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
- .
Read more about this topic: Gamma Matrices
Famous quotes containing the words mathematical and/or structure:
“It is by a mathematical point only that we are wise, as the sailor or the fugitive slave keeps the polestar in his eye; but that is sufficient guidance for all our life. We may not arrive at our port within a calculable period, but we would preserve the true course.”
—Henry David Thoreau (1817–1862)
“Man is more disposed to domination than freedom; and a structure of dominion not only gladdens the eye of the master who rears and protects it, but even its servants are uplifted by the thought that they are members of a whole, which rises high above the life and strength of single generations.”
—Karl Wilhelm Von Humboldt (1767–1835)