Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 46 / 1 / a

For the quantum rotor, the transition kernel is the path integralThe integral includes all paths on the circle, not just paths whose chosen real lifts have the same endpoint difference. Operator evolution and the stated energy-state normalization giveThe factor $1/(2\pi )$ comes from the completeness relation; it makes the zero-time limit the periodic Dirac delta distribution. For real time the oscillatory sum is understood through $T\to T-i0$, or as a distributional limit.