Gamma Matrices

Abstract

In mathematical physics, the gamma matrices, {γ 0 , γ 1 , γ 2 , γ 3 } , also known as the Dirac matri-ces, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cℓ₁,₃(R). It is also possible to define higher-dimensional gamma matrices. When interpreted as the matrices of the action of a set of orthogonal basis vectors for contravariant vectors in Minkowski space, the column vectors on which the matrices act become a space of spinors, on which the Clifford algebra of spacetime acts. This in turn makes it possible to represent infinitesimal spatial rotations and Lorentz boosts. Spinors facilitate spacetime computations in general, and in particular are fundamental to the Dirac equation for relativistic spin-½ particles. In Dirac representation, the four contravariant gamma matrices are γ 0 =     1 0 0 0 0 1 0 0 0 0 −1 0 0 0 0 −1     , γ 1 =     0 0 0 1 0 0 1 0 0 −1 0 0 −1 0 0 0     γ 2 =     0 0 0 −i 0 0 i 0 0 i 0 0 −i 0 0 0     , γ 3 =     0 0 1 0 0 0 0 −1 −1 0 0 0 0 1 0 0     . γ 0 is the time-like matrix and the other three are space-like matrices. Analogous sets of gamma matrices can be defined in any dimension and signature of the metric. For example, the Pauli matrices are a set of " gamma " matrices in dimen-sion 3 with metric of Euclidean signature (3,0). In 5 spacetime dimensions, the 4 gammas above together with the fifth gamma matrix to be presented below generate the Clifford algebra.