Part III Quantum Field Theory Example Sheet , Michælmas Jacob

anam = C1pnpm + C2qnqm + C3pnqm + C4qnpm, = C1pnpm + C2qnqm + C3pnqm + iC4δmn + C4pmqn, where Ci are arbitrary (commuting) functions of the indices m and n. This expression is clearly symmetric under the interchange of m,n so that we have ∴ [an, am] = 0. An identical argument—replacing an with an—shows that ∴ [an, am] = 0. This argument does not hold, however, for [an, am] because aman is not merely a permutation of the indices on anam. Nevertheless, looking at the definition of an, a † m, we see that aman = ama † n . Therefore, the commutator is just the 2i times the imaginary part of anam, which is just the coefficient of qnpm in the expansion of ana † m. That is, [an, am] = 2i ∗ ( − i 2 √ ωn ωm δmn )