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. The Weyl–Brauer matrices provide an explicit construction of higher-dimensional gamma matrices for Weyl spinors. Gamma matrices also appear in generic settings in Riemannian geometry, particularly when a spin structure can be defined.
Introduction.
Consider a space-time of dimension with the flat Minkowski metric, 
with formula_2 positive entries, formula_3 negative entries, formula_4 and . Set . The standard Dirac matrices correspond to taking and or . 
In higher (and lower) dimensions, one may define a group, the gamma group, behaving in the same fashion as the Dirac matrices. More precisely, if one selects a basis formula_5 for the (complexified) Clifford algebra formula_6, then the gamma group generated by formula_7 is isomorphic to the "multiplicative" subgroup generated by the basis elements formula_8 (ignoring the additive aspect of the Clifford algebra). 
By convention, the gamma group is realized as a collection of matrices, the gamma matrices, although the group definition does not require this. 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. Several matrix representations are possible, some given below, and others in the article on the Weyl–Brauer matrices. In the matrix representation, the spinors are formula_9-dimensional, with the gamma matrices acting on the spinors. 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.
Gamma group.
Most of the properties of the gamma matrices can be captured by a group, the gamma group. This group can be defined without reference to the real numbers, the complex numbers, or even any direct appeal to the Clifford algebra. 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. For formula_10 dimensions, the matrix products behave just as the conventional Dirac matrices. The Pauli group is a representation of the gamma group for formula_11 although the Pauli group has more relationships (is less free); see the note about the chiral element below for an example. The quaternions provide a representation for formula_12
The presentation of the gamma group formula_13 is as follows.
These generators completely define the gamma group. It can be shown that, for all formula_25 that formula_26 and so formula_27 Every element formula_25 can be uniquely written as a product of a finite number of generators placed in canonical order as
with the indexes in ascending order
The gamma group is a 2-group but not a regular p-group. The commutator subgroup (derived subgroup) is formula_32 therefore it is not a powerful p-group. 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" formula_33 corresponds to a combined P-symmetry and T-symmetry.
Transposition.
Given elements formula_34 of the generating set of the gamma group, the transposition or "reversal" is given by
If there are formula_36 elements formula_34 all distinct, then
Hermitian conjugation.
Another automorphism of the gamma group is given by conjugation, defined on the generators as
supplemented with formula_40 and formula_41 For general elements in the group, one takes the transpose: formula_42 From the properties of transposition, it follows that, for all elements formula_25 that either formula_44 or that formula_45 that is, all elements are either Hermitian or unitary. 
If one interprets the formula_2 dimensions as being "time-like", and the formula_3 dimensions as being "space-like", then this corresponds to P-symmetry in physics. That this is the "correct" identification follows from the conventional Dirac matrices, where formula_48 is associated with the time-like direction, and the formula_49 the spatial directions, with the "conventional" (+---) metric. Other metric and representational choices suggest other interpretations.
Main involution.
The main involution is the map that "flips" the generators: formula_50 but leaves formula_15 alone: formula_52 This map corresponds to the combined P-symmetry and T-symmetry in physics; all directions are reversed.
Chiral element.
Define the chiral element formula_53 as
where formula_55. The chiral element commutes with the generators as
It squares to
For the Dirac matrices, the chiral element corresponds to formula_58 thus its name, as it plays an important role in distinguishing the chirality of spinors.
For the Pauli group, the chiral element is formula_59 whereas for the gamma group formula_60, one cannot deduce any such relationship for formula_61 other than that it squares to formula_62 This is an example of where a representation may have more identities the represented group. For the quaternions, which provide a representation of formula_63 the chiral element is formula_64
Charge conjugation.
None of the above automorphisms (transpose, conjugation, main involution) are inner automorphisms; that is they "cannot" be represented in the form formula_65 for some existing element formula_66 in the gamma group, as presented above. Charge conjugation requires extending the gamma group with two new elements; by convention, these are
and
The above relations are not sufficient to define a group; formula_69 and other products are undetermined.
Matrix representation.
The gamma group has a matrix representation given by complex formula_70 matrices with formula_71 and formula_55 and formula_73 the floor function, the largest integer less than formula_74 The group presentation for the matrices can be written compactly in terms of the anticommutator relation from the Clifford algebra 
where the matrix is the identity matrix in dimensions. Transposition and Hermitian conjugation correspond to their usual meaning on matrices.
Charge conjugation.
For the remainder of this article,it is assumed that formula_76 and so formula_77. That is, the Clifford algebra is assumed. 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 formula_79 where the element on the left is the abstract group element, and the one on the right is the literal matrix transpose.
As before, the generators all generate the same group (the generated groups are all isomorphic; the operations are still involutions). However, since the are now matrices, it becomes plausible to ask whether there is a matrix that can act as a similarity transformation that embodies the automorphisms. In general, such a matrix can be found. 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. In even dimension both formula_82 exist, in odd dimension just one.
Note that formula_83 is a basis choice.
Symmetry properties.
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. Since distinct formula_18 anti-commute this motivates the introduction of an anti-symmetric "average". We introduce the anti-symmetrised products of distinct -tuples from 0...,−1:
where runs over all the permutations of symbols, and is the alternating character. There are 2"d" such products, but only 2 are independent, spanning the space of × matrices.
Typically, provide the (bi)spinor representation of the generators of the higher-dimensional Lorentz group, , generalizing the 6 matrices "σμν" of the spin representation of the Lorentz group in four dimensions.
For even , one may further define the hermitian "chiral matrix" 
such that = 0 and =1. (In odd dimensions, such a matrix would commute with all "a"s and would thus be proportional to the identity, so it is not considered.)
A matrix is called symmetric if
otherwise, for a − sign, it is called antisymmetric.
In the previous expression, can be either formula_89
or formula_90. In odd dimension, there is no ambiguity, but
in even dimension it is better to choose whichever one of formula_89
or formula_90 allows for Majorana spinors. In =6, there is no such
criterion and therefore we consider both.
Identities.
The proof of the trace identities for gamma matrices is independent of dimension. One therefore only needs to remember the 4D case and then change the overall factor of 4 to formula_93. For other identities (the ones that involve a contraction), explicit functions of formula_94 will appear.
Even when the number of physical dimensions is four, these more general identities are ubiquitous in loop calculations due to dimensional regularization.
Example of an explicit construction in the chiral basis.
The matrices can be constructed recursively, first in all even dimensions, = 2, and thence in odd ones, 2+1.
"d" = 2.
Using the Pauli matrices, take
and one may easily check that the charge conjugation matrices are
One may finally define the hermitian chiral chir to be
Generic even "d" = 2"k".
One may now construct the , matrices and the charge conjugations (±) in +2 dimensions, starting from the , (), and (±) matrices in dimensions.
Explicitly, 
One may then construct the charge conjugation matrices,
with the following properties,
Starting from the sign values for =2, (2,+)=+1 and (2,−)=−1, one may fix all subsequent signs ("d",±) which have periodicity 8; explicitly, one finds
Again, one may define the hermitian chiral matrix in +2 dimensions as
which is diagonal by construction and transforms under charge conjugation as
It is thus evident that = 0.
Generic odd "d" = 2"k" + 1.
Consider the previous construction for −1 (which is even) and simply take all matrices, to which append its . (The is required in order to yield an antihermitian matrix, and extend into the spacelike metric).
Finally, compute the charge conjugation matrix: choose between formula_89 and formula_90, in such a way that transforms as all the other matrices. Explicitly, require
As the dimension ranges, patterns typically repeat themselves with period 8. (cf. the Clifford algebra clock.)