Higher-dimensional gamma matrices

In mathematical physics, higher-dimensional gamma matrices generalize to arbitrary dimension the four-dimensional Gamma matrices of Dirac, which are a mainstay of relativistic quantum mechanics. They are utilized in relativistically invariant wave equations for fermions (such as spinors) in arbitrary space-time dimensions, notably in string theory and supergravity.

Consider a space-time of dimension $d$ with the flat Minkowski metric,

where $a,b = 0,1, ..., d -1$ . Set $N = 2 ⌊ d /2⌋$ . The standard Dirac matrices correspond to taking $d = N = 4$ .

The higher gamma matrices are a $d$ -long sequence of complex $N \times N$ matrices $\Gamma _{i},\ i=0,\ldots ,d-1$ which satisfy the anticommutator relation from the Clifford algebra $Cℓ 1,d-1 (R)$ (generating a representation for it),

where $I N$ is the identity matrix in $N$ dimensions. (The spinors acted on by these matrices have $N$ components in $d$ dimensions.) Such a sequence exists for all values of $d$ and can be constructed explicitly, as provided below.

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