The ten-dimensional case is explained in some detail in appendix 4.A of volume one of Green-Schwarz-Witten. Let me therefore consider the 4d case here. The calculation works essentially the same in 3d, 6d and 10d, though you can sometimes make use of Majorana and/or Weyl conditions of the spinors to simplify things.
We want to check that the expression
fabc(ˉλaγμλbˉϵγμλc−ˉλaγμλbˉλcγμϵ)
vanishes, where
λa and
ϵ are four-dimensional Majorana spinors. To show this the following identities involving four-dimensional gamma matrices and arbitrary spinors will be useful
ˉφψ=+ˉψφ,ˉφγμψ=−ˉψγμφ,ˉφγμνψ=−ˉψγμνφ,ˉφγμνρψ=+ˉψγμνρφ,ˉφγ0123ψ=+ˉψγ0123φ,
where, eg.,
γμν=12(γμγν−γνγμ).
Using the second identity we see that the two terms in (1) are equal.
Using the antisymmetry of fabc we can rewrite (1) as
(ˉλaγμλb)(ˉϵγμλc)+(ˉλcγμλa)(ˉϵγμλb)+(ˉλbγμλc)(ˉϵγμλa),
or, if we remove the spinors,
(γ0γμ)αβ(γ0γμ)γδ+(γ0γμ)δα(γ0γμ)γβ−(γ0γμ)βδ(γ0γμ)γα.
Finally we contract this expression with two arbitrary spinors
ψα and
φβ to obtain (here I have also removed an overall
γ0 and reordered the second term slightly)
(ˉψγμφ)(γμ)γδ+(ˉψγμ)δ(γμφ)γ−(ˉφγμ)δ(γμψ)γ.
The idea is to now think of this as a
4×4 matrix. A basis of such matrices is given by the 16 matrices
1,γμ,γμν,γμνρ,γ0123.
This basis is orthogonal in the sense that if we call the sixteen matrices
γ(I), with
I=1,…,16, then
Tr(γ(I)γ(J))=Tr(γ(I)γ(I))δIJ.
Hence, to prove that (3) vanishes we can check that we get zero when contracting with each of the matrices in (4).
It is easy to see that the first term in (3) is zero unless we contract with a single gamma matrix, so let us first contract with
γν. We then get
(ˉψγμφ)Tr(γμγν)+ˉψγμγνγμφ−ˉφγμγνγμψ=4ημνˉψγμφ−2ˉψγνφ+2ˉφγνψ=0,
where I have used equation (2b) as well the second of the following useful relations for four-dimensional gamma matrices
γμγμ=4,γμγνγμ=−2γν,γμγνργμ=0,γμγνρσγμ=+2γνρσ,γμγ0123γμ=−4γ0123.
If we instead contract (3) with
γνρ each term vanishes separately because of equation (5c). In the remaining three cases (contracting with
1,
γνρσ or
γ0123), the first term in (3) is zero and the second two terms cancel each other because of the positive sign in (2a), (2d) and (2e).
We have now shown that all components of the 4×4 matrix in (3) vanish. Since the spinors ψ and φ were arbitrary this means that (1) is zero as well.
The above procedure of expanding in a basis of (products of) gamma matrices is equivalent to, but more transparent than, using Fierz identities.
This post imported from StackExchange Physics at 2016-01-29 15:22 (UTC), posted by SE-user Olof