 Solution

ID: past-exam-of-the-theoretical-physics-ii-course-of-the-university-of-cambridge/2014/paper-1/2/d/solution

The angular action is

S = \int I \dot{θ}^{2} d s /2

. In each winding sector, write

θ = θ_{p} + ξ

with

ξ

zero at both endpoints. The cross term vanishes and the Gaussian fluctuation integral is the free-particle prefactor with mass replaced by

I

. Thus

K_{θ} (Δ, T) = \frac{I}{2 πi ℏ T} p \in Z \sum exp [\frac{i I ( Δ + 2 π p ) ^{2}}{2ℏ T}] .

Equivalently, multiply the line kernel for displacement

R (Δ + 2 π p)

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.

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