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

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.

The Euler-Lagrange equation gives $\ddot{\theta }=0$. Lift the path to the real line beginning at ${\theta }_i$. Its endpoint can be any ${\theta }_f+2\pi m$, where $m\in \mathbb{Z}$ labels the relative winding number. With $\Delta ={\theta }_f-{\theta }_i$, every classical path and its action areThese exhaust the constant-velocity paths in the different winding sectors.

Apply the Poisson summation formula in the convention supplied, takingCompleting the square in its Fourier transform and evaluating the analytically continued Gaussian integral givesThus the spectral sum becomesThe square-root branch is fixed by continuation from positive imaginary-time damping. Each term is the classical contribution of part (b), multiplied by the same fluctuation prefactor. Since the action is quadratic, writing $\theta ={\theta }_m+\eta$ with $\eta (0)=\eta (T)=0$ removes the cross term and leaves a free Gaussian fluctuation integral independent of $m$. This real-time winding representation of a quantum rotor kernel is exact, rather than only a leading semiclassical approximation.