 Solution

ID: past-exam-of-the-quantum-condensed-matter-field-theory-course-of-the-university-of-cambridge/2014/paper-1/2/solution

t = - i τ_{E}

changes real-time evolution into

e^{- H τ_{E} /ℏ}

. At

τ_{E} = β ℏ

, taking the trace integrates equal endpoints:

Z = \int d q K_{E} (q, q; β ℏ) = \int_{q (0) = q (β ℏ)} D q e^{- S_{E} [q] /ℏ}, S_{E} = \int_{0}^{β ℏ} [\frac{m}{2} \overset{q}{˙}^{2} + V (q)] d τ_{E} .

Thus temperature sets the imaginary-time circumference, and bosonic coordinate paths are periodic. The measure is fixed by the short-time propagator. The trace sums paths, not just a single stationary trajectory; a quadratic action is special because the saddle plus Gaussian fluctuations is exact.

 New to topics? Read the docs here!