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.
Gamma matrices also appear in generic settings in Riemannian geometry, particularly when a spin structure can be defined.
is isomorphic to the multiplicative subgroup generated by the basis elements
In particular, many important properties, including the C, P and T symmetries do not require a specific matrix representation, and one obtains a clearer definition of chirality in this way.
A detailed construction of spinors is given in the article on Clifford algebra.
Jost provides a standard reference for spinors in the general setting of Riemmannian geometry.
[1] The matrix representations of this group then provide a concrete realization that can be used to specify the action of the gamma matrices on spinors.
dimensions, the matrix products behave just as the conventional Dirac matrices.
although the Pauli group has more relationships (is less free); see the note about the chiral element below for an example.
In general, 2-groups have a large number of involutions; the gamma group does likewise.
Three particular ones are singled out below, as they have a specific interpretation in the context of Clifford algebras, in the context of the representations of the gamma group (where transposition and Hermitian conjugation literally correspond to those actions on matrices), and in physics, where the "main involution"
all distinct, then Another automorphism of the gamma group is given by conjugation, defined on the generators as supplemented with
This map corresponds to the combined P-symmetry and T-symmetry in physics; all directions are reversed.
The group presentation for the matrices can be written compactly in terms of the anticommutator relation from the Clifford algebra Cℓp,q(R) where the matrix IN is the identity matrix in N dimensions.
Transposition and Hermitian conjugation correspond to their usual meaning on matrices.
[a] In this case, the gamma matrices have the following property under Hermitian conjugation, Transposition will be denoted with a minor change of notation, by mapping
However, since the Γa are now matrices, it becomes plausible to ask whether there is a matrix that can act as a similarity transformation that embodies the automorphisms.
By convention, there are two of interest; in the physics literature, both referred to as charge conjugation matrices.
Explicitly, these are They can be constructed as real matrices in various dimensions, as the following table shows.
We denote a product of gamma matrices by and note that the anti-commutation property allows us to simplify any such sequence to one in which the indices are distinct and increasing.
We introduce the anti-symmetrised products of distinct n-tuples from 0, ..., d − 1: where π runs over all the permutations of n symbols, and ϵ is the alternating character.
Typically, Γab provide the (bi)spinor representation of the 1/2d(d − 1) generators of the higher-dimensional Lorentz group, SO+(1, d − 1), generalizing the 6 matrices σμν of the spin representation of the Lorentz group in four dimensions.
(In odd dimensions, such a matrix would commute with all Γas and would thus be proportional to the identity, so it is not considered.)
The proof of the trace identities for gamma matrices hold for all even dimension.
Even when the number of physical dimensions is four, these more general identities are ubiquitous in loop calculations due to dimensional regularization.
Explicitly, One may then construct the charge conjugation matrices, with the following properties, Starting from the sign values for d = 2, s(2,+) = +1 and s(2,−) = −1, one may fix all subsequent signs s(d,±) which have periodicity 8; explicitly, one finds Again, one may define the hermitian chiral matrix in d+2 dimensions as which is diagonal by construction and transforms under charge conjugation as It is thus evident that {Γchir , Γa} = 0.
Consider the previous construction for d − 1 (which is even) and simply take all Γa (a = 0, ..., d − 2) matrices, to which append its iΓchir ≡ Γd−1.
(The i is required in order to yield an antihermitian matrix, and extend into the spacelike metric).
Explicitly, require As the dimension d ranges, patterns typically repeat themselves with period 8.