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Higher dimensional gamma matrices

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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.

Contents

Consider a space-time of dimension d with the flat Minkowski metric,

η =∥ η a b ∥= diag ( + 1 , − 1 , … , − 1 )   ,

where a,b = 0,1, ..., d−1. Set N= 2⌊d/2⌋. The standard Dirac matrices correspond to taking d = N = 4.

The higher gamma matrices are a d-long sequence of complex N×N matrices Γ i ,   i = 0 , … , d − 1 which satisfy the anticommutator relation from the Clifford algebra Cℓ1,d−1(R) (generating a representation for it),

{ Γ a   ,   Γ b } = Γ a Γ b + Γ b Γ a = 2 η a b I N   ,

where IN is the identity matrix in N dimensions. (The spinors acted on by these matrices have N components in d dimensions.) Such a sequence exists for all values of d and can be constructed explicitly, as provided below.

The gamma matrices have the following property under hermitian conjugation,

Γ 0 † = + Γ 0   ,   Γ i † = − Γ i   ( i = 1 , … , d − 1 )   .

Charge conjugation

Since the groups generated by Γa, −ΓaT, ΓaT are the same, we can look for a similarity transformation which connects them all. This transformation is generated by a respective charge conjugation matrix.

Explicitly, we can introduce the following matrices

C ( + ) Γ a C ( + ) − 1 = + Γ a T C ( − ) Γ a C ( − ) − 1 = − Γ a T   .

They can be constructed as real matrices in various dimensions, as the following table shows. In even dimension both C ± exist, in odd dimension just one.

Symmetry properties

We denote a product of gamma matrices by

Γ a b c … = Γ a ⋅ Γ b ⋅ Γ c ⋯  

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 Γ a anti-commute this motivates the introduction of an anti-symmetric "average". We introduce the anti-symmetrised products of distinct n-tuples from 0,...,d−1:

Γ a 1 … a n = 1 n ! ∑ π ∈ S n ϵ ( π ) Γ a π ( 1 ) ⋯ Γ a π ( n )   ,

where π runs over all the permutations of n symbols, and ϵ is the alternating character. There are 2d such products, but only N2 are independent, spanning the space of N×N matrices.

Typically, Γab provide the (bi)spinor representation of the d(d−1)/2 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.

For even d, one may further define the hermitian chiral matrix

Γ chir = i d / 2 − 1 Γ 0 Γ 1 … Γ d − 1   ,

such that {Γchir , Γa} = 0 and Γchir2=1. (In odd dimensions, such a matrix would commute with all Γas and would thus be proportional to the identity, so it is not considered.)

A Γ matrix is called symmetric if

( C Γ a 1 … a n ) T = + ( C Γ a 1 … a n )   ;

otherwise, for a − sign, it is called antisymmetric. In the previous expression, C can be either C ( + ) or C ( − ) . In odd dimension, there is no ambiguity, but in even dimension it is better to choose whichever one of C ( + ) or C ( − ) allows for Majorana spinors. In d=6, there is no such criterion and therefore we consider both.

Example of an explicit construction in the chiral basis

The Γ matrices can be constructed recursively, first in all even dimensions, d= 2k, and thence in odd ones, 2k+1.

d = 2

Using the Pauli matrices, take

γ 0 = σ 1   ,   γ 1 = − i σ 2

and one may easily check that the charge conjugation matrices are

C ( + ) = σ 1 = C ( + ) ∗ = s ( 2 , + ) C ( + ) T = s ( 2 , + ) C ( + ) − 1 s ( 2 , + ) = + 1 C ( − ) = i σ 2 = C ( − ) ∗ = s ( 2 , − ) C ( − ) T = s ( 2 , − ) C ( − ) − 1 s ( 2 , − ) = − 1   .

One may finally define the hermitian chiral γchir to be

γ chir = γ 0 γ 1 = σ 3 = γ chir †   .

Generic even d = 2k

One may now construct the Γa , (a=0, ... , d+1), matrices and the charge conjugations C(±) in d+2 dimensions, starting from the γa' , ( a' =0, ... , d−1), and c(±) matrices in d dimensions.

Explicitly,

Γ a ′ = γ a ′ ⊗ σ 3       ( a ′ = 0 , … , d − 1 )   ,       Γ d = I ⊗ ( i σ 1 )   ,       Γ d + 1 = I ⊗ ( i σ 2 )   .

One may then construct the charge conjugation matrices,

C ( + ) = c ( − ) ⊗ σ 1   , C ( − ) = c ( + ) ⊗ ( i σ 2 )   ,

with the following properties,

C ( + ) = C ( + ) ∗ = s ( d + 2 , + ) C ( + ) T = s ( d + 2 , + ) C ( + ) − 1 s ( d + 2 , + ) = s ( d , − ) C ( − ) = C ( − ) ∗ = s ( d + 2 , − ) C ( − ) T = s ( d + 2 , − ) C ( − ) − 1 s ( d + 2 , − ) = − s ( d , + )   .

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

Γ chir = α d + 2 Γ 0 Γ 1 … Γ d + 1 = γ chir ⊗ σ 3   ,             α d = i d / 2 − 1   ,

which is diagonal by construction and transforms under charge conjugation as

C ( ± ) Γ chir C ( ± ) − 1 = β d + 2 Γ chir T                 β d = ( − ) d ( d − 1 ) / 2   .

It is thus evident that {Γchir , Γa} = 0.

Generic odd d = 2k + 1

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).

Finally, compute the charge conjugation matrix: choose between C ( + ) and C ( − ) , in such a way that Γd−1 transforms as all the other Γ matrices. Explicitly, require

C ( s ) Γ chir C ( s ) − 1 = β d Γ chir T = s Γ chir T   .

As the dimension d ranges, patterns typically repeat themselves with period 8. (cf. the Clifford algebra clock.)

References

Higher-dimensional gamma matrices Wikipedia
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