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Further applications concern the coherent superposition of "non-optical wave fields". In quantum mechanics for example one considers a probability field, which is related to the wave function formula_23 (interpretation: density of the probability amplitude). Here the applications concern, among others, the future technologies of quantum computing and the already available technology of quantum cryptography. Additionally the problems of the following subchapter are treated.
Coherence is used to check the quality of the transfer functions (FRFs) being measured. Low coherence can be caused by poor signal to noise ratio, and/or inadequate frequency resolution.
According to quantum mechanics, all objects can have wave-like properties (see de Broglie waves). For instance, in Young's double-slit experiment electrons can be used in the place of light waves. Each electron's wave-function goes through both slits, and hence has two separate split-beams that contribute to the intensity pattern on a screen. According to standard wave theory these two contributions give rise to an intensity pattern of bright bands due to constructive interference, interlaced with dark bands due to destructive interference, on a downstream screen. This ability to interfere and diffract is related to coherence (classical or quantum) of the waves produced at both slits. The association of an electron with a wave is unique to quantum theory.
When the incident beam is represented by a quantum pure state, the split beams downstream of the two slits are represented as a superposition of the pure states representing each split beam. The quantum description of imperfectly coherent paths is called a mixed state. A perfectly coherent state has a density matrix (also called the "statistical operator") that is a projection onto the pure coherent state and is equivalent to a wave function, while a mixed state is described by a classical probability distribution for the pure states that make up the mixture.
Macroscopic scale quantum coherence leads to novel phenomena, the so-called macroscopic quantum phenomena. For instance, the laser, superconductivity and superfluidity are examples of highly coherent quantum systems whose effects are evident at the macroscopic scale. The macroscopic quantum coherence (off-diagonal long-range order, ODLRO) for superfluidity, and laser light, is related to first-order (1-body) coherence/ODLRO, while superconductivity is related to second-order coherence/ODLRO. (For fermions, such as electrons, only even orders of coherence/ODLRO are possible.) For bosons, a Bose–Einstein condensate is an example of a system exhibiting macroscopic quantum coherence through a multiple occupied single-particle state.
The classical electromagnetic field exhibits macroscopic quantum coherence. The most obvious example is the carrier signal for radio and TV. They satisfy Glauber's quantum description of coherence.
Recently M. B. Plenio and co-workers constructed an operational formulation of quantum coherence as a resource theory. They introduced coherence monotones analogous to the entanglement monotones. Quantum coherence has been shown to be equivalent to quantum entanglement in the sense that coherence can be faithfully described as entanglement, and conversely that each entanglement measure corresponds to a coherence measure.
In analytic geometry, spatial transformations in the 3-dimensional Euclidean space formula_1 are distinguished into active or alibi transformations, and passive or alias transformations. An active transformation is a transformation which actually changes the physical position (alibi, elsewhere) of a point, or rigid body, which can be defined in the absence of a coordinate system; whereas a passive transformation is merely a change in the coordinate system in which the object is described (alias, other name) (change of coordinate map, or change of basis). By "transformation", mathematicians usually refer to active transformations, while physicists and engineers could mean either. Both types of transformation can be represented by a combination of a translation and a linear transformation.
Put differently, a "passive" transformation refers to description of the "same" object in two different coordinate systems.
On the other hand, an "active transformation" is a transformation of one or more objects with respect to the same coordinate system. For instance, active transformations are useful to describe successive positions of a rigid body. On the other hand, passive transformations may be useful in human motion analysis to observe the motion of the tibia relative to the femur, that is, its motion relative to a ("local") coordinate system which moves together with the femur, rather than a ("global") coordinate system which is fixed to the floor.
As an example, let the vector formula_2, be a vector in the plane. A rotation of the vector through an angle θ in counterclockwise direction is given by the rotation matrix:
which can be viewed either as an "active transformation" or a "passive transformation" (where the above matrix will be inverted), as described below.
Spatial transformations in the Euclidean space formula_1.
In general a spatial transformation formula_5 may consist of a translation and a linear transformation. In the following, the translation will be omitted, and the linear transformation will be represented by a 3×3-matrix formula_6.
As an active transformation, formula_6 transforms the initial vector formula_8 into a new vector formula_9.
If one views formula_10 as a new basis, then the coordinates of the new vector formula_11 in the new basis are the same as those of formula_12 in the original basis. Note that active transformations make sense even as a linear transformation into a different vector space. It makes sense to write the new vector in the unprimed basis (as above) only when the transformation is from the space into itself.
On the other hand, when one views formula_6 as a passive transformation, the initial vector formula_8 is left unchanged, while the coordinate system and its basis vectors are transformed in the opposite direction, that is, with the inverse transformation formula_15.
This gives a new coordinate system XYZ with basis vectors:
The new coordinates formula_17 of formula_18 with respect to the new coordinate system XYZ are given by:
From this equation one sees that the new coordinates are given by
As a passive transformation formula_6 transforms the old coordinates into the new ones.
Note the equivalence between the two kinds of transformations: the coordinates of the new point in the active transformation and the new coordinates of the point in the passive transformation are the same, namely
A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take up space. A point particle is an appropriate representation of any object whenever its size, shape, and structure are irrelevant in a given context. For example, from far enough away, any finite-size object will look and behave as a point-like object. A point particle can also be referred in the case of a moving body in terms of physics.
In the theory of gravity, physicists often discuss a ', meaning a point particle with a nonzero mass and no other properties or structure. Likewise, in electromagnetism, physicists discuss a ', a point particle with a nonzero charge.
Sometimes, due to specific combinations of properties, extended objects behave as point-like even in their immediate vicinity. For example, spherical objects interacting in 3-dimensional space whose interactions are described by the inverse square law behave in such a way as if all their matter were concentrated in their centers of mass. In Newtonian gravitation and classical electromagnetism, for example, the respective fields outside a spherical object are identical to those of a point particle of equal charge/mass located at the center of the sphere.
In quantum mechanics, the concept of a point particle is complicated by the Heisenberg uncertainty principle, because even an elementary particle, with no internal structure, occupies a nonzero volume. For example, the atomic orbit of an electron in the hydrogen atom occupies a volume of ~10−30 m3. There is nevertheless a distinction between elementary particles such as electrons or quarks, which have no known internal structure, versus composite particles such as protons, which do have internal structure: A proton is made of three quarks.
Elementary particles are sometimes called "point particles", but this is in a different sense than discussed above.
When a point particle has an additive property, such as mass or charge, concentrated at a single point in space, this can be represented by a Dirac delta function.
Point mass (pointlike mass) is the concept, for example in classical physics, of a physical object (typically matter) that has nonzero mass, and yet explicitly and specifically is (or is being thought of or modeled as) infinitesimal (infinitely small) in its volume or linear dimensions.
A common use for point mass lies in the analysis of the gravitational fields. When analyzing the gravitational forces in a system, it becomes impossible to account for every unit of mass individually. However, a spherically symmetric body affects external objects gravitationally as if all of its mass were concentrated at its center.
A point mass in probability and statistics does not refer to mass in the sense of physics, but rather refers to a finite nonzero probability that is concentrated at a point in the probability mass distribution, where there is a discontinuous segment in a probability density function. To calculate such a point mass, an integration is carried out over the entire range of the random variable, on the probability density of the continuous part. After equating this integral to 1, the point mass can be found by further calculation.
A point charge is an idealized model of a particle which has an electric charge. A point charge is an electric charge at a mathematical point with no dimensions.
The fundamental equation of electrostatics is Coulomb's law, which describes the electric force between two point charges. The electric field associated with a classical point charge increases to infinity as the distance from the point charge decreases towards zero making energy (thus mass) of point charge infinite.
Earnshaw's theorem states that a collection of point charges cannot be maintained in an equilibrium configuration solely by the electrostatic interaction of the charges.
In quantum mechanics, there is a distinction between an elementary particle (also called "point particle") and a composite particle. An elementary particle, such as an electron, quark, or photon, is a particle with no internal structure. Whereas a composite particle, such as a proton or neutron, has an internal structure (see figure).
However, neither elementary nor composite particles are spatially localized, because of the Heisenberg uncertainty principle. The particle wavepacket always occupies a nonzero volume. For example, see atomic orbital: The electron is an elementary particle, but its quantum states form three-dimensional patterns.
Nevertheless, there is good reason that an elementary particle is often called a point particle. Even if an elementary particle has a delocalized wavepacket, the wavepacket can be represented as a quantum superposition of quantum states wherein the particle is exactly localized. Moreover, the "interactions" of the particle can be represented as a superposition of interactions of individual states which are localized. This is not true for a composite particle, which can never be represented as a superposition of exactly-localized quantum states. It is in this sense that physicists can discuss the intrinsic "size" of a particle: The size of its internal structure, not the size of its wavepacket. The "size" of an elementary particle, in this sense, is exactly zero.
For example, for the electron, experimental evidence shows that the size of an electron is less than 10−18 m. This is consistent with the expected value of exactly zero. (This should not be confused with the classical electron radius, which, despite the name, is unrelated to the actual size of an electron.)
In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation. A Lorentz scalar may be generated from e.g., the scalar product of vectors, or from contracting tensors of the theory. While the components of vectors and tensors are in general altered under Lorentz transformations, Lorentz scalars remain unchanged.
A Lorentz scalar is not always immediately seen to be an invariant scalar in the mathematical sense, but the resulting scalar value is invariant under any basis transformation applied to the vector space, on which the considered theory is based. A simple Lorentz scalar in Minkowski spacetime is the "spacetime distance" ("length" of their difference) of two fixed events in spacetime. While the "position"-4-vectors of the events change between different inertial frames, their spacetime distance remains invariant under the corresponding Lorentz transformation. Other examples of Lorentz scalars are the "length" of 4-velocities (see below), or the Ricci curvature in a point in spacetime from General relativity, which is a contraction of the Riemann curvature tensor there.
In special relativity the location of a particle in 4-dimensional spacetime is given by
where formula_2 is the position in 3-dimensional space of the particle, formula_3 is the velocity in 3-dimensional space and formula_4 is the speed of light.
The "length" of the vector is a Lorentz scalar and is given by
where formula_6 is the proper time as measured by a clock in the rest frame of the particle and the Minkowski metric is given by
Often the alternate signature of the Minkowski metric is used in which the signs of the ones are reversed.
In the Minkowski metric the space-like interval formula_9 is defined as
We use the space-like Minkowski metric in the rest of this article.
The velocity in spacetime is defined as
The magnitude of the 4-velocity is a Lorentz scalar,
The inner product of acceleration and velocity.
The 4-acceleration is always perpendicular to the 4-velocity
Therefore, we can regard acceleration in spacetime as simply a rotation of the 4-velocity. The inner product of the acceleration and the velocity is a Lorentz scalar and is zero. This rotation is simply an expression of energy conservation:
where formula_17 is the energy of a particle and formula_18 is the 3-force on the particle.
Energy, rest mass, 3-momentum, and 3-speed from 4-momentum.
where formula_20 is the particle rest mass, formula_21 is the momentum in 3-space, and
Measurement of the energy of a particle.
Consider a second particle with 4-velocity formula_23 and a 3-velocity formula_24. In the rest frame of the second particle the inner product of formula_23 with formula_26 is proportional to the energy of the first particle
where the subscript 1 indicates the first particle.
Since the relationship is true in the rest frame of the second particle, it is true in any reference frame. formula_28, the energy of the first particle in the frame of the second particle, is a Lorentz scalar. Therefore,
in any inertial reference frame, where formula_30 is still the energy of the first particle in the frame of the second particle .
Measurement of the rest mass of the particle.
In the rest frame of the particle the inner product of the momentum is
Therefore, the rest mass (m) is a Lorentz scalar. The relationship remains true independent of the frame in which the inner product is calculated. In many cases the rest mass is written as formula_32 to avoid confusion with the relativistic mass, which is formula_33
Measurement of the 3-momentum of the particle.
The square of the magnitude of the 3-momentum of the particle as measured in the frame of the second particle is a Lorentz scalar.
Measurement of the 3-speed of the particle.
The 3-speed, in the frame of the second particle, can be constructed from two Lorentz scalars
Scalars may also be constructed from the tensors and vectors, from the contraction of tensors (such as formula_36), or combinations of contractions of tensors and vectors (such as formula_37).
In electromagnetism, the Lorenz gauge condition or Lorenz gauge, for Ludvig Lorenz, is a partial gauge fixing of the electromagnetic vector potential by requiring formula_1 The name is frequently confused with Hendrik Lorentz, who has given his name to many concepts in this field. The condition is Lorentz invariant. The condition does not completely determine the gauge: one can still make a gauge transformation formula_2 where formula_3 is a harmonic scalar function (that is, a scalar function satisfying formula_4 the equation of a massless scalar field). The Lorenz condition is used to eliminate the redundant spin-0 component in the representation theory of the Lorentz group. It is equally used for massive spin-1 fields where the concept of gauge transformations does not apply at all.
In electromagnetism, the Lorenz condition is generally used in calculations of time-dependent electromagnetic fields through retarded potentials. The condition is
where formula_6 is the four-potential, the comma denotes a partial differentiation and the repeated index indicates that the Einstein summation convention is being used. The condition has the advantage of being Lorentz invariant. It still leaves substantial gauge degrees of freedom.
In ordinary vector notation and SI units, the condition is
where formula_8 is the magnetic vector potential and formula_9 is the electric potential; see also gauge fixing.
A quick justification of the Lorenz gauge can be found using Maxwell's equations and the relation between the magnetic vector potential and the magnetic field:
Since the curl is zero, that means there is a scalar function formula_13 such that
This gives the well known equation for the electric field,
This result can be plugged into the Ampère–Maxwell equation,
To have Lorentz invariance, the time derivatives and spatial derivatives must be treated equally (i.e. of the same order). Therefore, it is convenient to choose the Lorenz gauge condition, which gives the result
A similar procedure with a focus on the electric scalar potential and making the same gauge choice will yield
These are simpler and more symmetric forms of the inhomogeneous Maxwell's equations. Note that the Coulomb gauge also fixes the problem of Lorentz invariance, but leaves a coupling term with first-order derivatives.
is the vacuum velocity of light, and formula_21 is the d'Alembertian operator. These equations are not only valid under vacuum conditions, but also in polarized media, if formula_22 and formula_23 are source density and circulation density, respectively, of the electromagnetic induction fields formula_24 and formula_25 calculated as usual from formula_13 and formula_27 by the equations
The explicit solutions for formula_13 and formula_8 – unique, if all quantities vanish sufficiently fast at infinity – are known as retarded potentials.
Time translation symmetry or temporal translation symmetry (TTS) is a mathematical transformation in physics that moves the times of events through a common interval. Time translation symmetry is the hypothesis that the laws of physics are unchanged, (i.e. invariant) under such a transformation. Time translation symmetry is a rigorous way to formulate the idea that the laws of physics are the same throughout history. Time translation symmetry is closely connected, via the Noether theorem, to conservation of energy. In mathematics, the set of all time translations on a given system form a Lie group.
There are many symmetries in nature besides time translation, such as spatial translation or rotational symmetries. These symmetries can be broken and explain diverse phenomena such as crystals, superconductivity, and the Higgs mechanism. However, it was thought until very recently that time translation symmetry could not be broken. Time crystals, a state of matter first observed in 2017, break time translation symmetry.
Symmetries are of prime importance in physics and are closely related to the hypothesis that certain physical quantities are only relative and unobservable. Symmetries apply to the equations that govern the physical laws (e.g. to a Hamiltonian or Lagrangian) rather than the initial conditions, values or magnitudes of the equations themselves and state that the laws remain unchanged under a transformation. If a symmetry is preserved under a transformation it is said to be "invariant". Symmetries in nature lead directly to conservation laws, something which is precisely formulated by the Noether theorem.
To formally describe time translation symmetry we say the equations, or laws, that describe a system at times formula_1 and formula_2 are the same for any value of formula_1 and formula_4.
One finds for its solutions formula_6 the combination:
The invariance of a Hamiltonian formula_10 of an isolated system under time translation implies its energy does not change with the passage of time. Conservation of energy implies, according to the Heisenberg equations of motion, that formula_11.
Where formula_14 is the time translation operator which implies invariance of the Hamiltonian under the time translation operation and leads to the conservation of energy.
In many nonlinear field theories like general relativity or Yang–Mills theories, the basic field equations are highly nonlinear and exact solutions are only known for ‘sufficiently symmetric’ distributions of matter (e.g. rotationally or axially symmetric configurations). Time translation symmetry is guaranteed only in spacetimes where the metric is static: that is, where there is a coordinate system in which the metric coefficients contain no time variable. Many general relativity systems are not static in any frame of reference so no conserved energy can be defined.
Time crystals, a state of matter first observed in 2017, break discrete time translation symmetry.
In physics and mathematics, an ansatz (; , meaning: "initial placement of a tool at a work piece", plural Ansätze ; ) is an educated guess or an additional assumption made to help solve a problem, and which may later be verified to be part of the solution by its results.
An ansatz is the establishment of the starting equation(s), the theorem(s), or the value(s) describing a mathematical or physical problem or solution. It typically provides an initial estimate or framework to the solution of a mathematical problem, and can also take into consideration the boundary conditions (in fact, an ansatz is sometimes thought of as a "trial answer" and an important technique in solving differential equations).
After an ansatz, which constitutes nothing more than an assumption, has been established, the equations are solved more precisely for the general function of interest, which then constitutes a confirmation of the assumption. In essence, an ansatz makes assumptions about the form of the solution to a problem so as to make the solution easier to find.
It has been demonstrated that machine learning techniques can be applied to provide initial estimates similar to those invented by humans and to discover new ones in case no ansatz is available.
Given a set of experimental data that looks to be clustered about a line, a linear ansatz could be made to find the parameters of the line by a least squares curve fit. Variational approximation methods use ansätze and then fit the parameters.
Another example could be the mass, energy, and entropy balance equations that, considered simultaneous for purposes of the elementary operations of linear algebra, are the "ansatz" to most basic problems of thermodynamics.
Another example of an ansatz is to suppose the solution of a homogeneous linear differential equation to take an exponential form, or a power form in the case of a difference equation. More generally, one can guess a particular solution of a system of equations, and test such an ansatz by directly substituting the solution into the system of equations. In many cases, the assumed form of the solution is general enough that it can represent arbitrary functions, in such a way that the set of solutions found this way is a full set of all the solutions.
In mathematics, a tensor is an algebraic object that describes a (multilinear) relationship between sets of algebraic objects related to a vector space. Objects that tensors may map between include vectors and scalars, and even other tensors. Tensors can take several different forms – for example: scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product. Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system.
Tensors are important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics (stress, elasticity, fluid mechanics, moment of inertia, ...), electrodynamics (electromagnetic tensor, Maxwell tensor, permittivity, magnetic susceptibility, ...), or general relativity (stress–energy tensor, curvature tensor, ...) and others. In applications, it is common to study situations in which a different tensor can occur at each point of an object; for example the stress within an object may vary from one location to another. This leads to the concept of a tensor field. In some areas, tensor fields are so ubiquitous that they are often simply called "tensors".