Dirac matrices

The Dirac matrices (after the British physicist Paul Dirac ), also called gamma matrices , are four matrices that satisfy the Dirac algebra . They appear in the Dirac equation .

definition

The Dirac matrices and , by definition, satisfy the Dirac algebra, that is, the algebraic conditions ${\displaystyle \gamma ^{0},\,\gamma ^{1}\,,\gamma ^{2}\,}$ ${\displaystyle \,\gamma ^{3}\,}$

${\displaystyle {\begin{aligned}\gamma ^{0}\gamma ^{0}&=1\,,&\gamma ^{1}\gamma ^{1}&=-1\,,&\gamma ^{2}\gamma ^{2}&=-1\,,&\gamma ^{3}\gamma ^{3}&=-1\,,\\\gamma ^{0}\gamma ^{1}&=-\gamma ^{1}\gamma ^{0}\,,&\gamma ^{0}\gamma ^{2}&=-\gamma ^{2}\gamma ^{0}\,,&\gamma ^{0}\gamma ^{3}&=-\gamma ^{3}\gamma ^{0}\,,&&\\\gamma ^{1}\gamma ^{2}&=-\gamma ^{2}\gamma ^{1}\,,&\gamma ^{1}\gamma ^{3}&=-\gamma ^{3}\gamma ^{1}\,,&\gamma ^{2}\gamma ^{3}&=-\gamma ^{3}\gamma ^{2}\,.&&\end{aligned}}}$

These conditions apply to anti-commutators , i.e. the sum of the products of two matrices in both orders,

${\displaystyle \{A,B\}=A\,B+B\,A\,.}$

In index notation, in which and stand for numbers , the conditions for the Dirac matrices are summarized as ${\displaystyle \mu }$ ${\displaystyle \nu }$ ${\displaystyle \{0,1,2,3\}}$

${\displaystyle \{\gamma ^{\mu },\gamma ^{\nu }\}=\gamma ^{\mu }\gamma ^{\nu }+\gamma ^{\nu }\gamma ^{\mu }=2\,\eta ^{\mu \nu }I\,.}$

Here are the components of the Minkowski metric with signature (1, -1, -1, -1) and is the unit matrix . ${\displaystyle \eta ^{\mu \nu }}$ ${\displaystyle I}$

The γ ⁵ matrix

In addition to the four gamma matrices, you also define the matrix

${\displaystyle \gamma ^{5}=\mathrm {i} \,\gamma ^{0}\gamma ^{1}\gamma ^{2}\gamma ^{3}\ .}$

It is its own inverse, it is Hermitian, anti-swapped with the gamma matrices, and therefore with every product of gamma matrices with an odd number of factors. ${\displaystyle \gamma ^{5}\gamma ^{5}=1\,,}$ ${\displaystyle \gamma ^{5}\gamma ^{\mu }=-\gamma ^{\mu }\gamma ^{5}\,,}$

properties

The gamma matrices generate a Clifford algebra . Every irreducible representation of this algebra by matrices consists of matrices. The elements of the vector space on which they act are called spinors . Different representations of Dirac algebra are equivalent to each other, that is, they differ only in the chosen basis. In particular, the negative transposed matrices and the Hermitian adjoint matrices are equivalent to the matrices because they also satisfy the Dirac algebra. There is therefore a matrix and a matrix such that ${\displaystyle 4\times 4}$ ${\displaystyle -\gamma ^{\mu \,{\text{T}}}}$ ${\displaystyle \gamma ^{\mu \,\dagger }}$ ${\displaystyle \,\gamma ^{\mu }\,}$ ${\displaystyle A}$ ${\displaystyle C}$

${\displaystyle C\gamma ^{\mu }C^{-1}=-\gamma ^{\mu \,{\text{T}}}\ ,\quad A\gamma ^{\mu }A^{-1}=\gamma ^{\mu \,\dagger }\,.}$

The matrix is important for the construction of scalars, vectors and tensors from spinors; the matrix occurs during charge conjugation . ${\displaystyle A}$ ${\displaystyle C}$

Each product of several Dirac matrices can be written down to a sign as the product of different Dirac matrices in lexographic order, because the product of two different gamma matrices can be rearranged at the expense of a sign. In addition, the square of each gamma matrix is ​​1 or −1. The products of different gamma matrices together with the one matrix and the negative matrices form a group with the 32 elements,

${\displaystyle \pm 1\,,\,\pm \gamma ^{\mu }\,,\,\pm \gamma ^{\mu }\gamma ^{\nu }\,,\,\mu <\nu \,,\,\pm \gamma ^{\lambda }\gamma ^{\mu }\gamma ^{\nu }\,,\,\lambda <\mu <\nu \,,\,\pm \gamma ^{0}\gamma ^{1}\gamma ^{2}\gamma ^{3}\,,\,{\text{wobei}}\,\lambda ,\mu ,\nu \in \{0,1,2,3\}\,.}$

Since every representation of a finite group is unitary given a suitable base choice, every representation of the gamma matrices is also unitary given a suitable base choice. Together with the Dirac algebra this means that Hermitian and the three other matrices are antihermitic, ${\displaystyle \gamma ^{0}}$ ${\displaystyle \gamma }$

${\displaystyle \gamma ^{0\,\dagger }=\gamma ^{0}\,,\,\gamma ^{1\,\dagger }=-\gamma ^{1}\,,\,\gamma ^{2\,\dagger }=-\gamma ^{2}\,,\,\gamma ^{3\,\dagger }=-\gamma ^{3}\,.}$

In unitary representations, the equivalence transformation leads to the adjoint matrices ${\displaystyle A=\gamma ^{0}}$

${\displaystyle \gamma ^{0}\gamma ^{\mu }\gamma ^{0}=\gamma ^{\mu \,\dagger }\,.}$

Using the properties of, it can be shown that the trace of any product of gamma matrices with an odd number of factors vanishes. ${\displaystyle \gamma ^{5}}$

${\displaystyle {\begin{aligned}{\text{Spur}}\,{\bigl (}\gamma ^{\mu _{1}}\dots \gamma ^{\mu _{2n+1}}{\bigr )}&={\text{Spur}}\,{\bigl (}\gamma ^{\mu _{1}}\dots \gamma ^{\mu _{2n+1}}\gamma ^{5}\gamma ^{5}{\bigr )}=-{\text{Spur}}\,{\bigl (}\gamma ^{5}\gamma ^{\mu _{1}}\dots \gamma ^{\mu _{2n+1}}\gamma ^{5}{\bigr )}\\&=-{\text{Spur}}\,{\bigl (}\gamma ^{\mu _{1}}\dots \gamma ^{\mu _{2n+1}}\gamma ^{5}\gamma ^{5}{\bigr )}=-{\text{Spur}}\,{\bigl (}\gamma ^{\mu _{1}}\dots \gamma ^{\mu _{2n+1}}{\bigr )}\end{aligned}}}$

In the penultimate step it was used that the track of a product does not change when the factors are cyclically interchanged and therefore applies. ${\displaystyle {\text{Spur}}\,(\gamma ^{5}\,B)={\text{Spur}}\,(B\,\gamma ^{5})}$

For the trace of a product of two gamma matrices we have (because the trace is cyclic)

${\displaystyle {\text{Spur}}\,\gamma ^{\mu }\,\gamma ^{\nu }={\frac {1}{2}}{\text{Spur}}(\,\gamma ^{\mu }\,\gamma ^{\nu }+\gamma ^{\nu }\,\gamma ^{\mu })={\frac {2\,\eta ^{\mu \nu }}{2}}{\text{Spur 1}}=4\,\eta ^{\mu \nu }\,.}$

The trace of four gamma matrices is reduced to the trace of two with Dirac algebra.

${\displaystyle {\begin{array}{rcl}2\,{\text{Spur}}\,\gamma ^{\kappa }\,\gamma ^{\lambda }\,\gamma ^{\mu }\,\gamma ^{\nu }&=&{\text{Spur}}(\,\gamma ^{\kappa }\,\gamma ^{\lambda }\,\gamma ^{\mu }\,\gamma ^{\nu }+\gamma ^{\lambda }\,\gamma ^{\mu }\,\gamma ^{\nu }\,\gamma ^{\kappa })\\&=&{\text{Spur}}(\,\gamma ^{\kappa }\,\gamma ^{\lambda }\,\gamma ^{\mu }\,\gamma ^{\nu }+\gamma ^{\lambda }\,\gamma ^{\kappa }\,\gamma ^{\mu }\,\gamma ^{\nu }\\&&\ \ \ \ -\gamma ^{\lambda }\,\gamma ^{\kappa }\,\gamma ^{\mu }\,\gamma ^{\nu }-\gamma ^{\lambda }\,\gamma ^{\mu }\,\gamma ^{\kappa }\,\gamma ^{\nu }\\&&\ \ \ \ +\gamma ^{\lambda }\,\gamma ^{\mu }\,\gamma ^{\kappa }\,\gamma ^{\nu }+\gamma ^{\lambda }\,\gamma ^{\mu }\,\gamma ^{\nu }\,\gamma ^{\kappa })\\&=&2\,\eta ^{\kappa \lambda }{\text{Spur}}(\gamma ^{\mu }\,\gamma ^{\nu })-2\,\eta ^{\kappa \mu }{\text{Spur}}(\gamma ^{\lambda }\,\gamma ^{\nu })+2\,\eta ^{\kappa \nu }{\text{Spur}}(\gamma ^{\lambda }\,\gamma ^{\mu })\end{array}}}$

Therefore:

${\displaystyle {\begin{array}{rcl}{\text{Spur}}\,\gamma ^{\kappa }\,\gamma ^{\lambda }\,\gamma ^{\mu }\,\gamma ^{\nu }&=&4\,(\eta ^{\kappa \lambda }\,\eta ^{\mu \nu }-\eta ^{\kappa \mu }\,\eta ^{\lambda \nu }+\eta ^{\kappa \nu }\,\eta ^{\lambda \mu })\end{array}}}$

So if different Dirac matrices do not appear in pairs in a product, the trace of the product disappears. It follows, among other things, that the sixteen matrices that are obtained as the product of zero to four different gamma matrices are linearly independent.

Dirac equation

Dirac introduced the gamma matrices to convert the Klein-Gordon equation , which is a second order differential equation, into a first order equation.

In natural units , the Dirac equation can be written as follows

${\displaystyle (i\gamma ^{\mu }\partial _{\mu }-m)\psi =0}$

where is a Dirac spinor . ${\displaystyle \psi }$

Multiplying both sides by gives you ${\displaystyle -(i\gamma ^{\nu }\partial _{\nu }+m)}$

${\displaystyle (\eta ^{\mu \nu }\partial _{\mu }\partial _{\nu }+m^{2})\psi =(\partial ^{2}+m^{2})\psi =0,}$

thus just the Klein-Gordon equation for a particle of mass . ${\displaystyle m}$

Relation to Lorentz transformations

The six matrices

${\displaystyle \Sigma ^{\mu \nu }={\frac {1}{4}}{\bigl (}\gamma ^{\mu }\gamma ^{\nu }-\gamma ^{\nu }\gamma ^{\mu }{\bigr )}}$

form the basis of a Lie algebra , which is isomorphic to the Lie algebra of the Lorentz transformations . They generate the transformations of the spinors belonging to Lorentz transformations (which are continuously related to the 1) . ${\displaystyle \psi }$

Chirality

From and it follows that the matrices ${\displaystyle (\gamma ^{5})^{2}=1}$ ${\displaystyle {\text{Spur}}\,\gamma ^{5}=0}$

${\displaystyle P_{L}={\frac {1-\gamma ^{5}}{2}}\,,\quad P_{R}={\frac {1+\gamma ^{5}}{2}}}$

Projectors are

${\displaystyle (P_{L})^{2}=P_{L}\,,\,(P_{R})^{2}=P_{R}\,,}$

which project onto complementary, two-dimensional subspaces,

${\displaystyle P_{L}\,P_{R}=0\,,\ {\text{Spur}}\,P_{L}={\text{Spur}}\,P_{R}=2\,,\quad P_{L}+P_{R}=1\,.}$

These subspaces distinguish between particles of different chirality .

Because swapped with the generators of spinor transformations, ${\displaystyle \gamma ^{5}}$

${\displaystyle \gamma ^{5}\Sigma ^{\mu \nu }=\Sigma ^{\mu \nu }\gamma ^{5}\,,}$

are the subspaces onto which and project, invariant under the Lorentz transformations generated by, in other words: The left and right-handed parts, and , of a spinor transform separately from each other. ${\displaystyle P_{L}}$ ${\displaystyle P_{R}}$ ${\displaystyle \Sigma ^{\mu \nu }}$ ${\displaystyle \psi _{L}=P_{L}\psi }$ ${\displaystyle \psi _{R}=P_{R}\psi }$ ${\displaystyle \psi }$

There and are Hermitian, because Hermitian is applies to ${\displaystyle P_{L}}$ ${\displaystyle P_{R}}$ ${\displaystyle \gamma ^{5}}$

${\displaystyle {\bar {\psi }}_{L}=(P_{L}\,\psi )^{\dagger }\gamma ^{0}=\psi ^{\dagger }P_{L}^{\dagger }\gamma ^{0}=\psi ^{\dagger }P_{L}\gamma ^{0}=\psi ^{\dagger }\gamma ^{0}P_{R}={\bar {\psi }}\,P_{R}}$,

where is generally defined as . The change results from swapping with . As with antikommutiert, the sign changes before the projection operator . Analogously, you get for . ${\displaystyle {\bar {\psi }}}$ ${\displaystyle {\bar {\psi }}=\psi ^{\dagger }\gamma ^{0}}$ ${\displaystyle P_{L}\rightarrow P_{R}}$ ${\displaystyle \gamma ^{5}}$ ${\displaystyle \gamma ^{0}}$ ${\displaystyle \gamma ^{5}}$ ${\displaystyle \gamma ^{0}}$ ${\displaystyle \gamma ^{5}}$ ${\displaystyle P_{L}={\frac {1-\gamma ^{5}}{2}}\rightarrow P_{R}={\frac {1+\gamma ^{5}}{2}}}$ ${\displaystyle {\bar {\psi }}_{R}={\bar {\psi }}\,P_{L}}$

parity

Because of a term that contains changes its sign under the parity transformation , it turns scalars into pseudoscalars and vectors into pseudo vectors . ${\displaystyle \gamma ^{0}\gamma ^{5}\gamma ^{0}=-\gamma ^{5}}$ ${\displaystyle \gamma ^{5}}$

In general, sizes that are composed of gamma matrices and one possibly from different spinors follow a law of transformation that can be read on the index image. Transform it ${\displaystyle {\overline {\psi }}=\psi ^{\dagger }A=\psi ^{\dagger }\gamma ^{0}}$ ${\displaystyle \psi }$ ${\displaystyle \chi }$

• ${\displaystyle {\overline {\psi }}\chi }$ like a scalar,
• ${\displaystyle {\overline {\psi }}\gamma ^{\mu }\chi }$like the components of a four-vector ,
• ${\displaystyle {\overline {\psi }}\Sigma ^{\mu \nu }\chi }$ like the components of an antisymmetric tensor,
• ${\displaystyle {\overline {\psi }}\gamma ^{\mu }\gamma ^{5}\chi }$ like the components of an axial four-vector,
• ${\displaystyle {\overline {\psi }}\gamma ^{5}\chi }$ like a pseudoscalar.

Feynman slash notation

Richard Feynman invented the slash notation named after him (also known as Feynman dagger or Feynman dagger). In this notation, the scalar product of a Lorentz vector with the vector of the gamma matrices is abbreviated as ${\displaystyle \textstyle \sum _{\mu =0}^{3}\,\gamma ^{\mu }A_{\mu }}$

${\displaystyle A\!\!\!/\ {\stackrel {\mathrm {def} }{=}}\ \sum _{\mu =0}^{3}\gamma ^{\mu }A_{\mu }}$.

This allows z. B. the Dirac equation can be written very clearly as

${\displaystyle {\Bigl (}i\partial \!\!\!/\ -{\frac {mc}{\hbar }}{\Bigr )}\,\psi (x)=0\ ,}$

or in natural units

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

Dirac representation

In a suitable basis, the gamma matrices have the form going back to Dirac (vanishing matrix elements not written out)

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

These matrices can be written more compactly with the help of the Pauli matrices (each entry here stands for a matrix): ${\displaystyle 2\times 2}$

${\displaystyle \gamma ^{0}={\begin{pmatrix}1&\\&-1\end{pmatrix}}\,,\quad \gamma ^{i}={\begin{pmatrix}&\sigma ^{i}\\-\sigma ^{i}&\end{pmatrix}}\,,\;i\in \{1,2,3\}\,,\quad \gamma ^{5}={\begin{pmatrix}&1\\1&\end{pmatrix}}\,.}$

The Dirac matrices can also be generated using the Kronecker product as follows:

${\displaystyle \gamma ^{0}=\sigma ^{3}\otimes 1,\quad \gamma ^{i}=\mathrm {i} \sigma ^{2}\otimes \sigma ^{i},\;i\in \{1,2,3\},\quad \gamma ^{5}=\sigma ^{1}\otimes 1}$

Weyl representation

After Hermann Weyl called Weyl representation is also chiral representation . In it is diagonal ${\displaystyle \gamma ^{5}}$

${\displaystyle \gamma ^{5}={\begin{pmatrix}-1&\\&1\end{pmatrix}}\,,\quad P_{L}={\frac {1-\gamma ^{5}}{2}}={\begin{pmatrix}1&\\&0\end{pmatrix}}\,,\quad P_{R}={\frac {1+\gamma ^{5}}{2}}={\begin{pmatrix}0&\\&1\end{pmatrix}}\,.}$

Compared to the Dirac representation, and are changed, the spatial matrices remain unchanged: ${\displaystyle \gamma ^{0}}$ ${\displaystyle \gamma ^{5}}$ ${\displaystyle \gamma }$

${\displaystyle \gamma ^{0}={\begin{pmatrix}&1\\1&\end{pmatrix}}\,,\quad \gamma ^{i}={\begin{pmatrix}&\sigma ^{i}\\-\sigma ^{i}&\end{pmatrix}}\,,\quad \gamma ^{5}={\begin{pmatrix}-1&\\&1\end{pmatrix}}}$

The Weyl representation results from a unitary base change from the Dirac representation,

${\displaystyle \gamma _{\text{Weyl}}^{\mu }=U\,\gamma _{\text{Dirac}}^{\mu }U^{-1}{\text{ mit }}U={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1&-1\\1&1\end{pmatrix}},\ U^{-1}=U^{\dagger }={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1&1\\-1&1\end{pmatrix}}\,.}$

Spinor transformations transform the first two and the last two components of the Dirac spinor separately in the Weyl basis.

The chiral representation is of particular importance in the Weyl equation , the massless Dirac equation.

Majorana representation

In the Majorana representation, all gamma matrices are imaginary. Then the Dirac equation is a real system of differential equations,

${\displaystyle {\begin{aligned}\gamma ^{0}&={\begin{pmatrix}&-\sigma ^{2}\\-\sigma ^{2}&\end{pmatrix}}\,,&\gamma ^{1}&={\begin{pmatrix}&\mathrm {i} \sigma ^{3}\\\mathrm {i} \sigma ^{3}&\end{pmatrix}}\,,&\\&\,&&\\\gamma ^{2}&={\begin{pmatrix}\mathrm {i} &\\&-\mathrm {i} \end{pmatrix}}\,,&\gamma ^{3}&={\begin{pmatrix}&-\mathrm {i} \sigma ^{1}\\-\mathrm {i} \sigma ^{1}&\end{pmatrix}}\,,&\gamma ^{5}&={\begin{pmatrix}&\mathrm {i} \\-\mathrm {i} &\end{pmatrix}}\,.\end{aligned}}}$

literature

• James Bjorken and Sidney Drell : Relativistic Quantum Mechanics , BI-Wissenschaftsverlag, Mannheim, 1990, (BI university paperback volume 98), ISBN 3-411-00098-8
• Michael Peskin and Daniel V. Schroeder: An Introduction to Quantum Field Theory , Addison-Wesley Publishing Co., New York, 1995, ISBN 0-201-50397-2
• Josef-Maria Jauch and Fritz Rohrlich : The theory of photons and electrons , Addison-Wesley Publishing Co., New York, 1955
• Ferdinando Gliozzi, Joel Sherk and David Olive, Supersymmetry, Supergravity Theories and the Dual Spinor Model , Nucl. Phys. B122, 253-290, 1977. (Dirac algebra in higher dimensions)
• Franz Schwabl, Quantum Mechanics for Advanced Students (QM II) , Springer, Heidelberg, ISBN 978-3-540-85076-2