Dirac equation

Quantum field theory

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Dirac equation Klein–Gordon equation Proca equations Wheeler-deWitt equation

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In physics, the Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928 and provides a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity. In quantum field theory (QFT the forces between particles are mediated by other particles Motivation and history When calculating Scattering cross sections in Particle physics, the interaction between particles can be described The history of quantum field theory starts with its creation by Dirac when he attempted to quantize the Electromagnetic field in the late 1920s The Klein–Gordon equation ( Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic version of the Schrödinger In Physics, in the area of field theory, the Proca action describes a Massive spin -1 field of mass m in Minkowski spacetime Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. This page is about the scientific concept of relativity for philosophical or sociological theories about relativity see Relativism. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons Year 1928 ( MCMXXVIII) was a Leap year starting on Sunday (link will display full calendar of the Gregorian calendar. In Particle physics, an elementary particle or fundamental particle is a particle not known to have substructure that is it is not known to be made In Quantum mechanics, spin is an intrinsic property of all elementary particles. The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial The equation demands the existence of antiparticles and actually predated their experimental discovery, making the discovery of the positron, the antiparticle of the electron, one of the greatest triumphs of modern theoretical physics. to most kinds of particles, there is an associated antiparticle with the same Mass and opposite Electric charge. The positrons or antielectron is the Antiparticle or the Antimatter counterpart of the Electron.

Mathematical formulation

The Dirac equation in the form originally proposed by Dirac is:

$\left(\beta mc^2 + \sum_{k = 1}^3 \alpha_k p_k \, c\right) \psi (\mathbf{x},t) = i \hbar \frac{\partial\psi}{\partial t}(\mathbf{x},t)$

where

m is the rest mass of the electron,

c is the speed of light,

p is the momentum operator,

$\hbar$ is the reduced Planck's constant,

x and t are the space and time coordinates. Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product The Planck constant (denoted h\ is a Physical constant used to describe the sizes of quanta. Space is the extent within which Matter is physically extended and objects and Events have positions relative to one another For other uses see Time (disambiguation Time is a component of a measuring system used to sequence events to compare the durations of

The new elements in this equation are the 4x4 matrices $\alpha_k\,$ and $\,\beta$ , and the four-component wavefunction $\,\psi$ . A wave function or wavefunction is a mathematical tool used in Quantum mechanics to describe any physical system The matrices are all Hermitian and have squares equal to the identity matrix:

$\alpha_i^2=I_4$

$\beta^2=I_4 \,$

and they all mutually anticommute:

$\alpha_i\alpha_j = -\alpha_j\alpha_i, \,$

$\alpha_i\beta = -\beta\alpha_i, \,$

where i and j are distinct and range from 1 to 3. A Hermitian matrix (or self-adjoint matrix) is a Square matrix with complex entries which is equal to its own Conjugate transpose &mdash that These matrices, and the form of the wavefunction, have a deep mathematical significance. The algebraic structure represented by the Dirac matrices had been created some 50 years earlier by the English mathematician W. K. Clifford, which in turn had been based on the mid-19th century work of the German mathematician Hermann Grassmann in his "Lineare Ausdehnungslehre" (Theory of Linear Extensions). W K Clifford may refer to William Kingdon Clifford, British mathematician and philosopher Lucy Clifford Mrs W Hermann Günther Grassmann ( April 15, 1809, Stettin ( Szczecin) &ndash September 26, 1877, Stettin) was a The latter had been regarded as well-nigh incomprehensible by most of his contemporaries. The appearance of something so seemingly abstract, at such a late date, in such a direct physical manner, amounts to one of the most remarkable chapters in the history of physics.

Comparison with the Schrödinger equation

The Dirac equation is superficially similar to the Schrödinger equation for a free mass:

$-\frac{\hbar^2}{2m}\nabla^2\phi = i\hbar\frac{\partial}{\partial t}\phi$

The left side represents the square of the momentum operator divided by twice the mass, which, classically speaking, is the kinetic energy. In Physics, especially Quantum mechanics, the Schrödinger equation is an equation that describes how the Quantum state of a Physical system If one wants to get a relativistic generalization of this equation, then the space and time derivatives must enter symmetrically, as they do in the Maxwell theory of the electromagnetic field, which is known to be relativistically invariant - that is, the derivatives must be of the same order in space and time. Now, in relativity, the momentum and the energy are each part of an invariant object, the 4-momentum, and they are connected by the relativistically invariant relation

$\frac{E^2}{c^2} - p^2 = m^2c^2$

with m now representing the rest mass. If we replace E and p by their operator equivalents in the Schrödinger theory, we get a differential equation that is a valid relativistic generalization of the Schrödinger equation:

$\left(\nabla^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2}\right)\phi = \frac{m^2c^2}{\hbar^2}\phi$

where it is assumed that the wave function is now a relativistic scalar. In fact Schrödinger, who was well acquainted with relativity, tried this equation before the one that bears his name, but found it unsuitable. Because the time derivative is second order, one must specify both the initial value of $\partial_t \phi\,$ as well as $\phi\,$ itself when solving the equation. This is typical in the solution of problems of wave propagation, as in electrodynamics. However, in quantum theory, one is interested not in the actual motion as such, rather, the energy spectrum - mathematically, what is needed is a well-defined eigenvalue problem. As in electrodynamics, there will be advanced waves that appear to be propagating backward in time toward the source - these can be safely discarded as unphysical in electrodynamics, but not here, because one needs all the solutions in order to be able to express any solution as an expansion in terms of energy eigenfunctions and the corresponding eigenvalues.

There was an even more serious objection to be raised - in the Schrödinger theory, the probability density is given by the positive definite expression

$\rho=\phi^*\phi\,$

and its current by

$J = -\frac{i\hbar}{2m}(\phi^*\nabla\phi - \phi\nabla\phi^*)$

with the conservation of probability density expressed as

$\nabla\cdot J + \frac{\partial\rho}{\partial t} = 0$

In a relativistic theory, the form of the probability density must match that of the current when we replace $\nabla$ by $\,\partial_t$ , and in order that the conservation of probability current be a relativistically invariant expression, must form the 0-component of a 4-vector - thus we must have

$\rho = \frac{i\hbar}{2m}(\phi^*\partial_t\phi - \phi\partial_t\phi^*)$

Everything is now perfectly relativistic, but the probability density is not positive definite, because one may freely choose the initial values of both $\phi\,$ and $\,\partial_t\phi$ . Such a theory would not have a simple, immediate physical interpretation, and so Schrodinger abandoned it. (Though it was short-lived as a single-particle equation, it is resurrected in quantum field theory, where it is known as the Klein–Gordon equation, and describes particles of spin-0. The Klein–Gordon equation ( Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic version of the Schrödinger )

Dirac's coup

What is needed, then, is an equation that is first-order in both space and time. One could formally take the relativistic expression for the energy $E = c\sqrt{p^2 + m^2c^2}$ , replace p by its operator equivalent, expand the square root in an infinite series of derivative operators, set up an eigenvalue problem, then solve the equation formally by iterations. Most physicists had little faith in such a process, even if it were technically possible.

As the story goes, Dirac was staring into the fireplace at Cambridge, pondering this problem, when he hit upon the idea of taking the square root of the wave operator thus:

$\nabla^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2} = (A \partial_x + B \partial_y + C \partial_z + \frac{i}{c}D \partial_t)(A \partial_x + B \partial_y + C \partial_z + \frac{i}{c}D \partial_t)$

On multiplying out the right side, we see that in order to get all the cross-terms such as $\partial_x\partial_y\,$ to vanish, we must assume

$AB + BA = 0, \;\ldots$

with

$A^2 = B^2 = \ldots = 1\,$

Dirac, who had just then been intensely involved with working out the foundations of Heisenberg's matrix mechanics, immediately understood that these conditions could be met if A, B. Matrix mechanics is a formulation of Quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925 . . are matrices, with the implication that the wave function has multiple components. This immediately explained the appearance of two-component wave functions in Pauli's phenomenological theory of spin, something that up until then had been regarded as mysterious, even to Pauli himself. In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin However, one needs at least 4x4 matrices to set up a system with the properties desired - so the wave function had four components, not two, as in the Pauli theory.

Given the factorization in terms of these matrices, one can now write down immediately an equation

$(A\partial_x + B\partial_y + C\partial_z + \frac{i}{c}D\partial_t)\psi = \kappa\psi$

with $\kappa\,$ to be determined. Applying again the matrix operator on either side yields

$(\nabla^2 - \frac{1}{c^2}\partial_t^2)\psi = \kappa^2\psi$

On taking $\kappa = {mc}/{\hbar}$ we find that all the components of the wave function individually satisfy the relativistic energy-momentum relation. Thus the sought-for equation that is first-order in both space and time is

$(A\partial_x + B\partial_y + C\partial_z + \frac{i}{c}D\partial_t - \frac{mc}{\hbar})\psi = 0$

With $(A,B,C) = i\beta \alpha_k\,$ and $\,D = \beta$ , we get the Dirac equation.

Comparison with the Pauli theory

The necessity of introducing half-integral spin goes back experimentally to the results of the Stern-Gerlach experiment. In Quantum mechanics, the Stern–Gerlach experiment, named after Otto Stern and Walther Gerlach, is an important 1922 experiment on the Deflection A beam of atoms is run through a strong inhomogeneous magnetic field, which then splits into N parts depending on the intrinsic angular momentum of the atoms. It was found that for silver atoms, the beam was split in two - the ground state therefore could not be integral, because even if the intrinsic angular momentum of the atoms were as small as possible, 1, the beam would be split into 3 parts, corresponding to atoms with $L z$ = -1, 0, and +1. The conclusion is that silver atoms have net intrinsic angular momentum of 1/2. Pauli set up a theory which explained this splitting by introducing a two-component wave function and a corresponding correction term in the Hamiltonian, representing a semi-classical coupling of this wave function to an applied magnetic field, as so:

$H = \frac{1}{2m}(\sigma\cdot(p - \frac{e}{c}A))^2 + e A^0$

Here $A μ$ is the applied electromagnetic field, and the three sigmas are Pauli matrices. In Physics, Hamilton's principle is William Rowan Hamilton 's formulation of the Principle of stationary action (see that article for historical formulations Hamiltonian mechanics is a re-formulation of Classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. The Pauli matrices are a set of 2 × 2 complex Hermitian and unitary matrices. $e$ is the charge of the particle, e. g. $e = - e 0$ for the electron. On squaring out the first term, a residual interaction with the magnetic field is found, along with the usual Hamiltonian of a charged particle interacting with an applied field:

$H = \frac{1}{2m}(p - \frac{e}{c}A)^2 + eA^0 - \frac{e\hbar}{2mc}\sigma\cdot B$

This Hamiltonian is now a 2x2 matrix, so the Schrödinger equation based on it,

$H \phi = i\hbar \frac{\partial\phi}{\partial t}$

must use a two-component wave function. Pauli had introduced the sigma matrices

$\sigma_k = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix},\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix},\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$

as pure phenomenology - Dirac now had a theoretical argument that implied that spin was somehow the consequence of the marriage of quantum theory to relativity. The term phenomenology in Science is used to describe a body of knowledge which relates several different empirical observations of phenomena to each other

The Pauli matrices share the same properties as the Dirac matrices—they are all Hermitian, square to 1, and anticommute. This allows one to immediately find a representation of the Dirac matrices in terms of the Pauli matrices:

$\alpha_k = \begin{pmatrix} 0 & \sigma_k \\ \sigma_k & 0 \end{pmatrix}$

$\beta = \begin{pmatrix} 1_2 & 0 \\ 0 & -1_2 \end{pmatrix}$

The Dirac equation now may be written as an equation coupling two-component spinors:

$\begin{pmatrix} mc^2 & c\sigma\cdot p \\ c\sigma\cdot p & -mc^2 \end{pmatrix} \begin{pmatrix} \phi_+ \\ \phi_- \end{pmatrix} = i\hbar\frac{\partial}{\partial t}\begin{pmatrix} \phi_+ \\ \phi_- \end{pmatrix}$

Notice that on the diagonal we find the rest energy of the particle. If we set the momentum to zero - that is, bring the particle to rest - then we have

$i\hbar\frac{\partial}{\partial t}\begin{pmatrix} \phi_+ \\ \phi_- \end{pmatrix} = \begin{pmatrix} mc^2 & 0 \\ 0 & -mc^2 \end{pmatrix} \begin{pmatrix} \phi_+ \\ \phi_- \end{pmatrix}$

The equations for the individual two-spinors are now decoupled, and we see that the "top" and "bottom" two-spinors are individually eigenfunctions of the energy with eigenvalues equal to plus and minus the rest energy, respectively. The appearance of this negative energy eigenvalue is completely consistent with relativity.

It should be strongly emphasized that this separation in the rest frame is not an invariant statement - the "bottom" two-spinor does not represent antimatter as such in general. The entire four-component spinor represents an irreducible whole - in general, states will have an admixture of positive and negative energy components. If we couple the Dirac equation to an electromagnetic field, as in the Pauli theory, then the positive and negative energy parts will be mixed together, even if they are originally decoupled. Dirac's main problem was to find a consistent interpretation of this mixing. As we shall see below, it brings a new phenomenon into physics - matter/antimatter creation and annihilation.

Covariant form and relativistic invariance

The explicitly covariant form of the Dirac equation is (employing the Einstein summation convention):

$i \hbar \gamma^\mu \partial_\mu \psi - m c \psi = 0 \,$

In the above, $\gamma_\mu\,$ are the Dirac matrices. In Mathematics, especially in applications of Linear algebra to Physics, the Einstein notation or Einstein summation convention is a notational In Mathematical physics, the gamma matrices, {γ0 γ1 γ2 γ3} also known as the Dirac matrices, form a matrix-valued $\gamma^0\,$ is Hermitian, and the $\gamma^k\,$ are anti-Hermitian, with the definition

$\gamma^0 = \beta \,$

$\gamma^k = \gamma^0 \alpha_k \,$

This may be summarized using the Minkowski metric on spacetime in the form

$\{\gamma^\mu,\gamma^\nu\} = 2 g^{\mu\nu} \,$

where the bracket expression $\{a, b\}\,$ means $\,ab + ba$ , the anticommutator. These are the defining relations of a Clifford algebra over a pseudo-orthogonal 4-d space with metric signature $( + - - - )$ . In Mathematics, Clifford algebras are a type of Associative algebra. Note that one may also employ the metric form $( - + + + )$ by multiplying all the gammas by a factor of $i$ . At an elementary level, the choice may be regarded as conventional, but there are specific reasons for preferring the former, both mathematically and for convenience in calculation and physical interpretation. In the literature, one almost always finds the convention $( + - - - )$ in use. The specific Clifford algebra employed in the Dirac equation is known as the Dirac algebra. In Mathematical physics, the Dirac algebra is the Clifford algebra C �( C) which is generated by Matrix multiplication

The Dirac equation may be interpreted as an eigenvalue expression, where the rest mass is proportional to an eigenvalue of the 4-momentum operator, the proportion being the speed of light in vacuo:

$P_{op}\psi = mc\psi \,$

In practice, physicists often use units of measure such that $\hbar$ and c are equal to 1, known as "natural" units. In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes The equation is then multiplied through by $- i$ and takes the simple form

$(\gamma^\mu\partial_\mu + im) \psi = 0\,$

or, if Feynman slash notation is employed,

$(\partial\!\!\!/ + im)\psi = 0$

A fundamental theorem states that if two distinct sets of matrices are given that both satisfy the Clifford relations, then they are connected to each other by a similarity transformation:

$\gamma^{\mu\prime} = S^{-1} \gamma^\mu S$

If in addition the matrices are all unitary, as are the Dirac set, then S itself is unitary;

$\gamma^{\mu\prime} = U^\dagger \gamma^\mu U$

The transformation U is unique up to a multiplicative factor of absolute value 1. In the study of Dirac fields in Quantum field theory, Richard Feynman invented the convenient Feynman slash notation (less commonly known as the '''Dirac''' Informally a unitary transformation is a transformation that respects the Dot product: the dot product of two vectors before the transformation is equal to their Let us now imagine a Lorentz transformation to have been performed on the derivative operators, which form a covariant vector. In order that the operator $\gamma^\mu\,\partial_\mu$ remain invariant, the gammas must transform among themselves as a contravariant vector with respect to their spacetime index. These new gammas will themselves satisfy the Clifford relations, because of the orthogonality of the Lorentz transformation. By the fundamental theorem, we may replace the new set by the old set subject to a unitary transformation. In the new frame, remembering that the rest mass is a relativistic scalar, the Dirac equation will then take the form

$( U^\dagger \gamma^\mu U\partial_\mu^\prime + im)\psi(x^\prime,t^\prime) = 0$

$U^\dagger(\gamma^\mu\partial_\mu^\prime + im)U \psi(x^\prime,t^\prime) = 0$

If we now define the transformed spinor

$\psi^\prime = U\psi$

then we have the transformed Dirac equation

$(\gamma^\mu\partial_\mu^\prime + im)\psi^\prime(x^\prime,t^\prime) = 0$

Thus, once we settle on a unitary representation of the gammas, it is final providing we transform the spinor according the unitary transformation that corresponds to the given Lorentz transformation.

These considerations reveal the origin of the gammas in geometry, hearkening back to Grassmann's original motivation - they represent a fixed basis of unit vectors in spacetime. Similarly, products of the gammas such as $\gamma_\mu\,\gamma_\nu$ represent oriented surface elements, and so on. With this in mind, we can find the form the unit volume element on spacetime in terms of the gammas as follows. By definition, it is

$V = \frac{1}{4!}\epsilon_{\mu\nu\alpha\beta}\gamma^\mu\gamma^\nu\gamma^\alpha\gamma^\beta$

In order that this be an invariant, the epsilon symbol must be a tensor, and so must contain a factor of $\sqrt{g}$ , where g is the determinant of the metric tensor. Since this is negative, that factor is imaginary. Thus

V = i γ 0 γ 1 γ 2 γ 3

This matrix is given the special symbol $\,\gamma_5$ , owing to its importance when one is considering improper transformations of spacetime, that is, those that change the orientation of the basis vectors. In the representation we are using for the gammas, it is

$\gamma_5 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$

Also note that could as easily have taken the negative square root of the determinant of g - the choice amounts to an initial handedness convention.

Lorentz Invariance of the Dirac equation

The Lorentz invariance of the Dirac equation follows from its covariant nature. In standard Physics, Lorentz covariance is a key property of Spacetime that follows from the Special theory of relativity, where it applies globally

Comparison with the Klein-Gordon equation

In Feynman slash notation the Klein-Gordon equation:

$(\partial^2 + m^2)\psi = 0\,$

can be factorised as:

$(i\partial\!\!\!/ + m)(i\partial\!\!\!/ - m)\psi = 0$

The last factor is simply the Dirac equation. In the study of Dirac fields in Quantum field theory, Richard Feynman invented the convenient Feynman slash notation (less commonly known as the '''Dirac''' The Klein–Gordon equation ( Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic version of the Schrödinger Hence any solution to the Dirac equation is automatically a solution to the Klein-Gordon equation, but the converse is not true: that is, not all solutions to the Klein–Gordon equation solve the Dirac equation.

Adjoint equation and Dirac current

By defining the adjoint spinor

$\bar{\psi} = \psi^\dagger\gamma^0$

and noticing that

$(\gamma^\mu)^\dagger\gamma^0 = \gamma^0\gamma^\mu \,$ ,

we obtain, by taking the Hermitian conjugate of the Dirac equation and multiplying from the right by $γ 0$ , the adjoint equation:

$\bar{\psi}(\gamma^\mu\partial_\mu - im) = 0 \,$

where $\partial_\mu$ is understood to act to the left. Multiplying the Dirac equation by $\bar{\psi}$ from the left, and the adjoint equation by $ψ$ from the right, and adding, produces the law of conservation of the Dirac current in covariant form:

$\partial_\mu \left( \bar{\psi}\gamma^\mu\psi \right) = 0$

Now we see the great advantage of the first-order equation over the one Schrödinger had tried - this is the conserved probability current density required by relativistic invariance, only now its 0-component is positive definite:

$J^0 = \bar{\psi}\gamma^0\psi = \psi^\dagger\psi$

The Dirac equation and its adjoint are the Euler–Lagrange equations of the 4-d invariant action integral

$S = \int L d^4\omega$

where the scalar L is the Dirac Lagrangian

$L = mc \bar{\psi}\psi - {i\hbar \over 2}(\bar{\psi}\gamma^\mu (\partial_\mu\psi) - (\partial_\mu\bar{\psi})\gamma^\mu \psi)$

and for the purposes of variation, $ψ$ and $\bar{\psi}$ are regarded as independent fields. In Calculus of variations, the Euler–Lagrange equation, or Lagrange's equation is a Differential equation whose solutions are the functions The relativistic invariance also follows immediately from the variational principle.

Coupling to an electromagnetic field

To consider problems in which an applied electromagnetic field interacts with the particles described by the Dirac equation, one uses the correspondence principle, and takes over into the theory the corresponding expression from classical mechanics, whereby the total momentum of a charged particle in an external field is modified as so:

$p \rightarrow p - \frac{e}{c}A$

(where e is the charge of the particle; for example, e<0 for an electron). This article discusses quantum theory For other uses see Correspondence principle (disambiguation. In natural units, the Dirac equation then takes the form

$[\gamma^\mu(\partial_\mu + ieA_\mu) + im_0]\psi = 0\,$

This validity of this prescription is confirmed experimentally with great precision. It is known as minimal coupling, and is found throughout particle physics. Indeed, while the introduction of the electromagnetic field in this way is essentially phenomenological in this context, it rises to a fundamental principle in quantum field theory. In quantum field theory (QFT the forces between particles are mediated by other particles <

Now as stated above, the transformation U is defined only up to a phase factor $e i θ$ . Also, the fundamental observable of the Dirac theory, the current, is unchanged if we multiply the wave function by an arbitrary phase. We may exploit this to get the form of the mutual interaction of a Dirac particle and the electromagnetic field, as opposed to simply considering a Dirac particle in an applied field, by assuming this arbitrary phase factor to depend continuously on position:

$\psi \rightarrow \psi^\prime = e^{i\theta(x,t)}\psi$

Notice now that

$\gamma^\mu(\partial_\mu + ieA_\mu)\psi^\prime = e^{i\theta}\gamma^\mu(\partial_\mu + ie(A_\mu + \frac{1}{e}\partial_\mu\theta))\psi = e^{i\theta}\gamma^\mu(\partial_\mu + ieA^\prime_\mu)\psi$

In order to preserve minimal coupling, we must add to the potential a term proportional to the gradient of the phase. But we know from electrodynamics that this leaves the electromagnetic field itself invariant. The value of the phase is arbitrary, but not how it changes from place to place. This is the starting point of gauge theory, which is the main principle on which quantum field theory is based. Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations The simplest such theory, and the one most thoroughly understood, is known as quantum electrodynamics. Quantum electrodynamics ( QED) is a relativistic Quantum field theory of Electrodynamics. The equations of field theory thus have invariance under both Lorentz transformations and gauge transformations.

Curved spacetime Dirac equation

The Dirac equation can be written in curved spacetime using vierbein fields. General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 This page covers applications of the Cartan formalism. For the general concept see Cartan connection. Vierbeins describe a local frame that enables to define Dirac matrices at every point. In General relativity, a frame field (also called a tetrad or Vierbein) is an Orthonormal set of four Vector fields, one In Mathematical physics, the gamma matrices, {γ0 γ1 γ2 γ3} also known as the Dirac matrices, form a matrix-valued Contracting these matrices with vierbeins give the right transformation properties. In Multilinear algebra, a tensor contraction is an operation on one or more Tensors that arises from the natural pairing of a finite- Dimensional This way Dirac equation takes the following form in curved spacetime ^[1]:

$\gamma^a e_a^\mu D_\mu \Psi + i m \Psi = 0$

Here $e_a^\mu$ is the vierbein and $D μ$ is the covariant derivative for fermion fields, defined as follows

$D_\mu = \partial_\mu - \frac{i}{4} \eta_{ac} \omega^c_{b\mu} \sigma^{ab}$

where $η a c$ is the Lorentzian metric, $σ a b$ is the commutator of Dirac matrices:

$\sigma^{ab}=\frac{i}{2} \left[\gamma^{a},\gamma^{b}\right]$

and $\omega^c_{b\mu}$ is the spin connection:

$\omega^c_{b\mu} = e^c_\nu \partial_\mu e^\nu_b + e^c_\nu e^\sigma_b \Gamma^\nu_{\sigma\mu}$

where $\Gamma^\nu_{\sigma\mu}$ is the Christoffel symbol. This page covers applications of the Cartan formalism. For the general concept see Cartan connection. In Mathematics, the covariant derivative is a way of specifying a Derivative along Tangent vectors of a Manifold. In Differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold. In Differential geometry and Mathematical physics, a spin connection is a connection on a Spinor bundle. In Mathematics and Physics, the Christoffel symbols, named for Elwin Bruno Christoffel (1829–1900 are coordinate-space expressions for the Levi-Civita Note that here, Latin letters denote the "Lorentzian" indices and Greek ones denote "Riemannian" indices.

Physical interpretation

The Dirac theory, while providing a wealth of information that is accurately confirmed by experiments, nevertheless introduces a new physical paradigm that appears at first difficult to interpret and even paradoxical. Some of these issues of interpretation must be regarded as open questions. Here we will see how the Dirac theory brilliantly answered some of the outstanding issues in physics at the time it was put forward, while posing others that are still the subject of debate.

Identification of observables

The critical physical question in a quantum theory is - what are the physically observable quantities defined by the theory? According to general principles, such quantities are defined by Hermitian operators that act on the Hilbert space of possible states of a system. The eigenvalues of these operators are then the possible results of measuring the corresponding physical quantity. In the Schrödinger theory, the simplest such object is the overall Hamiltonian, which represents the total energy of the system. If we wish to maintain this interpretation on passing to the Dirac theory, we must take the Hamiltonian to be

$H = \gamma^0 \left(mc^2 + c \sum_{k = 1}^3 \gamma^k (p_k-\frac{e}{c}A_k) \, c\right) + eA^0$

This looks promising, because we see by inspection the rest energy of the particle and, in case $A = 0$ , the energy of a charge placed in an electric potential $e A 0$ . What about the term involving the vector potential? In classical electrodynamics, the energy of a charge moving in an applied potential is

$H = c\sqrt{(p - \frac{e}{c}A)^2 + m^2c^2} + eA^0$

Thus the Dirac Hamiltonian is fundamentally distinguished from its classical counterpart, and we must take great care to correctly identify what is an observable in this theory. Much of the apparent paradoxical behavior implied by the Dirac equation amounts to a misidentification of these observables. Let us now describe one such effect. (cont'd)

History

Since the Dirac equation was originally invented to describe the electron, we will generally speak of "electrons" in this article. The equation also applies to quarks, which are also elementary spin-½ particles. In Physics, a quark (kwɔrk kwɑːk or kwɑːrk is a type of Subatomic particle. A modified Dirac equation can be used to approximately describe protons and neutrons, which are not elementary particles (they are made up of quarks), but have a net spin of ½. The proton ( Greek πρῶτον / proton "first" is a Subatomic particle with an Electric charge of one positive This article is a discussion of neutrons in general For the specific case of a neutron found outside the nucleus see Free neutron. Another modification of the Dirac equation, called the Majorana equation, is thought to describe neutrinos — also spin-½ particles. The Majorana equation is a Relativistic wave equation similar to the Dirac equation but includes the charge conjugate ψc of a Spinor ψ Neutrinos are Elementary particles that travel close to the Speed of light, lack an Electric charge, are able to pass through ordinary matter almost In Quantum mechanics, spin is an intrinsic property of all elementary particles.

The Dirac equation describes the probability amplitudes for a single electron. In Quantum mechanics, a probability amplitude is a complex -valued function that describes an uncertain or unknown quantity This is a single-particle theory; in other words, it does not account for the creation and destruction of the particles. It gives a good prediction of the magnetic moment of the electron and explains much of the fine structure observed in atomic spectral lines. In Atomic physics, the fine structure describes the splitting of the Spectral lines of Atoms due to first order relativistic corrections History See also Atomic theory, Atomism The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from an excess or deficiency of photons in a narrow frequency range compared It also explains the spin of the electron. Two of the four solutions of the equation correspond to the two spin states of the electron. The other two solutions make the peculiar prediction that there exist an infinite set of quantum states in which the electron possesses negative energy. In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός This strange result led Dirac to predict, via a remarkable hypothesis known as "hole theory," the existence of particles behaving like positively-charged electrons. Dirac thought at first these particles might be protons. He was chagrined when the strict prediction of his equation (which actually specifies particles of the same mass as the electron) was verified by the discovery of the positron in 1932. The positrons or antielectron is the Antiparticle or the Antimatter counterpart of the Electron. Year 1932 ( MCMXXXII) was a Leap year starting on Friday of the Gregorian calendar. When asked later why he hadn't actually boldly predicted the yet unfound positron with its correct mass, Dirac answered "Pure cowardice!" He shared the Nobel Prize anyway, in 1933.

Despite these successes, Dirac's theory is flawed by its neglect of the possibility of creating and destroying particles, one of the basic consequences of relativity. This difficulty is resolved by reformulating it as a quantum field theory. In quantum field theory (QFT the forces between particles are mediated by other particles Adding a quantized electromagnetic field to this theory leads to the theory of quantum electrodynamics (QED). The electromagnetic field is a physical field produced by electrically charged objects. Quantum electrodynamics ( QED) is a relativistic Quantum field theory of Electrodynamics. Moreover the equation cannot fully account for particles of negative energy but is restricted to positive energy particles.

A similar equation for spin 3/2 particles is called the Rarita-Schwinger equation. In Theoretical physics, the Rarita-Schwinger equation is the relativistic Field equation of spin -3/2 Fermions It is similar to the

Hole theory

The negative E solutions found in the preceding section 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 the electron and the electromagnetic field, any electron placed in a positive-energy eigenstate would decay into negative-energy eigenstates of successively lower energy by emitting excess energy in the form of photons. In Physics, the photon is the Elementary particle responsible for electromagnetic phenomena Real electrons obviously do not behave in this way.

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 vacuum means "absence of matter" or "an empty area or space" for the cleaning appliance see Vacuum cleaner. This description of the vacuum as a "sea" of electrons is called the Dirac sea. The Dirac sea is a theoretical model of the Vacuum as an infinite sea of particles possessing Negative energy. Since the Pauli exclusion principle forbids electrons from occupying the same state, any additional electron would be forced to occupy a positive-energy eigenstate, and positive-energy electrons would be forbidden from decaying into negative-energy eigenstates. The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925

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 initially thought that the hole might be the proton, but Hermann Weyl pointed out that the hole should behave as if it had the same mass as an electron, whereas the proton is over 1800 times heavier. Hermann Klaus Hugo Weyl ( 9 November 1885 – 8 December 1955) was a German Mathematician. The hole was eventually identified as the positron, experimentally discovered by Carl Anderson in 1932. The positrons or antielectron is the Antiparticle or the Antimatter counterpart of the Electron. Carl David Anderson ( 3 September 1905 &ndash 11 January 1991) was an American Physicist. Year 1932 ( MCMXXXII) was a Leap year starting on Friday of the Gregorian calendar.

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 has to be canceled by an infinite positive "bare" energy and the contribution to the charge density and current coming from the sea of negative-energy electrons is exactly canceled by an infinite positive "jellium" background so that the net electric charge density of the vacuum is zero. Jellium is the model of interacting Electrons in which a uniform background of Positive charge exists In quantum field theory, a Bogoliubov transformation on the creation and annihilation operators (turning an occupied negative-energy electron state into an unoccupied positive energy positron state and an unoccupied negative-energy electron state into an occupied positive energy positron state) allows us to bypass the Dirac sea formalism even though, formally, it is equivalent to it. In quantum field theory (QFT the forces between particles are mediated by other particles In Theoretical physics, the Bogoliubov transformation, named after Nikolay Bogolyubov, is a Unitary transformation from a Unitary representation

In certain applications of condensed matter physics, however, the underlying concepts of "hole theory" are valid. Condensed matter physics is the field of Physics that deals with the macroscopic physical properties of Matter. 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. Electrical conduction is the movement of electrically charged particles through a Transmission medium ( Electrical conductor) In Science and engineering, a conductor is a material which contains movable Electric charges. Fermi liquid is a generic term for a quantum mechanical Liquid of Fermions that arises under certain physical conditions when the Temperature In Thermodynamics and Chemistry, chemical potential, symbolized by μ, is a term introduced by the American engineer chemist and mathematical An unfilled state in the Fermi sea behaves like a positively-charged electron, though it is referred to as a "hole" rather than a "positron". The negative charge of the Fermi sea is balanced by the positively-charged ionic lattice of the material.

Dirac bilinears

There are five different (neutral) Dirac bilinear terms not involving any derivatives:

(S)calar: $\bar{\psi} \psi$ (scalar, P-even)
(P)seudoscalar: $\bar{\psi} \gamma^5 \psi$ (scalar, P-odd)
(V)ector: $\bar{\psi} \gamma^\mu \psi$ (vector, P-even)
(A)xial: $\bar{\psi} \gamma^\mu \gamma^5 \psi$ (vector, P-odd)
(T)ensor: $\bar{\psi} \sigma^{\mu\nu} \psi$ (antisymmetric tensor, P-even),

where $\sigma^{\mu\nu}=\frac{i}{2} \left[\gamma^{\mu},\gamma^{\nu}\right]$ and $\gamma^{5}=\gamma_{5}=\frac{i}{4!}\epsilon_{\mu\nu\rho\lambda}\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\lambda}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$ .

A Dirac mass term is an S coupling. A Yukawa coupling may be S or P. The electromagnetic coupling is V. The weak interactions are V-A.

References

^ Lawrie, Ian D. The Breit equation is a relativistic wave equation derived by Gregory Breit in 1929 based on the Dirac equation, which formally describes two The Klein–Gordon equation ( Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic version of the Schrödinger Quantum electrodynamics ( QED) is a relativistic Quantum field theory of Electrodynamics. In Theoretical physics, the Rarita-Schwinger equation is the relativistic Field equation of spin -3/2 Fermions It is similar to the The Feynman Checkerboard or Relativistic Chessboard model was Richard Feynman ’s Sum-over-paths formulation of the kernel for a free The theoretical and experimental justification for the Schrödinger equation motivates the discovery of the Schrödinger equation, the equation that describes the dynamics of . A Unified Grand Tour of Theoretical Physics.

Selected papers

P.A.M. Dirac "The Quantum Theory of the Electron", Proc. R. Soc. A117) link to the volume of the Proceedings of the Royal Society of London containing the article at page 610
P.A.M. Dirac "A Theory of Electrons and Protons", Proc. R. Soc. A126) link to the volume of the Proceedings of the Royal Society of London containing the article at page 360
C. D. Anderson, Phys. Rev. 43, 491 (1933)
R. Frisch and O. Stern, Z. Phys. 85, 4 (1933)

Textbooks

Halzen, Francis; Martin, Alan (1984). Quarks & Leptons: An Introductory Course in Modern Particle Physics. John Wiley & Sons. ISBN.
Dirac, P. A. M. , Principles of Quantum Mechanics, 4th edition (Clarendon, 1982)
Shankar, R. , Principles of Quantum Mechanics, 2nd edition (Plenum, 1994)
Bjorken, J D & Drell, S, Relativistic Quantum mechanics
Thaller, B. , The Dirac Equation, Texts and Monographs in Physics (Springer, 1992)
Schiff, L. I. , Quantum Mechanics, 3rd edition (McGraw-Hill, 1955)

External links

Dictionary

-noun

(physics) A relativistic wave equation that describes an electron (and similar particles); it predicted the existence of antiparticles

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