This note discusses the details of the -dimensional Dirac algebra as implemented in Matchete. At present Matchete operates with an anti-commuting implementation of γ₅ (Naive Dimensional Regularization), which implies that the Dirac trace is non-cyclic in order to avoid mathematical inconsistencies.

Thus, γ₅ is treated as a purely symbolic object and does not equate to an antisymmetric combination of four ordinary Dirac matrices.

There are primarily two methods relevant for the Dirac algebra:

Matrix multiplication in the space of Dirac algebra is done with NonCommutativeMultiply, which can be input with the in build Mathematica notation "**." This Dirac spinors in the form of fermion fields, can also multiply in to the spin chains, created with NonCommutativeMultiply. All Dirac matrices in a spin chain get merged into a DiracProduct head.

The charge conjugation matrices, CC/GammaCC, in -dimensions, are defined by the properties

These properties are a continuation of the four-dimensional identities (with the choice of a real C)

The basis for the -dimensional Dirac algebra is infinite dimensional. A suitable basis is formed by all anti-symmetrized products of the gamma matrices. We use the notation

for these basis tensors, and they can be entered conveniently as, e.g.,

Matchete, therefore, do not operator with the familiar sigma tensor, but rather uses Γ_μν= - σ_μν.

Dirac traces of a product of Dirac matrices can be performed with DiracTrace. Dirac tracing using anti-commuting γ₅ is known to cause mathematical ambiguities.

For traces with and even number of γ₅ matrices, the γ₅ matrices can be anti-commuted until they are next to each other and produce the identity. These traces are unambiguous and reproduce the Four-dimensional result, but with -dimensional metrics (unless some of the indices are contracted).

In traces with an odd number of γ₅ matrices, mathematical consistency can be obtained only by giving up on trace cyclicity; that is, in traces with six or more ordinary Dirac matrices in addition to γ₅, we cannot freely cycle matrices in the trace. This introduces an ambiguous (ϵ) piece of the trace, which depends on where the trace is started. A definite value is assigned to the γ₅-odd traces following from the KKS prescription. This prescription relies on the unique projection of gamma matrices onto the basis of anti-symmetric combinations, Γ_μ1μ2…. We define

which is consistent with other NDR manipulation of the Dirac matrices inside the trace (though of course not cyclicity!). As any product of Dirac matrices can be written in terms of the anti-symmetric products, the trace of γ₅-odd products follows.