 Solution

ID: past-exam-of-the-theoretical-physics-ii-course-of-the-university-of-cambridge/2012/paper-1/1/c/solution

In a finite-dimensional basis, interchange the two finite sums:

Tr (A B) = \sum_{j, k} A_{jk} B_{kj} = \sum_{k, j} B_{kj} A_{jk} = Tr (B A) .

A B = B A

or permit arbitrary reordering of three factors. For infinite-dimensional linear operators, the traces and rearrangements need hypotheses, for example one trace-class operator and one bounded operator; an arbitrary pair of unbounded linear operators is not covered.

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