General (anti-)commutators of gamma matrices

Commutators and anticommutators of gamma matrices with arbitrary numbers of (antisymmetrized) indices are derived. Gamma matrix algebra is ubiquitous in many calculations in high energy physics. Whereas it is a fairly simple business in four dimensions, in higher-dimensional applications such as supergravity or M-theory, it becomes quite involved, because the number of independent matrices grows quickly with the number of space-time dimensions. To ease such calculations, on the one hand, one can resort to the help of computer algebra packages [1, and references therein]. On the other hand, one can look in the literature for reference tables, such as the appendix of [2], where the commutators and anti-commutators of gamma matrices with up to four indices are listed. The table in [2] is, to my knowledge, the most complete such list, but it contains typographical errors, as has been noticed in [3]. The purpose of this short note is to derive general formulae for the commutators and anti-commutators of gamma matrices with any number of indices. A general treatment is possible, because the (anti-)commutators do not depend on the space-time dimension, d, except for the fact that the number of indices any gamma matrix can carry is limited by d. The final formulae take the form of explicit sums and do not involve recursion relations. Consider the d-dimensional Clifford algebra generated by the matrices γ (i = 1, . . . , d), which satisfy γγ + γγ = 2g , (1) where g is the inverse metric tensor. Throughout this paper, indices shall be raised and lowered with g and gij , respectively. Note that the metric gij can be curved or flat, and also its signature will be irrelevant for what follows. A useful basis of the Clifford algebra of matrices is given by the antisymmetrized products of the γi,1 γ1k = γ1γ2 · · · γk (1 ≤ k ≤ d) , (2) The antisymmetrization includes a factor 1/k! for normalization. For odd d, the basis such defined is overcomplete, but this will not influence our analysis. 1 and by the identity matrix, which we may include in the notation (2) by allowing also for k = 0. We will formally allow also for k > d implying that the corresponding matrix vanishes due to the antisymmetrization of the indices. We shall obtain general formulae for all commutators and anti-commutators of the γ-matrices (2). A useful notation we will employ is the generalized commutator bracket [a, b]x = ab+ xba , (3) where x = ±1. Our convention for the generalized Kronecker delta symbol is δ i1···ik j1···jk = δ [i1 j1 · · · δ ik ] jk . (4) Let us start with the easiest piece and write γjγ i1···ik = γjγ [i1 · · · γk = −γγjγ i2 · · · γk + 2δ [i1 j γ i2 · · · γk . After pulling γj through the other matrices, we end up with [ γj , γ i1···ik ] (−1)k+1 = 2kδ [i1 j γ i2···ik] . (5) This is a commutator for even k and an anti-commutator for odd k. Finding the other bracket (anti-commutator for even k, commutator for odd k) is best done using induction. Let us assume that, for some k, the following relation holds: [ γj , γ i1···ik ] (−1) = 2γj i1···ik . (6) Consider γj i1···ik+1 and rewrite it as 2γj i1···ik+1 = 2 k + 2 ( γjγ i1···ik+1 − (k + 1)γ1γj i2···ik+1] ) . (7) Applying the hypothesis (6) on the second term in the parentheses and using then (1) and (5), one obtains after a bit of algebra 2γj i1···ik+1 = γjγ i1···ik+1 + (−1)γ1k+1γj = [ γj , γ i1···ik+1 ] (−1)k+1 . (8) Thus, if the hypothesis (6) is valid for some k, then it will also hold for k + 1. Therefore, as (6) holds for k = 1 by the definition of γ , we have shown that it holds for any k. After this little exercise, we are ready to face the general cases [ γj1···jl , γ i1···ik ] ± . Our hypotheses, which we shall prove again by induction, are the following: [ γj1···jl, γ i1···ik ]