Dirac equation

The Dirac equation is a fundamental equation of relativistic quantum mechanics. It describes the properties and behavior of a fundamental fermion with spin 1/2 (for example electron, quark). It was developed by Paul Dirac in 1928 and, in contrast to the Schrödinger equation, fulfils the requirements of special relativity.

The Dirac equation is a first-order partial differential equation both in the three spatial coordinates and in time, in agreement with the invariance under Lorentz transformations required by special relativity. In the nonrelativistic limit case ( ${\tfrac {v}{c}}\to 0$ ), it transitions to the Pauli equation, which, unlike the Schrödinger equation, still contains the spin-orbit coupling and other terms. Each solution of the Dirac equation corresponds to a possible state of the particle in question, with the peculiarity that four spatial wave functions are needed to represent this state (see Dirac spinor), instead of two in the non-relativistic theory with spin or a single one in the case of spinless particles. For the particles described by the Dirac equation holds:

For a free particle, the relativistic energy-momentum relation is $E={\sqrt {m^{2}c^{4}+p^{2}c^{2}}}$ satisfied.

For a particle in the electrostatic field of a point charge, the hydrogen spectrum with its fine structure is obtained.

The particle has an intrinsic angular momentum (spin) which has the quantum number 1/2 and - because this does not occur in classical physics - cannot be traced back to the rotation of a mass distribution as in the case of a spinning top.

If the particle carries an electric charge, the spin is always associated with a magnetic dipole moment (→ spin magnetism). Compared to the magnetic dipole that the particle would cause by a rotational motion with the same amount of angular momentum (→ orbital magnetism), the moment associated with the spin has twice the strength (see Anomalous magnetic moment of the electron).

To the particle exists an antiparticle (to the electron a so-called positron) with the same mass and the same spin, but with opposite charge and magnetic moment.

All the properties mentioned correspond to experimental findings. At the time of the discovery of the Dirac equation in 1928, the first four were already known, but not their common basis. The latter property was predicted by the Dirac equation, and the first detection of an antiparticle was achieved by Carl David Anderson in 1932 (see positron).

The differential operator occurring in the Dirac equation also plays an important role in mathematics (differential geometry) (Dirac operator).

Dirac equation of an uncharged particle

The Dirac equation is a system of four coupled partial differential equations for the four component functions of the Dirac spinor ψ $\psi (x)$ . The variable here stands for $x=(x^{0},x^{1},x^{2},x^{3})\,,$ where the upper subscript 0 $(x^{1},x^{2},x^{3})$ denotes the time $t=x^{0}$ and the subscripts 1 to 3 denote the location coordinates

In natural units of measure with $c=1=\hbar$ , the Dirac equation for an uncharged particle of mass

$\left[{\mathrm i}\sum _{{\mu =0}}^{3}\gamma ^{{\mu }}{\frac {\partial }{\partial x^{{\mu }}}}-m\right]\,\psi (x)=0\,.$

The expression in square brackets is the standard form of a Dirac operator.

The constant gamma or Dirac matrices γ $\gamma ^{0},\gamma ^{1},\gamma ^{2}$ and γ $\gamma ^{3}$ act in the space of the four components of the spinor and couple them together. The products of two gamma matrices have the following properties:

$\gamma ^{{\mu }}\,\gamma ^{{\nu }}=-\gamma ^{{\nu }}\,\gamma ^{{\mu }}\quad (\,\mu ,\nu \in \{0,1,2,3\}\,,\mu \neq \nu ),$

$\gamma ^{{0}}\gamma ^{{0}}=-\gamma ^{{1}}\gamma ^{{1}}=-\gamma ^{{2}}\gamma ^{{2}}=-\gamma ^{{3}}\gamma ^{{3}}=1$

Thus they form a Clifford or Dirac algebra. If the Dirac operator

$\mathrm {i} \sum _{\mu =0}^{3}\gamma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}-m$

applied to both sides of the Dirac equation, the four differential equations decouple and the Klein-Gordon equation is obtained for each component of ψ : $\psi$

${\Bigl [}{\frac {\partial ^{2}}{(\partial {x^{{0}}})^{2}}}-{\frac {\partial ^{2}}{(\partial {x^{{1}}})^{2}}}-{\frac {\partial ^{2}}{(\partial {x^{{2}}})^{2}}}-{\frac {\partial ^{2}}{(\partial {x^{{3}}})^{2}}}+m^{2}{\Bigr ]}\,\psi (x)=0$

Thus, applying a Dirac operator twice leads to the Klein-Gordon equation, which is why the Dirac equation is also considered the "root" of the Klein-Gordon equation. For a particle in a momentum eigenstate, the Klein-Gordon equation yields (in the order of its terms) $-E^{2}+{\vec {p}}^{{\,2}}+m^{2}=0$ , that is, the relativistic energy-momentum relation of a particle of mass $m\,.$

Each irreducible representation of the Dirac algebra consists of $4\times 4$ matrices. In the standard or Dirac representation they have the following form (vanishing matrix elements with value zero are not written here):

${\begin{array}{c c}\gamma ^{0}={\begin{pmatrix}1&&&\\&1&&\\&&-1&\\&&&-1\end{pmatrix}}\,,&\gamma ^{1}={\begin{pmatrix}&&&1\\&&1&\\&-1&&\\-1&&&\end{pmatrix}}\,,\\\,&\,\\\gamma ^{2}={\begin{pmatrix}&&&-\mathrm {i} \\&&\mathrm {i} &\\&\mathrm {i} &&\\-\mathrm {i} &&&\end{pmatrix}}\,,&\gamma ^{3}={\begin{pmatrix}&&1&\\&&&-1\\-1&&&\\&1&&\end{pmatrix}}\,.\end{array}}$

Thus, the first two components of γ $\gamma _{0}$ form the two-component unit matrix, and the last two components form its negative. Similarly, the two upper components of the second, third, and fourth γ $\gamma$ matrices, respectively, yield the three 2×2 Pauli matrices σ $\sigma _{j}$ and the two last components of γ $\gamma _{j}$ their negatives. The latter go to zero in the nonrelativistic limit case like v/c . Thus this representation, the standard one, is particularly suitable for the treatment of slowly moving electrons. In the mathematically and physically equivalent Weyl representation, the spinor transformation behavior for Lorentz transformations is particularly simple, and in the likewise equivalent Majorana representation, the Dirac equation is a real system of equations. Further representations are obtained by equivalence transformations.

The four gamma matrices can be written symbolically as the contravariant 4-vector

$\gamma ^{\mu }={\begin{pmatrix}\gamma ^{0}\\\gamma ^{1}\\\gamma ^{2}\\\gamma ^{3}\end{pmatrix}}$

summarize. Then the first term of the Dirac equation has the form of a scalar product of the vectors γ $\gamma ^{{\mu }}$ and ∂ ${\frac {\partial }{\partial x^{{\mu }}}}$ . However, this is not invariant under Lorentz transformation, because γ $\gamma ^{{\mu }}$ remains constant. The Lorentz invariance of Dirac theory arises only because the Dirac operator $\psi$ acts on a spinor ψ whose four components are suitably co-transformed. In the final result a solution ψ $\psi (x)$ the Dirac equation is transformed by Lorentz transformation into a solution of the correspondingly transformed Dirac equation.

Momentum space and slash notation

In addition to the form just described in spatial space, the Dirac equation can also be written down in momentum space. It then reads

${\Bigl [}\gamma ^{\mu }p_{\mu }-m{\Bigr ]}\,\psi (p)=0\,,$

where for abbreviation the Einstein summation convention was used (which states that summation is done over equal indices). In the even further simplified Feynman-Slash notation, the scalar product with the gamma matrices is expressed by a slash symbol. In the space of places it results in

${\Bigl [}i\partial \!\!\!/\ -m{\Bigr ]}\,\psi (x)=0\,,$

and in momentum space

${\Bigl [}p\!\!\!/\ -m{\Bigr ]}\,\psi (p)=0\,.$

Dirac equation

Dirac equation of an uncharged particle

Momentum space and slash notation

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