Acta Physica Academiae Scientiarum Hungaricae, Tomus 30 (4), pp. 351 364 (1971) ON THE TRACE OF THE PRODUCT OF PAULI AND DIRAC MATRICES By S. SARKAR DEPAHTMENT OF TltEORETICAL PHYSICS IND1AN ASSOCIATION FOR THE CULTIVATION ab SCIENCE JADAVPUR, CALCUTTA - 32, INDIA (Received 5. V. 1970) Several methods for the determination of the trace of the product of an arbitrary num- ber of Pauli matrices are established. Formulae are derived lar the evaluation of various types of products of two traces when terms ~~ftheŸ ailai2... ~in occur in bothof them. Expressions are found for the product of two different traces and the square of the trace of an arbitrary number of Pauli matrices. Similar formulae are obtained when Dirac matrices occurring as ,~x~ 9,iAiare considered instead ofPauli matrices. From this all previous results in which ~~5 has i_l been considered separately ate recovered. A useful identity lar traces involving either Pauli of Dirac matrices is given. Introduction arte of the purposes of this paper is to reduce the problem of the calcula- tion of the trace of the product of any odd or even number of Pauli matrices, to one involving a smaller number -- in the final stage, two of three -- of Pauli matrices. First the formulae lar the determination of various types of products of two traces when a,~a,2.-" ai~ occur in both traces (summation ayer the dummy suffixes i~ is implied) are derived and then expressions for the product of two different traces and the square of the trace of an arbitrary number of Pauli matrices. Next, the five Dirac matrices 7i, 7_0, y~, 7~ and 7~ are considered simultaneously instead of the Pauli matrices and similar expres- sions are obtained. More explicitly, the Dirac matrices occur in the trace as 5 product of an arbitrary number of elements like~, AiT;. From the results i=1 obtained in this second stage all the results of the author's previous paper [1], 4 in which Dirac matrices occur in the form ~ Aiy i and ~~ may occur separately, i=l can be reproduced. An identity for traces involving either Pauli or Dirac matrices has been established. This is found to be useful in the reduction of the formulae and in demonstrating the equivalence of some of the results in our deduction. In this connection it should be mentioned that CHISHOLM [2] 3 3 has evaluated the sums ~ a~aaa~... ~da~ and ~,... ~~... Sp(~~~~~~...~r#). r=i r=i C~ISHOLM [2] has also solved the same problem for Dirac matrices. CAIANIELLO Acta Physica Academiae Scientiarum Hungaricae 30, 1971