Higher dimensional gamma matrices - Alchetron, the free social encyclopedia

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.

Charge conjugation

Since the groups generated by Γ_a, −Γ_a^T, Γ_a^T 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 2^d such products, but only N² are independent, spanning the space of N×N matrices.

Typically, Γ_ab 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 σ^μν 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 {Γ_chir , Γ_a} = 0 and Γ_chir²=1. (In odd dimensions, such a matrix would commute with all Γ_as 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 γ_chir to be

γ chir = γ 0 γ 1 = σ 3 = γ chir † .

Generic even d = 2k

One may now construct the Γ_a , (a=0, ... , d+1), matrices and the charge conjugations C_(±) in d+2 dimensions, starting from the γ_a' , ( 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 {Γ_chir , Γ_a} = 0.

Generic odd d = 2k + 1

Consider the previous construction for d−1 (which is even) and simply take all Γ_a (a=0, ..., d−2) matrices, to which append its iΓ_chir ≡ Γ_d−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 Γ_d−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

(Text) CC BY-SA

Contents