Physics – Mathematical Physics
Scientific paper
2011-06-08
Physics
Mathematical Physics
9 pages, no figures
Scientific paper
Let \Gamma_{a} be Dirac matrices in d-dimensional Minkowski spacetime, and let \beta_{i} = B_{i}^{ab} \Gamma_{ab}, where \Gamma_{ab} = \Gamma_{[a} \Gamma_{b]} and B_{i}^{ab} are arbitrary antisymmetric tensors. The trace of the symmetrized product of an odd number of \beta-matrices vanishes identically. The trace of the symmetrized product of 2n \beta-matrices can be written as a sum of certain B-contractions over the integer partitions of n, with every term being multiplied by a numerical factor \alpha. We provide a general algorithm to compute these \alpha-coefficients for any d and up to any desired value of n. The algorithm uses random matrices to generate a linear system of equations whose solution is the set of coefficients for a given n. A recurrence relation among these coefficients is shown to hold in all analyzed cases and is used to greatly simplify the computation for large values of n. Numerical values for the \alpha-coefficients are given for n = 1, ..., 7.
Izaurieta Fernando
Ramirez Ricardo
Rodríguez Eduardo
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