Past exam of the theoretical physics II course of the University of Cambridge / 2014 / Paper 1 / 2

The quantum propagator is the position-space kernel of the time-evolution operator: $\Psi (x,t)=\int K(x,t|x^{\prime },t^{\prime })\Psi (x^{\prime },t^{\prime })d{x}^{\prime }$. For a time-independent Quantum Hamiltonian, $U=e^{-iH(t-t^{\prime })/\hslash }$. Inserting a complete orthonormal energy eigenbasis givesA continuous spectrum requires the corresponding integral. At equal times, completeness makes this kernel a Dirac delta function.

Let $I=m{R}^2$, $T=t-t^{\prime }>0$, and $\Delta =\theta -{\theta }^{\prime }$. Periodic angular wave functions have modes $e^{il\theta }/\sqrt{2\pi }$ with integer $l$ and energies ${\hslash }^2{l}^2/(2I)$. Normalizing the kernel with respect to $d{\theta }^{\prime }$ givesIf the integration measure is arc length $Rd{\theta }^{\prime }$, the corresponding kernel is $K_{\theta }/R$. This measure choice accounts for the otherwise ambiguous radius prefactor.

Lift the circle to its real-valued angular covering coordinate. The free classical equation is $\ddot{\theta }=0$. For each winding number $p\in \mathbb{Z}$, the endpoint lift is $\theta +2\pi p$, andPositive and negative windings are distinct classical paths with the same physical endpoints. Reducing the angle modulo $2\pi$ returns their trajectories on the ring.

The angular action is $S=\int I{\dot{\theta }}^{2}ds/2$. In each winding sector, write $\theta ={\theta }_p+\xi$ with $\xi$ zero at both endpoints. The cross term vanishes and the Gaussian fluctuation integral is the free-particle prefactor with mass replaced by $I$. ThusEquivalently, multiply the line kernel for displacement $R(\Delta +2\pi p)$ by $R$ to change its measure from arc length to angle. The quadratic action makes the classical-path sum exact, rather than merely a stationary phase approximation. Real-time sums are understood with the usual negative imaginary part of $T$ and its boundary limit.

Take $\alpha =I/(i\hslash T)$ in the supplied Poisson summation identity, initially with a convergent regulator. Its left side is the winding sum. Multiplying its right side by the prefactor in part d givesConsequentlyRelabelling $q=-l$ gives exactly the spectral answer. The square-root branch is fixed by continuation from positive imaginary-time damping; the limiting kernel has the required periodic delta-function normalization.