The Dgamma[mu] command is used to represent the Dirac  matrix, where  ranges from 1 to the dimension d of spacetime; these are non-commutative objects satisfying

where the products in the above are noncommutative, constructed by using the `*` operator of the Physics package, and  is the metric tensor. When the spacetime is Euclidean (signature = `+`), g_[mu, nu] is replaced by delta[mu, nu], the Kronecker symbol. The properties of the Dirac matrices are derived from the defining relation above, which is invariant under a unitary transformation, thus these matrices are determined up to a transformation of that kind, and conventions are necessary to construct their representations. The Physics package's commands know about the properties of the Dirac matrices, and in particular the Trace command computes traces of products of gammas taking these properties into account.

The most common representations for the Dirac matrices are the standard (also known as Dirac), the chiral (also known as Weyl), and the Majorana representations. The conventions for the standard representation are uniform in the literature: in a Minkowski (also called pseudo-Euclidean) spacetime (signature = `-`), the contravariant Dirac matrices are:

Dgamma[0] = Matrix(2, [[Id, 0], [0, -Id]]);

where Id represents the 2 x 2 identity matrix,

Dgamma[j] = Matrix(2, [[0, sigma[j]], [-sigma[j], 0]]);

where  are the Pauli matrices, in the Physics package represented by Psigma, and  ranges from 1 to , where  is the dimension of the spacetime.

The conventions for the chiral and Majorana representations are not uniform in the literature. The conventions adopted here are the same ones shown in Wikipedia, so that in the chiral representation, the Dgamma[j] are the same as in the standard representation, while Dgamma[0] changes to

Dgamma[0] = Matrix(2, [[0, Id], [Id, 0]]);

The convention implemented for the Majorana representation, that is, a representation where all the components of the Dirac matrices are imaginary, is

Dgamma[0] = Matrix(2, [[0, sigma[2]], [sigma[2], 0]]),
Dgamma[1] = Matrix(2, [[I*sigma[3], 0], [0, I*sigma[3]]]),
Dgamma[2] = Matrix(2, [[0, -sigma[2]], [sigma[2], 0]]),
Dgamma[3] = Matrix(2, [[-I*sigma[1], 0], [0, -I*sigma[1]]]);

Note that the form of the Dirac matrices depends on the signature of spacetime: in a Euclidean spacetime (signature = `+`), for the standard, chiral, and Majorana representations, that form is obtained from the formulas above by performing a Wick rotation, equivalent to multiplying the Dgamma[j] by , while Dgamma[0] = Dgamma[d] remains unchanged, where  is the dimension of spacetime.

In all of these representations, in a four dimensional spacetime, a matrix Dgamma[5] satisfying

is given in the case of a Minkowski (signature = `-`) spacetime by:

where  is the imaginary unit (to represent it with a lowercase , see interface). In a four dimensional Euclidean spacetime, Dgamma[5] is given by

where .

When the Physics package is loaded, no particular representation is enforced, and the symbol Dgamma[mu] just represents the corresponding Dirac matrix. To set a representation and concretely make Dgamma[mu] be the corresponding matrix, use the Setup command. For example, enter Physics[Setup](Dgammarepresentation = standard); at the Maple prompt.

The following are some representation-free frequently used identities for the Dirac matrices, valid provided the dimension, , is greater than 1, expressed by using the sum rule for repeated indices:

Some representation-free identities for the traces of products of Dirac matrices in four dimensions are:

where in these formulas, Trace is the Physics command to compute traces, g_ is the metric, epsilon is the Levi-Civita totally antisymmetric symbol, and  is the imaginary unit.