Past exam of the field theories and collective phenomena in condensed matter course of the University of Oxford / 2025 / 1 / b / Solution

For a free particle, vary the Euclidean action with a variation $h$ obeying $h(0)=h(T_E)=0$. Integration by parts givesThe Euler-Lagrange equation is therefore ${\ddot{x}}_0=0$. The fixed boundary conditions determine its unique solution:This classical path is also the unique minimum among paths with finite Euclidean action: the second variation is $m{\int }_0^{T_E}{\dot{h}}^{2}d\tau >0$ for every nonzero admissible variation. Its Euclidean action is