Gamma-Matrix Identities

This is a compact lookup card. The fixed basis tables, trace signs, source translations, and complete convention ledger are owned by Gamma-Matrix Conventions; the representation-independent derivation is on Gamma Matrices.

Formula

With metric signature

\eta^{\mu\nu} = \operatorname{diag}(1,-1,-1,-1),

gamma matrices satisfy the Clifford algebra

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

Slash notation is

p\!\!\!/ \equiv \gamma^\mu p_\mu, \qquad A\!\!\!/ \equiv \gamma^\mu A_\mu.

Because $p_\mu p_\nu$ is symmetric,

(p\!\!\!/)^2 = p^\mu p_\mu I.

The antisymmetric sigma matrices are

\sigma^{\mu\nu} = \frac{i}{2} [\gamma^\mu,\gamma^\nu].

With the transformation convention $S(\Lambda)=\exp[-i\omega_{\mu\nu}\sigma^{\mu\nu}/4]$ , the corresponding spinor-representation generators are $\sigma^{\mu\nu}/2$ .

The chirality matrix convention used here is

\gamma^5 = i\gamma^0\gamma^1\gamma^2\gamma^3, \qquad \{\gamma^5,\gamma^\mu\}=0, \qquad (\gamma^5)^2=I.

Useful trace identities include

\operatorname{Tr}(\gamma^\mu\gamma^\nu) = 4\eta^{\mu\nu},

and

\begin{aligned} \operatorname{Tr} (\gamma^\mu\gamma^\nu\gamma^\rho\gamma^\sigma) &= 4( \eta^{\mu\nu}\eta^{\rho\sigma} - \eta^{\mu\rho}\eta^{\nu\sigma} \\ &\qquad + \eta^{\mu\sigma}\eta^{\nu\rho} ). \end{aligned}

Assumptions

The metric signature is $\eta=\operatorname{diag}(1,-1,-1,-1)$ .
Gamma matrices are four-dimensional Dirac gamma matrices.
Repeated Lorentz indices are summed.
Trace identities are representation independent within the same Clifford-algebra convention.

Validity

Gamma-matrix identities are algebraic. Their physical meaning depends on how spinors, Lorentz transformations, Fourier conventions, and adjoints are defined. In another metric signature, signs and definitions such as $\gamma^5$ may need translation.

Common Mistakes

Using identities from a $(-+++)$ convention without translating signs.
Forgetting that $p_\mu=(p^0,-\mathbf p)$ in the $(+---)$ convention.
Writing unsupported slash commands instead of explicit compact slash notation.
Confusing $\sigma^{\mu\nu}$ with the three Pauli matrices $\sigma_i$ .
Forgetting that trace identities involving $\gamma^5$ require additional convention choices.

Quick Check

Show that $(p\!\!\!/)^2=p^\mu p_\mu I$ .

Solution

Use

(p\!\!\!/)^2 = \gamma^\mu\gamma^\nu p_\mu p_\nu.

Since $p_\mu p_\nu$ is symmetric, only the anticommutator contributes:

\gamma^\mu\gamma^\nu p_\mu p_\nu = \frac12 \{\gamma^\mu,\gamma^\nu\} p_\mu p_\nu.

Using $\{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu}I$ gives

(p\!\!\!/)^2 = \eta^{\mu\nu}p_\mu p_\nu I = p^\mu p_\mu I.

Canonical owner and released companions

References

J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics, McGraw-Hill, 1964.
M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, 1995.
M. Srednicki, Quantum Field Theory, Cambridge University Press, 2007.