On the trace of the product of Pauli matrices occurring asa _r =a _{r i} σ _i +ia _{r 4} and that of the product of dirac matrices, and the interconnection between them

Abstract

In this paper we have evaluatedσ _iuσ_i, σ_iσ_juσ_iσ_j, Tr(σ _i,u)Tr(σ _iu), Tr(σ _iσ_jv) and Tr(u) whereu andv involve Pauli matricesσ _i and the 2×2 unit matrix in the form of products of elements of the typea _r=a _{r i}σ_i+ia _{r 4} with the help of the results of the trace calculation involving Dirac matrices. We have evaluated γ_v Uγ, γ_μ Sγ _μ, γ_μγ_v Uγ _μγ_v, Tr(γ ₅ U)Tr(γ ₅ V), Tr(γ ₅ U) and Tr(U). HereU,V are products of an even number of elements andS, S′are products of an odd number of elements of the typeA _r(=A _{r μ}γ_μ. We have also dealt with the cases in which the dummy suffixesi andμ occurring in some of the above expressions are replaced by ‘a’ which assume any specific value instead of implying a summation. We have considered also the evaluation of the above-mentioned traces when the term, 1 ±γ ₅, occurs within the trace brackets; this is required in the calculation of the traces involvingσ _i and the unit 2×2 matrix. It has been shown that the problem of the trace calculation involving Dirac matrices can be reduced to one involving three Pauli matricesσ _i and the unit 2×2 matrix.