Temperature of quantum fields and periodicity

Question

I have read this PSE post Finite Temperature Quantum Field Theory, saying that

In a QFT at finite temperature, we consider the Euclidean time to be periodic, i.e. we consider a theory on the manifold $\mathbb{R}^{d−1}×S^1$, where the spatial coordinates are in $\mathbb{R}^{d−1}$ and the time coordinate in $S^1$. The temperature of the system is then the inverse of the circumstance $T=1/\beta,\ \tau\in[0,\beta)$.

Why are we allowed to identify the imaginary time as the inverse of the temperature and why do we enforce periodicity on the circle of circumference $\beta$, i.e. why does the condition $\phi(0,\vec{x})=\pm\phi(\beta,\vec{x})$ (for $\vec{x}$ being a vector in $\mathbb{R}^{d-1}$), where $\phi$ is some scalar field, has to hold?

I have also read another PSE post Why is Euclidean Time Periodic?, saying that

it turns out that thermal averages of operators are periodic with respect to the Wick rotated time $\tau$

Is this motivation enough for us to impose this periodicity in thermal field theories?

I realize that this may be a textbook question, so if it is, please point me to the right direction. If not, please answer this post.

Answer 1

I will develop my comment. Consider the general amplitude term $$\tag{1}\label{1}\langle\phi(t_f)| e^{-it_f H} |\phi(t_f)\rangle.$$ The technique to get a path integral form of this expression is to consider $N$ time slices. We rewrote the previous expression $$\langle\phi(t_f)| e^{-it_f H} |\phi(t_f)\rangle = \langle\phi(t_f)| e^{-i\Delta t H} \cdots e^{-i\Delta t H} |\phi(t_f)\rangle$$ with $t_f = N \Delta t$. Then you insert $N$ identity relations of the form $$\int d\phi_i |\phi_i\rangle \langle\phi_i|$$ inside the amplitude term (you also add identities for the conjugate field $\pi = \dot{\phi}$). In the end you get

$$\langle\phi(t_f)| e^{-i\Delta t H} \cdots e^{-i\Delta t H} |\phi(t_f)\rangle = \int \Pi_i d\phi_i d\pi_i\langle\phi(t_f)|\pi_N\rangle \langle \pi_N|e^{-i\Delta t H} \cdots e^{-i\Delta t H} \langle\phi_1|\phi(t_f)\rangle . $$ Taking $N \rightarrow \infty$ you get the path integral for the amplitude term. But because $\langle \phi_1|\phi(t_f)\rangle = \delta(\phi_1 - \phi_{t_f})$, the fields in the path integral will be periodic, and so you will have to consider periodic path in the path integral of the partition function. When you take the trace over $e^{-\beta H}$, you will simply redo the calculus but with imaginary time.

EDIT :

In thermal field theory, $$Z = Tr [e^{-\beta H}] = \int \phi \langle \phi| e^{-\beta H} |\phi\rangle ,$$ where the integrand is equivalent to ($\ref{1}$), where $\beta$ plays the role of the complex time.