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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2017/iii/paper-301/4/a/solution

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Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 301 / 4 / a / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-05

For ordinary bosonic operators at distinct times, time ordering places the later operator to the left:

T {O_{1} (t_{1}) O_{2} (t_{2})} = {O_{1} (t_{1}) O_{2} (t_{2}), O_{2} (t_{2}) O_{1} (t_{1}), t_{1} > t_{2}, t_{2} > t_{1} .

(1)

Equivalently, multiply these two products by the corresponding Heaviside step functions. For two odd fermionic operators the second product receives a minus sign; more generally a graded interchange has sign

(- 1)^{F_{1} F_{2}}

. This sign is the fermionic sign convention for time-ordered field products, not a minus sign for arbitrary bosonic operators. No equal-time convention is needed for the two distinct times in this clause.

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