Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 304 / 1

Insert a complete set of momentum eigenstates into the quantum-mechanical propagator:Completing the square and evaluating the resulting Gaussian integral, with the usual i-epsilon prescription, gives the free-particle propagatorThe square-root branch is fixed by requiring $K(x_f,T;x_i,0)\to \delta (x_f-x_i)$ as $T\downarrow 0$.

The Euler-Lagrange equation is $m\ddot{x}=0$. The unique path with the prescribed endpoints is thereforeIt is a minimum of the Euclidean action and a stationary point of the real-time action. Its classical action isThe principle of stationary action consequently fixes the position-dependent phase of the semiclassical propagator asBecause the action is quadratic, the stationary-phase evaluation of the path integral is exact. Composition of propagators, or the Van Vleck determinant, gives $C(T)=\sqrt{m/(2\pi i\hslash T)}$, reproducing part i.

The particle on a circle has the complete orthonormal basis of energy eigenstates $\{|n⟩:n\in \mathbb{Z}\}$. Its spectral representation givesThe integer $n$ is the quantized angular momentum in units of $\hslash$.

Choose a real lift $\Delta \theta ={\theta }_f-{\theta }_i$. Classical paths fall into winding number sectors $j\in \mathbb{Z}$ and areEach sector has the same fluctuation determinant, so the image-sum form of the propagator isSet $a=\hslash T/(2m{R}^2)$. Applying the Poisson summation formula to the Gaussian $e^{i{x}^2/(4a)}$ giveswhich is exactly the spectral propagator found in part i. Thus the angular-momentum sum is dual to a sum over homotopy classes of classical paths.

With a nonconstant potential energy, momentum no longer diagonalizes the Hamiltonian operator. The operator derivation must use the energy eigenfunctions of the full Hamiltonian, a Dyson series, or a time-sliced Trotter product formula. In the classical derivation, the straight paths are replaced by every solution of the nonlinear Euler-Lagrange equation with the specified endpoints. The semiclassical propagator becomes a sumwhere the prefactor is the Van Vleck determinant and ${\mu }_{\mathrm{cl}}$ is a Maslov index. Unlike a quadratic theory, the classical-path sum is generally only an asymptotic approximation: the exact path integral includes fluctuations of every order. On the circle, the sum must still include all winding number sectors.