Problem
Compute the overlap of the ground-state wave functions of a harmonic oscillator with two different frequencies. A free-bosonic field theory is just a collection of oscillators. Use your calculation to show that the overlap of the ground states for two different values of the mass is zero for any field theory, in any number of dimensions and infinite volume. Show that the overlap is zero even in finite volume if the number of space dimensions is two or greater. This is symptomatic of a more general problem. The states of two field theories, containing the same fields but with different parameters in the Lagrangian, do not live in the same Hilbert space. The formulation of field theory in terms of Green functions and functional integrals avoids this problem.
Question Does anyone have a hint on how to solve the finite volume case? I know that if we put the theory in a box of length $L$ with periodic boundary conditions, the momenta become quantized; however, I'm not sure how to proceed after that. Thank you.
