Gamma Matrices - Mathematical Structure

Mathematical Structure

The defining property for the gamma matrices to generate a Clifford algebra is the anticommutation relation

where is the anticommutator, is the Minkowski metric with signature (+ − − −) and is the 4x4 unit matrix.

This defining property is considered to be more fundamental than the numerical values used in the gamma matrices. Covariant gamma matrices are defined by

and Einstein notation is assumed.

Note that the other sign convention for the metric, (− + + +) necessitates either a change in the defining equation:

or a multiplication of all gamma matrices by, which of course changes their hermiticity properties detailed below. Under the alternative sign convention for the metric the covariant gamma matrices are then defined by

.