 Solution

ID: past-exam-of-the-field-theories-and-collective-phenomena-in-condensed-matter-course-of-the-university-of-oxford/2025/1/b/solution

h

obeying

h (0) = h (T_{E}) = 0

δ S_{Euc} = m \int_{0}^{T_{E}} \overset{x}{˙} \dot{h} d τ = - m \int_{0}^{T_{E}} \overset{x}{¨} h d τ .

\overset{x}{¨}_{0} = 0

. The fixed boundary conditions determine its unique solution:

x_{0} (τ) = x_{i} + \frac{x _{f} - x _{i}}{β ℏ} τ, 0 \leq τ \leq β ℏ.

This classical path is also the unique minimum among paths with finite Euclidean action: the second variation is

m \int_{0}^{T_{E}} \dot{h}^{2} d τ > 0

for every nonzero admissible variation. Its Euclidean action is

S_{Euc} [x_{0}] = \frac{m ( x _{f} - x _{i} ) ^{2}}{2 β ℏ} .

 New to topics? Read the docs here!