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 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 positive entries, negative entries, and. Set. The standard Dirac matrices correspond to taking and or.
In higher 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 for the Clifford algebra, then the gamma group generated by is isomorphic to the multiplicative subgroup generated by the basis elements .
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 -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 dimensions, the matrix products behave just as the conventional Dirac matrices. The Pauli group is a representation of the gamma group for although the Pauli group has more relationships ; see the note about the chiral element below for an example. The quaternions provide a representation for
The presentation of the gamma group is as follows.
These generators completely define the gamma group. It can be shown that, for all that and so Every element can be uniquely written as a product of a finite number of generators placed in canonical order as
with the indexes in ascending order
and The gamma group is finite, and has at most elements in it.
The gamma group is a 2-group but not a regular p-group. The commutator subgroup is 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, and in physics, where the "main involution" corresponds to a combined P-symmetry and T-symmetry.

Transposition

Given elements of the generating set of the gamma group, the transposition or reversal is given by
If there are elements all distinct, then

Hermitian conjugation

Another automorphism of the gamma group is given by conjugation, defined on the generators as
supplemented with and For general elements in the group, one takes the transpose: From the properties of transposition, it follows that, for all elements that either or that that is, all elements are either Hermitian or unitary.
If one interprets the dimensions as being "time-like", and the 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 is associated with the time-like direction, and the 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: but leaves alone: This map corresponds to the combined P-symmetry and T-symmetry in physics; all directions are reversed.

Chiral element

Define the chiral element as
where. The chiral element commutes with the generators as
It squares to
For the Dirac matrices, the chiral element corresponds to thus its name, as it plays an important role in distinguishing the chirality of spinors.
For the Pauli group, the chiral element is whereas for the gamma group, one cannot deduce any such relationship for other than that it squares to This is an example of where a representation may have more identities than the represented group. For the quaternions, which provide a representation of the chiral element is

Charge conjugation

None of the above automorphisms are inner automorphisms; that is they cannot be represented in the form for some existing element 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; and other products are undetermined.

Matrix representation

The gamma group has a matrix representation given by complex matrices with and and the floor function, the largest integer less than or equal to 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 and so. 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 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. 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 exist, in odd dimension just one.
d

Note that 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 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 2d such products, but only 2 are independent, spanning the space of × matrices.
Typically, provide the 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 and.
A matrix is called symmetric if
otherwise, for a − sign, it is called antisymmetric.
In the previous expression, can be either or. In odd dimension, there is no ambiguity, but in even dimension it is better to choose whichever one of or allows for Majorana spinors. In = 6, there is no such criterion and therefore we consider both.
dCSymmetricAntisymmetric

Identities

The proof of the trace identities for gamma matrices hold for all even dimension. One therefore only needs to remember the 4D case and then change the overall factor of 4 to. For other identities, explicit functions of 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

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, = +1 and = −1, one may fix all subsequent signs 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. Once a permutation is applied to make the +1 and -1 eigenvalues of the chiral matrix consecutive, this choice becomes the direct analogue of the chiral basis in four dimensions.

Generic odd ''d'' = 2''k'' + 1

Consider the previous construction for − 1 and simply take all matrices, to which append its. .
Finally, compute the charge conjugation matrix: choose between and, in such a way that transforms as all the other matrices. Explicitly, require
As the dimension ranges, patterns typically repeat themselves with period 8.

General reading

  • de Wit, Bryce and Smith, J.. Field Theory in Particle Physics, Volume 1, Paperback,,
  • Murayama, H..
  • Pietro Giuseppe Frè. "Gravity, a Geometrical Course: Volume 1: Development of the Theory and Basic Physical Applications." Springer-Verlag.. ''See pp 315ff.''