Principles of Quantum Mechanics · Problem 20.16

The Dirac Equation — R. Shankar

Physics — Quantum Mechanics · dirac-equation-bilinear-covariants

Problem

The bilinear covariants of the Dirac theory are quantities of the form

\bar{\psi}\Gamma\psi

where

\Gamma

is a

4\times4

matrix. Classify the 16 independent bilinear covariants according to their transformation properties under Lorentz transformations (scalar, pseudoscalar, vector, axial vector, antisymmetric tensor) by considering

\Gamma \in \{1, \gamma^5, \gamma^\mu, \gamma^\mu\gamma^5, \sigma^{\mu\nu}\}

where

\gamma^5 = i\gamma^0\gamma^1\gamma^2\gamma^3

and

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

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