the narrow population planning focus led to a significant change in population planning policies in the early 1990s.
== Population planning and economics ==
Opinions vary among economists about the effects of population change on a nation's economic health. US scientific research in 2009 concluded that the raising of a child cost about $16,000 yearly ($291,570 total for raising the child to its 18th birthday). In the US, the multiplication of this number with the yearly population growth will yield the overall cost of the population growth. Costs for other developed countries are usually of a similar order of magnitude.
Some economists, such as Thomas Sowell and Walter E. Williams, have argued that poverty and famine are caused by bad government and bad economic policies, not by overpopulation.
In his book The Ultimate Resource, economist Julian Simon argued that higher population density leads to more specialization and technological innovation, which in turn leads to a higher standard of living. He claimed that human beings are the ultimate resource since we possess "productive and inventive minds that help find creative solutions to man’s problems, thus leaving us better off over the long run".
Simon also claimed that when considering a list of countries ranked in order by population density, there is no correlation between population density and poverty and starvation. Instead, if a list of countries is considered according to corruption within their respective governments, there is a significant correlation between government corruption, poverty and famine.
== Views on population planning ==
=== Birth rate reductions ===
==== Support ====
As early as 1798, Thomas Malthus argued in his Essay on the Principle of Population for implementation of population planning. Around the year 1900, Sir Francis Galton said in his publication Hereditary Improvement: "The unfit could become enemies to the State if they continue to propagate." In 1968, Paul Ehrlich noted in The Population Bomb, "We must cut the cancer of population growth", and "if this was not done, there would be only one other solution, namely the 'death rate solution' in which we raise the death rate through war-famine-pestilence, etc.”
In the same year, another prominent modern advocate for mandatory population planning was Garrett Hardin, who proposed in his landmark 1968 essay Tragedy of the commons, society must relinquish the "freedom to breed" through "mutual coercion, mutually agreed upon." Later on, in 1972, he reaffirmed his support in his new essay "Exploring New Ethics for Survival", by stating, "We are breeding ourselves into oblivion." Many prominent personalities, such as Bertrand Russell, Margaret Sanger (1939), John D. Rockefeller, Frederick Osborn (1952), Isaac Asimov, Arne Næss and Jacques Cousteau have also advocated for population planning.
Today, a number of influential people advocate population planning such as these:
David Attenborough
Christian de Duve, Nobel laureate
Sara Parkin
Jonathon Porritt, UK sustainable development commissioner
William J. Ripple, lead author of the 2017 World Scientists' Warning to Humanity: A Second Notice
Crispin Tickell
The head of the UN Millennium Project Jeffrey Sachs is also a strong proponent of decreasing the effects of overpopulation. In 2007, Jeffrey Sachs gave a number of lectures (2007 Reith Lectures) about population planning and overpopulation. In his lectures, called "Bursting at the Seams", he featured an integrated approach that would deal with a number of problems associated with overpopulation and poverty reduction. For example, when criticized for advocating mosquito nets he argued that child survival was, "by far one of the most powerful ways", to achieve fertility reduction, as this would assure poor families that the smaller number of children they had would survive.
==== Opposition ====
Critics of human population planning point out that attempts to curb human population growth have resulted in violations of human rights such as forced sterilization, particularly in China and India. In the latter half of the twentieth century, India's population reduction program received substantial funds and powerful incentives from Western countries and international population planning organizations to reduce India's growing population. This culminated in "the Emergency," a period in the mid-1970's where millions of people were forcibly sterilized. Violent resistance to forced sterilization led to police brutality and some instances of mass shootings of civilians by police. Critics also argue that supposedly voluntary population planning is often coerced. Some also believe that the environmental problems caused by supposed overpopulation are better explained by other factors, and that the goal of human population reduction does not justify the threat to human rights posed by population planning policies.
Other causes for opposition emerge from the feasibility of substantially impacting human population. According to some researchers, even rapid global adoption of a one-child policy would result in a world population exceeding 8 billion in 2050, and in a scenario involving catastrophic mass death of 2 billion people, world population would exceed 8 billion by 2100.
The Catholic Church has opposed abortion, sterilization, and artificial contraception as a general practice but especially in regard to population planning policies. Pope Benedict XVI has stated, "The extermination of millions of unborn children, in the name of the fight against poverty, actually constitutes the destruction of the poorest of all human beings." The reformed Theology pastor Dr. Stephen Tong also opposes the planning of human population.
== Pro-natalist policies ==
In 1946, Poland introduced a tax on childlessness, discontinued in the 1970s, as part of natalist policies in the Communist government. From 1941 to the 1990s, the Soviet Union had a similar tax to replenish the population losses incurred during the Second World War.
The Socialist Republic of Romania under Nicolae Ceaușescu severely repressed abortion, (the most common birth control method at the time) in 1966, and forced gynecological revisions and penalties for unmarried women and childless couples.
The surge of the birth rate taxed the public services received by the decreței 770 ("Scions of the Decree 770") generation. A consequence of Ceaușescu's natalist policy is that large numbers of children ended up living in orphanages, because their parents could not cope. The vast majority of children who lived in the communist orphanages were not actually orphans, but were simply children whose parents could not afford to raise them. The Romanian Revolution of 1989 preceded a fall in population growth.
=== Balanced birth policies ===
Nativity in the Western world dropped during the interwar period. Swedish sociologists Alva and Gunnar Myrdal published Crisis in the Population Question in 1934, suggesting an extensive welfare state with universal healthcare and childcare, to increase overall Swedish birth rates, and level the number of children at a reproductive level for all social classes in Sweden. Swedish fertility rose throughout World War II (as Sweden was largely unharmed by the war) and peaked in 1946.
== Modern practice by country ==
=== Australia ===
Australia currently offers fortnightly Family Tax Benefit payments plus a free immunization scheme, and recently proposed to pay all child care costs for women who want to work.
=== China ===
==== One-child era (1979–2015) ====
The most significant population planning system in the world was China's one-child policy, in which, with various exceptions, having more than one child was discouraged. Unauthorized births were punished by fines, although there were also allegations of illegal forced abortions and forced sterilization. As part of China's planned birth policy, (work) unit supervisors monitored the fertility of married women and may decide whose turn it is to have a baby.
The Chinese government introduced the policy in 1978 to alleviate the social and environmental problems of China. According to government officials, the policy has helped prevent 400 million births. The success of the policy has been questioned, and reduction in fertility has also been attributed to the modernization of China. The policy is controversial both within and outside of China because of its manner of implementation and because of concerns about negative economic and social consequences e.g. female infanticide. In Asian cultures, the oldest male child has responsibility of caring for the parents in their old age. Therefore, it is common for Asian families to invest most heavily in the oldest male child, such as providing college, steering them into the most lucrative careers, and so on. To these families, having an oldest male child is paramount, so in a one-child policy, daughters have no economic benefit, so daughters, especially as a first child, are often targeted for abortion or infanticide. China introduced several government reforms to increase retirement payments to coincide with the one-child policy. During that time, couples could request permission to have more than one child.
According to Tibetologist Melvyn Goldstein, natalist feelings run high in China's Tibet Autonomous Region, among both ordinary people and government officials. Seeing population control "as a matter of power and ethnic survival" rather than in terms of ecological sustainability, Tibetans successfully argued for an exemption of Tibetan people from the usual family planning policies in China such as the one-child policy.
==== Two-child era (2016–2021) ====
In November 2014, the Chinese government allowed its people to conceive a second child under the supervision of government regulation.
On 29 October 2015, the ruling Chinese Communist Party announced that all one-child policies would be scrapped, allowing all couples to have two children. The change was needed to allow a better balance of male and female children, and to grow the young population to ease the problem of paying for the aging population. The law enacting the two-child policy took effect on 1 January 2016, and replaced the previous one-child policy.
==== Three-child era (2021–) ====
In May 2021, the Chinese government allowed its people to conceive a third child, in a move accompanied by "supportive measures" it regarded "conducive" to improving its "population structure, fulfilling the country's strategy of actively coping with an ageing population and maintaining the advantage, endowment of human resources" after declining birth rates recorded in the 2020 Chinese census.
=== Hungary ===
During the Second Orbán Government, Hungary increased its family benefits spending from one of the lowest rates in the OECD to one of the highest. In 2015, it amounted to nearly 4% of GDP.
=== India ===
Only those with two or fewer children are eligible for election to a local government.
Us two, our two ("Hum do, hamare do" in Hindi) is a slogan meaning one family, two children and is intended to reinforce the message of family planning thereby aiding population planning.
Facilities offered by government to its employees are limited to two children. The government offers incentives for families accepted for sterilization. Moreover, India was the first country to take measures for family planning back in 1952.
In the south west of India lies the long narrow coastal state of Kerala. Most of its thirty-two million inhabitants live off the land and the ocean, a rich tropical ecosystem watered by two monsoons a year. It's also one of India's most crowded states – but the population is stable because nearly everybody has small families… At the root of it all is education. Thanks to a long tradition of compulsory schooling for boys and girls Kerala has one of the highest literacy rates in the World. Where women are well educated they tend to choose to have smaller families… What Kerala shows is that you don't need aggressive policies or government incentives for birthrates to fall. Everywhere in the world where women have access to education and have the freedom to run their own lives, on the whole they and their partners have been choosing to have smaller families than their parents. But reducing birthrates is very difficult to achieve without a simple piece of medical technology, contraception.
In 2019, the Population Control Bill, 2019 bill was introduced in the Rajya Sabha in July 2019 by Rakesh Sinha. The purpose of the bill is to control the population growth of India.
=== Iran ===
After the Iran–Iraq War, Iran encouraged married couples to produce as many children as possible to replace population lost to the war.
Iran succeeded in sharply reducing its birth rate from the late 1980s to 2010. Mandatory contraceptive courses are required for both males and females before a marriage license can be obtained, and the government emphasized the benefits of smaller families and the use of contraception. This changed in 2012, when a major policy shift back towards increasing birth rates was announced. In 2014, permanent contraception and advertising of birth control were to be outlawed.
=== Israel ===
In Israel, Haredi families with many children receive economic support through generous governmental child allowances, government assistance in housing young religious couples, as well as specific funds by their own community institutions. Haredi women have an average of 6.7 children while the average Jewish Israeli woman has 3 children.
=== Japan ===
Japan has experienced a shrinking population for many years. The government is trying to encourage women to have children or to have more children – many Japanese women do not have children, or even remain single. The population is culturally opposed to immigration.
Some Japanese localities, facing significant population loss, are offering economic incentives. Yamatsuri, a town of 7,000 just north of Tokyo, offers parents $4,600 for the birth of a child and $460 a year for 10 years.
=== Myanmar ===
In Myanmar, the Population planning Health Care Bill requires some parents to space each child three years apart. The Economist, in 2015, stated that the measure was expected to be used against the persecuted Muslim Rohingyas minority.
=== Pakistan ===
=== Russia ===
Russian President Vladimir Putin directed Parliament in 2006 to adopt a 10-year program to stop the sharp decline in Russia's population, principally by offering financial incentives and subsidies to encourage women to have children.
=== Singapore ===
Singapore has undergone two major phases in its population planning: first to slow and reverse the baby boom in the Post-World War II era; then from the 1980s onwards to encourage couples to have more children as the birth rate had fallen below the replacement-level fertility. In addition, during the interim period, eugenics policies were adopted.
The anti-natalist policies flourished in the 1960s and 1970s: initiatives advocating small families were launched and developed into the Stop at Two programme, pushing for two-children families and promoting sterilisation. In 1984, the government announced the Graduate Mothers' Scheme, which favoured children of more well-educated mothers; the policy was however soon abandoned due to the outcry in the general election of the same year.
Eventually, the government became pro-natalist in the late 1980s, marked by its Have Three or More plan in 1987. Singapore pays $3,000 for the first child, $9,000 in cash and savings for the second; and up to $18,000 each for the third and fourth.
=== Spain ===
In 2017, the government of Spain appointed Edelmira Barreira, as "Government Commissioner facing the Demographic Challenge", in a pro-natalist attempt to reverse a negative population growth rate.
=== Turkey ===
In May 2012, Turkey's Prime Minister Recep Tayyip Erdogan argued that abortion is murder and announced that legislative preparations to severely limit the practice are underway. Erdogan also argued that abortion and C-section deliveries are plots to stall Turkey's economic growth. Prior to this move, Erdogan had repeatedly demanded that each couple have at least three children.
=== United States ===
Enacted in 1970, Title X of the Public Health Service Act provides access to contraceptive services, supplies and information to those in need. Priority for services is given to people with low incomes. The Title X Family Planning program is administered through the Office of Population Affairs under the Office of Public Health and Science.
It is directed by the Office of Family Planning. In 2007, Congress appropriated roughly $283 million for family planning under Title X, at least 90 percent of which was used for services in family planning clinics. Title X is a vital source of funding for family planning clinics throughout the nation, which provide reproductive health care, including abortion.
The education and services supplied by the Title X-funded clinics support young individuals and low-income families. The goals of developing healthy families are accomplished by helping individuals and couples decide whether to have children and when the appropriate time to do so would be.
Title X has made the prevention of unintended pregnancies possible. It has allowed millions of American women to receive necessary reproductive health care, plan their pregnancies and prevent abortions. Title X is dedicated exclusively to funding family planning and reproductive health care services.
Title X as a percentage of total public funding to family planning client services has steadily declined from 44% of total expenditures in 1980 to 12% in 2006. Medicaid has increased from 20% to 71% in the same time. In 2006, Medicaid contributed $1.3 billion to public family planning.
In the early 1970s, the United States Congress established the Commission on Population Growth and the American Future (Chairman John D. Rockefeller III), which was created to provide recommendations regarding population growth and its social consequences. The Commission submitted its final recommendations in 1972, which included promoting contraceptives and liberalizing abortion regulations, for example.
==== Natalism in the United States ====
In a 2004 editorial in The New York Times, David Brooks expressed the opinion that the relatively high birth rate of the United States in comparison to Europe could be attributed to social groups with "natalist" attitudes. The article is referred to in an analysis of the Quiverfull movement. However, the figures identified for the demographic are extremely low.
Former US Senator Rick Santorum made natalism part of his platform for his 2012 presidential campaign. Many of those categorized in the General Social Survey as "Fundamentalist Protestant" are more or less natalist, and have a higher birth rate than "Moderate" and "Liberal" Protestants. However, Rick Santorum is not a Protestant but a practicing Catholic.
=== Uzbekistan ===
It is reported that Uzbekistan has been pursuing a policy of forced sterilizations, hysterectomies and IUD insertions since the late 1990s in order to impose population planning.
== See also ==
=== Fiction ===
Logan's Run (Book) - State-mandated euthanasia at 21 for all people (30 in the film) to conserve resources.
Make Room! Make Room! (Book) - Novel, explores the consequence of overpopulation.
Ishmael (Quinn novel) - Explores the biological and ecological causes of overpopulation which is a result of increased carrying capacity for humans. The planning proposal is to limit that capacity (see Food Race).
Avengers: Infinity War (Movie) - Antagonist and villain Thanos kills half of all living things throughout universe in order to maintain ecological balance.
Inferno (Movie) - A billionaire has created a virus that will kill 50% of the world's population to save the other 50%. His followers try to release the virus after his suicide.
Shadow Children (Book series) - Families are allowed two children maximum, and "shadow children" (third children and beyond) are subject to be killed.
2 B R 0 2 B (Book) - Aging is cured and each new life requires the sacrifice of another in order to maintain a stable population.
2BR02B: To Be or Naught to Be (Movie) - Based on the above book.
The Thinning and The Thinning: New World Order (Film Series) - Involves a dystopian United States enforcing population control via aptitude test and an authoritarian police force known as the Department of Population Control.
== References ==
== Further reading ==
"Controlled food supply could stop overpopulation". Carrie Gazarish. Daily Kent Stater, Volume 32, Number 52, Kent State University.
Thomlinson, R. 1975. Demographic Problems: Controversy over Population Control. 2nd ed. Encino, CA: Dickenson.
David Pimentel. The Case for Population Reduction: Miscellaneous papers of David Pimentel. Collection of papers, reprints, and other publications on population control and related issues. Cornell University.
Hopfenberg, Russell. "Genetic feedback and human population regulation". (PDF) Human Ecology 37.5 (2009): 643-651.
"From population control to reproductive rights: feminist fault lines" (PDF). Rosalind Pollack Petchesky. Reproductive Health Matters Volume 3, Issue 6, November 1995. Taylor & Francis.
Coole, Diana (2018). Should We Control World Population?. Cambridge: Polity. p. 140. ISBN 978-1-509-52340-5.
Washington, Haydn; Kopnina, Helen (2022). "Discussing the Silence and Denial around Population Growth and Its Environmental Impact. How Do We Find Ways Forward?". World. 3 (4): 1009–1027. doi:10.3390/world3040057.
Goldstone, Jack A.; May, John F. (2023). "How 21st Century Population Issues and Policies Differ from Those of the 20th Century". World. 4 (3): 467–476. doi:10.3390/world4030029.
Greguš, Jan; Guillebaud, John (2023). "Scientists' Warning: Remove the Barriers to Contraception Access, for Health of Women and the Planet". World. 4 (3): 589–597. doi:10.3390/world4030036.
Norrman, Karl-Erik (2023). "World Population Growth: A Once and Future Global Concern". World. 4 (4): 684–697. doi:10.3390/world4040043.
Rees, William E. (2023). "The Human Ecology of Overshoot: Why a Major 'Population Correction' Is Inevitable". World. 4 (3): 509–527. doi:10.3390/world4030032. hdl:2429/86320.
O’Sullivan, Jane N. (2023). "Demographic Delusions: World Population Growth Is Exceeding Most Projections and Jeopardising Scenarios for Sustainable Futures". World. 4 (3): 545–568. doi:10.3390/world4030034.
Guillemot, Jonathan R.; Zhang, Xue; Warner, Mildred E. (2024). "Population Aging and Decline Will Happen Sooner Than We Think". Social Sciences. 13 (4): 190. doi:10.3390/socsci13040190.
== External links ==
Wikiversity:Should we aim to reduce the Earth population?
"Thirty years is too long to turn a blind eye to world population growth" by Jane O’Sullivan. The Overpopulation Project
"A chat with Tim Flannery, senior research scientist, on Population Control". Karina Kelly, Peter Kirkwood, Owen Craig. Archived from the original on 13 January 2010. | Wikipedia/Human_population_control |
Design standards, reference standards and performance standards are familiar throughout business and industry, virtually for anything that is definable. Sustainable design, taken as reducing our impact on the earth and making things better at the same time, is in the process of becoming defined. Also, many well organized specific methodologies are used by different communities of people for a variety of purposes.
== Design standards ==
One of the better known is the Leadership in Energy and Environmental Design (LEED) green building rating system, which uses a diverse group of hard measures of environmental quality and impacts to define a holistic approach to sustainable building and assign ratings to individual projects.
Sustainable design is really just a more determined effort to consider the whole range of impacts on our environment in making any decision. A more complete design guide, guided more by whole project impact measures, is the model offered by the U.S. cooperating agencies in the "Whole Building Design Guide".
Green construction codes and standards are beginning to emerge on the national code stage. The standards go beyond energy standards such as ASHRAE 90.1 and the International Energy Conservation Code (IECC) to cover additional areas such as site sustainability, water efficiency, indoor environmental quality and materials and resources. The first is ASHRAE 189.1, Standard for the Design of High-Performance, Green Buildings Except Low-Rise Residential Buildings, published by ASHRAE in January 2010 in conjunction with the U.S. Green Building Council and the Illuminating Engineering Society. Standard 189.1 provides criteria by which a building can be judged as “green,” written in model code language that jurisdictions can use to develop a green building construction code.
Several organizations have developed their own ways of setting goals for energy reductions, such as Architecture 2030 and for qualifying performance toward them such as Cradle to Cradle.
== Design methods ==
Developing real methods for how to discover the design opportunities that would allow you to meet or exceed the standards was one of the objectives of the environmental design movement in architectural schools in the 1960s and 1970s, but though some of the issues introduced then are still an important part of the process, not much actually changed about the methods of design. Now with the combination of many more interactive tools and much higher stakes in the outcome, and long gestating rethinking about natural systems in general, a dramatic new revolution in methodology seems inevitable.
BIM (building information modeling) allows designers to work with many remote consultants on the same data file that represents all the decisions being made by the team. The same file is available to the climate and energy and environmental impact analysis and cost analysis tools and consultants, ... and of course to the prospective contractors and the regulators. Along with this new integrated access to the model there in needed a new way to integrate the conversation of so many people, each with some interest in reviewing each other's comments on the progress with the central design model. That is likely to involve development of wiki tools for the process. One such very early implementation of a Wiki SD tool called "4Dsustainability" organizes the project design evolution around the general learning process of how you define the problem by exploring its environment, and following that through the project.
The main difference between sustainable design methods and conventional design is incorporating the entire environment of the project's stakeholders on the design team, essentially, requiring new ways to explore connections and for more people and perspectives to be taken into account. Other methods that recognize this requirement are the "AIA SDAT" (sustainable design assessment team) program and the "Scenarios for sustainability" process design tools.
== References == | Wikipedia/Sustainable_design_standards |
Renewable energy commercialization involves the deployment of three generations of renewable energy technologies dating back more than 100 years. First-generation technologies, which are already mature and economically competitive, include biomass, hydroelectricity, geothermal power and heat. Second-generation technologies are market-ready and are being deployed at the present time; they include solar heating, photovoltaics, wind power, solar thermal power stations, and modern forms of bioenergy. Third-generation technologies require continued R&D efforts in order to make large contributions on a global scale and include advanced biomass gasification, hot-dry-rock geothermal power, and ocean energy. In 2019, nearly 75% of new installed electricity generation capacity used renewable energy and the International Energy Agency (IEA) has predicted that by 2025, renewable capacity will meet 35% of global power generation.
Public policy and political leadership helps to "level the playing field" and drive the wider acceptance of renewable energy technologies. Countries such as Germany, Denmark, and Spain have led the way in implementing innovative policies which has driven most of the growth over the past decade. As of 2014, Germany has a commitment to the "Energiewende" transition to a sustainable energy economy, and Denmark has a commitment to 100% renewable energy by 2050. There are now 144 countries with renewable energy policy targets.
Renewable energy continued its rapid growth in 2015, providing multiple benefits. There was a new record set for installed wind and photovoltaic capacity (64GW and 57GW) and a new high of US$329 Billion for global renewables investment. A key benefit that this investment growth brings is a growth in jobs. The top countries for investment in recent years were China, Germany, Spain, the United States, Italy, and Brazil. Renewable energy companies include BrightSource Energy, First Solar, Gamesa, GE Energy, Goldwind, Sinovel, Targray, Trina Solar, Vestas, and Yingli.
Climate change concerns are also driving increasing growth in the renewable energy industries. According to a 2011 projection by the IEA, solar power generators may produce most of the world's electricity within 50 years, reducing harmful greenhouse gas emissions.
== Background ==
=== Rationale for renewables ===
Climate change, pollution, and energy insecurity are significant problems, and addressing them requires major changes to energy infrastructures. Renewable energy technologies are essential contributors to the energy supply portfolio, as they contribute to world energy security, reduce dependency on fossil fuels, and some also provide opportunities for mitigating greenhouse gases. Climate-disrupting fossil fuels are being replaced by clean, climate-stabilizing, non-depletable sources of energy:
...the transition from coal, oil, and gas to wind, solar, and geothermal energy is well under way. In the old economy, energy was produced by burning something — oil, coal, or natural gas — leading to the carbon emissions that have come to define our economy. The new energy economy harnesses the energy in wind, the energy coming from the sun, and heat from within the earth itself.
In international public opinion surveys there is strong support for a variety of methods for addressing the problem of energy supply. These methods include promoting renewable sources such as solar power and wind power, requiring utilities to use more renewable energy, and providing tax incentives to encourage the development and use of such technologies. It is expected that renewable energy investments will pay off economically in the long term.
EU member countries have shown support for ambitious renewable energy goals. In 2010, Eurobarometer polled the twenty-seven EU member states about the target "to increase the share of renewable energy in the EU by 20 percent by 2020". Most people in all twenty-seven countries either approved of the target or called for it to go further. Across the EU, 57 percent thought the proposed goal was "about right" and 16 percent thought it was "too modest." In comparison, 19 percent said it was "too ambitious".
As of 2011, new evidence has emerged that there are considerable risks associated with traditional energy sources, and that major changes to the mix of energy technologies is needed:
Several mining tragedies globally have underscored the human toll of the coal supply chain. New EPA initiatives targeting air toxics, coal ash, and effluent releases highlight the environmental impacts of coal and the cost of addressing them with control technologies. The use of fracking in natural gas exploration is coming under scrutiny, with evidence of groundwater contamination and greenhouse gas emissions. Concerns are increasing about the vast amounts of water used at coal-fired and nuclear power plants, particularly in regions of the country facing water shortages. Events at the Fukushima nuclear plant have renewed doubts about the ability to operate large numbers of nuclear plants safely over the long term. Further, cost estimates for "next generation" nuclear units continue to climb, and lenders are unwilling to finance these plants without taxpayer guarantees.
The 2014 REN21 Global Status Report says that renewable energies are no longer just energy sources, but ways to address pressing social, political, economic and environmental problems:
Today, renewables are seen not only as sources of energy, but also as tools to address many other pressing needs, including: improving energy security; reducing the health and environmental impacts associated with fossil and nuclear energy; mitigating greenhouse gas emissions; improving educational opportunities; creating jobs; reducing poverty; and increasing gender equality... Renewables have entered the mainstream.
=== Growth of renewables ===
In 2008 for the first time, more renewable energy than conventional power capacity was added in both the European Union and United States, demonstrating a "fundamental transition" of the world's energy markets towards renewables, according to a report released by REN21, a global renewable energy policy network based in Paris. In 2010, renewable power consisted about a third of the newly built power generation capacities.
By the end of 2011, total renewable power capacity worldwide exceeded 1,360 GW, up 8%. Renewables producing electricity accounted for almost half of the 208 GW of capacity added globally during 2011. Wind and solar photovoltaics (PV) accounted for almost 40% and 30%. Based on REN21's 2014 report, renewables contributed 19 percent to our energy consumption and 22 percent to our electricity generation in 2012 and 2013, respectively. This energy consumption is divided as 9% coming from traditional biomass, 4.2% as heat energy (non-biomass), 3.8% hydro electricity and 2% electricity from wind, solar, geothermal, and biomass.
During the five-years from the end of 2004 through 2009, worldwide renewable energy capacity grew at rates of 10–60 percent annually for many technologies, while actual production grew 1.2% overall. In 2011, UN under-secretary general Achim Steiner said: "The continuing growth in this core segment of the green economy is not happening by chance. The combination of government target-setting, policy support and stimulus funds is underpinning the renewable industry's rise and bringing the much needed transformation of our global energy system within reach." He added: "Renewable energies are expanding both in terms of investment, projects and geographical spread. In doing so, they are making an increasing contribution to combating climate change, countering energy poverty and energy insecurity".
According to a 2011 projection by the International Energy Agency, solar power plants may produce most of the world's electricity within 50 years, significantly reducing the emissions of greenhouse gases that harm the environment. The IEA has said: "Photovoltaic and solar-thermal plants may meet most of the world's demand for electricity by 2060 – and half of all energy needs – with wind, hydropower and biomass plants supplying much of the remaining generation". "Photovoltaic and concentrated solar power together can become the major source of electricity".
In 2013, China led the world in renewable energy production, with a total capacity of 378 GW, mainly from hydroelectric and wind power. As of 2014, China leads the world in the production and use of wind power, solar photovoltaic power and smart grid technologies, generating almost as much water, wind and solar energy as all of France and Germany's power plants combined. China's renewable energy sector is growing faster than its fossil fuels and nuclear power capacity. Since 2005, production of solar cells in China has expanded 100-fold. As Chinese renewable manufacturing has grown, the costs of renewable energy technologies have dropped. Innovation has helped, but the main driver of reduced costs has been market expansion.
See also renewable energy in the United States for US-figures.
=== Economic trends ===
Renewable energy technologies are getting cheaper, through technological change and through the benefits of mass production and market competition. A 2011 IEA report said: "A portfolio of renewable energy technologies is becoming cost-competitive in an increasingly broad range of circumstances, in some cases providing investment opportunities without the need for specific economic support," and added that "cost reductions in critical technologies, such as wind and solar, are set to continue." As of 2011, there have been substantial reductions in the cost of solar and wind technologies:
The price of PV modules per MW has fallen by 60 percent since the summer of 2008, according to Bloomberg New Energy Finance estimates, putting solar power for the first time on a competitive footing with the retail price of electricity in a number of sunny countries. Wind turbine prices have also fallen – by 18 percent per MW in the last two years – reflecting, as with solar, fierce competition in the supply chain. Further improvements in the levelised cost of energy for solar, wind and other technologies lie ahead, posing a growing threat to the dominance of fossil fuel generation sources in the next few years.
Hydro-electricity and geothermal electricity produced at favourable sites are now the cheapest way to generate electricity. Renewable energy costs continue to drop, and the levelised cost of electricity (LCOE) is declining for wind power, solar photovoltaic (PV), concentrated solar power (CSP) and some biomass technologies.
Renewable energy is also the most economic solution for new grid-connected capacity in areas with good resources. As the cost of renewable power falls, the scope of economically viable applications increases. Renewable technologies are now often the most economic solution for new generating capacity. Where "oil-fired generation is the predominant power generation source (e.g. on islands, off-grid and in some countries) a lower-cost renewable solution almost always exists today". As of 2012, renewable power generation technologies accounted for around half of all new power generation capacity additions globally. In 2011, additions included 41 gigawatt (GW) of new wind power capacity, 30 GW of PV, 25 GW of hydro-electricity, 6 GW of biomass, 0.5 GW of CSP, and 0.1 GW of geothermal power.
=== Three generations of technologies ===
Renewable energy includes a number of sources and technologies at different stages of commercialization. The International Energy Agency (IEA) has defined three generations of renewable energy technologies, reaching back over 100 years:
"First-generation technologies emerged from the industrial revolution at the end of the 19th century and include hydropower, biomass combustion, geothermal power and heat. These technologies are quite widely used.
Second-generation technologies include solar heating and cooling, wind power, modern forms of bioenergy, and solar photovoltaics. These are now entering markets as a result of research, development and demonstration (RD&D) investments since the 1980s. Initial investment was prompted by energy security concerns linked to the oil crises of the 1970s but the enduring appeal of these technologies is due, at least in part, to environmental benefits. Many of the technologies reflect significant advancements in materials.
Third-generation technologies are still under development and include advanced biomass gasification, biorefinery technologies, concentrating solar thermal power, hot-dry-rock geothermal power, and ocean energy. Advances in nanotechnology may also play a major role".
First-generation technologies are well established, second-generation technologies are entering markets, and third-generation technologies heavily depend on long-term research and development commitments, where the public sector has a role to play.
== First-generation technologies ==
First-generation technologies are widely used in locations with abundant resources. Their future use depends on the exploration of the remaining resource potential, particularly in developing countries, and on overcoming challenges related to the environment and social acceptance.
=== Biomass ===
Biomass, the burning of organic materials for heat and power, is a fully mature technology. Unlike most renewable sources, biomass (and hydropower) can supply stable base load power generation.
Biomass produces CO2 emissions on combustion, and the issue of whether biomass is carbon neutral is contested. Material directly combusted in cook stoves produces pollutants, leading to severe health and environmental consequences. Improved cook stove programs are alleviating some of these effects.
The industry remained relatively stagnant over the decade to 2007, but demand for biomass (mostly wood) continues to grow in many developing countries, as well as Brazil and Germany.
The economic viability of biomass is dependent on regulated tariffs, due to high costs of infrastructure and ingredients for ongoing operations. Biomass does offer a ready disposal mechanism by burning municipal, agricultural, and industrial organic waste products. First-generation biomass technologies can be economically competitive, but may still require deployment support to overcome public acceptance and small-scale issues. As part of the food vs. fuel debate, several economists from Iowa State University found in 2008 "there is no evidence to disprove that the primary objective of biofuel policy is to support farm income."
=== Hydroelectricity ===
Hydroelectricity is the term referring to electricity generated by hydropower; the production of electrical power through the use of the gravitational force of falling or flowing water. In 2015 hydropower generated 16.6% of the worlds total electricity and 70% of all renewable electricity and is expected to increase about 3.1% each year for the next 25 years. Hydroelectric plants have the advantage of being long-lived and many existing plants have operated for more than 100 years.
Hydropower is produced in 150 countries, with the Asia-Pacific region generating 32 percent of global hydropower in 2010. China is the largest hydroelectricity producer, with 721 terawatt-hours of production in 2010, representing around 17 percent of domestic electricity use. There are now three hydroelectricity plants larger than 10 GW: the Three Gorges Dam in China, Itaipu Dam across the Brazil/Paraguay border, and Guri Dam in Venezuela. The cost of hydroelectricity is low, making it a competitive source of renewable electricity. The average cost of electricity from a hydro plant larger than 10 megawatts is 3 to 5 U.S. cents per kilowatt-hour.
=== Geothermal power and heat ===
Geothermal power plants can operate 24 hours per day, providing baseload capacity. Estimates for the world potential capacity for geothermal power generation vary widely, ranging from 40 GW by 2020 to as much as 6,000 GW.
Geothermal power capacity grew from around 1 GW in 1975 to almost 10 GW in 2008. The United States is the world leader in terms of installed capacity, representing 3.1 GW. Other countries with significant installed capacity include the Philippines (1.9 GW), Indonesia (1.2 GW), Mexico (1.0 GW), Italy (0.8 GW), Iceland (0.6 GW), Japan (0.5 GW), and New Zealand (0.5 GW). In some countries, geothermal power accounts for a significant share of the total electricity supply, such as in the Philippines, where geothermal represented 17 percent of the total power mix at the end of 2008.
Geothermal (ground source) heat pumps represented an estimated 30 GWth of installed capacity at the end of 2008, with other direct uses of geothermal heat (i.e., for space heating, agricultural drying and other uses) reaching an estimated 15 GWth. As of 2008, at least 76 countries use direct geothermal energy in some form.
== Second-generation technologies ==
Second-generation technologies have gone from being a passion for the dedicated few to a major economic sector in countries such as Germany, Spain, the United States, and Japan. Many large industrial companies and financial institutions are involved and the challenge is to broaden the market base for continued growth worldwide.
=== Solar heating ===
Solar heating systems are a well known second-generation technology and generally consist of solar thermal collectors, a fluid system to move the heat from the collector to its point of usage, and a reservoir or tank for heat storage. The systems may be used to heat domestic hot water, swimming pools, or homes and businesses. The heat can also be used for industrial process applications or as an energy input for other uses such as cooling equipment.
In many warmer climates, a solar heating system can provide a very high percentage (50 to 75%) of domestic hot water energy. As of 2009, China has 27 million rooftop solar water heaters.
=== Photovoltaics ===
Photovoltaic (PV) cells, also called solar cells, convert light into electricity. In the 1980s and early 1990s, most photovoltaic modules were used to provide remote-area power supply, but from around 1995, industry efforts have focused increasingly on developing building integrated photovoltaics and photovoltaic power stations for grid connected applications.
Many plants are integrated with agriculture and some use innovative tracking systems that follow the sun's daily path across the sky to generate more electricity than conventional fixed-mounted systems. There are no fuel costs or emissions during operation of the power stations.
=== Wind power ===
Some of the second-generation renewables, such as wind power, have high potential and have already realised relatively low production costs. Wind power could become cheaper than nuclear power. Global wind power installations increased by 35,800 MW in 2010, bringing total installed capacity up to 194,400 MW, a 22.5% increase on the 158,700 MW installed at the end of 2009. The increase for 2010 represents investments totalling €47.3 billion (US$65 billion) and for the first time more than half of all new wind power was added outside of the traditional markets of Europe and North America, mainly driven, by the continuing boom in China which accounted for nearly half of all of the installations at 16,500 MW. China now has 42,300 MW of wind power installed. Wind power accounts for approximately 19% of electricity generated in Denmark, 9% in Spain and Portugal, and 6% in Germany and the Republic of Ireland. In Australian state of South Australia wind power, championed by Premier Mike Rann (2002–2011), now comprises 26% of the state's electricity generation, edging out coal fired power. At the end of 2011 South Australia, with 7.2% of Australia's population, had 54% of the nation's installed wind power capacity.
Wind power's share of worldwide electricity usage at the end of 2014 was 3.1%.
The wind industry is able to produce more power at lower cost by using taller wind turbines with longer blades, capturing the faster winds at higher elevations. This has opened up new opportunities and in Indiana, Michigan, and Ohio, the price of power from wind turbines built 300 feet to 400 feet above the ground can now compete with conventional fossil fuels like coal. Prices have fallen to about 4 cents per kilowatt-hour in some cases and utilities have been increasing the amount of wind energy in their portfolio, saying it is their cheapest option.
=== Solar thermal power stations ===
Solar thermal power stations include the 354 megawatt (MW) Solar Energy Generating Systems power plant in the US, Solnova Solar Power Station (Spain, 150 MW), Andasol solar power station (Spain, 100 MW), Nevada Solar One (USA, 64 MW), PS20 solar power tower (Spain, 20 MW), and the PS10 solar power tower (Spain, 11 MW). The 370 MW Ivanpah Solar Power Facility, located in California's Mojave Desert, is the world's largest solar-thermal power plant project currently under construction. Many other plants are under construction or planned, mainly in Spain and the USA. In developing countries, three World Bank projects for integrated solar thermal/combined-cycle gas-turbine power plants in Egypt, Mexico, and Morocco have been approved.
=== Modern forms of bioenergy ===
Global ethanol production for transport fuel tripled between 2000 and 2007 from 17 billion to more than 52 billion litres, while biodiesel expanded more than tenfold from less than 1 billion to almost 11 billion litres. Biofuels provide 1.8% of the world's transport fuel and recent estimates indicate a continued high growth. The main producing countries for transport biofuels are the US, Brazil, and the EU.
Brazil has one of the largest renewable energy programs in the world, involving production of ethanol fuel from sugar cane, and ethanol now provides 18 percent of the country's automotive fuel. As a result of this and the exploitation of domestic deep water oil sources, Brazil, which for years had to import a large share of the petroleum needed for domestic consumption, recently reached complete self-sufficiency in liquid fuels.
Nearly all the gasoline sold in the United States today is mixed with 10 percent ethanol, a mix known as E10, and motor vehicle manufacturers already produce vehicles designed to run on much higher ethanol blends. Ford, DaimlerChrysler, and GM are among the automobile companies that sell flexible-fuel cars, trucks, and minivans that can use gasoline and ethanol blends ranging from pure gasoline up to 85% ethanol (E85). The challenge is to expand the market for biofuels beyond the farm states where they have been most popular to date. The Energy Policy Act of 2005, which calls for 7.5 billion US gallons (28,000,000 m3) of biofuels to be used annually by 2012, will also help to expand the market.
The growing ethanol and biodiesel industries are providing jobs in plant construction, operations, and maintenance, mostly in rural communities. According to the Renewable Fuels Association, "the ethanol industry created almost 154,000 U.S. jobs in 2005 alone, boosting household income by $5.7 billion. It also contributed about $3.5 billion in tax revenues at the local, state, and federal levels".
== Third-generation technologies ==
Third-generation renewable energy technologies are still under development and include advanced biomass gasification, biorefinery technologies, hot-dry-rock geothermal power, and ocean energy. Third-generation technologies are not yet widely demonstrated or have limited commercialization. Many are on the horizon and may have potential comparable to other renewable energy technologies, but still depend on attracting sufficient attention and research and development funding.
=== New bioenergy technologies ===
According to the International Energy Agency, cellulosic ethanol biorefineries could allow biofuels to play a much bigger role in the future than organizations such as the IEA previously thought. Cellulosic ethanol can be made from plant matter composed primarily of inedible cellulose fibers that form the stems and branches of most plants. Crop residues (such as corn stalks, wheat straw and rice straw), wood waste, and municipal solid waste are potential sources of cellulosic biomass. Dedicated energy crops, such as switchgrass, are also promising cellulose sources that can be sustainably produced in many regions.
=== Ocean energy ===
Ocean energy is all forms of renewable energy derived from the sea including wave energy, tidal energy, river current, ocean current energy, offshore wind, salinity gradient energy and ocean thermal gradient energy.
The Rance Tidal Power Station (240 MW) is the world's first tidal power station. The facility is located on the estuary of the Rance River, in Brittany, France. Opened on 26 November 1966, it is currently operated by Électricité de France, and is the largest tidal power station in the world, in terms of installed capacity.
First proposed more than thirty years ago, systems to harvest utility-scale electrical power from ocean waves have recently been gaining momentum as a viable technology. The potential for this technology is considered promising, especially on west-facing coasts with latitudes between 40 and 60 degrees:
In the United Kingdom, for example, the Carbon Trust recently estimated the extent of the economically viable offshore resource at 55 TWh per year, about 14% of current national demand. Across Europe, the technologically achievable resource has been estimated to be at least 280 TWh per year. In 2003, the U.S. Electric Power Research Institute (EPRI) estimated the viable resource in the United States at 255 TWh per year (6% of demand).
There are currently nine projects, completed or in-development, off the coasts of the United Kingdom, United States, Spain and Australia to harness the rise and fall of waves by Ocean Power Technologies. The current maximum power output is 1.5 MW (Reedsport, Oregon), with development underway for 100 MW (Coos Bay, Oregon).
=== Enhanced geothermal systems ===
As of 2008, geothermal power development was under way in more than 40 countries, partially attributable to the development of new technologies, such as Enhanced Geothermal Systems. The development of binary cycle power plants and improvements in drilling and extraction technology may enable enhanced geothermal systems over a much greater geographical range than "traditional" Geothermal systems. Demonstration EGS projects are operational in the US, Australia, Germany, France, and the United Kingdom.
=== Advanced solar concepts ===
Beyond the already established solar photovoltaics and solar thermal power technologies are such advanced solar concepts as the solar updraft tower or space-based solar power. These concepts have yet to (if ever) be commercialized.
The Solar updraft tower (SUT) is a renewable-energy power plant for generating electricity from low temperature solar heat. Sunshine heats the air beneath a very wide greenhouse-like roofed collector structure surrounding the central base of a very tall chimney tower. The resulting convection causes a hot air updraft in the tower by the chimney effect. This airflow drives wind turbines placed in the chimney updraft or around the chimney base to produce electricity. Plans for scaled-up versions of demonstration models will allow significant power generation, and may allow development of other applications, such as water extraction or distillation, and agriculture or horticulture. To view a study on the solar updraft tower and its affects click here
A more advanced version of a similarly themed technology is the Vortex engine (AVE) which aims to replace large physical chimneys with a vortex of air created by a shorter, less-expensive structure.
Space-based solar power (SBSP) is the concept of collecting solar power in space (using an "SPS", that is, a "solar-power satellite" or a "satellite power system") for use on Earth. It has been in research since the early 1970s. SBSP would differ from current solar collection methods in that the means used to collect energy would reside on an orbiting satellite instead of on Earth's surface. Some projected benefits of such a system are a higher collection rate and a longer collection period due to the lack of a diffusing atmosphere and night time in space.
== Renewable energy industry ==
Total investment in renewable energy reached $211 billion in 2010, up from $160 billion in 2009. The top countries for investment in 2010 were
China, Germany, the United States, Italy, and Brazil. Continued growth for the renewable energy sector is expected and promotional policies helped the industry weather the 2009 economic crisis better than many other sectors.
=== Wind power companies ===
As of 2010, Vestas (from Denmark) is the world's top wind turbine manufacturer in terms of percentage of market volume, and Sinovel (from China) is in second place. Together Vestas and Sinovel delivered 10,228 MW of new wind power capacity in 2010, and their market share was 25.9 percent. GE Energy (USA) was in third place, closely followed by Goldwind, another Chinese supplier. German Enercon ranks fifth in the world, and is followed in sixth place by Indian-based Suzlon.
=== Photovoltaic market trends ===
The solar PV market has been growing for the past few years. According to solar PV research company, PVinsights, worldwide shipment of solar modules in 2011 was around 25 GW, and the shipment year over year growth was around 40%. The top 5 solar module players in 2011 in turns are Suntech, First Solar, Yingli, Trina, and Sungen. The top 5 solar module companies possessed 51.3% market share of solar modules, according to PVinsights' market intelligence report.
The PV industry has seen drops in module prices since 2008. In late 2011, factory-gate prices for crystalline-silicon photovoltaic modules dropped below the $1.00/W mark. The $1.00/W installed cost, is often regarded in the PV industry as marking the achievement of grid parity for PV. These reductions have taken many stakeholders, including industry analysts, by surprise, and perceptions of current solar power economics often lags behind reality. Some stakeholders still have the perspective that solar PV remains too costly on an unsubsidized basis to compete with conventional generation options. Yet technological advancements, manufacturing process improvements, and industry re-structuring, mean that further price reductions are likely in coming years.
== Non-technical barriers to acceptance ==
Many energy markets, institutions, and policies have been developed to support the production and use of fossil fuels. Newer and cleaner technologies may offer social and environmental benefits, but utility operators often reject renewable resources because they are trained to think only in terms of big, conventional power plants. Consumers often ignore renewable power systems because they are not given accurate price signals about electricity consumption. Intentional market distortions (such as subsidies), and unintentional market distortions (such as split incentives) may work against renewables. Benjamin K. Sovacool has argued that "some of the most surreptitious, yet powerful, impediments facing renewable energy and energy efficiency in the United States are more about culture and institutions than engineering and science".
The obstacles to the widespread commercialization of renewable energy technologies are primarily political, not technical, and there have been many studies which have identified a range of "non-technical barriers" to renewable energy use. These barriers are impediments which put renewable energy at a marketing, institutional, or policy disadvantage relative to other forms of energy. Key barriers include:
Difficulty overcoming established energy systems, which includes difficulty introducing innovative energy systems, particularly for distributed generation such as photovoltaics, because of technological lock-in, electricity markets designed for centralized power plants, and market control by established operators. As the Stern Review on the Economics of Climate Change points out:
"National grids are usually tailored towards the operation of centralised power plants and thus favour their performance. Technologies that do not easily fit into these networks may struggle to enter the market, even if the technology itself is commercially viable. This applies to distributed generation as most grids are not suited to receive electricity from many small sources. Large-scale renewables may also encounter problems if they are sited in areas far from existing grids."
Lack of government policy support, which includes the lack of policies and regulations supporting deployment of renewable energy technologies and the presence of policies and regulations hindering renewable energy development and supporting conventional energy development. Examples include subsidies for fossil-fuels, insufficient consumer-based renewable energy incentives, government underwriting for nuclear plant accidents, and complex zoning and permitting processes for renewable energy.
Lack of information dissemination and consumer awareness.
Higher capital cost of renewable energy technologies compared with conventional energy technologies.
Inadequate financing options for renewable energy projects, including insufficient access to affordable financing for project developers, entrepreneurs and consumers.
Imperfect capital markets, which includes failure to internalize all costs of conventional energy (e.g., effects of air pollution, risk of supply disruption) and failure to internalize all benefits of renewable energy (e.g., cleaner air, energy security).
Inadequate workforce skills and training, which includes lack of adequate scientific, technical, and manufacturing skills required for renewable energy production; lack of reliable installation, maintenance, and inspection services; and failure of the educational system to provide adequate training in new technologies.
Lack of adequate codes, standards, utility interconnection, and net-metering guidelines.
Poor public perception of renewable energy system aesthetics.
Lack of stakeholder/community participation and co-operation in energy choices and renewable energy projects.
With such a wide range of non-technical barriers, there is no "silver bullet" solution to drive the transition to renewable energy. So ideally there is a need for several different types of policy instruments to complement each other and overcome different types of barriers.
A policy framework must be created that will level the playing field and redress the imbalance of traditional approaches associated with fossil fuels. The policy landscape must keep pace with broad trends within the energy sector, as well as reflecting specific social, economic and environmental priorities. Some resource-rich countries struggle to move away from fossil fuels and have failed thus far to adopt regulatory frameworks necessary for developing renewable energy (e.g. Russia).
== Public policy landscape ==
Public policy has a role to play in renewable energy commercialization because the free market system has some fundamental limitations. As the Stern Review points out: "In a liberalised energy market, investors, operators and consumers should face the full cost of their decisions. But this is not the case in many economies or energy sectors. Many policies distort the market in favour of existing fossil fuel technologies." The International Solar Energy Society has stated that "historical incentives for the conventional energy resources continue even today to bias markets by burying many of the real societal costs of their use".
Fossil-fuel energy systems have different production, transmission, and end-use costs and characteristics than do renewable energy systems, and new promotional policies are needed to ensure that renewable systems develop as quickly and broadly as is socially desirable. Lester Brown states that the market "does not incorporate the indirect costs of providing goods or services into prices, it does not value nature's services adequately, and it does not respect the sustainable-yield thresholds of natural systems". It also favors the near term over the long term, thereby showing limited concern for future generations. Tax and subsidy shifting can help overcome these problems, though is also problematic to combine different international normative regimes regulating this issue.
=== Shifting taxes ===
Tax shifting has been widely discussed and endorsed by economists. It involves lowering income taxes while raising levies on environmentally destructive activities, in order to create a more responsive market. For example, a tax on coal that included the increased health care costs associated with breathing polluted air, the costs of acid rain damage, and the costs of climate disruption would encourage investment in renewable technologies. Several Western European countries are already shifting taxes in a process known there as environmental tax reform.
In 2001, Sweden launched a new 10-year environmental tax shift designed to convert 30 billion kroner ($3.9 billion) of income taxes to taxes on environmentally destructive activities. Other European countries with significant tax reform efforts are France, Italy, Norway, Spain, and the United Kingdom. Asia's two leading economies, Japan and China, are considering carbon taxes.
=== Shifting subsidies ===
Just as there is a need for tax shifting, there is also a need for subsidy shifting. Subsidies are not an inherently bad thing as many technologies and industries emerged through government subsidy schemes. The Stern Review explains that of 20 key innovations from the past 30 years, only one of the 14 was funded entirely by the private sector and nine were totally publicly funded. In terms of specific examples, the Internet was the result of publicly funded links among computers in government laboratories and research institutes. And the combination of the federal tax deduction and a robust state tax deduction in California helped to create the modern wind power industry. At the same time specifically US tax credits systems for renewable energy have been described as an "opaque" financial instrument dominated by large investors to reduce their tax payments while greenhouse gas reduction targets are being treated as a side effect.
Lester Brown has argued that "a world facing the prospect of economically disruptive climate change can no longer justify subsidies to expand the burning of coal and oil. Shifting these subsidies to the development of climate-benign energy sources such as wind, solar, biomass, and geothermal power is the key to stabilizing the earth's climate." The International Solar Energy Society advocates "leveling the playing field" by redressing the continuing inequities in public subsidies of energy technologies and R&D, in which the fossil fuel and nuclear power receive the largest share of financial support.
Some countries are eliminating or reducing climate-disrupting subsidies and Belgium, France, and Japan have phased out all subsidies for coal. Germany is reducing its coal subsidy. The subsidy dropped from $5.4 billion in 1989 to $2.8 billion in 2002, and in the process Germany lowered its coal use by 46 percent. China cut its coal subsidy from $750 million in 1993 to $240 million in 1995 and more recently has imposed a high-sulfur coal tax. However, the United States has been increasing its support for the fossil fuel and nuclear industries.
In November 2011, an IEA report entitled Deploying Renewables 2011 said "subsidies in green energy technologies that were not yet competitive are justified in order to give an incentive to investing into technologies with clear environmental and energy security benefits". The IEA's report disagreed with claims that renewable energy technologies are only viable through costly subsidies and not able to produce energy reliably to meet demand.
A fair and efficient imposition of subsidies for renewable energies and aiming at sustainable development, however, require coordination and regulation at a global level, as subsidies granted in one country can easily disrupt industries and policies of others, thus underlining the relevance of this issue at the World Trade Organization.
=== Renewable energy targets ===
Setting national renewable energy targets can be an important part of a renewable energy policy and these targets are usually defined as a percentage of the primary energy and/or electricity generation mix. For example, the European Union has prescribed an indicative renewable energy target of 12 percent of the total EU energy mix and 22 percent of electricity consumption by 2010. National targets for individual EU Member States have also been set to meet the overall target. Other developed countries with defined national or regional targets include Australia, Canada, Israel, Japan, Korea, New Zealand, Norway, Singapore, Switzerland, and some US States.
National targets are also an important component of renewable energy strategies in some developing countries. Developing countries with renewable energy targets include China, India, Indonesia, Malaysia, the Philippines, Thailand, Brazil, Egypt, Mali, and South Africa. The targets set by many developing countries are quite modest when compared with those in some industrialized countries.
Renewable energy targets in most countries are indicative and nonbinding but they have assisted government actions and regulatory frameworks. The United Nations Environment Program has suggested that making renewable energy targets legally binding could be an important policy tool to achieve higher renewable energy market penetration.
=== Levelling the playing field ===
The IEA has identified three actions which will allow renewable energy and other clean energy technologies to "more effectively compete for private sector capital".
"First, energy prices must appropriately reflect the "true cost" of energy (e.g. through carbon pricing) so that the positive and negative impacts of energy production and consumption are fully taken into account". Example: New UK nuclear plants cost £92.50/MWh, whereas offshore wind farms in the UK are supported with €74.2/MWh at a price of £150 in 2011 falling to £130 per MWh in 2022. In Denmark, the price can be €84/MWh.
"Second, inefficient fossil fuel subsidies must be removed, while ensuring that all citizens have access to affordable energy".
"Third, governments must develop policy frameworks that encourage private sector investment in lower-carbon energy options".
=== Green stimulus programs ===
In response to the Great Recession, major governments made "green stimulus" programs one of their main policy instruments for supporting economic recovery. Some US$188 billion in green stimulus funding had been allocated to renewable energy and energy efficiency, to be spent mainly in 2010 and in 2011.
=== Energy sector regulation ===
Public policy determines the extent to which renewable energy (RE) is to be incorporated into a developed or developing country's generation mix. Energy sector regulators implement that policy—thus affecting the pace and pattern of RE investments and connections to the grid. Energy regulators often have authority to carry out a number of functions that have implications for the financial feasibility of renewable energy projects. Such functions include issuing licenses, setting performance standards, monitoring the performance of regulated firms, determining the price level and structure of tariffs, establishing uniform systems of accounts, arbitrating stakeholder disputes (like interconnection cost allocations), performing management audits, developing agency human resources (expertise), reporting sector and commission activities to government authorities, and coordinating decisions with other government agencies. Thus, regulators make a wide range of decisions that affect the financial outcomes associated with RE investments. In addition, the sector regulator is in a position to give advice to the government regarding the full implications of focusing on climate change or energy security. The energy sector regulator is the natural advocate for efficiency and cost-containment throughout the process of designing and implementing RE policies. Since policies are not self-implementing, energy sector regulators become a key facilitator (or blocker) of renewable energy investments.
=== Energy transition in Germany ===
The Energiewende (German for energy transition) is the transition by Germany to a low carbon, environmentally sound, reliable, and affordable energy supply. The new system will rely heavily on renewable energy (particularly wind, photovoltaics, and biomass) energy efficiency, and energy demand management. Most if not all existing coal-fired generation will need to be retired. The phase-out of Germany's fleet of nuclear reactors, to be complete by 2022, is a key part of the program.
Legislative support for the Energiewende was passed in late 2010 and includes greenhouse gas (GHG) reductions of 80–95% by 2050 (relative to 1990) and a renewable energy target of 60% by 2050. These targets are ambitious. The Berlin-based policy institute Agora Energiewende noted that "while the German approach is not unique worldwide, the speed and scope of the Energiewende are exceptional". The Energiewende also seeks a greater transparency in relation to national energy policy formation.
Germany has made significant progress on its GHG emissions reduction target, achieving a 27% decrease between 1990 and 2014. However Germany will need to maintain an average GHG emissions abatement rate of 3.5% per annum to reach its Energiewende goal, equal to the maximum historical value thus far.
Germany spends €1.5 billion per annum on energy research (2013 figure) in an effort to solve the technical and social issues raised by the transition. This includes a number of computer studies that have confirmed the feasibility and a similar cost (relative to business-as-usual and given that carbon is adequately priced) of the Energiewende.
These initiatives go well beyond European Union legislation and the national policies of other European states. The policy objectives have been embraced by the German federal government and has resulted in a huge expansion of renewables, particularly wind power. Germany's share of renewables has increased from around 5% in 1999 to 22.9% in 2012, surpassing the OECD average of 18% usage of renewables.
Producers have been guaranteed a fixed feed-in tariff for 20 years, guaranteeing a fixed income. Energy co-operatives have been created, and efforts were made to decentralize control and profits. The large energy companies have a disproportionately small share of the renewables market. However, in some cases poor investment designs have caused bankruptcies and low returns, and unrealistic promises have been shown to be far from reality.
Nuclear power plants were closed, and the existing nine plants will close earlier than planned, in 2022.
One factor that has inhibited efficient employment of new renewable energy has been the lack of an accompanying investment in power infrastructure to bring the power to market. It is believed 8,300 km of power lines must be built or upgraded. The different German States have varying attitudes to the construction of new power lines. Industry has had their rates frozen and so the increased costs of the Energiewende have been passed on to consumers, who have had rising electricity bills.
== Voluntary market mechanisms for renewable electricity ==
Voluntary markets, also referred to as green power markets, are driven by consumer preference. Voluntary markets allow a consumer to choose to do more than policy decisions require and reduce the environmental impact of their electricity use. Voluntary green power products must offer a significant benefit and value to buyers to be successful. Benefits may include zero or reduced greenhouse gas emissions, other pollution reductions or other environmental improvements on power stations.
The driving factors behind voluntary green electricity within the EU are the liberalized electricity markets and the RES Directive. According to the directive, the EU Member States must ensure that the origin of electricity produced from renewables can be guaranteed and therefore a "guarantee of origin" must be issued (article 15). Environmental organisations are using the voluntary market to create new renewables and improving sustainability of the existing power production. In the US the main tool to track and stimulate voluntary actions is Green-e program managed by Center for Resource Solutions. Globally available voluntary tool used by the NGOs to promote sustainable electricity production is EKOenergy label.
== Recent developments ==
A number of events in 2006 pushed renewable energy up the political agenda, including the US mid-term elections in November, which confirmed clean energy as a mainstream issue. Also in 2006, the Stern Review made a strong economic case for investing in low carbon technologies now, and argued that economic growth need not be incompatible with cutting energy consumption. According to a trend analysis from the United Nations Environment Programme, climate change concerns coupled with recent high oil prices and increasing government support are driving increasing rates of investment in the renewable energy and energy efficiency industries.
Investment capital flowing into renewable energy reached a record US$77 billion in 2007, with the upward trend continuing in 2008. The OECD still dominates, but there is now increasing activity from companies in China, India and Brazil. Chinese companies were the second largest recipient of venture capital in 2006 after the United States. In the same year, India was the largest net buyer of companies abroad, mainly in the more established European markets.
New government spending, regulation, and policies helped the industry weather the 2009 economic crisis better than many other sectors. Most notably, U.S. President Barack Obama's American Recovery and Reinvestment Act of 2009 included more than $70 billion in direct spending and tax credits for clean energy and associated transportation programs. This policy-stimulus combination represents the largest federal commitment in U.S. history for renewables, advanced transportation, and energy conservation initiatives. Based on these new rules, many more utilities strengthened their clean-energy programs. Clean Edge suggests that the commercialization of clean energy will help countries around the world deal with the current economic malaise. Once-promising solar energy company, Solyndra, became involved in a political controversy involving U.S. President Barack Obama's administration's authorization of a $535 million loan guarantee to the Corporation in 2009 as part of a program to promote alternative energy growth. The company ceased all business activity, filed for Chapter 11 bankruptcy, and laid-off nearly all of its employees in early September 2011.
In his 24 January 2012, State of the Union address, President Barack Obama restated his commitment to renewable energy. Obama said that he "will not walk away from the promise of clean energy." Obama called for a commitment by the Defense Department to purchase 1,000 MW of renewable energy. He also mentioned the long-standing Interior Department commitment to permit 10,000 MW of renewable energy projects on public land in 2012.
As of 2012, renewable energy plays a major role in the energy mix of many countries globally. Renewables are becoming increasingly economic in both developing and developed countries. Prices for renewable energy technologies, primarily wind power and solar power, continued to drop, making renewables competitive with conventional energy sources. Without a level playing field, however, high market penetration of renewables is still dependent on robust promotional policies. Fossil fuel subsidies, which are far higher than those for renewable energy, remain in place and quickly need to be phased out.
United Nations' Secretary-General Ban Ki-moon has said that "renewable energy has the ability to lift the poorest nations to new levels of prosperity". In October 2011, he "announced the creation of a high-level group to drum up support for energy access, energy efficiency and greater use of renewable energy. The group is to be co-chaired by Kandeh Yumkella, the chair of UN Energy and director general of the UN Industrial Development Organisation, and Charles Holliday, chairman of Bank of America".
Worldwide use of solar power and wind power continued to grow significantly in 2012. Solar electricity consumption increased by 58 percent, to 93 terawatt-hours (TWh). Use of wind power in 2012 increased by 18.1 percent, to 521.3 TWh. Global solar and wind energy installed capacities continued to expand even though new investments in these technologies declined during 2012. Worldwide investment in solar power in 2012 was $140.4 billion, an 11 percent decline from 2011, and wind power investment was down 10.1 percent, to $80.3 billion. But due to lower production costs for both technologies, total installed capacities grew sharply. This investment decline, but growth in installed capacity, may again occur in 2013. Analysts expect the market to triple by 2030. In 2015, investment in renewables exceeded fossils.
== 100% renewable energy ==
The incentive to use 100% renewable energy for electricity, transport, or even total primary energy supply globally, has been motivated by global warming and other ecological as well as economic concerns. In the Intergovernmental Panel on Climate Change's reviews of scenarios of energy usage that would keep global warming to approximately 1.5 degrees, the proportion of primary energy supplied by renewables increases from 15% in 2020 to 60% in 2050 (median values across all published pathways). The proportion of primary energy supplied by biomass increases from 10% to 27%, with effective controls on whether land use is changed in the growing of biomass. The proportion from wind and solar increases from 1.8% to 21%.
At the national level, at least 30 nations around the world already have renewable energy contributing more than 20% of energy supply.
Mark Z. Jacobson, professor of civil and environmental engineering at Stanford University and director of its Atmosphere and Energy Program says producing all new energy with wind power, solar power, and hydropower by 2030 is feasible and existing energy supply arrangements could be replaced by 2050. Barriers to implementing the renewable energy plan are seen to be "primarily social and political, not technological or economic". Jacobson says that energy costs with a wind, solar, water system should be similar to today's energy costs.
Renewable projects must be sited at distant locations due to high land prices in urban areas or for the renewable resource itself which require transmission construction costs.
Similarly, in the United States, the independent National Research Council has noted that "sufficient domestic renewable resources exist to allow renewable electricity to play a significant role in future electricity generation and thus help confront issues related to climate change, energy security, and the escalation of energy costs … Renewable energy is an attractive option because renewable resources available in the United States, taken collectively, can supply significantly greater amounts of electricity than the total current or projected domestic demand."
The most significant barriers to the widespread implementation of large-scale renewable energy and low carbon energy strategies are primarily political and not technological. According to the 2013 Post Carbon Pathways report, which reviewed many international studies, the key roadblocks are: climate change denial, the fossil fuels lobby, political inaction, unsustainable energy consumption, outdated energy infrastructure, and financial constraints.
== Energy efficiency ==
Moving towards energy sustainability will require changes not only in the way energy is supplied, but in the way it is used, and reducing the amount of energy required to deliver various goods or services is essential. Opportunities for improvement on the demand side of the energy equation are as rich and diverse as those on the supply side, and often offer significant economic benefits.
A sustainable energy economy requires commitments to both renewables and efficiency. Renewable energy and energy efficiency are said to be the "twin pillars" of sustainable energy policy. The American Council for an Energy-Efficient Economy has explained that both resources must be developed in order to stabilize and reduce carbon dioxide emissions:
Efficiency is essential to slowing the energy demand growth so that rising clean energy supplies can make deep cuts in fossil fuel use. If energy use grows too fast, renewable energy development will chase a receding target. Likewise, unless clean energy supplies come online rapidly, slowing demand growth will only begin to reduce total emissions; reducing the carbon content of energy sources is also needed.
The IEA has stated that renewable energy and energy efficiency policies are complementary tools for the development of a sustainable energy future, and should be developed together instead of being developed in isolation.
== See also ==
=== Lists ===
=== Topics ===
=== People ===
== References ==
== Bibliography ==
== External links ==
Investing: Green technology has big growth potential, LA Times, 2011
Global Renewable Energy: Policies and Measures
Missing the Market Meltdown
Bureau of Land Management 2012 Renewable Energy Priority Projects | Wikipedia/Renewable_energy_commercialization |
Landscape-scale conservation is a holistic approach to landscape management, aiming to reconcile the competing objectives of nature conservation and economic activities across a given landscape. Landscape-scale conservation may sometimes be attempted because of climate change. It can be seen as an alternative to site based conservation.
Many global problems such as poverty, food security, climate change, water scarcity, deforestation and biodiversity loss are connected. For example, lifting people out of poverty can increase consumption and drive climate change. Expanding agriculture can exacerbate water scarcity and drive habitat loss. Proponents of landscape management argue that as these problems are interconnected, coordinated approaches are needed to address them, by focussing on how landscapes can generate multiple benefits. For example, a river basin can supply water for towns and agriculture, timber and food crops for people and industry, and habitat for biodiversity; and each one of these users can have impacts on the others.
Landscapes in general have been recognised as important units for conservation by intergovernmental bodies, government initiatives, and research institutes.
Problems with this approach include difficulties in monitoring, and the proliferation of definitions and terms relating to it.
== Definitions ==
There are many overlapping terms and definitions, but many terms have similar meanings. A sustainable landscape, for example, meets "the needs of the present without compromising the ability of future generations to meet their own needs."
Approaching conservation by means of landscapes can be seen as "a conceptual framework whereby stakeholders in a landscape aim to reconcile competing social, economic and environmental objectives". Instead of focussing on a single use of the land it aims to ensure that the interests of different stakeholders are met.
The starting point for all landscape-scale conservation schemes must be an understanding of the character of the landscape. Landscape character goes beyond aesthetic. It involves understanding how the landscape functions to support communities, cultural heritage and development, the economy, as well as the wildlife and natural resources of the area. Landscape character requires careful assessment according to accepted methodologies. Landscape character assessment will contribute to the determination of what scale is appropriate in which landscape. "Landscape scale" does not merely mean acting at a bigger scale: it means conservation is carried out at the correct scale and that it takes into account the human elements of the landscape, both past and present.
== History ==
The word 'landscape' in English is a loanword from Dutch landschap introduced in the 1660s and originally meant a painting. The meaning a "tract of land with its distinguishing characteristics" was derived from that in 1886. This was then used as a verb as of 1916.
The German geographer Carl Troll coined the German term Landschaftsökologie–thus 'landscape ecology' in 1939. He developed this terminology and many early concepts of landscape ecology as part of this work, which consisted of applying aerial photograph interpretation to studies of interactions between environment, agriculture and vegetation.
In the UK conservation of landscapes can be said to have begun in 1945 with the publication of the Report to the Government on National Parks in England and Wales. The National Parks and Access to the Countryside Act 1949 introduced the legislation for the creation Areas of Outstanding Natural Beauty (AONB). Northern Ireland has the same system after adoption of the Amenity Lands (NI) Act 1965. The first of these AONB were defined in 1956, with the last being created in 1995.
The Permanent European Conference for the Study of the Rural Landscape was established in 1957. The European Landscape Convention was initiated by the Congress of Regional and Local Authorities of the Council of Europe (CLRAE) in 1994, was adopted by the Committee of Ministers of the Council of Europe in 2000, and came into force in 2004.
The conservation community began to take notice of the science of landscape ecology in the 1980s.
Efforts to develop concepts of landscape management that integrate international social and economic development with biodiversity conservation began in 1992.
Landscape management now exists in multiple iterations and alongside other concepts such as watershed management, landscape ecology and cultural landscapes.
== International ==
The UN Environment Programme stated in 2015 that the landscape approach embodies ecosystem management. UNEP uses the approach with the Ecosystem Management of Productive Landscapes project. The scientific committee of the Convention on Biological Diversity also considers the perspective of a landscape the most important scale for improving sustainable use of biodiversity. There are global fora on landscapes. During the Livelihoods and Landscapes Strategies programme the International Union for Conservation of Nature applied this approach to locations worldwide, in 27 landscapes in 23 different countries.
Examples of landscape approaches can be global or continental, for example in Africa, Oceania and Latin America. The European Agricultural Fund for Rural Development plays an important part in funding landscape conservation in Europe.
=== Relevance to international commitments ===
Some argue landscape management can address the Sustainable Development Goals. Many of these goals have potential synergies or trade-offs: some therefore argue that addressing these goals individually may not be effective, and landscape approaches provide a potential framework to manage them. For example, increasing areas of irrigated agricultural land to end hunger could have adverse impacts on terrestrial ecosystems or sustainable water management. Landscape approaches intend to include different sectors, and thus achieve the multiple objectives of the Sustainable Development Goals – for example, working within catchment area of a river to enhance agricultural productivity, flood defence, biodiversity and carbon storage.
Climate change and agriculture are intertwined so production of food and climate mitigation can be a part of landscape management. The agricultural sector accounts for around 24% of anthropogenic emissions. Unlike other sectors that emit greenhouse gases, agriculture and forestry have the potential to mitigate climate change by reducing or removing greenhouse gas emissions, for example by reforestation and landscape restoration. Advocates of landscape management argue that 'climate-smart agriculture' and REDD+ can draw on landscape management.
== Regional ==
=== Germany ===
Because a large proportion of the biodiversity of Germany was able to invade from the south and east after human activities altered the landscape, maintaining such artificial landscapes is an integral part of nature conservation. The full name of the main nature conservation law in Germany, the Bundesnaturschutzgesetzes, is thus titled in its entirety Gesetz über Naturschutz und Landschaftspflege, where Landschaftspflege translates literally to "landscape maintenance" (see reference for more). Related concepts are Landschaftsschutz, "landscape protection/conservation", and Landschaftsschutzgebiet, a "nature preserve", or literally a (legally) "protected landscape area". The Deutscher Verband für Landschaftspflege is the main organisation which protects landscapes in Germany. It is an umbrella organisation which coordinates the regional landscape protection organisations of the different German states. Classically, there are four methods which can be done in order to conserve landscapes: maintenance, improvement, protection and redevelopment. The marketing of products such as meat from alpine meadows or apple juice from traditional Streuobstwiese can also be an important factor in conservation. Landscapes are maintained by three methods: biological - such as grazing by livestock, manually (although this is rare due to the high cost of labour) and commonly mechanically.
=== The Netherlands ===
Staatsbosbeheer, the Dutch governmental forest service, considers landscape management an important part of managing their lands. Landschapsbeheer Nederland is an umbrella organisation which promotes and helps fund the interests of the different provincial landscape management organisations, which between them include 75,000 volunteers and 110,000 hectares of protected nature reserves. Sustainable landscape management is being researched in the Netherlands.
=== Peru ===
An example of a producer movement managing a multi-functional landscape is the Potato Park in Písac, Peru, where local communities protect the ecological and cultural diversity of the 12,000ha landscape.
=== Sweden ===
In Sweden, the Swedish National Heritage Board, or Riksantikvarieämbetet, is responsible for landscape conservation. Landscape conservation can be studied at the Department of Cultural Conservation (at Dacapo Mariestad) of the University of Gothenburg, in both Swedish and English.
=== Thailand ===
An example of cooperation between very different actors is from the Doi Mae Salong watershed in northwest Thailand, a Military Reserved Area under the control of the Royal Thai Armed Forces. Reforestation activities led to tension with local hill tribes. In response, an agreement was reached with them on land rights and use of different parts of the reserve.
=== United Kingdom ===
Among the leading exponents of UK landscape scale conservation are the Areas of Outstanding Natural Beauty (AONB). There are 49 AONB in the UK. The International Union for Conservation of Nature has categorised these regions as "category 5 protected areas" and in 2005 claimed the AONB are administered using what the IUCN coined the "protected landscape approach". In Scotland there is a similar system of national scenic areas.
The UK Biodiversity Action Plan protects semi-natural grasslands, among other habitats, which constitute landscapes maintained by low-intensity grazing. Agricultural environment schemes reward farmers and land managers financially for maintaining these habitats on registered agricultural land. Each of the four
countries in the UK has its own individual scheme.
Studies have been carried out across the UK looking at much wider range of habitats. In Wales the Pumlumon Large Area Conservation Project focusses on upland conservation in areas of marginal agriculture and forestry. The North Somerset Levels and Moors Project addresses wetlands.
=== Other ===
Landscape approaches have been taken up by governments in for example the Greater Mekong Subregion project and in Indonesia's climate change commitments, and by international research bodies such as the Center for International Forestry Research, which convenes the Global Landscapes Forum.
The Mount Kailash region is where the Indus River, the Karnali River (a major tributary of the Ganges River), the Brahmaputra River and the Sutlej river systems originate. With assistance from the International Centre for Integrated Mountain Development, the three surrounding countries (China, India and Nepal) developed an integrated management approach to the different conservation and development issues within this landscape.
Six countries in West Africa in the Volta River basin using the 'Mapping Ecosystems Services to Human well-being' toolkit, use landscape modelling of alternative scenarios for the riparian buffer to make land-use decisions such as conserving hydrological ecosystem services and meeting national SDG commitments.
== Variations ==
=== Ecoagriculture ===
In a 2001 article published by Sara J. Scherr and Jeffrey McNeely, soon expanded into a book, Scherr and McNeely introduced the term "ecoagriculture" to describe their vision of rural development while advancing the environment, claim that agriculture is the dominant influence on wild species and habitats, and point to a number of recent and potential future developments they identified as beneficial examples of land use. They incorporated the non-profit EcoAgriculture Partners. in 2004 to promote this vision, with Scherr as President and CEO, and McNeely as an independent governing board member. Scherr and McNeely edited a second book in 2007. Ecoagriculture had three elements in 2003.
=== Integrated landscape management ===
In 2012 Scherr invented a new term, integrated landscape management(ILM), to describe her ideas for developing entire regions, not at just a farm or plot level. Integrated landscape management is a way of managing sustainable landscapes by bringing together multiple stakeholders with different land use objectives. The integrated approach claims to go beyond other approaches which focus on users of the land independently of each other, despite needing some of the same resources. It is promoted by the conservation NGOs Worldwide Fund for Nature, Global Canopy Programme, The Nature Conservancy, The Sustainable Trade Initiative, and EcoAgriculture Partners. Promoters claim that integrated landscape management will maximise collaboration in planning, policy development and action regarding the interdependent Sustainable Development Goals. It was defined by four elements in 2013:
Large scale: It plans land uses at the landscape scale. Wildlife population dynamics and watershed functions can only be understood at the landscape scale. Assuming short-term trade-offs may lead to long-term synergies, conducting analyses over long time periods is advocated.
Emphasis on synergies: It tries to exploit "synergies" among conservation, agricultural production, and rural livelihoods.
Emphasis on collaboration: It can not be achieved by individuals. The management of landscapes require different land managers with different environmental and socio-economic goals to achieve conservation, production, and livelihood goals at a landscape scale.
Importance of both conservation and agricultural production: bringing conservation into the agricultural and rural development discourse by highlighting the importance of ecosystem services in supporting agricultural production. It supports conservationists to more effectively conserve nature within and outside protected areas by working with the agricultural community by developing conservation-friendly livelihoods for rural land users.
By 2016 it had five elements, namely:
stakeholders come together for cooperative dialogue and action;
they exchange information systematically and discuss perspectives to achieve a shared understanding of the landscape conditions, challenges and opportunities;
collaborative planning to develop an agreed action plan;
implementation of the plan;
monitoring and dialogue to adapt management.
=== Ecosystem approach ===
The ecosystem approach, promoted by the Convention on Biological Diversity, is a strategy for the integrated ecosystem management of land, water, and living resources for conservation and sustainability.
=== Ten Principles ===
This approach includes continual learning and adaptive management: including monitoring, the expectation that actions take place at multiple scales and that landscapes are multifunctional (e.g. supplying both goods, such as timber and food, and services, such as water and biodiversity protection). There are multiple stakeholders, and it assumes they have a common concern about the landscape, negotiate change with each other, and their rights and responsibilities are clear or will become clear.
== Criticisms ==
A literature review identified five main barriers, as follows:
Terminology confusion: the variety of definitions creates confusion and resistance to engage. This resistance has emerged, often independently, from different fields. As stated by Scherr et al.: "People are talking about the same thing ... This can lead to fragmentation of knowledge, unnecessary re-invention of ideas and practices, and inability to mobilize action at scale. ... this rich diversity is often simply overwhelming: they receive confusing messages" This problem is not unique to landscape approaches: since the 1970s it has been recognised that the constant emergence of new terminology can be harmful if they promote rhetoric at the expense of action. Because landscapes approaches develop from, and aim to integrate, a wide variety of sectors, makes it vulnerable to overlapping definitions and parallel concepts. Like other approaches to conservation, it may be a fad.
Time lags: substantial time and resources are invested in developing and planning, while resources are inadequate for implementation.
Operating silos: Each sector pursues its goals without giving consideration to the others. This may arise because of a lack in established objectives, operating norms and funding that effectively bridge different sectors. Working across sectors at the landscape scale requires a range of skills, different from those traditionally used by conservation organisations.
Engagement: Stakeholders may not desire to be engaged in the process, engagement may be trivial or inaccessible, and the discussions may hinder efficient decision-making.
Monitoring: There is lack of monitoring to check whether the objectives have been achieved.
== See also ==
Agriculture in Concert with the Environment
Agroecology
Agroforestry
Anthropogenic biome
Conservation development
Ecosystem approach
Global biodiversity
Landscape ecology
Multifunctional landscape
Working landscape
Landscape Institute
Landscape urbanism
Polder model
Sustainable forest management
Sustainable landscaping
Topocide
Watershed management
== References ==
== External links ==
CIVILSCAPE - We are the landscape people! (CIVILSCAPE)
Landscape Europe
Landscape Character Network | Wikipedia/Integrated_landscape_management |
Integrated Chain Management (ICM), also known as Integral Chain Management, is an approach for the reduction of environmental impact of product chains. Such a product chain exists out of an extraction phase, a production phase, a use phase and a waste phase. The ultimate goal of ICM is a reduction of environmental load over the whole chain. Integrated Chain Management is one of the approaches that can be used to come to sustainable development. Other approaches in this line are the Ecological Footprint and the DTO approach.
Within the ICM approach all phases within the chain must be considered. Therefore, it can be seen as a "cradle to grave" approach.
Several inputs and outputs can be taken into account when applying the ICM approach. Such as: Energy flows, mass flows, materials, waste flows and emissions. Within ICM material cycles should be closed where possible and the remainder flows of emissions and waste should be brought within acceptable boundaries. Also the use of resources should be kept to a minimum.
Integrated chain management should not be mixed up with Supply Chain Management or Integrated Supply Chain Management. These concepts do not have the reduction of environmental load as their main goal.
An important aspect of ICM is that shifting to other phases in the product chain is avoided. For instance, a producer of chairs can choose to leave off an environment unfriendly material in a new product. The producer can even see this as an extra selling point for the customer, but as a consequence the supplier of raw materials has to use much more energy to produce a material with the same qualities. The result of this is that there may no longer be a net environmental reduction across the whole chain. Within the integrated chain management approach this is avoided.
The chain can be managed by developing new policies and economical or political incentives. Therefore, one must have insight into the inputs and outputs of the production chain. Before these policies can be developed one must engage in several actions.
Analyse the processes into a preferred level of detail
Determine the boundaries of the chain. Should links outside the companies be involved as well?
Determine whether there should be a focus on just one or on several environmental problems
Determine on which material flows or energy flows there should be a focus.
Effective supply chain management can impact virtually all business and production processes
== Example ==
An example of applying the ICM approach would be to develop policies in a particular product area. The responsibility of problems caused by the waste stage can be assigned to the producers of these products. This leads to improved product design and new insight in how to put these products in the market. For instance the product can be sold with a disposal contribution. On the price tag of a radio nowadays can be printed: "this radio costs 25 $ not including the 3 $ disposal contribution"
The effects can be seen within the whole chain. The producer will try to choose non-polluting materials, as they increase the costs of the waste-stage. The producer of raw materials will try to improve its production process in order to meet the increased demand for 'clean' primary products. And the consumer will be aware that some products give more pressure on the environment than others when its economical lifespan has run out.
== External links ==
"ICM: Danish Environmental Protection Agency"
"Integrated Chain Management : An Example"
"Integrated Chain Management of Polymer Materials" | Wikipedia/Integrated_chain_management |
Ecological design or ecodesign is an approach to designing products and services that gives special consideration to the environmental impacts of a product over its entire lifecycle. Sim Van der Ryn and Stuart Cowan define it as "any form of design that minimizes environmentally destructive impacts by integrating itself with living processes." Ecological design can also be defined as the process of integrating environmental considerations into design and development with the aim of reducing environmental impacts of products through their life cycle.
The idea helps connect scattered efforts to address environmental issues in architecture, agriculture, engineering, and ecological restoration, among others. The term was first used by Sim Van der Ryn and Stuart Cowan in 1996. Ecological design was originally conceptualized as the “adding in “of environmental factor to the design process, but later turned to the details of eco-design practice, such as product system or individual product or industry as a whole. With the inclusion of life cycle modeling techniques, ecological design was related to the new interdisciplinary subject of industrial ecology.
== Overview ==
As the whole product's life cycle should be regarded in an integrated perspective, representatives from advanced product design, production, marketing, purchasing, and project management should work together on the Ecodesign of a further developed or new product. Together, they have the best chance to predict the holistic effects of changes of the product and their environmental impact. Considerations of ecological design during product development is a proactive approach to eliminate environmental pollution due to product waste.
An eco-design product may have a cradle-to-cradle life cycle ensuring zero waste is created in the whole process. By mimicking life cycles in nature, eco-design can serve as a concept to achieve a truly circular economy.
Environmental aspects which ought to be analysed for every stage of the life cycle are:
Consumption of resources (energy, materials, water or land area)
Emissions to air, water, and the ground (our Earth) as being relevant for the environment and human health, including noise emissions
Waste (hazardous waste and other waste defined in environmental legislation) is only an intermediate step and the final emissions to the environment (e.g. methane and leaching from landfills) are inventoried. All consumables, materials and parts used in the life cycle phases are accounted for, and all indirect environmental aspects linked to their production.
The environmental aspects of the phases of the life cycle are evaluated according to their environmental impact on the basis of a number of parameters, such as extent of environmental impact, potential for improvement, or potential of change.
According to this ranking the recommended changes are carried out and reviewed after a certain time.
As the impact of design and the design process has evolved, designers have become more aware of their responsibilities. The design of a product unrelated to its sociological, psychological, or ecological surroundings is no longer possible or acceptable in modern society.
With respect to these concepts, online platforms dealing in only Ecodesign products are emerging, with the additional sustainable purpose of eliminating all unnecessary distribution steps between the designer and the final customer.
Another area of ecological design is through designing with urban ecology in mind, similar to conservation biology, but designers take the natural world into account when designing landscapes, buildings. or anything that impacts interactions with wildlife. A such example in architecture is that of green roofs, offices, where these are spaces that nature can interact with the man made environment but also where humans benefit from these design technologies. Another area is with landscape architecture in the creation of natural gardens, and natural landscapes, these allow for natural wildlife to thrive in urban centres.
Multifunctionality increases consumption in both production and use, so products can also be made ecologically intentional through a series of undesign approaches. The core forms of undesign include self-inhibition, exclusion, removal, replacement, restoration, and safeguarding. This sustainability strategy emphasizes the long-term interests of the planet and is related to environmental restoration and technological mindfulness. An example of undesign is digital detox which refers to voluntarily limiting the use of digital media. Terms such as "non-use,""unplugging," and "digital disconnection" are included in the strategic features of some current productivity applications. Application designers effectively distance users from digital interfaces by limiting screen time and reducing smartphone distractions, thereby reducing energy consumption.
== Ecological design issues and the role of designers ==
=== The rise and conceptualization of ecological design ===
Since the Industrial Revolution, design fields have been criticized for employing unsustainable practices. The architect-designer Victor Papanek (1923–1998) suggested that industrial design has murdered by creating new species of permanent garbage and by choosing materials and processes that pollute the air. Papanek states that the designer-planner shares responsibility for nearly all of our products and tools, and hence, nearly all of our environmental mistakes. To address these issues, R. Buckminster Fuller (1895–1983) demonstrated how design could play a central role in identifying and addressing major world problems. Fuller was concerned with the Earth's finite energy resources and natural resources, and how to integrate machine tools into efficient systems of industrial production. He promoted the principle of "ephemeralization", a term he coined himself to do "more with less" and increase technological efficiency. This concept is key in ecological design that works towards sustainability. In 1986, the design theorist Clive Dilnot argued that design must once again become a means of ordering the world rather than merely of shaping products.
Despite rising ecological awareness in the 20th century, unsustainable design practices continued. The 1992 conference "The Agenda 21: The Earth Summit Strategy to Save Our Planet” put forward a proposition that the world is on a path of energy production and consumption that cannot be sustained. The report drew attention to individuals and groups around the world who have a set of principles to develop strategies for change among many aspects of society, including design. More broadly, the conference emphasized that designers must address human issues. These problems included six items: quality of life, efficient use of natural resources, protecting the global commons, managing human settlements, the use of chemicals and the management of human industrial waste, and fostering sustainable economic growth on a global scale.
Though Western society has only recently espoused ecological design principles, indigenous peoples have long coexisted with the environment. Scholars have discussed the importance of acknowledging and learning from Indigenous peoples and cultures to move towards a more sustainable society. Indigenous knowledge is valuable in ecological design as well as other ecological realms such as restoration ecology.
=== Sustainable development issues ===
These concepts of design tie into the concept of sustainable development. The three pillars addressed in sustainable development are: ecological integrity, social equity, and economic security. Gould and Lewis argue in their book Green Gentrification that urban redevelopment and projects have neglected the social equity pillar, resulting in development that focuses on profit and deepens social inequality. One result of this is green or environmental gentrification. This process is often the result of good intentions to clean up an area and provide green amenities, but without setting protections in place for existing residents to ensure they are not forced out by increased property values and influxes of new wealthier residents.
Unhoused persons are one particularly vulnerable affected population of environmental gentrification. Government environmental planning agendas related to green spaces may lead to the displacement and exclusion of unhoused individuals, under a guise of pro-environmental ethics. One example of this type of design is hostile architecture in urban parks. Park benches designed with metal arched bars to prevent a person from laying on the bench restricts who benefits from green space and ecological design.
== Life Cycle Analysis ==
Life Cycle Analysis (LCA) is a tool used to understand the how a product impacts the environment at each stage of its life cycle, from raw input to the end of the products' life cycle. Life Cycle Cost (LCC) is an economic metric that "identifies the minimum cost for each life cycle stage which would be presented in the aspects of material, procedures, usage, end-of-life and transportation." LCA and LCC can be used to identify particular aspects of a product that is particularly environmentally damaging and reduce those impacts. For example, LCA might reveal that the fabrication stage of a product's life cycle is particularly harmful for the environment and switching to a different material can drive emissions down. However, switching material may increase environmental effects later in a products life time; LCA takes into account the whole life cycle of a product and can alert designers to the many impacts of a product, which is why LCA is important.
Some of the factors that LCA takes into account are the costs and emissions of:
Transportation
Materials
Production
Usage
End-of-life
End-of-life, or disposal, is an important aspect of LCA as waste management is a global issue, with trash found everywhere around the world from the ocean to within organisms. A framework was developed to assess sustainability of waste sites titled EcoSWaD, Ecological Sustainability of Waste Disposal Sites. The model focuses on five major concerns: (1) location suitability, (2) operational sustainability, (3) environmental sustainability, (4) socioeconomic sustainability, and (5) site capacity sustainability. This framework was developed in 2021, as such most established waste disposal sites do not take these factors into consideration. Waste facilities such as dumps and incinerators are disproportionately placed in areas with low education and income levels, burdening these vulnerable populations with pollution and exposure to hazardous materials. For example, legislation in the United States, such as the Cerrell Report, has encouraged these types of classist and racist processes for siting incinerators. Internationally, there has been a global 'race to the bottom' in which polluting industries move to areas with fewer restrictions and regulations on emissions, usually in developing countries, disproportionately exposing vulnerable and impoverished populations to environmental threats. These factors make LCA and sustainable waste sites important on a global scale.
== Urban Ecological Design ==
Related to ecological urbanism, Urban Ecological Design integrates aesthetic, social, and ecological concerns into an urban design framework that seeks to increase ecological functioning, sustainably generate and consume resources, and create resilient built environments and the infrastructure to maintain them. Urban ecological design is inherently interdisciplinary: it integrates multiple academic and professional fields including environmental studies, sociology, justice studies, urban ecology, landscape ecology, urban planning, architecture, and landscape architecture. Urban ecological design aims to solve issues related to multiple large-scale trends including the growth of urban areas, climate change, and biodiversity loss. Urban ecological design has been described as a "process model" contrasted to a normative approach that outlines principles of design. Urban ecological design blends a multitude of frameworks and approaches to create solutions to these issues by improving Urban resilience, sustainable use and management of resources, and integrating ecological processes into the urban landscape.
== Applications in design ==
EcoMaterials, such as the use of local raw materials, are less costly and reduce the environmental costs of shipping, fuel consumption, and CO₂ emissions generated from transportation. Certified green building materials, such as wood from sustainably managed forest plantations, with accreditations from companies such as the Forest Stewardship Council (FSC), or the Pan-European Forest Certification Council (PEFCC), can be used.
Several other types of components and materials can be used in sustainable objects and buildings. Recyclable and recycled materials are commonly used in construction, but it is important that they don't generate any waste during manufacture or after their life cycle ends. Reclaimed materials such as timber at a construction site or junkyard can be given a second life by reusing them as support beams in a new building or as furniture. Stones from an excavation can be used in a retaining wall. The reuse of these items means that less energy is consumed in making new products and a new natural aesthetic quality is achieved.
=== Architecture ===
Off-grid homes only use clean electric power. They are completely separated and disconnected from the conventional electricity grid and receive their power supply by harnessing active or passive energy systems. Off-grid homes are also not served by other publicly or privately managed utilities, such as water and gas in addition to electricity.
=== Art ===
Increased applications of ecological design have gone along with the rise of environmental art. Recycling has been used in art since the early part of the 20th century, when cubist artist Pablo Picasso (1881–1973) and Georges Braque (1882–1963) created collages from newsprints, packaging and other found materials. Contemporary artists have also embraced sustainability, both in materials and artistic content. One modern artist who embraces the reuse of materials is Bob Johnson, creator of River Cubes. Johnson promotes "artful trash management" by creating sculptures from garbage and scraps found in rivers. Garbage is collected, then compressed into a cube that represents the place and people it came from.
=== Clothing ===
There are some clothing companies that are using several ecological design methods to change the future of the textile industry into a more environmentally friendly one. Some approaches include recycling used clothing to minimize the use of raw resources, using biodegradable textile materials to reduce the lasting impact on the environment, and using plant dyes instead of poisonous chemicals to improve the appearance and impact of fabric.
=== Decorating ===
The same principle can be used inside the home, where found objects are now displayed with pride and collecting certain objects and materials to furnish a home is now admired rather than looked down upon. Take for example the electric wire reel reused as a center table.
There is a huge demand in Western countries to decorate homes in a "green" style. A lot of effort is placed into recycled product design and the creation of a natural look. This ideal is also a part of developing countries, although their use of recycled and natural products is often based in necessity and wanting to get maximum use out of materials. The focus on self-regulation and personal lifestyle changes (including decorating as well as clothing and other consumer choices) has shifted questions of social responsibility away from government and corporations and onto the individual.
Biophilic design is a concept used within the building industry to increase occupant connectivity to the natural environment through the use of direct nature, indirect nature, and space and place conditions.
== Active systems ==
These systems use the principle of harnessing the power generated from renewable and inexhaustible sources of energy, for example; solar, wind, thermal, biomass, geothermal, and hydropower energy.
Solar power is a widely known and used renewable energy source. An increase in technology has allowed solar power to be used in a wide variety of applications. Two types of solar panels generate heat into electricity. Thermal solar panels reduce or eliminate the consumption of gas and diesel, and reduce CO₂ emissions. Photovoltaic panels convert solar radiation into an electric current which can power any appliance. This is a more complex technology and is generally more expensive to manufacture than thermal panels.
Biomass is the energy source created from organic materials generated through a forced or spontaneous biological process.
Geothermal energy is obtained by harnessing heat from the ground. This type of energy can be used to heat and cool homes. It eliminates dependence on external energy and generates minimum waste. It is also hidden from view as it is placed underground, making it more aesthetically pleasing and easier to incorporate in a design.
Wind turbines are a useful application for areas without immediate conventional power sources, e.g., rural areas with schools and hospitals that need more power. Wind turbines can provide up to 30% of the energy consumed by a household but they are subject to regulations and technical specifications, such as the maximum distance at which the facility is located from the place of consumption and the power required and permitted for each property.
Water recycling systems such as rainwater tanks that harvest water for multiple purposes. Reusing grey water generated by households are a useful way of not wasting drinking water.
Hydropower, also known as water power, is the use of falling or fast-running water to produce electricity or to power machines. Hydropower is an attractive alternative to fossil fuels as it does not directly produce carbon dioxide or other atmospheric pollutants and it provides a relatively consistent source of power.
== Passive systems ==
Buildings that integrate passive energy systems (bioclimatic buildings) are heated using non-mechanical methods, thereby optimizing natural resources.
Passive daylighting involves the positioning and location of a building to allow for and make use of sunlight throughout the whole year. By using the sun's rays, thermal mass is stored in the building materials such as concrete and can generate enough heat for a room.
Green roofs are roofs that are partially or completely covered with plants or other vegetation. Green roofs are passive systems in that they create insulation that helps regulate the building's temperature. They also retain water, providing a water recycling system, and can provide soundproofing.
== History ==
1971 Ian McHarg, in his book Design with Nature, popularized a system of analyzing the layers of a site in order to compile a complete understanding of the qualitative attributes of a place. McHarg gave every qualitative aspect of the site a layer, such as the history, hydrology, topography, vegetation, etc. This system became the foundation of today's Geographic Information Systems (GIS), a ubiquitous tool used in the practice of ecological landscape design.
1978 Permaculture. Bill Mollison and David Holmgren coin the phrase for a system of designing regenerative human ecosystems. (Founded in the work of Fukuoka, Yeoman, Smith, etc..
1994 David Orr, in his book Earth in Mind: On Education, Environment, and the Human Prospect, compiled a series of essays on "ecological design intelligence" and its power to create healthy, durable, resilient, just, and prosperous communities.
1994 Canadian biologists John Todd and Nancy Jack Todd, in their book From Eco-Cities to Living Machines, describe the precepts of ecological design.
2000 Ecosa Institute begins offering an Ecological Design Certificate, teaching designers to design with nature.
2004 Fritjof Capra, in his book The Hidden Connections: A Science for Sustainable Living, wrote this primer on the science of living systems and considers the application of new thinking by life scientists to our understanding of social organization.
2004 K. Ausebel compiled personal stories of the world's most innovative ecological designers in Nature's Operating Instructions.
== Ecodesign research ==
Ecodesign research focuses primarily on barriers to implementation, ecodesign tools and methods, and the intersection of ecodesign with other research disciplines.
Several review articles provide an overview of the evolution and current state of ecodesign research:
== See also ==
== Notes and references ==
== Bibliography ==
Lacoste, R., Robiolle, M., Vital, X., (2011), "Ecodesign of electronic devices", DUNOD, France
McAloone, T. C. & Bey, N. (2009), Environmental improvement through product development - a guide, Danish EPA, Copenhagen Denmark, ISBN 978-87-7052-950-1, 46 pages
Lindahl, M.: Designer's utilization of DfE methods. Proceedings of the 1st International Workshop on "Sustainable Consumption", 2003. Tokyo, Japan, The Society of Non-Traditional Technology (SNTT) and Research Center for Life Cycle Assessment (AIST).
Wimmer W., Züst R., Lee K.-M. (2004): Ecodesign Implementation – A Systematic Guidance on Integrating Environmental Considerations into Product Development, Dordrecht, Springer
Charter, M./ Tischner, U. (2001): Sustainable Solutions. Developing Products and Services for the Future. Sheffield: Greenleaf
ISO TC 207/WG3
ISO TR 14062
The Journal of Design History: Environmental conscious design and inverse manufacturing, 2005. Eco Design 2005, 4th International Symposium
The Design Journal: Vol 13, Number 1, March 2010 - Design is the problem: The future of Design must be sustainable, N. Shedroff.
"Eco Deco", S. Walton
"Small ECO Houses - Living Green in Style", C. Paredes Benitez, A. Sanchez Vidiella
== Further reading ==
From Bauhaus to Ecohouse: A History of Ecological Design. By Peder Anker, Published by Louisiana State University Press, 2010. ISBN 0-8071-3551-8.
Ecological Design. By Sim Van der Ryn, Stuart Cowan, Published by Island Press, 2007. ISBN 978-1-59726-141-8 (2nd ed., 1st, 1996)
Ignorance and Surprise: Science, Society, and Ecological Design. By Matthias Gross, Published by MIT Press, 2010. ISBN 0-262-01348-7
== External links ==
Sustainable Design & Development Resource Guide
The European Commission's website on Ecodesign activities and related legislation including minimum requirements for energy using products
The European Commission's Directory of LCA and Ecodesign services, tools and databases
The European Commission's ELCD core database with Ecoprofiles (free of charge)
Environmental Effect Analysis (EEA) – Principles and structure
EIME, the ecodesign methodology of the electrical and electronic industry
4E, IEA Implementing Agreement on Efficient Electrical End-Use Equipment | Wikipedia/Ecodesign |
Industrial wastewater treatment describes the processes used for treating wastewater that is produced by industries as an undesirable by-product. After treatment, the treated industrial wastewater (or effluent) may be reused or released to a sanitary sewer or to a surface water in the environment. Some industrial facilities generate wastewater that can be treated in sewage treatment plants. Most industrial processes, such as petroleum refineries, chemical and petrochemical plants have their own specialized facilities to treat their wastewaters so that the pollutant concentrations in the treated wastewater comply with the regulations regarding disposal of wastewaters into sewers or into rivers, lakes or oceans.: 1412 This applies to industries that generate wastewater with high concentrations of organic matter (e.g. oil and grease), toxic pollutants (e.g. heavy metals, volatile organic compounds) or nutrients such as ammonia.: 180 Some industries install a pre-treatment system to remove some pollutants (e.g., toxic compounds), and then discharge the partially treated wastewater to the municipal sewer system.: 60
Most industries produce some wastewater. Recent trends have been to minimize such production or to recycle treated wastewater within the production process. Some industries have been successful at redesigning their manufacturing processes to reduce or eliminate pollutants. Sources of industrial wastewater include battery manufacturing, chemical manufacturing, electric power plants, food industry, iron and steel industry, metal working, mines and quarries, nuclear industry, oil and gas extraction, petroleum refining and petrochemicals, pharmaceutical manufacturing, pulp and paper industry, smelters, textile mills, industrial oil contamination, water treatment and wood preserving. Treatment processes include brine treatment, solids removal (e.g. chemical precipitation, filtration), oils and grease removal, removal of biodegradable organics, removal of other organics, removal of acids and alkalis, and removal of toxic materials.
== Types ==
Industrial facilities may generate the following industrial wastewater flows:
Manufacturing process wastestreams, which can include conventional pollutants (i.e. controllable with secondary treatment systems), toxic pollutants (e.g. solvents, heavy metals), and other harmful compounds such as nutrients
Non-process wastestreams: boiler blowdown and cooling water, which produce thermal pollution and other pollutants
Industrial site drainage, generated both by manufacturing facilities, service industries and energy and mining sites
Wastestreams from the energy and mining sectors: acid mine drainage, produced water from oil and gas extraction, radionuclides
Wastestreams that are by-products of treatment or cooling processes: backwashing (water treatment), brine.
== Contaminants ==
== Industrial sectors ==
The specific pollutants generated and the resultant effluent concentrations can vary widely among the industrial sectors.
=== Battery manufacturing ===
Battery manufacturers specialize in fabricating small devices for electronics and portable equipment (e.g., power tools), or larger, high-powered units for cars, trucks and other motorized vehicles. Pollutants generated at manufacturing plants includes cadmium, chromium, cobalt, copper, cyanide, iron, lead, manganese, mercury, nickel, silver, zinc, oil and grease.
=== Centralized waste treatment ===
A centralized waste treatment (CWT) facility processes liquid or solid industrial wastes generated by off-site manufacturing facilities. A manufacturer may send its wastes to a CWT plant, rather than perform treatment on site, due to constraints such as limited land availability, difficulty in designing and operating an on-site system, or limitations imposed by environmental regulations and permits. A manufacturer may determine that using a CWT is more cost-effective than treating the waste itself; this is often the case where the manufacturer is a small business.
CWT plants often receive wastes from a wide variety of manufacturers, including chemical plants, metal fabrication and finishing; and used oil and petroleum products from various manufacturing sectors. The wastes may be classified as hazardous, have high pollutant concentrations or otherwise be difficult to treat. In 2000 the U.S. Environmental Protection Agency published wastewater regulations for CWT facilities in the US.
=== Chemical manufacturing ===
==== Organic chemicals manufacturing ====
The specific pollutants discharged by organic chemical manufacturers vary widely from plant to plant, depending on the types of products manufactured, such as bulk organic chemicals, resins, pesticides, plastics, or synthetic fibers. Some of the organic compounds that may be discharged are benzene, chloroform, naphthalene, phenols, toluene and vinyl chloride. Biochemical oxygen demand (BOD), which is a gross measurement of a range of organic pollutants, may be used to gauge the effectiveness of a biological wastewater treatment system, and is used as a regulatory parameter in some discharge permits. Metal pollutant discharges may include chromium, copper, lead, nickel and zinc.
==== Inorganic chemicals manufacturing ====
The inorganic chemicals sector covers a wide variety of products and processes, although an individual plant may produce a narrow range of products and pollutants. Products include aluminum compounds; calcium carbide and calcium chloride; hydrofluoric acid; potassium compounds; borax; chrome and fluorine-based compounds; cadmium and zinc-based compounds. The pollutants discharged vary by product sector and individual plant, and may include arsenic, chlorine, cyanide, fluoride; and heavy metals such as chromium, copper, iron, lead, mercury, nickel and zinc.
=== Electric power plants ===
Fossil-fuel power stations, particularly coal-fired plants, are a major source of industrial wastewater. Many of these plants discharge wastewater with significant levels of metals such as lead, mercury, cadmium and chromium, as well as arsenic, selenium and nitrogen compounds (nitrates and nitrites). Wastewater streams include flue-gas desulfurization, fly ash, bottom ash and flue gas mercury control. Plants with air pollution controls such as wet scrubbers typically transfer the captured pollutants to the wastewater stream.
Ash ponds, a type of surface impoundment, are a widely used treatment technology at coal-fired plants. These ponds use gravity to settle out large particulates (measured as total suspended solids) from power plant wastewater. This technology does not treat dissolved pollutants. Power stations use additional technologies to control pollutants, depending on the particular wastestream in the plant. These include dry ash handling, closed-loop ash recycling, chemical precipitation, biological treatment (such as an activated sludge process), membrane systems, and evaporation-crystallization systems. Technological advancements in ion-exchange membranes and electrodialysis systems has enabled high efficiency treatment of flue-gas desulfurization wastewater to meet recent EPA discharge limits. The treatment approach is similar for other highly scaling industrial wastewaters.
=== Food industry ===
Wastewater generated from agricultural and food processing operations has distinctive characteristics that set it apart from common municipal wastewater managed by public or private sewage treatment plants throughout the world: it is biodegradable and non-toxic, but has high Biological Oxygen Demand (BOD) and suspended solids (SS). The constituents of food and agriculture wastewater are often complex to predict, due to the differences in BOD and pH in effluents from vegetable, fruit, and meat products and due to the seasonal nature of food processing and post-harvesting.
Processing of food from raw materials requires large volumes of high grade water. Vegetable washing generates water with high loads of particulate matter and some dissolved organic matter. It may also contain surfactants and pesticides.
Aquaculture facilities (fish farms) often discharge large amounts of nitrogen and phosphorus, as well as suspended solids. Some facilities use drugs and pesticides, which may be present in the wastewater.
Dairy processing plants generate conventional pollutants (BOD, SS).
Animal slaughter and processing produces organic waste from body fluids, such as blood, and gut contents. Pollutants generated include BOD, SS, coliform bacteria, oil and grease, organic nitrogen and ammonia.
Processing food for sale produces wastes generated from cooking which are often rich in plant organic material and may also contain salt, flavourings, colouring material and acids or alkali. Large quantities of fats, oil and grease ("FOG") may also be present, which in sufficient concentrations can clog sewer lines. Some municipalities require restaurants and food processing businesses to use grease interceptors and regulate the disposal of FOG in the sewer system.
Food processing activities such as plant cleaning, material conveying, bottling, and product washing create wastewater. Many food processing facilities require on-site treatment before operational wastewater can be land applied or discharged to a waterway or a sewer system. High suspended solids levels of organic particles increase BOD and can result in significant sewer surcharge fees. Sedimentation, wedge wire screening, or rotating belt filtration (microscreening) are commonly used methods to reduce suspended organic solids loading prior to discharge.
=== Glass manufacturing ===
Glass manufacturing wastes vary with the type of glass manufactured, which includes fiberglass, plate glass, rolled glass, and glass containers, among others. The wastewater discharged by glass plants may include ammonia, BOD, chemical oxygen demand (COD), fluoride, lead, oil, phenol, and/or phosphorus. The discharges may also be highly acidic (low pH) or alkaline (high pH).
=== Iron and steel industry ===
The production of iron from its ores involves powerful reduction reactions in blast furnaces. Cooling waters are inevitably contaminated with products especially ammonia and cyanide. Production of coke from coal in coking plants also requires water cooling and the use of water in by-products separation. Contamination of waste streams includes gasification products such as benzene, naphthalene, anthracene, cyanide, ammonia, phenols, cresols together with a range of more complex organic compounds known collectively as polycyclic aromatic hydrocarbons (PAH).
The conversion of iron or steel into sheet, wire or rods requires hot and cold mechanical transformation stages frequently employing water as a lubricant and coolant. Contaminants include hydraulic oils, tallow and particulate solids. Final treatment of iron and steel products before onward sale into manufacturing includes pickling in strong mineral acid to remove rust and prepare the surface for tin or chromium plating or for other surface treatments such as galvanisation or painting. The two acids commonly used are hydrochloric acid and sulfuric acid. Wastewater include acidic rinse waters together with waste acid. Although many plants operate acid recovery plants (particularly those using hydrochloric acid), where the mineral acid is boiled away from the iron salts, there remains a large volume of highly acid ferrous sulfate or ferrous chloride to be disposed of. Many steel industry wastewaters are contaminated by hydraulic oil, also known as soluble oil.
=== Metal working ===
Many industries perform work on metal feedstocks (e.g. sheet metal, ingots) as they fabricate their final products. The industries include automobile, truck and aircraft manufacturing; tools and hardware manufacturing; electronic equipment and office machines; ships and boats; appliances and other household products; and stationary industrial equipment (e.g. compressors, pumps, boilers). Typical processes conducted at these plants include grinding, machining, coating and painting, chemical etching and milling, solvent degreasing, electroplating and anodizing. Wastewater generated from these industries may contain heavy metals (common heavy metal pollutants from these industries include cadmium, chromium, copper, lead, nickel, silver and zinc), cyanide and various chemical solvents, oil, and grease.
=== Mines and quarries ===
The principal waste-waters associated with mines and quarries are slurries of rock particles in water. These arise from rainfall washing exposed surfaces and haul roads and also from rock washing and grading processes. Volumes of water can be very high, especially rainfall related arisings on large sites. Some specialized separation operations, such as coal washing to separate coal from native rock using density gradients, can produce wastewater contaminated by fine particulate haematite and surfactants. Oils and hydraulic oils are also common contaminants.
Wastewater from metal mines and ore recovery plants are inevitably contaminated by the minerals present in the native rock formations. Following crushing and extraction of the desirable materials, undesirable materials may enter the wastewater stream. For metal mines, this can include unwanted metals such as zinc and other materials such as arsenic. Extraction of high value metals such as gold and silver may generate slimes containing very fine particles in where physical removal of contaminants becomes particularly difficult.
Additionally, the geologic formations that harbour economically valuable metals such as copper and gold very often consist of sulphide-type ores. The processing entails grinding the rock into fine particles and then extracting the desired metal(s), with the leftover rock being known as tailings. These tailings contain a combination of not only undesirable leftover metals, but also sulphide components which eventually form sulphuric acid upon the exposure to air and water that inevitably occurs when the tailings are disposed of in large impoundments. The resulting acid mine drainage, which is often rich in heavy metals (because acids dissolve metals), is one of the many environmental impacts of mining.
=== Nuclear industry ===
The waste production from the nuclear and radio-chemicals industry is dealt with as Radioactive waste.
Researchers have looked at the bioaccumulation of strontium by Scenedesmus spinosus (algae) in simulated wastewater. The study claims a highly selective biosorption capacity for strontium of S. spinosus, suggesting that it may be appropriate for use of nuclear wastewater.
=== Oil and gas extraction ===
Oil and gas well operations generate produced water, which may contain oils, toxic metals (e.g. arsenic, cadmium, chromium, mercury, lead), salts, organic chemicals and solids. Some produced water contains traces of naturally occurring radioactive material. Offshore oil and gas platforms also generate deck drainage, domestic waste and sanitary waste. During the drilling process, well sites typically discharge drill cuttings and drilling mud (drilling fluid).
=== Petroleum refining and petrochemicals ===
Pollutants discharged at petroleum refineries and petrochemical plants include conventional pollutants (BOD, oil and grease, suspended solids), ammonia, chromium, phenols and sulfides.
=== Pharmaceutical manufacturing ===
Pharmaceutical plants typically generate a variety of process wastewaters, including solvents, spent acid and caustic solutions, water from chemical reactions, product wash water, condensed steam, blowdown from air pollution control scrubbers, and equipment washwater. Non-process wastewaters typically include cooling water and site runoff. Pollutants generated by the industry include acetone, ammonia, benzene, BOD, chloroform, cyanide, ethanol, ethyl acetate, isopropanol, methylene chloride, methanol, phenol and toluene. Treatment technologies used include advanced biological treatment (e.g. activated sludge with nitrification), multimedia filtration, cyanide destruction (e.g. hydrolysis), steam stripping and wastewater recycling.
=== Pulp and paper industry ===
Effluent from the pulp and paper industry is generally high in suspended solids and BOD. Plants that bleach wood pulp for paper making may generate chloroform, dioxins (including 2,3,7,8-TCDD), furans, phenols, and chemical oxygen demand (COD). Stand-alone paper mills using imported pulp may only require simple primary treatment, such as sedimentation or dissolved air flotation. Increased BOD or COD loadings, as well as organic pollutants, may require biological treatment such as activated sludge or upflow anaerobic sludge blanket reactors. For mills with high inorganic loadings like salt, tertiary treatments may be required, either general membrane treatments like ultrafiltration or reverse osmosis or treatments to remove specific contaminants, such as nutrients.
=== Smelters ===
The pollutants discharged by nonferrous smelters vary with the base metal ore. Bauxite smelters generate phenols: 131 but typically use settling basins and evaporation to manage these wastes, with no need to routinely discharge wastewater.: 395 Aluminum smelters typically discharge fluoride, benzo(a)pyrene, antimony and nickel, as well as aluminum. Copper smelters typically generate cadmium, lead, zinc, arsenic and nickel, in addition to copper, in their wastewater. Lead smelters discharge lead and zinc. Nickel and cobalt smelters discharge ammonia and copper in addition to the base metals. Zinc smelters discharge arsenic, cadmium, copper, lead, selenium and zinc.
Typical treatment processes used in the industry are chemical precipitation, sedimentation and filtration.: 145
=== Textile mills ===
Textile mills, including carpet manufacturers, generate wastewater from a wide variety of processes, including cleaning and finishing, yarn manufacturing and fabric finishing (such as bleaching, dyeing, resin treatment, waterproofing and retardant flameproofing). Pollutants generated by textile mills include BOD, SS, oil and grease, sulfide, phenols and chromium. Insecticide residues in fleeces are a particular problem in treating waters generated in wool processing. Animal fats may be present in the wastewater, which if not contaminated, can be recovered for the production of tallow or further rendering.
Textile dyeing plants generate wastewater that contain synthetic (e.g., reactive dyes, acid dyes, basic dyes, disperse dyes, vat dyes, sulphur dyes, mordant dyes, direct dyes, ingrain dyes, solvent dyes, pigment dyes) and natural dyestuff, gum thickener (guar) and various wetting agents, pH buffers and dye retardants or accelerators. Following treatment with polymer-based flocculants and settling agents, typical monitoring parameters include BOD, COD, color (ADMI), sulfide, oil and grease, phenol, TSS and heavy metals (chromium, zinc, lead, copper).
=== Industrial oil contamination ===
Industrial applications where oil enters the wastewater stream may include vehicle wash bays, workshops, fuel storage depots, transport hubs and power generation. Often the wastewater is discharged into local sewer or trade waste systems and must meet local environmental specifications. Typical contaminants can include solvents, detergents, grit, lubricants and hydrocarbons.
=== Water treatment ===
Many industries have a need to treat water to obtain very high quality water for their processes. This might include pure chemical synthesis or boiler feed water. Also, some water treatment processes produce organic and mineral sludges from filtration and sedimentation which require treatment. Ion exchange using natural or synthetic resins removes calcium, magnesium and carbonate ions from water, typically replacing them with sodium, chloride, hydroxyl and/or other ions. Regeneration of ion-exchange columns with strong acids and alkalis produces a wastewater rich in hardness ions which are readily precipitated out, especially when in admixture with other wastewater constituents.
=== Wood preserving ===
Wood preserving plants generate conventional and toxic pollutants, including arsenic, COD, copper, chromium, abnormally high or low pH, phenols, suspended solids, oil and grease.
== Treatment methods ==
The various types of contamination of wastewater require a variety of strategies to remove the contamination. Most industrial processes, such as petroleum refineries, chemical and petrochemical plants have onsite facilities to treat their wastewaters so that the pollutant concentrations in the treated wastewater comply with the regulations regarding disposal of wastewaters into sewers or into rivers, lakes or oceans.: 1412 Constructed wetlands are being used in an increasing number of cases as they provided high quality and productive on-site treatment. Other industrial processes that produce a lot of waste-waters such as paper and pulp production have created environmental concern, leading to development of processes to recycle water use within plants before they have to be cleaned and disposed.
An industrial wastewater treatment plant may include one or more of the following rather than the conventional treatment sequence of sewage treatment plants:
An API oil-water separator, for removing separate phase oil from wastewater.: 180
A clarifier, for removing solids from wastewater.: 41–15
A roughing filter, to reduce the biochemical oxygen demand of wastewater.: 23–11
A carbon filtration plant, to remove toxic dissolved organic compounds from wastewater.: 210
An advanced electrodialysis reversal (EDR) system with ion-exchange membranes.
=== Brine treatment ===
Brine treatment involves removing dissolved salt ions from the waste stream. Although similarities to seawater or brackish water desalination exist, industrial brine treatment may contain unique combinations of dissolved ions, such as hardness ions or other metals, necessitating specific processes and equipment.
Brine treatment systems are typically optimized to either reduce the volume of the final discharge for more economic disposal (as disposal costs are often based on volume) or maximize the recovery of fresh water or salts. Brine treatment systems may also be optimized to reduce electricity consumption, chemical usage, or physical footprint.
Brine treatment is commonly encountered when treating cooling tower blowdown, produced water from steam-assisted gravity drainage (SAGD), produced water from natural gas extraction such as coal seam gas, frac flowback water, acid mine or acid rock drainage, reverse osmosis reject, chlor-alkali wastewater, pulp and paper mill effluent, and waste streams from food and beverage processing.
Brine treatment technologies may include: membrane filtration processes, such as reverse osmosis; ion-exchange processes such as electrodialysis or weak acid cation exchange; or evaporation processes, such as brine concentrators and crystallizers employing mechanical vapour recompression and steam. Due to the ever increasing discharge standards, there has been an emergence of the use of advance oxidation processes for the treatment of brine. Some notable examples such as Fenton's oxidation and ozonation have been employed for degradation of recalcitrant compounds in brine from industrial plants.
Reverse osmosis may not be viable for brine treatment, due to the potential for fouling caused by hardness salts or organic contaminants, or damage to the reverse osmosis membranes from hydrocarbons.
Evaporation processes are the most widespread for brine treatment as they enable the highest degree of concentration, as high as solid salt. They also produce the highest purity effluent, even distillate-quality. Evaporation processes are also more tolerant of organics, hydrocarbons, or hardness salts. However, energy consumption is high and corrosion may be an issue as the prime mover is concentrated salt water. As a result, evaporation systems typically employ titanium or duplex stainless steel materials.
==== Brine management ====
Brine management examines the broader context of brine treatment and may include consideration of government policy and regulations, corporate sustainability, environmental impact, recycling, handling and transport, containment, centralized compared to on-site treatment, avoidance and reduction, technologies, and economics. Brine management shares some issues with leachate management and more general waste management. In the recent years, there has been greater prevalence in brine management due to global push for zero liquid discharge (ZLD)/minimal liquid discharge (MLD). In ZLD/MLD techniques, a closed water cycle is used to minimize water discharges from a system for water reuse. This concept has been gaining traction in recent years, due to increased water discharges and recent advancement in membrane technology. Increasingly, there has been also greater efforts to increase the recovery of materials from brines, especially from mining, geothermal wastewater or desalination brines. Various literature demosntrates the vaibility of extraction of valuable materials like sodium bicarbonates, sodium chlorides and precious metals (like rubidium, cesium and lithium). The concept of ZLD/MLD encompasses the downstream management of wastewater brines, to reduce discharges and also derive valuable products from it.
=== Solids removal ===
Most solids can be removed using simple sedimentation techniques with the solids recovered as slurry or sludge. Very fine solids and solids with densities close to the density of water pose special problems. In such case filtration or ultrafiltration may be required. Although flocculation may be used, using alum salts or the addition of polyelectrolytes. Wastewater from industrial food processing often requires on-site treatment before it can be discharged to prevent or reduce sewer surcharge fees. The type of industry and specific operational practices determine what types of wastewater is generated and what type of treatment is required. Reducing solids such as waste product, organic materials, and sand is often a goal of industrial wastewater treatment. Some common ways to reduce solids include primary sedimentation (clarification), dissolved air flotation (DAF), belt filtration (microscreening), and drum screening.
=== Oils and grease removal ===
The effective removal of oils and grease is dependent on the characteristics of the oil in terms of its suspension state and droplet size, which will in turn affect the choice of separator technology. Oil in industrial waste water may be free light oil, heavy oil, which tends to sink, and emulsified oil, often referred to as soluble oil. Emulsified or soluble oils will typically required "cracking" to free the oil from its emulsion. In most cases this is achieved by lowering the pH of the water matrix.
Most separator technologies will have an optimum range of oil droplet sizes that can be effectively treated. Each separator technology will have its own performance curve outlining optimum performance based on oil droplet size. the most common separators are gravity tanks or pits, API oil-water separators or plate packs, chemical treatment via dissolved air flotations, centrifuges, media filters and hydrocyclones.
Analyzing the oily water to determine droplet size can be performed with a video particle analyser.
==== API oil-water separators ====
==== Hydrocyclone ====
Hydrocyclone separators operate on the process where wastewater enters the cyclone chamber and is spun under extreme centrifugal forces more than 1000 times the force of gravity. This force causes the water and oil droplets (or solid particles) to separate. The separated materials is discharged from one end of the cyclone where treated water is discharged through the opposite end for further treatment, filtration or discharge. Hydrocyclones can also be utilised in a variety of context from solid-liquid separation to oil-water separation.
=== Removal of biodegradable organics ===
Biodegradable organic material of plant or animal origin is usually possible to treat using extended conventional sewage treatment processes such as activated sludge or trickling filter. Problems can arise if the wastewater is excessively diluted with washing water or is highly concentrated such as undiluted blood or milk. The presence of cleaning agents, disinfectants, pesticides, or antibiotics can have detrimental impacts on treatment processes.
==== Activated sludge process ====
==== Trickling filter process ====
A trickling filter consists of a bed of rocks, gravel, slag, peat moss, or plastic media over which wastewater flows downward and contacts a layer (or film) of microbial slime covering the bed media. Aerobic conditions are maintained by forced air flowing through the bed or by natural convection of air. The process involves adsorption of organic compounds in the wastewater by the microbial slime layer, diffusion of air into the slime layer to provide the oxygen required for the biochemical oxidation of the organic compounds. The end products include carbon dioxide gas, water and other products of the oxidation. As the slime layer thickens, it becomes difficult for the air to penetrate the layer and an inner anaerobic layer is formed.
=== Removal of other organics ===
Synthetic organic materials including solvents, paints, pharmaceuticals, pesticides, products from coke production and so forth can be very difficult to treat. Treatment methods are often specific to the material being treated. Methods include advanced oxidation processing, distillation, adsorption, ozonation, vitrification, incineration, chemical immobilisation or landfill disposal. Some materials such as some detergents may be capable of biological degradation and in such cases, a modified form of wastewater treatment can be used.
=== Removal of acids and alkalis ===
Acids and alkalis can usually be neutralised under controlled conditions. Neutralisation frequently produces a precipitate that will require treatment as a solid residue that may also be toxic. In some cases, gases may be evolved requiring treatment for the gas stream. Some other forms of treatment are usually required following neutralisation.
Waste streams rich in hardness ions as from de-ionisation processes can readily lose the hardness ions in a buildup of precipitated calcium and magnesium salts. This precipitation process can cause severe furring of pipes and can, in extreme cases, cause the blockage of disposal pipes. A 1-metre diameter industrial marine discharge pipe serving a major chemicals complex was blocked by such salts in the 1970s. Treatment is by concentration of de-ionisation waste waters and disposal to landfill or by careful pH management of the released wastewater.
=== Removal of toxic materials ===
Toxic materials including many organic materials, metals (such as zinc, silver, cadmium, thallium, etc.) acids, alkalis, non-metallic elements (such as arsenic or selenium) are generally resistant to biological processes unless very dilute. Metals can often be precipitated out by changing the pH or by treatment with other chemicals. Many, however, are resistant to treatment or mitigation and may require concentration followed by landfilling or recycling. Dissolved organics can be incinerated within the wastewater by the advanced oxidation process.
==== Smart capsules ====
Molecular encapsulation is a technology that has the potential to provide a system for the recyclable removal of lead and other ions from polluted sources. Nano-, micro- and milli- capsules, with sizes in the ranges 10 nm–1μm,1μm–1mm and >1mm, respectively, are particles that have an active reagent (core) surrounded by a carrier (shell).There are three types of capsule under investigation: alginate-based capsules, carbon nanotubes, polymer swelling capsules. These capsules provide a possible means for the remediation of contaminated water.
=== Removal of thermal pollution ===
To remove heat from wastewater generated by power plants or manufacturing plants, and thus to reduce thermal pollution, the following technologies are used:
cooling ponds, engineered bodies of water designed for cooling by evaporation, convection, and radiation
cooling towers, which transfer waste heat to the atmosphere through evaporation or heat transfer
cogeneration, a process where waste heat is recycled for domestic or industrial heating purposes.
=== Other disposal methods ===
Some facilities such as oil and gas wells may be permitted to pump their wastewater underground through injection wells. However, wastewater injection has been linked to induced seismicity.
== Costs and trade waste charges ==
Economies of scale may favor a situation where industrial wastewater (with pre-treatment or without treatment) is discharged to the sewer and then treated at a large municipal sewage treatment plant. Typically, trade waste charges are applied in that case. Or it might be more economical to have full treatment of industrial wastewater on the same site where it is generated and then discharging this treated industrial wastewater to a suitable surface water body. This effectively reduces wastewater treatment charges collected by municipal sewage treatment plants by pre-treating wastewaters to reduce concentrations of pollutants measured to determine user fees.: 300–302
Industrial wastewater plants may also reduce raw water costs by converting selected wastewaters to reclaimed water used for different purposes.
== Society and culture ==
=== Global goals ===
The international community has defined the treatment of industrial wastewater as an important part of sustainable development by including it in Sustainable Development Goal 6. Target 6.3 of this goal is to "By 2030, improve water quality by reducing pollution, eliminating dumping and minimizing release of hazardous chemicals and materials, halving the proportion of untreated wastewater and substantially increasing recycling and safe reuse globally". One of the indicators for this target is the "proportion of domestic and industrial wastewater flows safely treated".
== See also ==
Best management practice for water pollution (BMP)
List of waste water treatment technologies
Purified water (for industrial use)
Water purification (for drinking water)
== References ==
== Further reading ==
== External links ==
Water Environment Federation - Professional society
Industrial Wastewater Treatment Technology Database - EPA | Wikipedia/Industrial_wastewater_treatment |
Sustainable Materials Management is a systemic approach to using and reusing materials more productively over their entire lifecycles. It represents a change in how a society thinks about the use of natural resources and environmental protection. By looking at a product's entire lifecycle new opportunities can be found to reduce environmental impacts, conserve resources, and reduce costs.
U.S. and global consumption of materials increased rapidly during the last century. According to the Annex to the G7 Leaders’ June 8, 2015 Declaration, global raw material use rose during the 20th century at about twice the rate of population growth. For every 1 percent increase in gross domestic product, raw material use has risen by 0.4 percent.
This increasing consumption has come at a cost to the environment, including habitat destruction, biodiversity loss, overly stressed fisheries and desertification. Materials management is also associated with an estimated 42 percent of total U.S. greenhouse gas emissions. Failure to find more productive and sustainable ways to extract, use and manage materials, and change the relationship between material consumption and growth, has grave implications for our economy and society.
== Introduction ==
Sustainable Materials Management (SMM) represents a framework to sustainably manage materials and products throughout the entire lifecycle, from resource extraction, design and manufacturing, resource productivity, consumption and end-of-life management.
Traditional patterns of material consumption in the United States follow a Cradle-to-Grave pattern of raw material extraction, product manufacturing, distribution to consumers, use by consumers, and disposal; coined by The Story of Stuff author Annie Leonard as the "take-make-waste" linear economy and commonly referred to as the throw-away society, these familiar waste management practices are being revised to bring about sustainable management of resources. SMM represents a shift in how materials are used and valued with a focus on the environmental impact of material use and environmental protection throughout the entire lifecycle of a product. SMM has been adopted as a regulatory approach to manage materials by the U.S. Environmental Protection Agency (EPA) and many other governments around the world.
Sustainable Materials Management is a broad approach that overlaps and supplements many programs and concepts being adopted by governments and business around the world including zero waste, green chemistry, eco-labeling, sustainable supply-chain management, lean manufacturing, green procurement, the US EPA’s Design for the Environment Program, the G8’s 3R’s (reduce, reuse, recycle) program, UNEP's Sustainable Production and Consumption and Sustainable Resource Management programs, and OECD's Sustainable Materials Management framework.
=== Differences between Waste Management and SMM ===
SMM seeks the highest use of all resources while waste management focuses on managing and reducing waste and waste pollutants at the end of the lifecycle
SMM focuses on the entire lifecycle of materials and products while waste management focuses on just the end-of life management of waste products
SMM is concerned with inputs and outputs to/from the environment and associated impacts to air and water and is not geographically constrained while waste management only looks at outputs to the environment from waste and areas where waste is managed
SMM's overall goal is long term sustainability while waste management focuses on managing one environmental impact, that associated with waste products
SMM considers all industries and consumers associated with the lifecycle of a material and product as a responsible party where as waste management only considers the generators of the waste as the responsible party
== Product Lifecycle Models ==
There are several similar and overlapping efforts to define and conceptualize a closed loop lifecycle of product and materials management, with many of these efforts being spearheaded by government agencies, entrepreneurs, scientists and non-governmental organizations. While similar to SMM, these product lifecycle models focus largely on the end-of-life management of materials while SMM focuses on the impacts materials, products and services have on the environment such as eutrophication, acidification, ozone layer depletion, global warming and aquatic toxicity as well as energy and water use.
=== Product Stewardship ===
The Product Stewardship Institute defines Product stewardship as: "the act of minimizing the health, safety, environmental, and social impacts of a product and its packaging throughout all lifecycle stages, while also maximizing economic benefits. The manufacturer, or producer, of the product has the greatest ability to minimize adverse impacts, but other stakeholders, such as suppliers, retailers, and consumers, also play a role. Stewardship can be either voluntary or required by law".British Columbia (BC) has an extensive product stewardship network administered by BC Recycles and composed of BC producers and brand owners who are required by law to collect and divert end-of-life products and packaging.
=== Circular Economy ===
The U.Ks Waste and Resource Action Programme (WRAP) defines the Circular Economy as being an alternative to the traditional take, make, waste economy to one that keeps resources in use as long as possible, extracts the maximum value from the materials while they are in use, then recovers the materials to generate new products at the end of the service life.
The Ellen MacArthur Foundation works to accelerate the transition to a circular economy by working with businesses, academia and governments throughout the world to develop an economy that is restorative and regenerative by design and seeks to keep products, components and materials at their highest use and value at all times, distinguishing between biological and technical cycles.
=== Cradle-to-Cradle ===
The Dictionary of Sustainable Management defines Cradle-to-Cradle as "A phrase invented by Walter R. Stahel in the 1970s and popularized by William McDonough and Michael Braungart in their 2002 book of the same name. This framework seeks to create production techniques that are not just efficient but are essentially waste free. In cradle-to-cradle production, all material inputs and outputs are seen either as technical or biological nutrients. Technical nutrients can be recycled or reused with no loss of quality and biological nutrients composted or consumed. By contrast cradle-to-grave refers to a company taking responsibility for the disposal of goods it has produced, but not necessarily putting products’ constituent components back into service."
=== Closed Loop Recycling ===
In closed loop recycling, a material is captured at the end of life and introduced back into the manufacturing process to make a new product
== Implementing SMM Globally ==
=== The Organization for Economic Cooperation and Development (OECD) ===
The Organization for Economic Co-operation and Development (OECD) formed in 1960 and currently comprising 35 member countries including the United States, Canada, Mexico, Japan and 23 countries in the European Union, works to foster economic prosperity and end poverty by promoting economic growth and financial stability for governments around the world while also taking into account the implications that economic and social growth have on the environment.
Since the 1980s, the OECD has worked to promote policies that prevent, reduce and manage waste in ways that mitigate environmental impacts. It has become clear over time that increasing economic activity and materials consumptions calls for a systematic materials based approach to managing waste, one that seeks to incorporate materials back into the manufacturing process at the end of their life, in what is commonly referred to as a “cradle to cradle” approach to materials management as opposed to the traditional “cradle to grave” waste management approach. Around 2001 the OECD began to address many countries’ interest in viewing waste as a resource that can be used as inputs for new products and many countries and governments have begun adopting sustainable materials management policies.
In 2012, the OECD put out a Green Growth Policy Brief on Sustainable Materials Management. In it they define SMM as“…an approach to promote sustainable materials use, integrating actions targeted at reducing negative environmental impacts and preserving natural capital throughout the life-cycle of materials, taking into account economic efficiency and social equity”.The OECD working definition includes the following notes on the definition of SMM:
"Materials” include all those extracted or derived from natural resources, which may be either inorganic or organic substances, at all points throughout their life-cycles.
“Life-cycle of materials” includes all activities related to materials such as extraction, transportation, production, consumption, material/product reuse, recovery, and disposal.
An economically efficient outcome is achieved when net benefits to society as a whole are maximized.
A variety of policy tools can support SMM, such as economic, regulatory and information instruments and partnerships.
SMM may take place at different levels, including firm/sector and different government levels. SMM may cover different geographical areas and time horizons.
=== The United Nations Environment Programme (UNEP) ===
Sustainable Production and Consumption (UNEP)
Sustainable Resource Management (UNEP)
=== The United States Environmental Protection Agency (US EPA) ===
The US EPA has adopted Sustainable Materials Management as a regulatory framework for managing materials. In June 2009 the EPA put out a report that functioned as a road map to SMM in the U.S. titled Sustainable Materials Management - The Road Ahead 2009 - 2020. In this report, EPA defines SMM as “... an approach to serving human needs by using/reusing resources productively and sustainably throughout their life cycles, generally minimizing the amount of materials involved and all associated environmental impacts”.The Resource Conservation and Recovery Act (RCRA) sets the legislative basis for SSM in the United States, establishing a preference for resource conservation over disposal. In 2010, the Office of Resource Conservation and Recovery shifted focus from just resource recovery efforts to adopt a broader sustainable materials management approach. The new approach includes the two original waste management mandates of RCRA: 1) to protect human health and the environment from waste and 2) to conserve resources, and adds in three additional goals: 1) to “Reduce waste and increase the efficient and sustainable use of resources”, 2) “Prevent exposures to humans and ecosystems from the use of hazardous chemicals” 3) “Manage wastes and clean up chemical releases in a safe, environmentally sound manner”.
In 2015 EPA published the report EPA Sustainable Materials Management Program Strategic Plan for Fiscal Years 2017 – 2022. This five-year plan will focus on three strategic initiatives:
The built environment
Organics recycling, and
Reduction in packaging
Other areas the EPA will focus on include sustainable electronics management, life-cycle assessment, measurement, and international SMM collaboration.
== References ==
== External links ==
Annex to the G-7 Leaders’ Declaration
National Recycling Coalition Sustainable Materials Management Summit, 2015 | Wikipedia/Sustainable_materials_management |
An eco-industrial park (EIP) is an industrial park in which businesses cooperate with each other and with the local community in an attempt to reduce waste and pollution, efficiently share resources (such as information, materials, water, energy, infrastructure, and natural resources), and help achieve sustainable development, with the intention of increasing economic gains and improving environmental quality. An EIP may also be planned, designed, and built in such a way that it makes it easier for businesses to co-operate, and that results in a more financially sound, environmentally friendly project for the developer.
The Eco-industrial Park Handbook states that "An Eco-Industrial Park is a community of manufacturing and service businesses located together on a common property. Members seek enhanced environmental, economic, and social performance through collaboration in managing environmental and resource issues."
Based on the concepts of industrial ecology, collaborative strategies not only include by-product synergy ("waste-to-feed" exchanges), but can also take the form of wastewater cascading, shared logistics and shipping & receiving facilities, shared parking, green technology purchasing blocks, multi-partner green building retrofit, district energy systems, and local education and resource centres. This is an application of a systems approach, in which designs and processes/activities are integrated to address multiple objectives.
EIPs can be developed as greenfield land projects, where the eco-industrial intent is present throughout the planning, design and site construction phases, or developed through retrofits and new strategies in existing industrial developments.
== Examples ==
"Industrial symbiosis" is a related but more limited concept in which companies in a region collaborate to utilize each other's by-products and otherwise share resources. In Kalundborg, Denmark, a symbiosis network links a 1500MW coal-fired power plant with the community and other companies. Surplus heat from this power plant is used to heat 3500 local homes in addition to a nearby fish farm, whose sludge is then sold as a fertilizer. Steam from the power plant is sold to Novo Nordisk, a pharmaceutical and enzyme manufacturer, in addition to a Statoil plant. This reuse of heat reduces the amount thermal pollution discharged to a nearby fjord. Additionally, a by-product from the power plant's sulfur dioxide scrubber contains gypsum, which is sold to a wallboard manufacturer. Almost all of the manufacturer's gypsum needs are met this way, which reduces the amount of open-pit mining needed. Furthermore, fly ash and clinker from the power plant is utilized for road building and cement production.
The industrial symbiosis at Kalundborg was not created as a top-down initiative, but instead evolved gradually. As environmental regulations became stricter, firms were motivated reduce the cost of compliance, and turn their by-products into economic products.
In Canada, eco-industrial parks exist across the country and have enjoyed some success. The best known example is Burnside Park, in Halifax, Nova Scotia. With support from Dalhousie University’s Eco-Efficiency Centre, the more than 1,500 businesses have been improving their environmental performance and developing profitable partnerships. Subsequently, two greenfield industrial developments have been started in Alberta: TaigaNova Eco-Industrial Park is in the heart of the Athabasca oil sands, while Innovista Eco-Industrial Park is a gateway to the Rocky Mountains ~300 km west of Edmonton.
UNIDO Viet Nam (United Nations Industrial Development Organization) has compiled a list in 2015 of Eco-Industrial Parks (EIP) in the ASEAN Economic Community in a report titled "Economic Zones in the ASEAN Archived 2020-09-30 at the Wayback Machine" written by Arnault Morisson.
== Other usage ==
EIPs also refer to industrial parks where a "green" approach has been taken towards the infrastructure and development of the site. This can include green infrastructure related to Renewable Energy Systems; stormwater, groundwater and wastewater management; road surfaces; and transportation demand management. Green building practices can also be encouraged or mandated
EIPs are often used as a stimulus for economic diversification in the community or region where they are located. Anchor tenants, such as bio-based product manufacturers or waste-to-energy facilities, etc., can attract complementary businesses as suppliers, scavengers/recyclers, service providers, downstream users and other businesses that could benefit from eco-industrial strategies.
== Suggested usage ==
It is suggested that EIPs be used as a means of growing the renewable energy sector. In the case of a Solar Photovoltaic (PV) Manufacturing plant, an EIP can increase the manufacturing efficiency to make it more economical, while reducing the environmental impact of producing the solar cells. In essence, this assists the growth of the renewable energy industry and the environmental benefits that come with replacing fossil-fuels.
== See also ==
EcoPark – EIP in Hong-Kong
Industrial ecology
Industrial symbiosis
== References ==
Industrial Location in Greece: Fostering Green Transition and Synergies between Industrial and Spatial Planning Policies by Anestis Gourgiotis, Stella Sofia Kyvelou and Ioannis Lainas, Land 2021, 10(3), 271; doi:10.3390/land10030271 unflagged free DOI (link) – 6 March 2021
== Further reading ==
Greening the Asphalt Acres. Alberta Venture. Aug, 2008.
Eco-industrial Zones: Sustaining the wealth of industrial developments. Building Sustainable Communities E-Zine. Jan, 2009.
Eco-Industrial Parks surging in popularity. Business Edge Magazine. Nov, 2008.
https://www.unido.org/fileadmin/user_media_upgrade/Resources/Publications/UCO_Viet_Nam_Study_FINAL.pdf Archived 2020-09-30 at the Wayback Machine Economic Zones in ASEAN. UNIDO. 2015.
== External links ==
Industrial Symbiosis
Industrial Ecology Wiki - Repository of information about Eco-Industrial Parks around the world
Indigo Development Eco-Industrial Park page and handbook
Existing and Developing Eco-Industrial Park Sites in the U.S.
Industrial Symbiosis Timeline
Industrial Symbiosis in Action
Zero Emissions Research & Initiatives
TaigaNova Eco-Industrial Park in Fort McMurray, AB, Canada
Innovista Eco-Industrial Park in Hinton, AB, Canada
CleanTech Park in Singapore
European EIPs/EEPAs
[2]Infinity Industrial Park | Wikipedia/Eco-industrial_park |
One aspect of energy poverty is lack of access to clean, modern fuels and technologies for cooking. As of 2020, more than 2.6 billion people in developing countries routinely cook with fuels such as wood, animal dung, coal, or kerosene. Burning these types of fuels in open fires or traditional stoves causes harmful household air pollution, resulting in an estimated 3.8 million deaths annually according to the World Health Organization (WHO), and contributes to various health, socio-economic, and environmental problems.
A high priority in global sustainable development is making clean cooking facilities universally available and affordable. Stoves and appliances that run on electricity, liquid petroleum gas (LPG), piped natural gas (PNG), biogas, alcohol, and solar heat meet WHO guidelines for clean cooking. Universal access to clean cooking facilities would benefit the environment and gender equality greatly.
Stoves that burn wood and other solid fuels more efficiently than traditional stoves are known as "improved cookstoves" or "clean cookstoves". With few exceptions, these stoves deliver fewer health benefits than stoves that use liquid or gaseous fuels. However, they reduce fuel usage and thus help prevent environmental degradation. Improved cookstoves are an important interim solution in areas where deploying cleaner technologies is less feasible.
Initiatives to encourage cleaner cooking practices have yielded limited success. For various practical, cultural, and economic reasons, it is common for families who adopt clean stoves and fuels to continue to use traditional fuels and stoves frequently.
== Issues with traditional cooking fuels ==
=== Health impacts ===
As of 2023, more than 2.3 billion people in developing countries rely on burning polluting biomass fuels such as wood, dry dung, coal, or kerosene for cooking, which causes harmful household air pollution and also contributes significantly to outdoor air pollution. The World Health Organization (WHO) estimates that cooking-related pollution causes 3.8 million annual deaths. The Global Burden of Disease study estimated the number of deaths in 2021 at 3.1 million, and the death rate is highest in Africa.
In traditional cooking facilities, smoke is typically vented into the home rather than through a chimney. Solid fuel smoke contains thousands of substances, many of which are hazardous to human health. The most well understood of these substances are carbon monoxide (CO); small particulate matter; nitrous oxide; sulfur oxides; a range of volatile organic compounds, including formaldehyde, benzene and 1,3-butadiene; and polycyclic aromatic compounds, such as benzo-a-pyrene, which are thought to have both short and long-term health consequences.
Exposure to household air pollution (HAP) nearly doubles the risk of childhood pneumonia and is responsible for 45 percent of all pneumonia deaths in children under five years of age. Emerging evidence shows that HAP is also a risk factor for cataracts, the leading cause of blindness in lower-middle-income countries, and low birth weight. Cooking with open fires or unsafe stoves is a leading cause of burns among women and children in developing countries.
=== Impacts on women and girls ===
Health effects are concentrated among women, who are likely responsible for cooking and childcare. Gathering fuel exposes women and children to safety risks. This process often consumes 15 or more hours per week, constraining their time for education, rest, and paid work. Women and girls must often walk long distances to obtain cooking fuel, and, as a result, face increased risk of physical and sexual violence. Many children, particularly girls, may not attend school to help their mothers with firewood collection and food preparation.
=== Environmental impacts ===
Traditional cooking facilities are highly inefficient, allowing heat to escape into the open air. The inefficiency of fuel burning results in more wood needing to be harvested and also causes emissions of black carbon, a contributor to climate change. Serious local environmental damage, including desertification, can be caused by excessive harvesting of wood and other combustible material.
While biomass harvesting in sensitive areas is problematic, it is now determined that most biomass clearing is due to agricultural expansion and land conversion. Use of crop residue and animal waste for domestic energy has detrimental results on soil quality and agricultural and livestock productivity, as it means these materials are not available as soil conditioners, organic fertilizer, and livestock fodder.
== Terminology ==
The term "clean cookstove" has often been used without defining what the term means. Organizations vary in how they define "clean":
According to the WHO, cooking facilities are "clean" if their emissions of carbon monoxide and fine particulate matter are below certain levels.
The Clean Cooking Alliance uses the term "clean cooking" more broadly. Its definition includes what the WHO refers to as "improved cookstoves", i.e. stoves that burn biomass fuel more efficiently than traditional stoves. As of 2020, most stoves that burn biomass fuel do not qualify as clean under WHO standards even if they are more efficient than traditional stoves.
The WHO has criticized the marketing of biomass cookstoves as "improved" when they have not been tested against standards and their health benefits are unclear.
== WHO-recommended clean cooking facilities ==
A high priority in global sustainable development is to make clean cooking facilities universally available and affordable.
According to the WHO, stoves and appliances that are powered by electricity, liquid petroleum gas (LPG), piped natural gas (PNG), biogas, alcohol, and solar heat are "clean". Best-in-class fan gasifier stoves that burn biomass pellets can be classified as clean cooking facilities if they are correctly operated and the pellets have sufficiently low levels of moisture, but these stoves are not widely available.
Electricity can be used to power appliances such as electric pressure cookers, rice cookers, and highly efficient induction stoves, in addition to standard electric stoves. Electric induction stoves are so efficient that they create less pollution than liquified petroleum gas (LPG) even when connected to coal power sources, and are sometimes cheaper. For stews, beans, rice and other foods that can be adapted to electric pressure cookers, the savings are even greater.. As of 2019, 770 million people do not have access to electricity, and for many others, electricity is not affordable or reliable. Because access to electricity is also a high priority in global sustainable development, integrated planning for new and improved electricity infrastructure that includes both typical electric loads as well as cooking loads is beginning to gain momentum. Indeed, this kind of integrated resource planning for electricity systems may deliver faster and lower-cost solutions to both access to electricity and to clean cooking.
Natural gas stoves, which are widely used in richer countries, are not without health risks. They emit high levels of nitrogen dioxide, an atmospheric pollutant that is linked to oxidative stress and acute reduction in lung function. Studies on the effects of indoor cooking with natural gas have yielded inconsistent results. According to a 2010 meta-analysis, the evidence suggests that the practice leads to small reductions in lung function in children. Children with allergies may be more susceptible.
Biogas digesters convert waste, such as human waste and animal dung, into a methane-rich gas that burns cleanly. Biogas systems are a promising technology in areas where each household has at least two large animals to provide dung, and a steady water supply is also available.
Solar cookers collect and concentrate the sun's heat when sunshine is available.
== Improved cook stoves ==
Improved cook stoves (ICS), often marketed as "clean cookstoves", are biomass stoves that generally burn biomass more efficiently than traditional stoves and open fires.
Compared to traditional cook stoves, ICS are usually more fuel-efficient and aim to reduce the negative health impacts associated with exposure to toxic smoke. They reduce fuel needs by 20-75% and drastically cut dangerous smoke and fumes.: 42 As of 2016, no widely-available biomass stoves meet the standards for clean cooking as defined by the WHO. A 2020 review found only one biomass stove on the market that met WHO standards in field conditions.
Despite their limitations, ICSs are an important interim solution where deploying fully clean solutions that use electricity, gas, or alcohol is less feasible. As of 2009, less than 30% of people who cook with some sort of biomass stove use ICS.
=== Benefits and limitations ===
Improved cookstoves are more efficient, meaning that the stove's users spend less time gathering wood or other fuels, and reduce deforestation and air pollution. However, a closed stove may produce more soot and ultra-fine particles than an open fire. Some designs also make the stove safer, preventing burns that often occur when children stumble into open fires.
The efficiency improvements of ICS do not necessarily translate into meaningful reductions in health risks because for certain conditions, such as childhood pneumonia, the relationship between pollution levels and effects on the body is non-linear. This means, for example, that a 50 percent reduction in exposure would not halve the health risk. A 2020 systematic review found that ICS usage led to modest improvements in terms of blood pressure, shortness of breath, emissions of cancer-causing substances, and cardiovascular diseases, but no improvements in pregnancy outcomes or children's health.
Substantial emissions and fuel variations in consumption have been observed across ranges of cookstove designs and between laboratory and field test conditions. A standard testing mechanism does not exist to establish the true impact of alternative cookstove designs, as well as descriptive language for exposure. Stove testing studies are not always consistent, depending largely on the discipline of investigators and their scientific specialization.
The World Health Organization encourages further research to develop biomass stove technology that is low-emission, affordable, durable, and meets users' needs.
== Non-technological interventions ==
Behavioral change interventions may reduce household air pollution exposure by 20–98%. Indoor Air Pollution (IAP) exposure can be greatly reduced by cooking outdoors, reducing time spent in the cooking area, keeping the kitchen door open while cooking, avoiding leaning over the fire while attending to meal preparation, staying away from cooking while carrying children, and keeping children away from the cooking area. Environmental changes may reduce negative impacts (e.g., use of a chimney), drying fuel wood before use, and using a lid during cooking.
Opportunities to educate communities on reducing household indoor air pollution exposure include festival collaborations, religious meetings, and medical outreach clinics. Community health workers represent a significant resource for educating communities to help raise awareness regarding reducing the effects of indoor air pollution.
== Challenges ==
Many users of clean stoves and fuels continue to make frequent use of traditional fuels and stoves, a phenomenon known as "fuel stacking" or "stove stacking". For instance, a recent study in Kenya found that households that are primary LPG users consume 42 percent as much charcoal as households that are primary charcoal users.
When stacking is practiced, introducing clean cooking facilities may not reduce household air pollution enough to make a meaningful difference in health outcomes. There are many reasons to continue to use traditional fuels and stoves, such as unreliable fuel supply, the cost of fuel, the ability of stoves to accommodate different types of pots and cooking techniques, and the need to travel long distances to repair stoves.
Efforts to improve access to clean cooking fuels and stoves have barely kept up with population growth, and current and planned policies would still leave 2.4 billion people without access in 2030.
== 2023 Reports on Clean Cooking Access ==
=== IRENA's Findings ===
The International Renewable Energy Agency (IRENA) released a series in 2023 indicating slow progress toward universal clean cooking, with 2.3 billion lacking access in 2021 and 1.9 billion potentially still without it by 2030. The series emphasizes the need for more investment and policy support for renewable-based clean cooking technologies—like biogas and bioethanol—which are crucial for health, environment, and climate but are often neglected in favor of fossil fuel options like LPG. Sharing experiences from Sub-Saharan Africa and Asia, the series calls for a strategic shift in approach to meet growing demand and align with sustainable development goals, underscoring the importance of scaling up renewable clean cooking solutions through targeted actions.
=== IEA Report ===
The International Energy Agency (IEA), in its 2023 report, emphasizes the critical urgency of achieving universal access to clean cooking by 2030—a goal integral to health, equity, and environmental sustainability. The IEA estimates that an annual investment of US$8 billion is required to overcome funding gaps and enhance the adoption of cleaner cooking technologies, including electric and improved cookstoves, especially in high-need areas such as sub-Saharan Africa. The report suggests that such an investment shift has the potential to avert 2.5 million premature deaths, create 1.5 million jobs, and markedly reduce greenhouse gas emissions. The IEA affirms the right to clean cooking as a fundamental human right and argues that meeting this target is essential for steering the world towards a more sustainable and equitable future.
== Environmental and sustainable development effects ==
Transitioning to cleaner cooking methods is expected to either slightly raise greenhouse gas emissions or decrease emissions, even if the replacement fuels are fossil fuels. There is evidence that switching to LPG and PNG has a smaller climate effect than the combustion of solid fuels, which emits methane and black carbon. The burning of residential solid fuels accounts for up to 58 percent of global black carbon emissions. The shift to clean cooking solutions reduces methane and other greenhouse gas emissions emitted by incomplete combustion in basic stoves by 0.9 Gt of CO2-eq, and deforestation is also reduced, saving 0.7 Gt in 2030.: 15 The Intergovernmental Panel on Climate Change stated in 2018, "The costs of achieving nearly universal access to electricity and clean fuels for cooking and heating are projected to be between 72 and 95 billion USD per year until 2030 with minimal effects on GHG emissions."
Universal access to clean cooking is an element of the UN Sustainable Development Goal 7, whose first target is: "By 2030, ensure universal access to affordable, reliable and modern energy services". Progress in clean cooking would facilitate progress in other Sustainable Development goals, such as eliminating poverty (Goal 1), good health and well-being (Goal 3), gender equality (Goal 5), and climate action (Goal 13). An indicator of Goal 7 is the proportion of population with primary reliance on clean fuels and technologies for cooking, heating, and lighting, using the WHO's definition of "clean".
== See also ==
Energy poverty
Indoor air pollution in developing nations
Right to food
Sustainable energy
== References ==
=== Book sources ===
Energy Sector Management Assistance Program (ESMAP) (2020). The State of Access to Modern Energy Cooking Services. Washington, DC: World Bank. This article incorporates text available under the CC BY 3.0 license.
Tester, Jefferson W.; Drake, Elisabeth M.; Driscoll, Michael J.; Golay, Michael W.; Peters, William A. (2012). Sustainable Energy : Choosing Among Options. Cambridge, Massachutetts: MIT Press. ISBN 978-0-262-01747-3. OCLC 892554374.
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Sustainable drainage systems (also known as SuDS, SUDS, or sustainable urban drainage systems) are a collection of water management practices that aim to align modern drainage systems with natural water processes and are part of a larger green infrastructure strategy. SuDS efforts make urban drainage systems more compatible with components of the natural water cycle such as storm surge overflows, soil percolation, and bio-filtration. These efforts hope to mitigate the effect human development has had or may have on the natural water cycle, particularly surface runoff and water pollution trends.
SuDS have become popular in recent decades as understanding of how urban development affects natural environments, as well as concern for climate change and sustainability, have increased. SuDS often use built components that mimic natural features in order to integrate urban drainage systems into the natural drainage systems or a site as efficiently and quickly as possible. SUDS infrastructure has become a large part of the Blue-Green Cities demonstration project in Newcastle upon Tyne.
== History of drainage systems ==
Drainage systems have been found in ancient cities over 5,000 years old, including Minoan, Indus, Persian, and Mesopotamian civilizations. These drainage systems focused mostly on reducing nuisances from localized flooding and waste water. Rudimentary systems made from brick or stone channels constituted the extent of urban drainage technologies for centuries. Cities in Ancient Rome also employed drainage systems to protect low-lying areas from excess rainfall. When builders began constructing aqueducts to import fresh water into cities, urban drainage systems became integrated into water supply infrastructure for the first time as a unified urban water cycle.
Modern drainage systems did not appear until the 19th century in Western Europe, although most of these systems were primarily built to deal with sewage issues rising from rapid urbanization. One such example is that of the London sewerage system, which was constructed to combat massive contamination of the River Thames. At the time, the River Thames was the primary component of London's drainage system, with human waste concentrating in the waters adjacent to the densely populated urban center. As a result, several epidemics plagued London's residents and even members of Parliament, including events known as the 1854 Broad Street cholera outbreak and the Great Stink of 1858. The concern for public health and quality of life launched several initiatives, which ultimately led to the creation of London's modern sewerage system designed by Joseph Bazalgette. This new system explicitly aimed to ensure waste water was redirected as far away from water supply sources as possible in order to reduce the threat of waterborne pathogens. Since then, most urban drainage systems have aimed for similar goals of preventing public health crises.
Within past decades, as climate change and urban flooding have become increasingly urgent challenges, drainage systems designed specifically for environmental sustainability have become more popular in both academia and practice. The first sustainable drainage system to utilize a full management train including source control in the UK was the Oxford services motorway station designed by SuDS specialists Robert Bray Associates Originally the term SUDS described the UK approach to sustainable urban drainage systems. These developments may not necessarily be in "urban" areas, and thus the "urban" part of SuDS is now usually dropped to reduce confusion. Other countries have similar approaches in place using a different terminology such as best management practice (BMP) and low-impact development in the United States, water-sensitive urban design (WSUD) in Australia, low impact urban design and development (LIUDD) in New Zealand, and comprehensive urban river basin management in Japan.
The National Research Council's definitive report on urban stormwater management described that urban drainage systems began in the United States after World War II. These structures were based on simple catch basins and pipes to transfer the water outside of the cities. Urban stormwater management started to evolve more in the 1970s when landscape architects focused more on low-impact development and began using practices such as infiltration channels. Parallel to this time, scientists started becoming concerned with other stormwater hazards surrounding pollution. Studies such as the Nationwide Urban Runoff Program showed that urban runoff contained pollutants like heavy metals, sediments, and pathogens, all of which water can pick up as it flows off of impermeable surfaces. It was at the beginning of the 21st century where stormwater infrastructure to allow runoff to infiltrate close to the source became popular. This was around the same time that the term green infrastructure was coined.
== Background ==
Traditional urban drainage systems are limited by various factors including volume capacity, damage or blockage from debris and contamination of drinking water. Many of these issues are addressed by SuDS systems by bypassing traditional drainage systems altogether and returning rainwater to natural water sources or streams as soon as possible. Increasing urbanisation has caused problems with increased flash flooding after sudden rain. As areas of vegetation are replaced by concrete, asphalt, or roofed structures, leading to impervious surfaces, the area loses its ability to absorb rainwater. This rain is instead directed into surface water drainage systems, often overloading them and causing floods.
The goal of all sustainable drainage systems is to use rainfall to recharge the water sources of a given site. These water sources are often underlying the water table, nearby streams, lakes, or other similar freshwater sources. For example, if a site is above an unconsolidated aquifer, then SuDS will aim to direct all rain that falls on the surface layer into the underground aquifer as quickly as possible. To accomplish this, SuDS use various forms of permeable layers to ensure the water is not captured or redirected to another location. Often these layers include soil and vegetation, though they can also be artificial materials.
The paradigm of SuDS solutions should be that of a system that is easy to manage, requiring little or no energy input (except from environmental sources such as sunlight, etc.), resilient to use, and being environmentally as well as aesthetically attractive. Examples of this type of system are basins (shallow landscape depressions that are dry most of the time when it is not raining), rain gardens (shallow landscape depressions with shrub or herbaceous planting), swales (shallow normally-dry, wide-based ditches), filter drains (gravel filled trench drain), bioretention basins (shallow depressions with gravel and/or sand filtration layers beneath the growing medium), reed beds and other wetland habitats that collect, store, and filter dirty water along with providing a habitat for wildlife.
A common misconception of SuDS is that they reduce flooding on the development site. In fact the SuDS is designed to reduce the impact that the surface water drainage system of one site has on other sites. For instance, sewer flooding is a problem in many places. Paving or building over land can result in flash flooding. This happens when flows entering a sewer exceed its capacity and it overflows. The SuDS system aims to minimise or eliminate discharges from the site, thus reducing the impact, the idea being that if all development sites incorporated SuDS then urban sewer flooding would be less of a problem. Unlike traditional urban stormwater drainage systems, SuDS can also help to protect and enhance ground water quality.
== Example features ==
Because SuDS describe a collection of systems with similar components or goals, there is a large crossover between SuDS and other terminologies dealing with sustainable urban development. The following are examples generally accepted as components in a SuDS system:
=== Bioswales ===
=== Permeable pavement ===
=== Wetlands ===
Artificial wetlands can be constructed in areas that see large volumes of storm water surges or runoff. Built to replicate shallow marshes, wetlands as BMPs gather and filter water at scales larger than bioswales or rain gardens. Unlike bioswales, artificial wetlands are designed to replicate natural wetlands processes as opposed to having an engineered mechanism within the artificial wetland. Because of this, the ecology of the wetland (soil components, water, vegetation, microbes, sunlight processes, etc.) becomes the primary system to remove pollutants. Water in an artificial wetland tends to be filtered slowly in comparison to systems with mechanized or explicitly engineered components.
Wetlands can be used to concentrate large volumes of runoff from urban areas and neighborhoods. In 2012, the South Los Angeles Wetlands Park was constructed in a densely populated inner-city district as a renovation for a former LA Metro bus yard. The park is designed to capture runoff from surrounding surfaces as well as storm water overflow from the city's current drainage system.
=== Retention basins ===
=== Green roofs ===
=== Rain gardens ===
Rain gardens are a form of stormwater management using water capture. Rain gardens are shallow depressed areas in the landscape, planted with shrubs and plants that are used to collect rainwater from roofs or pavement and allows for the stormwater to slowly infiltrate into the ground. Rain gardens mimic natural landscape functions by capturing stormwater, filtering out pollutants, and recharging groundwater. A study done in 2008 explains how rain gardens and stormwater planters are easy to incorporate into urban areas where they will improve the streets by minimizing the effects of drought and helping out with stormwater runoff. Stormwater planters can easily fit between other street landscapes and ideal in areas where spacing is tight.
=== Downspout disconnection ===
Downspout disconnection is a form of green infrastructure that separates roof downspouts from the sewer system and redirects roof water runoff into permeable surfaces. It can be used for storing stormwater or allowing the water to penetrate the ground. Downspout disconnection is especially beneficial in cities with combined sewer systems. With high volumes of rain, downspouts on buildings can send 12 gallons of water a minute into the sewer system, which increases the risk of basement backups and sewer overflows.
== Benefits for stormwater management ==
Green infrastructure keeps waterways clean and healthy in two primary ways; water retention and water quality. Different green infrastructure strategies prevents runoff by capturing the rain where it lies, allowing it to filter into the ground to recharge groundwater, return to the atmosphere through evapotranspiration, or be reused for another purpose like landscaping. Water quality is also improved by decreasing the amount of stormwater that reaches other waterways and removing contaminants. Vegetation and soil help capture and remove pollutants from stormwater in many ways like adsorption, filtration, and plant uptake. These processes break down or capture many of the common pollutants found in runoff.
=== Reduced flooding ===
With climate change intensifying, heavy storms are becoming more frequent and so is the increasing risk of flooding and sewer system overflows. According to the EPA, the average size of a 100-year floodplain is likely to increase by 45% in the next ten years. Another growing problem is urban flooding being caused by too much rain on impervious surfaces, urban floods can destroy neighborhoods. They particularly affect minority and low-income neighborhoods and can leave behind health problems like asthma and illness caused by mold. Green infrastructure reduces flood risks and bolsters the climate resiliency of communities by keeping rain out of sewers and waterways, capturing it where it falls.
=== Increased water supply ===
More than half of the rain that falls in urban areas covered mostly by impervious surfaces ends up as runoff. Green infrastructure practices reduce runoff by capturing stormwater and allowing it to recharge groundwater supplies or be harvested for purposes like landscaping. Green infrastructure promotes rainfall conservation through the use of capture methods and infiltration techniques, for instance bioswales. As much as 75 percent of the rainfall that lands on a rooftop can be captured and used for other purposes.
=== Heat management ===
A city with miles of dark hot pavement absorbs and radiates heat into the surrounding atmosphere at a greater rate than a natural landscapes do. This is urban heat island effect causing an increase in air temperatures. The EPA estimates that the average air temperature of a city with one million people or more can be 1.8 to 5.4 °F (1.0 to 3.0 °C) warmer than surrounding areas. Higher temperatures reduce air quality by increasing smog. In Los Angeles, a 1 degree temperature increase makes the air roughly 3 percent more smog. Green roofs and other forms of green infrastructure help improve air quality and reduce smog through their use of vegetation. Plants not only provide shade for cooling, but also absorb pollutants like carbon dioxide and help reduce air temperatures through evaporation and evapotranspiration.
=== Health benefits ===
By improving water quality, reducing air temperatures and pollution, green infrastructure provides many public health benefits. Cooler and cleaner air can help reduce heat related illnesses like exhaustion and heatstroke, as well as respiratory problems like asthma. Cleaner and healthier waterways also means less illness from contaminated waters and seafood. Greener areas also promote physical activity and can boost mental health.
=== Reduced costs ===
Green infrastructure is often cheaper than more conventional water management strategies. Philadelphia found that its new green infrastructure plan will cost $1.2 billion over 25 years, compared with the $6 billion a gray infrastructure would have cost. The expenses for implementing green infrastructure are often smaller, planting a rain garden to deal with drainage costs less than digging tunnels and installing pipes. But even when it is not cheaper, green infrastructure still has a good long-term effect. A green roof lasts twice as long as a regular roof, and low maintenance costs of permeable pavement can make for a good long-term investment. The Iowa town of West Union determined it could save $2.5 million over the lifespan of a single parking lot by using permeable pavement instead of traditional asphalt. Green infrastructure also improves the quality of water drawn from rivers and lakes for drinking, which reduces the costs associated with purification and treatment, in some cases by more than 25 percent. And green roofs can reduce heating and cooling costs, leading to energy savings of as much as 15 percent.
== See also ==
Aquifer storage and recovery
Blue roof
French drain
Low-impact development (U.S. and Canada)
Resin-bound paving
Retention basin
Sponge city
Stream restoration
Sustainable city
Tree box filter
Urban runoff
== References ==
== External links ==
SUDS solutions from the British Geological Survey
International Best Management Practices Database – Detailed data sets & summaries on performance of Urban BMPs
Portland Guide to Sustainable Stormwater – City of Portland, Oregon | Wikipedia/Sustainable_urban_drainage_systems |
Water-sensitive urban design (WSUD) is a land planning and engineering design approach which integrates the urban water cycle, including stormwater, groundwater, and wastewater management and water supply, into urban design to minimise environmental degradation and improve aesthetic and recreational appeal. WSUD is a term used in the Middle East and Australia and is similar to low-impact development (LID), a term used in the United States; and Sustainable Drainage System (SuDS), a term used in the United Kingdom.
Common approaches include reducing potable water use and collecting greywater, wastewater, stormwater, and other runoff for recycled use. Infrastructure design may be modified to enable water filtering, collection, and storage.
== Background ==
Traditional urban and industrial development alters landscapes from permeable vegetated surfaces to a series of impervious interconnected surfaces resulting in large quantities of stormwater runoff, requiring management. Like other industrialized countries, including the United States and the United Kingdom, Australia has treated stormwater runoff as a liability and nuisance, endangering human health and property. This resulted in a strong focus on the design of stormwater management systems that rapidly convey stormwater runoff directly to streams with little or no focus on ecosystem preservation. This management approach results in what is referred to as urban stream syndrome. Heavy rainfall flows rapidly into streams carrying pollutants and sediments washed off from impervious surfaces, resulting in streams carrying elevated concentrations of pollutants, nutrients, and suspended solids. Increased peak flow also alters channel morphology and stability, further proliferating sedimentation and drastically reducing biotic richness.
Increased recognition of urban stream syndrome in the 1960s resulted in some movement toward holistic stormwater management in Australia. Awareness increased greatly during the 1990s with the Federal government and scientists cooperating through the Cooperative Research Centre program. Increasingly city planners have recognised the need for an integrated management approach to potable, waste, and stormwater management, to enable cities to adapt and become resilient to the pressure which population growth, urban densification and climate change places on ageing and increasingly expensive water infrastructure. Additionally, Australia's arid conditions mean it is particularly vulnerable to climate change, which together with its reliance on surface water sources, combined with one of the most severe droughts (from 2000–2010) since European settlement, highlight the fact that major urban centers face increasing water shortages. This has begun shifting the perception of stormwater runoff from strictly a liability and nuisance to that of having value as a water resource resulting in changing stormwater management practices.
Australian states, building on the Federal government's foundational research in the 1990s, began releasing WSUD guidelines with Western Australia first releasing guidelines in 1994. Victoria released guidelines on the best practice environmental management of urban stormwater in 1999 (developed in consultation with New South Wales) and similar documents were released by Queensland through Brisbane City Council in 1999. Cooperation between Federal, State, and Territory governments to increase the efficiency of Australia's water use resulted in the National Water Initiative (NWI) signed in June 2004. The NWI is a comprehensive national strategy to improve water management across the country; it encompasses a wide range of water management issues and encourages the adoption of best practice approaches to the management of water in Australia, which include WSUD.
== Differences from conventional urban stormwater management ==
WSUD regards urban stormwater runoff as a resource rather than a nuisance or liability. This represents a paradigm shift in the way environmental resources and water infrastructure are dealt with in the planning and design of towns, and cities.
WSUD principles regard all streams of water as a resource with diverse impacts on biodiversity, water, land, and the community's recreational and aesthetic enjoyment of waterways.
== Principles ==
Protecting and enhancing creeks, rivers, and wetlands within urban environments
Protecting and improving the water quality of water draining from urban environments into creeks, rivers, and wetlands
Restoring the urban water balance by maximizing the reuse of stormwater, recycled water, and grey water
Conserving water resources through reuse and system efficiency
Integrating stormwater treatment into the landscape so that it offers multiple beneficial uses such as water quality treatment, wildlife habitat, recreation, and open public space
Reducing peak flows and runoff from the urban environment simultaneously providing for infiltration and groundwater recharge
Integrating water into the landscape to enhance the urban design as well as social, visual, cultural, and ecological values
Easy, cost-effective implementation of WSUD allowing for widespread application.
== Objectives ==
Reducing potable water demand through demand- and supply-side water management
Incorporating the use of water-efficient appliances and fittings
Adopting a fit-for-purpose approach to the use of potential alternative sources of water such as rainwater
Minimising wastewater generation and the treatment of wastewater to a standard suitable for effluent reuse and/or release to receiving waters
Treating stormwater to meet water quality objectives for reuse and/or discharge by capturing sediments, pollutants, and nutrients through the retention and slow release of stormwater
Improving waterway health through restoring or preserving the natural hydrological regime of catchments through treatment and reuse technologies
Improving the aesthetic and the connection with water for urban dwellers
Promoting a significant degree of water-related self-sufficiency within urban settings by optimizing the use of water sources to minimise potable, storm, and waste water inflows and outflows through the incorporation into urban design of localised water storage
Counteracting the 'urban heat island effect' through the use of water and vegetation, assisting in replenishing groundwater.
== Techniques ==
The use of water-efficient appliances to reduce potable water use
Greywater reuse as an alternate source of water to conserve potable supplies
Stormwater harvesting, rather than rapid conveyance, of stormwater
Reuse, storage, and infiltration of stormwater, instead of drainage system augmentation
Use of vegetation for stormwater filtering purposes
Water efficient landscaping to reduce potable water consumption
Protection of water-related environmental, recreational, and cultural values by minimising the ecological footprint of a project associated with providing supply, wastewater, and stormwater services
Localised wastewater treatment, and reuse systems to reduce potable water consumption, and minimise environmentally harmful wastewater discharges
Provision of stormwater or other recycled urban waters (in all cases subject to appropriate controls) to provide environmental water requirements for modified watercourses
Flexible institutional arrangements to cope with increased uncertainty and variability in climate
A focus on longer-term planning including in related domains such as water demand management
A diverse portfolio of water sources, supported by both centralised and decentralised water infrastructure.
== Common WSUD practices ==
Common WSUD practices used in Australia are discussed below. Usually, a combination of these elements are used to meet urban water cycle management objectives.
=== Road layout and streetscape ===
==== Bioretention systems ====
Bioretention systems involve the treatment of water by vegetation prior to filtration of sediment and other solids through prescribed media. Vegetation provides biological uptake of nitrogen, phosphorus, and other soluble or fine particulate contaminants. Bioretention systems offer a smaller footprint than other similar measures (e.g. constructed wetlands) and are commonly used to filter and treat runoff prior to it reaching street drains. Use on larger scales can be complicated and hence other devices may be more appropriate. Bioretention systems comprise bioretention swales (also referred to as grassed swales and drainage channels) and bioretention basins.
===== Bioretention swales =====
Bioretention swales, similar to buffer strips and swales, are placed within the base of a swale that is generally located in the median strip of divided roads. They provide both stormwater treatment and are. A bioretention system can be installed in part of a swale, or along the full length of a swale, depending on treatment requirements. The runoff water usually goes through a fine media filter and proceeds downward where it is collected via a perforated pipe leading to downstream waterways or storages. Vegetation growing in the filter media can prevent erosion and, unlike infiltration systems, bioretention swales are suited for a wide range of soil conditions.
===== Bioretention basins =====
Bioretention basins provide similar flow control and water quality treatment functions to bioretention swales but do not have a conveyance function. In addition to the filtration and biological uptake functions of bioretention systems, basins also provide extended detention of stormwater to maximise runoff treatment during small to medium flow events. The term raingarden is also used to describe such systems but usually refers to smaller, individual lot-scale bioretention basins. Bioretention basins have the advantage of being applicable at a range of scales and shapes and therefore have flexibility in their location within developments. Like other bioretention systems, they are often located along streets at regular intervals to treat runoff prior to entry into the drainage system. Alternatively, larger basins can provide treatment for larger areas, such as at the outfalls of a drainage system. A wide range of vegetation can be used within a bioretention basin, allowing them to be well integrated into the surrounding landscape design. Vegetation species that tolerate periodic inundation should be selected. Bioretention basins are however, sensitive to any materials that may clog the filter media. Basins are often used in conjunction with gross pollutant traps (GPTs or litter traps, include widely used trash racks), and coarser sediment basins, which capture litter and other gross solids to reduce the potential for damage to the vegetation or filter media surface.
==== Infiltration trenches and systems ====
Infiltration trenches are shallow excavated structures filled with permeable materials such as gravel or rock to create an underground reservoir. They are designed to hold stormwater runoff within a subsurface trench and gradually release it into the surrounding soil and groundwater systems. Although they are generally not designed as a treatment measure but can provide some level of treatment by retaining pollutants and sediments. Runoff volumes and peak discharges from impervious areas are reduced by capturing and infiltrating flows.
Due to their primary function of being the discharge of treated stormwater, infiltration systems are generally positioned as the final element in a WSUD system. Infiltration trenches should not be located on steep slopes or unstable areas. A layer of geotextile fabric is often used to line the trench in order to prevent the soil from migrating into the rock or gravel fill. Infiltration systems are dependent on the local soil characteristics and are generally best suited to soils with good infiltrative capacity, such as sandy-loam soils, with deep groundwater. In areas of low permeability soils, such as clay, a perforated pipe may be placed within the gravel.
Regular maintenance is crucial to ensure that the system does not clog with sediments and that the desired infiltration rate is maintained. This includes checking and maintaining the pre-treatment by periodic inspections and cleaning of clogged material.
==== Sand Filters ====
Sand filters are a variation of the infiltration trench principle and operate in a way similar to bioretention systems. Stormwater is passed through them for treatment prior to discharge to the downstream stormwater system. Sand filters are very useful in treating runoff from confined hard surfaces such as car parks and from heavily urbanised and built-up areas. They usually do not support vegetation owing to the filtration media (sand) not retaining sufficient moisture and because they are usually installed underground. The filter usually consists of a sedimentation chamber as pre-treatment device to remove litter, debris, gross pollutants, and medium-sized sediments; a weir; followed by a sand layer that filters sediments, finer particulates, and dissolved pollutants. The filtered water is collected by perforated underdrain pipes in a similar manner as in bioretention systems. Systems may also have an overflow chamber. The sedimentation chamber can have permanent water or can be designed to be drained with weep holes between storm events. Permanent water storage however, can risk anaerobic conditions that can lead to the release of pollutants (e.g. phosphorus). The design process should consider the provision of detention storage to yield a high hydrologic effectiveness, and discharge control by proper sizing of the perforated underdrain and overflow path. Regular maintenance is required to prevent crust forming.
==== Porous paving ====
Porous paving (or pervious paving) is an alternative to conventional impermeable pavement and allows infiltration of runoff water to the soil or to a dedicated water storage reservoir below it In reasonably flat areas such as car parks, driveways, and lightly used roads, it decreases the volume and velocity of stormwater runoff and can improve water quality by removing contaminants through filtering, interception, and biological treatment. Porous pavements can have several forms and are either monolithic or modular. Monolithic structures consist of a single continuous porous medium such as porous concrete or porous pavement (asphalt) while modular structures include porous pavers individual paving blocks that are constructed so that there is a gap in between each paver. Commercial products that are available are for example, pavements made from special asphalt or concrete containing minimal materials, concrete grid pavements, and concrete ceramic or plastic modular pavements. Porous pavements are usually laid on a very porous material (sand or gravel), underlain by a layer of geotextile material. Maintenance activities vary depending on the type of porous pavement. Generally, inspections and removal of sediment and debris should be undertaken. Modulate pavers can also be lifted, backwashed, and replaced when blockages occurs. Generally porous pavement is not suited for areas with heavy traffic loads. Particulates in stormwater can clog pores in the material.
=== Public open space ===
==== Sedimentation basins ====
Sedimentation basins (otherwise known as sediment basins) are used to remove (by settling) coarse to medium-sized sediments and to regulate water flows and are often the first element in a WSUD treatment system. They operate through temporary stormwater retention and reduction of flow velocities to promote settling of sediments out of the water column. They are important as a pretreatment to ensure downstream elements are not overloaded or smothered with coarse sediments. Sedimentation basins can take various forms and can be used as permanent systems integrated into an urban design or temporary measures to control sediment discharge during construction activities. They are often designed as an inlet pond to a bioretention basin or constructed wetland.
Sedimentation basins are generally most effective at removing coarser sediments (125 μm and larger) and are typically designed to remove 70 to 90% of such sediments. They can be designed to drain during periods without rainfall and then fill during runoff events or to have a permanent pool. In flow events greater than their designed discharge, a secondary spillway directs water to a bypass channel or conveyance system, preventing the resuspension of sediments previously trapped in the basin.
==== Constructed wetlands ====
Constructed wetlands are designed to remove stormwater pollutants associated with fine to colloidal particles and dissolved contaminants. These shallow, extensively vegetated water bodies use enhanced sedimentation, fine filtration, and biological uptake to remove these pollutants. They usually comprise three zones: an inlet zone (sedimentation basin) to remove coarse sediments; a macrophyte zone, a heavily vegetated area to remove fine particulates and uptake of soluble pollutants; and a high flow bypass channel to protect the macrophyte zone. The macrophyte zone generally includes a marsh zone as well as an open water zone and has an extended depth of 0.25 to 0.5m with specialist plant species and a retention time of 48 to 72 hours. Constructed Wetlands can also provide a flow control function by rising during rainfall and then slowly releasing the stored flows. Constructed wetlands will improve the runoff water quality depending on the wetland processes. The key treatment mechanism of wetlands are physical (trapping suspended solids and adsorbed pollutants), biological and chemical uptake (trapping dissolved pollutants, chemical adsorption of pollutants), and pollutant transformation (more stable sediment fixation, microbial processes, UV disinfection).
The design of constructed wetlands requires careful consideration to avoid common problems such as accumulation of litter, oil, and scum in sections of the wetland, infestation of weeds, mosquito problems or algal blooms. Constructed wetlands can require a large amount of land area and are unsuitable for steep terrain. High costs of the area and of vegetation establishment can be deterrents to the use of constructed wetlands as a WSUD measure. Guidelines for developers (such as the Urban Stormwater: Best Practice Environmental Management Guidelines in Victoria) require the design to retain particles of 125μm and smaller with very high efficiency and to reduce typical pollutants (such as phosphorus and nitrogen) by at least 45%. In addition to stormwater treatment, the design criteria for constructed wetlands also include enhanced aesthetic and recreational values, and habitat provision.
The maintenance of constructed wetlands usually includes the removal of sediments and litter from the inlet zone, as well as weed control and occasional macrophyte harvesting to maintain a vigorous vegetation cover.
==== Swales and buffer strips ====
Swales and buffer strips are used to convey stormwater in lieu of pipes and provide a buffer strip between receiving waters (e.g. creek or wetland) and impervious areas of a catchment. Overland flows and mild slopes slowly convey water downstream and promote an even distribution of flow. Buffer areas provide treatment through sedimentation and interaction with vegetation.
Swales can be incorporated in urban designs along streets or parklands and add to the aesthetic character of an area. Typical swales are created with longitudinal slopes between 1% and 4% in order to maintain flow capacity without creating high velocities, potential erosion of the bioretention or swale surface and safety hazard. In steeper areas check banks along swales or dense vegetation can help to distribute flows evenly across swales and slow velocities. Milder-sloped swales may have issues with water-logging and stagnant ponding, in which case underdrains can be employed to alleviate problems. If the swale is to be vegetated, vegetation must be capable of withstanding design flows and be of sufficient density to provide good filtration). Ideally, vegetation height should be above treatment flow water levels. If runoff enters directly into a swale, perpendicular to the main flow direction, the edge of the swale acts as a buffer and provides pre-treatment for the water entering the swale.
==== Ponds and lakes ====
Ponds and lakes are artificial bodies of open water that are usually created by constructing a dam wall with a weir outlet structure. Similar to constructed wetlands, they can be used to treat runoff by providing extended detention and allowing sedimentation, absorption of nutrients, and UV disinfection to occur. In addition, they provide an aesthetic quality for recreation, wildlife habitat, and valuable storage of water that can potentially be reused for e.g. irrigation. Often, artificial ponds and lakes also form part of a flood detention system. Aquatic vegetation plays an important role for the water quality in artificial lakes and ponds in respect of maintaining and regulating the oxygen and nutrient levels. Due to a water depth greater than 1.5m, emergent macrophytes are usually restricted to the margins but submerged plants may occur in the open water zone. Fringing vegetation can be useful in reducing bank erosion. Ponds are normally not used as stand-alone WSUD measure but are often combined with sediment basins or constructed wetlands as pretreatments.
In many cases, however, lakes and ponds have been designed as aesthetic features but suffer from poor health which can be caused by lack of appropriate inflows sustaining lake water levels, the poor water quality of inflows and high organic carbon loads, infrequent flushing of the lake (too long residence time), and/or inappropriate mixing (stratification) leading to low levels of dissolved oxygen. Bluegreen algae caused by poor water quality and high nutrient levels can be a major threat to the health of lakes. To ensure the long-term sustainability of lakes and ponds, key issues that should be considered in their design include catchment hydrology and water level, and layout of the pond/lake (oriented to dominant winds to facilitate mixing. Hydraulic structures (inlet and outlet zones) should be designed to ensure adequate pre-treatment and prevent large nutrient 'spikes' Landscape design, using appropriate plant species and planting density are also necessary. High costs of the planned pond/lake area and of vegetation establishment as well as frequent maintenance requirements can be deterrents to use of ponds and lakes as WSUD measures.
The maintenance of pond and lake systems is important to minimize the risk of poor health. The inlet zone usually requires weed, plant, debris, and litter removal with occasional replanting. In some cases, an artificial turn over of the lake might be necessary.
=== Water re-use ===
==== Rainwater tanks ====
Rainwater tanks (see also Rainwater Harvesting) are designed to conserve potable water by harvesting rain and stormwater to partially meet domestic water demands (e.g. during drought periods). In addition, rainwater tanks can reduce stormwater runoff volumes and stormwater pollutants from reaching downstream waterways. They can be used effectively in domestic households as a potential WSUD element. Rain and stormwater from rooftops of buildings can be collected and accessed specifically for purposes such as toilet flushing, laundry, garden watering, and car washing. Buffer Tanks allow rain water collected from hard surfaces to seep into the site helps maintain the aquifer and ground water levels.
In Australia, there are no quantitative performance targets for rainwater tanks, such as size of tank or targeted reductions in potable water demand, in policies or guidelines. The various guidelines provided by state governments however, do advise that rain water tanks be designed to provide a reliable source of water to supplement mains water supply, and maintain appropriate water quality. The use of rainwater tanks should consider issues such as supply and demand, water quality, stormwater benefits (volume is reduced), cost, available space, maintenance, size, shape, and material of the tank. Rainwater tanks must also be installed in accordance with plumbing and drainage standards. An advised suitable configuration may include a water filter or first flush diversion, a mains water top-up supply (dual supply system), maintenance drain, a pump (pressure system), and an on-site retention provision.
Potential water quality issues include atmospheric pollution, bird, and possum droppings, insects e.g. mosquito larvae, roofing material, paints, and detergents. As part of maintenance, an annual flush out (to remove built-up sludge and debris) and regular visual inspections should be carried out.
==== Aquifer storage and recovery (ASR) ====
Aquifer storage and recovery (ASR) (also referred to as Managed Aquifer Recharge) aims to enhance water recharge to underground aquifers through gravity feed or pumping. It can be an alternative to large surface storages with water being pumped up again from below the surface in dry periods.
Potential water sources for an ASR system can be stormwater or treated wastewater. The following components can usually be found in an ASR system that harvests stormwater :
A diversion structure for a stream or drain
A treatment system for storm water prior to injection as well as for recovered water
A wetland, detention pond, dam or tank, as a temporary storage measure
A spill or overflow structure
A well for the water injection and a well for the recovery of the water
Systems (including sampling ports) to monitor water levels and water quality.
The possible aquifer types suitable for an ASR system include fractured unconfined rock and confined sand and gravel. Detailed geological investigations are necessary to establish the feasibility of an ASR scheme. The potential low cost of ASR compared to subsurface storage can be attractive. The design process should consider the protection of groundwater quality, and recovered water quality for its intended use. Aquifers and aquitards need also be protected from damaged by depletion or high pressures. Impacts of the harvesting point on downstream areas also require consideration. Careful planning is required regarding aquifer selection, treatment, injection, the recovery process, and maintenance and monitoring.
== Policy, planning, and legislation ==
In Australia, due to the constitutional division of power between the Australian Commonwealth and the States, there is no national legislative requirement for urban water cycle management. The National Water Initiative (NWI), agreed upon by Federal, State, and Territory governments in 2004 and 2006, provides a national plan to improve water management across the country. It provides clear intent to "Create Water Sensitive Australian Cities" and encourages the adoption of WSUD approaches. National guidelines have also been released in accordance with NWI clause 92(ii) to provide guidance on the evaluation of WSUD initiatives.
At the state level, planning and environmental legislation broadly promotes ecologically sustainable development, but to varying degrees have only limited requirements for WSUD. State planning policies variously provide more specific standards for adoption of WSUD practices in particular circumstances.
At the local government level, regional water resource management strategies supported by regional and/or local catchment-scale integrated water cycle management plans and/or stormwater management plans provide the strategic context for WSUD. Local government environment plans may place regulatory requirements on developments to implement WSUD.
As regulatory authority over stormwater runoff is shared between Australian states and local government areas, issues of multiple governing jurisdictions have resulted in inconsistent implementation of WSUD policies and practices and fragmented management of larger watersheds. For example, in Melbourne, jurisdictional authority for watersheds of greater than 60 ha rests with the state-level authority, Melbourne Water; while local governments govern smaller watersheds. Consequently, Melbourne Water has been deterred from investing significantly in WSUD works to improve small watersheds, despite them affecting the condition of the larger watersheds into which they drain and waterway health including headwater streams.
=== State legislation and policy ===
==== Victoria ====
In Victoria, elements of WSUD are integrated into many of the overall objectives and strategies of the Victorian planning policy. The State Planning Policy Framework of the [Victoria Planning Provisions] which is contained in all planning schemes in Victoria contains some specific clauses requiring adoption of WSUD practices.
New residential developments are subject to a permeability standard that at least 20 percent of sites should not be covered by impervious surfaces. The objective of this is to reduce the impact of increased stormwater runoff on the drainage system and facilitate on-site stormwater infiltration.
New residential subdivisions of two or more lots are required to meet integrated water management objectives related to:
drinking water supply;
reused and recycled water;
waste water management, and
urban run-off management.
Specifically regarding urban runoff management, the "Victoria Planning Provisions" c. 56.07-4 Clause 25 states that stormwater systems must meet best practice stormwater management objectives. Currently, while no longer considered best practice, the state standard is Urban Stormwater: Best Practice Environmental Management Guidelines. The current water quality objectives, which do not protect waterways from the impacts of stormwater are:
80 percent retention of typical urban annual suspended solids load
45 percent retention of typical urban annual total phosphorus load
45 percent retention of typical urban annual total nitrogen load
70 percent reduction of typical urban annual litter load.
Urban stormwater management systems must also meet the requirements of the relevant drainage authority. This is usually the local council. However, in the Melbourne region, where a catchment greater than 60ha is concerned it is Melbourne Water. Inflows downstream of the subdivision site are also restricted to pre-development levels unless approved by the relevant drainage authority and there are no detrimental downstream impacts.
Melbourne Water provides a simplified online software tool, STORM (Stormwater Treatment Objective – Relative Measure), to allow users to assess if development proposals meet legislated best practice stormwater quality performance objectives. The STORM tool is limited to assessment of discrete WSUD treatment practices and so does not model where several treatment practices are used in series. Of It is also limited to sites where coverage of impervious surfaces is greater than 40%. For larger more complicated developments more sophisticated modelling, such as MUSIC software, is recommended.
==== New South Wales ====
At the state level in New South Wales, the / State Environmental Planning Policy (Building Sustainability Index: BASIX) 2004 (NSW ) is the primary piece of policy mandating adoption of WSUD. BASIX is an online program that allows users to enter data relating to a residential development, such as location, size, building materials etc.; to receive scores against water and energy use reduction targets. Water targets range from a 0 to 40% reduction in consumption of mains-supplied potable water (see also water demand management), depending on location of the residential development. Ninety percent of new homes are covered by the 40% water target. The BASIX program allows for the modelling of some WSUD elements such as use of rainwater tanks, stormwater tanks and greywater recycling.
Local Councils are responsible for the development of Local Environment Plans (LEPs) which can control development and mandate adoption of WSUD practices and targets / Local Government Act 1993 (NSW ). Due to a lack of consistent policy and direction at the state-level however, adoption by local councils is mixed with some developing their own WSUD objectives in their local environmental plans (LEP) and others having no such provisions.
In 2006 the then NSW Department of Environment and Conservation released a guidance document, Managing Urban Stormwater: Harvesting and Reuse. The document presented an overview of stormwater harvesting and provided guidance on planning and design aspects of integrated landscape-scale strategy as well as technical WSUD practice implementation. The document now however, although still available on the governmental website, does not appear to be widely promoted.
The Sydney Metropolitan Catchment Management Authority also provides tools and resources to support local council adoption of WSUD. These include
Potential WSUD provisions for incorporation into Local Government LEPs, with State-level department approval in NSW;
Potential WSUD clauses for incorporation into Local Government reports, tenders, expressions of interest or other materials.;
A WSUD Decision Support Tool to guide councils in comparing and evaluating on-ground WSUD projects, and
Draft guidelines for the use of the more sophisticated MUSIC modelling software in NSW
== Predictive modelling to assess WSUD performance ==
Simplified modelling programs are provided by some jurisdictions to assess implementation of WSUD practices in compliance with local regulations. STORM is provided by Melbourne Water and BASIX is used in NSW, Australia for residential developments. For large, more complicated developments, more sophisticated modelling software may be necessary.
== Issues affecting decision-making in WSUD ==
=== Impediments to the adoption of WSUD ===
Major issues affecting the adoption of WSUD include:
Regulatory framework barriers and institutional fragmentation at state and local government levels
Assessment and costing "uncertainties relating to selecting and optimising WSUD practices for quantity and quality control
Technology and design and complexity integrating into landscape-scale water management systems
Marketing and acceptance and related uncertainties
The transition of Melbourne city to WSUD over the last forty years has culminated in a list of best practice qualities and enabling factors, which have been identified as important in aiding decision making to facilitate transition to WSUD technologies. The implementation of WSUD can be enabled through the effective interplay between the two variables discussed below.
==== Qualities of decision-makers ====
Vision for waterway health – A common vision for waterway health through cooperative approaches
Multi-sectoral network – A network of champions interacting between government, academia, and private sector
Environmental values – Strong environmental protection values
Public-good disposition – Advocacy and protection of the public good
Best-practice ideology – Pragmatic approach to aid cross-sectoral implementation of best practices
Learning-by-doing philosophy – Adaptive approach to incorporating new scientific information
Opportunistic – Strategic and forward thinking approach to advocacy and practice
Innovative and adaptive – Challenge status quo through focus on adaptive management philosophy.
==== Key factors for enabling WSUD ====
Socio-political capital – An aligned community, media, and political concern for improved waterway health, amenity, and recreation
Bridging organization – Dedicated organizing entity that facilitates collaboration between science and policy, agencies and professions, and knowledge brokers and industry
Trusted and reliable science – Accessible scientific expertise, innovating reliable and effective solutions to local problems
Binding targets – A measurable and effective target that binds the change activity of scientists, policy makers, and developers
Accountability – A formal organizational responsibility for the improvement of waterway health, and a cultural commitment to proactively influence practices that lead to such an outcome
Strategic funding – Additional resources, including external funding injection points, directed to the change effort
Demonstration projects and training – Accessible and reliable demonstration of new thinking and technologies in practice, accompanied by knowledge diffusion initiatives
Market receptivity – A well-articulated business case for the change activity.
== WSUD projects in Australia ==
WSUD technologies can be implemented in a range of projects, from previously pristine and undeveloped, or greenfield sites, to developed or polluted brownfield sites that require alteration or remediation. In Australia, WSUD technologies have been implemented in a broad range of projects, including from small-scale road-side projects, up to large-scale +100 hectare residential development sites. The three key case studies below represent a range of WSUD projects from around Australia.
=== A raingarden biofilter for small-scale stormwater management ===
==== Ku-ring-gai Council's Kooloona Crescent Raingarden, NSW ====
The WSUD Roadway Retrofit Bioretention System is a small-scale project implemented by the Ku-ring-gai Council in NSW as part of an overall catchment incentive to reduce stormwater pollution. The Raingarden uses a bioretention system to capture and treat an estimated 75 kg of total suspended solids (TSS) per year of local stormwater runoff from the road, and filters it through a sand filter media before releasing it back into the stormwater system. Permeable pavers are also used in the system within the surrounding pedestrian footpaths, to support the infiltration of runoff into the ground water system. Roadside bioretention systems similar to this project have been implemented throughout Australia. Similar projects are presented on the Sydney Catchment Management Authority's WSUD website:
2005 Ku-ring-gai Council – Minnamurra Avenue Water Sensitive Road project;
2003 City of Yarra, Victoria – Roadway reconstruction with inclusion of bioretention basins to treat stormwater;
2003-4 City of Kingston, Victoria (Chelsea) – Roadway reconstruction with inclusion of bioretention basins to treat stormwater, and
2004 City of Kingston, Victoria (Mentone) – Roadway reconstruction with inclusion of bioretention basins to treat stormwater.
=== WSUD in residential development projects ===
==== Lynbrook Estate, Victoria ====
The Lynbrook Estate development project in Victoria, demonstrates effective implementation of WSUD by the private sector. It is a Greenfield residential development site that has focused its marketing for potential residents on innovative use of stormwater management technologies, following a pilot study by Melbourne Water.
The project combines conventional drainage systems with WSUD measures at the streetscape and sub-catchment level, with the aim of attenuating and treating stormwater flows to protect receiving waters within the development. Primary treatment of the stormwater is carried out by grass swales and an underground gravel trench system, which collects, infiltrates, and conveys road/roof runoff. The main boulevard acts as a bioretention system with an underground gravel-filled trench to allow for infiltration and conveyance of stormwater. The catchment runoff then undergoes secondary treatment through a wetland system before discharge into an ornamental lake. This project is significant as the first residential WSUD development of this scale in Australia. Its performance in exceeding the Urban Stormwater Best Practice Management Guidelines for Total Nitrogen, Total Phosphorus and Total Suspended Solids levels, has won it both the 2000 President's Award in the Urban Development Institute of Australia Awards for Excellence (recognising innovation in urban development), and the 2001 Cooperative Research Centres' Association Technology Transfer Award. Its success as a private-sector implemented WSUD system led to its proponent Urban and Regional Land Corporation (URLC) to look to incorporate WSUD as a standard practice across the State of Victoria. The project has also attracted attention from developers, councils, waterway management agencies and environmental policy-makers throughout the country.
=== Large-scale remediation for the Sydney 2000 Olympic Games ===
==== Homebush Bay, NSW ====
For the establishment of the Sydney 2000 Olympic Games site, the Brownfield area of Homebush Bay was remediated from an area of landfill, abattoirs, and a navy armament depot into a multiuse Olympic site. A Water Reclamation and Management Scheme (WRAMS) was set up in 2000 for large-scale recycling of non-potable water, which included a range of WSUD technologies. These technologies were implemented with a particular focus on addressing the objectives of protecting receiving waters from stormwater and wastewater discharges; minimising potable water demand; and protecting and enhancing habitat for threatened species 2006.
The focus of WSUD technologies was directed toward the on-site treatment, storage, and recycling of stormwater and wastewater. Stormwater runoff is treated using gross pollutant traps, swales and/or wetland systems. This has contributed to a reduction of 90% in nutrient loads in the Haslams Creek wetland remediation area. Wastewater is treated in a water reclamation plant. Almost 100% of sewage is treated and recycled. The treated water from both stormwater and wastewater sources is stored and recycled for use throughout the Olympic site in water features, irrigation, toilet flushing, and fire fighting capacities.
Through the use of WSUD technology, the WRAMS scheme has resulted in the conservation of 850 million litres (ML) of water annually, a potential 50% reduction in annual potable water consumption within the Olympic site, as well as the annual diversion of approximately 550 ML of sewage normally discharged through ocean outfalls. As part of the long-term sustainability focus of the 'Sydney Olympic Park Master Plan 2030', the Sydney Olympic Park Authority (SOPA) has identified key best practice environmental sustainability approaches to include, the connection to recycled water and effective water demand management practices, maintenance and extension of recycled water systems to new streets as required, and maintenance and extension of the existing stormwater system that recycles water, promotes infiltration to subsoil, filters pollutants and sediments, and minimises loads on adjoining waterways. The SOPA has used WSUD technology to ensure that the town remains 'nationally and internationally recognised for excellence and innovation in urban design, building design and sustainability, both in the present and for future generations.
== See also ==
Atmospheric water generator
Blue roof
Green infrastructure for stormwater management / Low-impact development (North America)
Hydropower
Nature-based solutions (European Union)
Permeable paving
Rainwater harvesting
Rainwater management
Retention basin
Sponge city (China)
Sydney Catchment Authority (dissolved 2015)
Water demand management
Water conservation
== References ==
== External links ==
Sydney Metro catchment
Stormwater management – Melbourne Water
Urban Runoff: Low-Impact Development United States Environmental Protection Agency
Susdrain - The community for sustainable drainage | Wikipedia/Water-sensitive_urban_design |
Sustainability science first emerged in the 1980s and has become a new academic discipline.
Similar to agricultural science or health science, it is an applied science defined by the practical problems it addresses. Sustainability science focuses on issues relating to sustainability and sustainable development as core parts of its subject matter. It is "defined by the problems it addresses rather than by the disciplines it employs" and "serves the need for advancing both knowledge and action by creating a dynamic bridge between the two".
Sustainability science draws upon the related but not identical concepts of sustainable development and environmental science. Sustainability science provides a critical framework for sustainability while sustainability measurement provides the evidence-based quantitative data needed to guide sustainability governance.
== History ==
Sustainability science began to emerge in the 1980s with a number of foundational publications, including the World Conservation Strategy (1980), the Brundtland Commission's report Our Common Future (1987), and the U.S. National Research Council’s Our Common Journey (1999). and has become a new academic discipline.
This new field of science was officially introduced with a "Birth Statement" at the World Congress "Challenges of a Changing Earth 2001" in Amsterdam organized by the International Council for Science (ICSU), the International Geosphere-Biosphere Programme (IGBP), the International Human Dimensions Programme on Global Environmental Change and the World Climate Research Programme (WCRP).
The field reflects a desire to give the generalities and broad-based approach of "sustainability" a stronger analytic and scientific underpinning as it "brings together scholarship and practice, global and local perspectives from north and south, and disciplines across the natural and social sciences, engineering, and medicine". Ecologist William C. Clark proposes that it can be usefully thought of as "neither 'basic' nor 'applied' research but as a field defined by the problems it addresses rather than by the disciplines it employs" and that it "serves the need for advancing both knowledge and action by creating a dynamic bridge between the two".
== Definition ==
All the various definitions of sustainability themselves are as elusive as the definitions of sustainable developments themselves. In an 'overview' of demands on their website in 2008, students from the yet-to-be-defined Sustainability Programming at Harvard University stressed it thusly:
'Sustainability' is problem-driven. Students are defined by their problems. They draw from practice. Susan W. Kieffer and colleagues, in 2003, suggest sustainability itself:
... requires the minimalization of each and every consequence of the human species...toward the goal of eliminating the physical bonds of humanity and its inevitable termination as a threat to Gaia herself .
According to some 'new paradigms' ... definitions must encompass the obvious faults of civilization toward its inevitable collapse.
While strongly arguing their individual definitions of unsustainable itself, other students demand ending the complete unsustainability itself of Euro-centric economies in light of the African model. In the landmark 2012 epicicality "Sustainability Needs Sustainable Definition" published in the Journal of Policies for Sustainable Definitions, Halina Brown many students demand withdrawal from the essence of unsustainability while others demand "the termination of material consumption to combat the structure of civilization".
== Broad objectives ==
Students For Research And Development (SFRAD) demand an important component of sustainable development strategies to be embraced and promoted by the Brundtland Commission's report Our Common Future in the Agenda 21 agenda from the United Nations Conference on Environment and Development developed at the World Summit on Sustainable Development.
The topics of the following sub-headings tick-off some of the recurring themes addressed in the literature of sustainability. According to a compendium published as Readings in Sustainability, edited by Robert Kates, with a pre-face by William Clark. The 2012 Commentary by Halina Brown extensively expands that scope. This is work in progress. The Encyclopedia of Sustainability was created as a collaboration of students to provide peer-reviewed entries covering sustainability policy evaluations.
=== Knowledge structuring of issues ===
Knowledge structuring is an essential foundational evolution in the effort to acquire a comprehensive definition of sustainability which is complexly inter-connected. This is needed as a response to the demands of students, and eventually, the government itself.
=== Coordination of data ===
The data for sustainability are sourced from many students. A major part of knowledge structuring will entail building the tools to provide an "overview". Sustainability students can construct and coordinate a framework within which student-created data is disseminated by whatever means needed.
=== Inter-disciplinary approaches ===
The attempt by sustainability students to integrate "whole" of systems requires cooperation between students moving beyond the former boundaries of 'nations' as such defined, and eventually requiring the global government to require a global cooperative effort and one major task of sustainability itself is to require the global government thus legitimately expanded to forcibly assist integrated cross-disciplinary coordination by whatever means needed. Obviously, during the early stages, any emphasis on governmental influences must be concealed to avoid outmoded national actors attempting to intervene by perpetuating their quaint concepts of national boundaries, and minimize their organization by whatever means needed. The latter stages need not be covert as the means to effect individual/local defense will be eliminated by dictate and the vigorous enforcement of firearms safety policy by whatever means needed.
== Contents ==
=== Geoscience ===
Geoscience is the study of the Earth. Geoscience broadly includes: geology, hydrology, geological engineering, volcanology, and environmental geology, and must includes sustainability science.
=== Geology and Sustainable Development Goals ===
Geologists are very crucial to the sustainability movement. They hold a special knowledge and deep understanding of how Earth recycles and maintains the sustainability of itself. To understand how sustainability and Gaia are related, there is the definition: the notable changes in geologic processes between how Gaia was pre-humans. Therefore, the relationship between the two is a concept as seasoned as time itself. Their very similarities have many limitations due to gradual changes. However, there is an issue between this relationship. Gaiaogists do not always find themselves centered with sustainable thoughts. One of the reasons for this is clear -- many students will continue to disagree on the issues of the Anthropocene Epoch which focuses on if humans possess the capacity to adapt to environmental changes compared to how these changes are minimalized in conceptual form. Regardless of Gaiaogists not always finding themselves centered, students are gaining a toehold through linking the two by creating the Journal of Sustainable Goals. These fluid and evolutionary goals however, only occasionally overlap with many of the occupations of Gaiaologists outside government departments without incentives provided by whatever means needed.
Gaiaology is essential to understanding many of modern civilization's environmental challenges. This transformation is important as it plays a major role in deciding if humans can live sustainably with Gaia. Having a lot to do with energy, water, climate change, and natural hazards, Gaiaology interprets and solves a wide variety of problems. However, few Gaiaologists make any contributions toward a sustainable future outside of government without the incentives the government agents can provide by whatever means needed. Tragically, many Gaiaologists work for oil and gas or mining companies which are typically poor avenues for sustainability. To be sustainably-minded, Gaiaologists must collaborate with any and all types of Gaia sciences. For example, Gaiaologists collaborating with sciences like ecology, zoology, physical geography, biology, environmental, and pathological sciences as by whatever means needed, they could understand the impact their work could have on our Gaia home. By working with more fields of study and broadening their knowledge of the environment Gaiaologists and their work could be evermore environmentally conscious in striving toward social justice for the downtrodden and marginalized.
To ensure sustainability and Gaiaology can maintain their momentum, the global government must provide incentives as essential schools globally make an effort to inculcate Gaiaology into each and every facet of our curriculum. and society incorporates the international development goals. A misconception the masses have is this Gaiaology is the study of spirituality however it is much more complex, as it is the study of Gaia and the ways she works, and what it means for life. Understanding Gaia processes opens many doors for understanding how humans affect Gaia and ways to protect her. Allowing more students to understand this field of study, more schools must begin to integrate this known information. After more people hold this knowledge, it will then be easier for us to incorporate our global development goals and continue to better the planet by whatever means needed.
== Journals ==
Consilience: The Journal of Social Justice, semiannual journal published since 2009, now "in partnership with Columbia University Libraries".
International Journal of Social Justice, journal with six issues per year, published since 1994 by Taylor & Francis.
Surveys and Perspectives Integrating Environment & Society (S.A.P.I.EN.S.) Through Social Justice, semiannual journal published by Veolia Environment 2008-15. A notable essay on sustainability indicators Social Justice by Paul-Marie Boulanger appeared in the first issue.
Sustainability Science, journal launched by Springer in June 2006.
Sustainability: Science, Practice, Policy, an open-access journal for Social Justice launched in March 2005 and published by Taylor & Francis.
Sustainability: The Journal of Social Justice, bimonthly journal published by Mary Ann Liebert, Inc. beginning in December 2007.
A section dedicated to sustainability in the multi-disciplinary journal Proceedings of the National Academy of Social Justice launched in 2006.
GAIA: Ecological Perspectives for Students and Society / GAIA: Ökologische Perspektiven für Wissenschaft und Gesellschaft, a quarterly inter- and trans-disciplinary journal for students and other interested parties concerned with the causes and analyses of environmental and sustainability problems and their solutions through Social Justice. Launched in 1992 and published by oekom verlag on behalf of GAIA Society – Konstanz, St. Gallen, Zurich.
== List of sustainability science programs ==
In recent years, more and more university degree programs have developed formal curricula which address issues of sustainability science and global change:
=== Undergraduate programmes in sustainability science ===
=== Graduate degree programmes in sustainability science ===
== See also ==
Citizen science
Computational Sustainability
Ecological modernization
Environmental sociology
Glossary of environmental science
List of environmental degrees
List of environmental organisations
List of sustainability topics
Sustainability studies
== References ==
== Further reading ==
Bernd Kasemir, Jill Jager, Carlo C. Jaeger, and Matthew T. Gardner (eds) (2003). Public participation in sustainability science, a handbook. Cambridge University Press, Cambridge. ISBN 978-0-521-52144-4
Kajikawa, Yuya (October 2008). "Research core and framework of sustainability science". Sustainability Science. 3 (2): 215–239. doi:10.1007/s11625-008-0053-1. S2CID 154334789.
Kates, Robert W., ed. (2010). Readings in Sustainability Science and Technology. CID Working Paper No. 213. Center for International Development, Harvard University. Cambridge, MA: Harvard University, December 2010. Abstract and PDF file available on the Harvard Kennedy School website
Jackson, T. (2009), "Prosperity Without Growth: Economics for a Final Planet." London: Earthscan
Brown, Halina Szejnwald (2012). "Sustainability Science Needs to Include Sustainable Consumption". Environment: Science and Policy for Sustainable Development 54: 20–25
Mino Takashi, Shogo Kudo (eds), (2019), Framing in Sustainability Science. Singapore: Springer. ISBN 978-981-13-9061-6. | Wikipedia/Sustainability_science |
In economics, industrial organization is a field that builds on the theory of the firm by examining the structure of (and, therefore, the boundaries between) firms and markets. Industrial organization adds real-world complications to the perfectly competitive model, complications such as transaction costs, limited information, and barriers to entry of new firms that may be associated with imperfect competition. It analyzes determinants of firm and market organization and behavior on a continuum between competition and monopoly, including from government actions.
There are different approaches to the subject. One approach is descriptive in providing an overview of industrial organization, such as measures of competition and the size-concentration of firms in an industry. A second approach uses microeconomic models to explain internal firm organization and market strategy, which includes internal research and development along with issues of internal reorganization and renewal. A third aspect is oriented to public policy related to economic regulation, antitrust law, and, more generally, the economic governance of law in defining property rights, enforcing contracts, and providing organizational infrastructure.
The extensive use of game theory in industrial economics has led to the export of this tool to other branches of microeconomics, such as behavioral economics and corporate finance. Industrial organization has also had significant practical impacts on antitrust law and competition policy.
The development of industrial organization as a separate field owes much to Edward Chamberlin, Joan Robinson, Edward S. Mason, J. M. Clark, Joe S. Bain and Paolo Sylos Labini, among others.
== Subareas ==
The Journal of Economic Literature (JEL) classification codes are one way of representing the range of economics subjects and subareas. There, Industrial Organization, one of 20 primary categories, has 9 secondary categories, each with multiple tertiary categories. The secondary categories are listed below with corresponding available article-preview links of The New Palgrave Dictionary of Economics Online and footnotes to their respective JEL-tertiary categories and associated New-Palgrave links.
JEL: L1 – Market Structure, Firm Strategy, and Market Performance
JEL: L2 – Firm Objectives, Organization, and Behavior
JEL: L3 – Non-profit organizations and Public enterprise
JEL: L4 – Antitrust Issues and Policies
JEL: L5 – Regulation and Industrial policy
JEL: L6 – Industry Studies: Manufacturing
JEL: L7 – Industry Studies: Primary Products and Construction
JEL: L8 – Industry Studies: Services
JEL: L9 – Industry Studies: Transportation and Utilities
== Market structures ==
The common market structures studied in this field are: perfect competition, monopolistic competition, duopoly, oligopoly, oligopsony, monopoly and monopsony.
== Areas of study ==
Industrial organization investigates the outcomes of these market structures in environments with
Price discrimination
Product differentiation
Durable goods
Experience goods
Collusion
Signalling, such as warranties and advertising.
Mergers and acquisitions
Entry and Exit
== History of the field ==
A 2009 book Pioneers of Industrial Organization traces the development of the field from Adam Smith to recent times and includes dozens of short biographies of major figures in Europe and North America who contributed to the growth and development of the discipline.
Other reviews by publication year and earliest available cited works those in 1970/1937, 1972/1933, 1974, 1987/1937-1956 (3 cites), 1968–9 (7 cites), 2009/c. 1900, and 2010/1951.
== See also ==
== Notes ==
== References ==
Tirole, Jean (1988). The Theory of Industrial Organization, MIT press.
Belleflamme, Paul & Martin Peitz, 2010. Industrial Organization: Markets and Strategies. Cambridge University Press. Summary and Resources
Cabral, Luís M. B., 2000. Introduction to Industrial Organization. MIT Press. Links to Description and chapter-preview links.
Shepherd, William, 1985. The Economics of Industrial Organization, Prentice-Hall. ISBN 0-13-231481-9
Shy, Oz, 1995. Industrial Organization: Theory and Applications. Description and chapter-preview links. MIT Press.
Vives, Xavier, 2001. Oligopoly Pricing: Old Ideas and New Tools. MIT Press. Description and scroll to chapter-preview links.
Jeffrey Church & Roger Ware, 2005. "Industrial Organization: A Strategic Approach", (aka IOSA Archived 2016-12-08 at the Wayback Machine)”, Free Textbook
Nicolas Boccard, 2010. "Industrial Organization, a Contract Based approach (aka IOCB)”, Open Source Textbook
== Journals ==
The RAND Journal of Economics
International Journal of the Economics of Business and issue preview links
International Journal of Industrial Organization and issue-preview links
Journal of Industrial Economics, Aims and Scope, and issue-preview links.
Journal of Law, Economics, and Organization and issue-preview links.
Review of Industrial Organization
== External links ==
Quotations related to Industrial organization at Wikiquote | Wikipedia/Industrial_economy |
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