Page "Spinor" Paragraph 75

Given a vector space V and a quadratic form g an explicit matrix representation of the Clifford algebra can be defined as follows.

Choose an orthonormal basis for V i. e. where and for.

Let.

Fix a set of matrices such that ( i. e. fix a convention for the gamma matrices ).

Then the assignment extends uniquely to an algebra homomorphism by sending the monomial in the Clifford algebra to the product of matrices and extending linearly.

The space on which the gamma matrices act is a now a space of spinors.

One needs to construct such matrices explicitly, however.

In dimension 3, defining the gamma matrices to be the Pauli sigma matrices gives rise to the familiar two component spinors used in non relativistic quantum mechanics.

Likewise using the 4 × 4 Dirac gamma matrices gives rise to the 4 component Dirac spinors used in 3 + 1 dimensional relativistic quantum field theory.

In general, in order to define gamma matrices of the required kind, one can use the Weyl-Brauer matrices.