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It takes a leading role when questions of "parity" arise because the volume element as a directed magnitude changes sign under a space-time reflection. Taking the positive square root above thus amounts to choosing a handedness convention on spacetime. |
where is the conjugate transpose of , and noticing that |
we obtain, by taking the Hermitian conjugate of the Dirac equation and multiplying from the right by , the adjoint equation: |
where is understood to act to the left. Multiplying the Dirac equation by from the left, and the adjoint equation by from the right, and adding, produces the law of conservation of the Dirac current: |
Now we see the great advantage of the first-order equation over the one Schrödinger had tried – this is the conserved current density required by relativistic invariance, only now its 4th component is "positive definite" and thus suitable for the role of a probability density: |
Because the probability density now appears as the fourth component of a relativistic vector and not a simple scalar as in the Schrödinger equation, it will be subject to the usual effects of the Lorentz transformations such as time dilation. Thus, for example, atomic processes that are observed as rates, will necessar... |
See Dirac spinor for details of solutions to the Dirac equation. Note that since the Dirac operator acts on 4-tuples of square-integrable functions, its solutions should be members of the same Hilbert space. The fact that the energies of the solutions do not have a lower bound is unexpected – see the hole theory sectio... |
Here and formula_52 represent the components of the electromagnetic four-potential in their standard SI units, and the three sigmas are the Pauli matrices. On squaring out the first term, a residual interaction with the magnetic field is found, along with the usual classical Hamiltonian of a charged particle interactin... |
This Hamiltonian is now a matrix, so the Schrödinger equation based on it must use a two-component wave function. On introducing the external electromagnetic 4-vector potential into the Dirac equation in a similar way, known as minimal coupling, it takes the form: |
A second application of the Dirac operator will now reproduce the Pauli term exactly as before, because the spatial Dirac matrices multiplied by , have the same squaring and commutation properties as the Pauli matrices. What is more, the value of the gyromagnetic ratio of the electron, standing in front of Pauli's new ... |
Assuming the field is weak and the motion of the electron non-relativistic, we have the total energy of the electron approximately equal to its rest energy, and the momentum going over to the classical value, |
and so the second equation may be written |
which is of order – thus at typical energies and velocities, the bottom components of the Dirac spinor in the standard representation are much suppressed in comparison to the top components. Substituting this expression into the first equation gives after some rearrangement |
It should be strongly emphasized that this separation of the Dirac spinor into large and small components depends explicitly on a low-energy approximation. The entire Dirac spinor represents an "irreducible" whole, and the components we have just neglected to arrive at the Pauli theory will bring in new phenomena in th... |
In the limit , the Dirac equation reduces to the Weyl equation, which describes relativistic massless spin- particles. |
Both the Dirac equation and the Adjoint Dirac equation can be obtained from (varying) the action with a specific Lagrangian density that is given by: |
If one varies this with respect to one gets the Adjoint Dirac equation. Meanwhile, if one varies this with respect to one gets the Dirac equation. |
The critical physical question in a quantum theory is—what are the physically observable quantities defined by the theory? According to the postulates of quantum mechanics, such quantities are defined by Hermitian operators that act on the Hilbert space of possible states of a system. The eigenvalues of these operators... |
where, as always, there is an implied summation over the twice-repeated index . This looks promising, because we see by inspection the rest energy of the particle and, in the case of , the energy of a charge placed in an electric potential . What about the term involving the vector potential? In classical electrodynami... |
Thus, the Dirac Hamiltonian is fundamentally distinguished from its classical counterpart, and we must take great care to correctly identify what is observable in this theory. Much of the apparently paradoxical behavior implied by the Dirac equation amounts to a misidentification of these observables. |
The negative solutions to the equation are problematic, for it was assumed that the particle has a positive energy. Mathematically speaking, however, there seems to be no reason for us to reject the negative-energy solutions. Since they exist, we cannot simply ignore them, for once we include the interaction between th... |
To cope with this problem, Dirac introduced the hypothesis, known as hole theory, that the vacuum is the many-body quantum state in which all the negative-energy electron eigenstates are occupied. This description of the vacuum as a "sea" of electrons is called the Dirac sea. Since the Pauli exclusion principle forbids... |
Dirac further reasoned that if the negative-energy eigenstates are incompletely filled, each unoccupied eigenstate – called a hole – would behave like a positively charged particle. The hole possesses a "positive" energy since energy is required to create a particle–hole pair from the vacuum. As noted above, Dirac init... |
It is not entirely satisfactory to describe the "vacuum" using an infinite sea of negative-energy electrons. The infinitely negative contributions from the sea of negative-energy electrons have to be canceled by an infinite positive "bare" energy and the contribution to the charge density and current coming from the se... |
In certain applications of condensed matter physics, however, the underlying concepts of "hole theory" are valid. The sea of conduction electrons in an electrical conductor, called a Fermi sea, contains electrons with energies up to the chemical potential of the system. An unfilled state in the Fermi sea behaves like a... |
In quantum field theories such as quantum electrodynamics, the Dirac field is subject to a process of second quantization, which resolves some of the paradoxical features of the equation. |
The Dirac equation is Lorentz covariant. Articulating this helps illuminate not only the Dirac equation, but also the Majorana spinor and Elko spinor, which although closely related, have subtle and important differences. |
Understanding Lorentz covariance is simplified by keeping in mind the geometric character of the process. Let formula_65 be a single, fixed point in the spacetime manifold. Its location can be expressed in multiple coordinate systems. In the physics literature, these are written as formula_66 and formula_67, with the u... |
One can imagine formula_65 as having a fiber of different coordinate frames above it. In geometric terms, one says that spacetime can be characterized as a fiber bundle, and specifically, the frame bundle. The difference between two points formula_66 and formula_67 in the same fiber is a combination of rotations and Lo... |
The presentation here follows that of Itzykson and Zuber. It is very nearly identical to that of Bjorken and Drell. A similar derivation in a general relativistic setting can be found in Weinberg. Under a Lorentz transformation formula_74 the Dirac spinor to transform as |
It can be shown that an explicit expression for formula_76 is given by |
where formula_78 parameterizes the Lorentz transformation, and formula_79 is the 4x4 matrix |
This matrix can be interpreted as the intrinsic angular momentum of the Dirac field. That it deserves this interpretation arises by contrasting it to the generator formula_81 of Lorentz transformations, having the form |
This can be interpreted as the total angular momentum. It acts on the spinor field as |
Note the formula_66 above does "not" have a prime on it: the above is obtained by transforming formula_85 obtaining the change to formula_86 and then returning to the original coordinate system formula_87. |
The geometrical interpretation of the above is that the frame field is affine, having no preferred origin. The generator formula_81 generates the symmetries of this space: it provides a relabelling of a fixed point formula_89 The generator formula_79 generates a movement from one point in the fiber to another: a moveme... |
Let formula_95 be a Lorentz transformation. The Dirac equation is |
If the Dirac equation is to be covariant, then it should have exactly the same form in all Lorentz frames: |
The two spinors formula_98 and formula_99 should both describe the same physical field, and so should be related by a transformation that does not change any physical observables (charge, current, mass, "etc.") The transformation should encode only the change of coordinate frame. It can be shown that such a transformat... |
Inserting this into the transformed equation, the result is |
The original Dirac equation is then regained if |
An explicit expression for formula_104 (equal to the expression given above) can be obtained by considering an infinitessimal Lorentz transformation |
where formula_106 is the metric tensor and formula_107 is antisymmetric. After plugging and chugging, one obtains |
which is the (infinitessimal) form for formula_76 above. To obtain the affine relabelling, write |
After properly antisymmetrizing, one obtains the generator of symmetries formula_81 given earlier. Thus, both formula_81 and formula_79 can be said to be the "generators of Lorentz transformations", but with a subtle distinction: the first corresponds to a relabelling of points on the affine frame bundle, which forces ... |
The Dirac equation can be formulated in a number of other ways. |
This article has developed the Dirac equation in flat spacetime according to special relativity. It is possible to formulate the Dirac equation in curved spacetime. |
This article developed the Dirac equation using four vectors and Schrödinger operators. The Dirac equation in the algebra of physical space uses a Clifford algebra over the real numbers, a type of geometric algebra. |
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 i... |
Two common examples are the massive Thirring model and the Soler model. |
The Thirring model was originally formulated as a model in (1 + 1) space-time dimensions and is characterized by the Lagrangian density |
where is the spinor field, is the Dirac adjoint spinor, |
(Feynman slash notation is used), is the coupling constant, is the mass, and are the "two"-dimensional gamma matrices, finally is an index. |
The Soler model was originally formulated in (3 + 1) space-time dimensions. It is characterized by the Lagrangian density |
is now the four-gradient operator contracted with the "four"-dimensional Dirac gamma matrices , so therein . |
In Einstein-Cartan theory the Lagrangian density for a Dirac spinor field is given by (formula_5) |
is the Fock-Ivanenko covariant derivative of a spinor with respect to the affine connection, formula_8 is the spin connection, formula_9 is the determinant of the metric tensor formula_10, and the Dirac matrices satisfy |
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 ... |
where formula_13 is the general-relativistic covariant derivative of a spinor, and formula_14 is the Einstein gravitational constant, formula_15. The cubic term in this equation becomes significant at densities on the order of formula_16. |
In mathematical physics, the Dirac equation in curved spacetime generalizes the original Dirac equation to curved space. |
It can be written by using vierbein fields and the gravitational spin connection. The vierbein defines a local rest frame, allowing the constant Gamma matrices to act at each spacetime point. In this way, Dirac's equation takes the following form in curved spacetime: |
Here is the vierbein and is the covariant derivative for fermionic fields, defined as follows |
where is the commutator of Gamma matrices: |
Note that here Latin indices denote the "Lorentzian" vierbein labels while Greek indices denote manifold coordinate indices. |
The optical response of a semiconductor follows if one can determine its macroscopic polarization formula_1 as a function of the electric field formula_2 that excites it. The connection between formula_1 and the microscopic polarization formula_4 is given by |
where the sum involves crystal-momenta formula_6 of all relevant electronic states. In semiconductor optics, one typically excites transitions between a valence and a conduction band. In this connection, formula_7 is the dipole matrix element between the conduction and valence band and formula_8 defines the correspondi... |
The derivation of the SBEs starts from a system Hamiltonian that fully includes the free-particles, Coulomb interaction, dipole interaction between classical light and electronic states, as well as the phonon contributions. Like almost always in many-body physics, it is most convenient to apply the second-quantization ... |
Due to the many-body interactions within formula_9, the dynamics of the observable formula_10 couples to new observables and the equation structure cannot be closed. This is the well-known BBGKY hierarchy problem that can be systematically truncated with different methods such as the cluster-expansion approach. |
At operator level, the microscopic polarization is defined by an expectation value for a single electronic transition between a valence and a conduction band. In second quantization, conduction-band electrons are defined by fermionic creation and annihilation operators formula_14 and formula_15, respectively. An analog... |
that describe transition amplitudes for moving an electron from conduction to valence band (formula_19 term) or vice versa (formula_8 term). At the same time, an electron distribution follows from |
It is also convenient to follow the distribution of electronic vacancies, i.e., the holes, |
that are left to the valence band due to optical excitation processes. |
The quantum dynamics of optical excitations yields an integro-differential equations that constitute the SBEs |
+ \mathrm{i} \hbar \left. \frac{\partial}{\partial t} P_{\mathbf{k}} \right|_{\mathrm{scatter}}\, |
as well as the renormalized carrier energy |
where formula_27 corresponds to the energy of free electron–hole pairs and formula_28 is the Coulomb matrix element, given here in terms of the carrier wave vector formula_29. |
The symbolically denoted formula_30 contributions stem from the hierarchical coupling due to many-body interactions. Conceptually, formula_31, formula_32, and formula_33 are single-particle expectation values while the hierarchical coupling originates from two-particle correlations such as polarization-density correlat... |
All these correlation effects can be systematically included by solving also the dynamics of two-particle correlations. At this level of sophistication, one can use the SBEs to predict optical response of semiconductors without phenomenological parameters, which gives the SBEs a very high degree of predictability. Inde... |
The presented SBEs are formulated in the momentum space since carrier's crystal momentum follows from formula_36. An equivalent set of equations can also be formulated in position space. However, especially, the correlation computations are much simpler to be performed in the momentum space. |
The formula_31 dynamic shows a structure where an individual formula_31 is coupled to "all" other microscopic polarizations due to the Coulomb interaction formula_28. Therefore, the transition amplitude formula_31 is collectively modified by the presence of other transition amplitudes. Only if one sets formula_28 to ze... |
Conceptually, formula_31 is just a transition amplitude for exciting an electron from valence to conduction band. At the same time, the homogeneous part of formula_31 dynamics yields an eigenvalue problem that can be expressed through the generalized Wannier equation. The eigenstates of the Wannier equation is analogou... |
However, a real exciton is a true two-particle correlation because one must then have a correlation between one electron to another hole. Therefore, the appearance of exciton resonances in the polarization does not signify the presence of excitons because formula_31 is a single-particle transition amplitude. The excito... |
Therefore, it is often customary to specify optical resonances as exciton"ic" instead of exciton resonances. The actual role of excitons on optical response can only be deduced by quantitative changes to induce to the linewidth and energy shift of excitonic resonances. |
The solutions of the Wannier equation can produces valuable insight to the basic properties of a semiconductor's optical response. In particular, one can solve the steady-state solutions of the SBEs to predict optical absorption spectrum analytically with the so-called Elliott formula. In this form, one can verify that... |
The SBEs are particularly useful when solving the light propagation through a semiconductor structure. In this case, one needs to solve the SBEs together with the Maxwell's equations driven by the optical polarization. This self-consistent set is called the Maxwell–SBEs and is frequently applied to analyze present-day ... |
At this level, the SBEs provide an extremely versatile method that describes linear as well as nonlinear phenomena such as excitonic effects, propagation effects, semiconductor microcavity effects, four-wave-mixing, polaritons in semiconductor microcavities, gain spectroscopy, and so on. One can also generalize the SBE... |
The Cauchy momentum equation is a vector partial differential equation put forth by Cauchy that describes the non-relativistic momentum transport in any continuum. |
In convective (or Lagrangian) form the Cauchy momentum equation is written as: |
Note that only we use column vectors (in the Cartesian coordinate system) above for clarity, but the equation is written using physical components (which are neither covariants ("column") nor contravariants ("row") ). However, if we chose a non-orthogonal curvilinear coordinate system, then we should calculate and writ... |
After an appropriate change of variables, it can also be written in conservation form: |
where is the momentum density at a given space-time point, is the flux associated to the momentum density, and contains all of the body forces per unit volume. |
Let us start with the generalized momentum conservation principle which can be written as follows: "The change in system momentum is proportional to the resulting force acting on this system". It is expressed by the formula: |
where formula_13 is momentum in time t, formula_14 is force averaged over formula_15. After dividing by formula_15 and passing to the limit formula_17 we get (derivative): |
Let us analyse each side of the equation above. |
We split the forces into body forces formula_19 and surface forces formula_20 |
Surface forces act on walls of the cubic fluid element. For each wall, the X component of these forces was marked in the figure with a cubic element (in the form of a product of stress and surface area e.g. formula_22). |
Adding forces (their X components) acting on each of the cube walls, we get: |
After ordering formula_24 and performing similar reasoning for components formula_25 (they have not been shown in the figure, but these would be vectors parallel to the Y and Z axes, respectively) we get: |
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