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Appendix D
The Dirac Algebra is a complex matrix representation of the
Clifford Algebra ${\text{Cl(}}1,3)$ generated by a set of 4x4
matrices $\left\{ {{\gamma ^a},{\gamma _a};a = 0,1,2,3} \right\}$.
The gamma matrices formally constitute an orthonormal basis
of a (local) Minkowski spacetime defined by the inner product
(anticommutator) relation:
\[{\gamma ^a} \cdot {\gamma ^b}
: = \frac{1}{2}\left\{ {{\gamma ^a},{\gamma ^b}} \right\} = {\eta
^{ab}}{I_4}\]
where ${\eta ^{ab}}$ is the Minkowski metric with signature $( + ,
- , - , - )$. It is conventional to suppress component indices of
the gamma matrices and spinors. Usually, the identity matrix at the
right-hand side is also omitted. The wedge product (commutator)
\[{\gamma ^a} \wedge {\gamma
^b} : = \frac{1}{2}\left[ {{\gamma ^a},{\gamma ^b}} \right]: =
2{\sigma ^{ab}}\]
defines the spin matrices $\sigma ^{ab}$. These form the
spin representation of the Lorentz group, so that $S(\Lambda ) =
\exp \left( {\frac{1}{2}{\varepsilon _{ab}}{\sigma ^{ab}}} \right)$
are Lorentz transformations in spinor space; see Gauge
Theory.
There are many matrix representations that conform to the
anticommutator rule (D.1), amongst which the Standard (Dirac-Pauli)
representation and the Chiral (Weyl) representation.
[Wikipedia: Gamma Matrices]
Common Properties
- $\quad {\left( {{\gamma ^0}}
\right)^2} = {\left( {{\gamma _0}} \right)^2} = 1,\quad
{\left( {{\gamma ^0}} \right)^\dagger } = {\gamma ^0} $
- $\quad {\left( {{\gamma ^i}} \right)^2}{\text{ = }}{\left(
{{\gamma _i}} \right)^2} {\text{ = }} - 1,{\text{ }}{\left(
{{\gamma ^i}} \right)^\dagger } = - {\gamma ^i}, \quad i =
1,2,3 $
- $\quad {\left( {{\gamma ^a}} \right)^\dag } = {\gamma
^0}{\gamma ^a}{\gamma ^0},\quad {\left( {{\sigma ^{ab}}}
\right)^\dag } = - {\gamma ^0}{\sigma ^{ab}}{\gamma ^0},
{\text{ }}\;\, {\left( {{\gamma ^0}{\gamma ^a}} \right)^\dag }
= {\gamma ^0}{\gamma ^a}$
- $ \quad {\gamma ^5}: = i{\gamma ^0}{\gamma ^1}{\gamma
^2}{\gamma ^3},\quad {\gamma _5}: = - i{\gamma _0}{\gamma
_1}{\gamma _2}{\gamma _3},\quad{\left( {{\gamma ^5}}
\right)^2} = 1$
- $\quad {\left( {{\gamma ^5}}\right)^\dag } = {\gamma ^5} =
{\gamma _5} = {({\gamma _5})^{ - 1}}, \quad {\left( {{\gamma
^5}{\gamma ^a}} \right)^\dag } = {\gamma ^0}{\gamma ^5}{\gamma
^a}{\gamma ^0}$
The dagger operator implements the complex conjugation of the
transpose.
The matrix representations of the unit pseudo-scalar elements of the
Dirac Algebra are conventionally denoted as the ‘fifth’ matrices $\{
{\gamma ^5},{\gamma _5}\} $; they have an imaginary unit included in
their definition so as to have real eigenvalues. These pseudo
scalars anti-commute with the four gamma matrices $\left\{ {{\gamma
^5},{\gamma ^a}} \right\} = 0$ and commute with the spin matrices:
$\left[ {{\gamma ^5},{\sigma ^{ab}}} \right] = 0$.