We had established an intriguing relationship between the path integral of the previous chapter and the partition function here through "complexification" of the time coordinate. In this problem, we will make this relationship a bit more precise and attempt to understand what complexification of time actually accomplishes.

(a) Show that up to a normalization factor, the path integral \(Z_{\text {path }}\) is related to the partition function \(Z_{\text {part }}\) via: \(Z_{\text {path }} Z_{\text {path }}^{\dagger}=Z_{\text {part }}\). For this to be true, what must the relationship between path integral time and partition function temperature be? What would we describe the trajectory of the particle to be in the path integral given this relationship?

(b) Now, use the position space energy eigenbasis representation of the path integral of Eq. (11.79) and the result of part (a) to derive the partition function from the path integral.

(c) Show that these relationships hold for the harmonic oscillator, with path integral given in Eq. (11.137) and partition function in Eq. (12.147).