Gamma matrices

Generators of the Clifford algebra for relativistic quantum mechanics
In mathematical physics, the gamma matrices, also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra 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 particles. Gamma matrices were introduced by Paul Dirac in 1928. In Dirac representation, the four contravariant gamma matrices are

is the time like, Hermitian matrix. The other three are space like, anti Hermitian matrices. More compactly, and where denotes the Kronecker product and the (for ) denote the Pauli matrices. In addition, for discussions of group theory the identity matrix (I) is sometimes included with the four gamma matricies, and there is an auxiliary, "fifth" traceless matrix used in conjunction with the regular gamma matrices

The "fifth matrix" is not a proper member of the main set of four; it is used for separating nominal left and right chiral representations. The gamma matrices have a group structure, the gamma group, that is shared by all matrix representations of the group, in any dimension, for any signature of the metric. For example, the 2×2 Pauli matrices are a set of "gamma" matrices in three dimensional space with metric of Euclidean signature (3, 0). In five spacetime dimensions, the four gammas, above, together with the fifth gamma matrix to be presented below generate the Clifford algebra.

The Clifford algebra in odd dimensions behaves like two copies of the Clifford algebra of one less dimension, a left copy and a right copy. Thus, one can employ a bit of a trick to repurpose as one of the generators of the Clifford algebra in five dimensions. In this case, the set therefore, by the last two properties (keeping in mind that ) and those of the ‘old’ gammas, forms the basis of the Clifford algebra in 5 spacetime dimensions for the metric signature (1,4). In metric signature (4,1), the set {γ^( 0), γ^( 1), γ^( 2), γ^( 3), γ^( 5)} is used, where the γ^( μ) are the appropriate ones for the (3,1) signature. This pattern is repeated for spacetime dimension 2n even and the next odd dimension 2n + 1 for all n ≥ 1. For more detail, see higher dimensional gamma matrices.

The following identities follow from the fundamental anticommutation relation, so they hold in any basis (although the last one depends on the sign choice for ).

In quantum field theory one can Wick rotate the time axis to transit from Minkowski space to Euclidean space. This is particularly useful in some renormalization procedures as well as lattice gauge theory. In Euclidean space, there are two commonly used representations of Dirac matrices: