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### How exact is the analogy between statistical mechanics and quantum field theory?

Famously, the path integral of quantum field theory is related to the partition function of statistical mechanics via a Wick rotation and there is therefore a formal analogy between the two. I have a ...
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### partition function for Wightman and Haag-Kastler QFT

From what I hear, some modern mathematical approach quantum field theory uses the following definition "A $d$-dimensional $S$-structured quantum field theory $Q$ is a mathematical object, ...
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I'll start with a few definitions: $$\beta \equiv \frac{1}{k_bT}$$ Where T is the temperature of a system. And the partition function: $$Z \equiv \sum_{j}e^{-\beta \epsilon_j}=\int D(\epsilon)d\... 1answer 81 views ### Einstein model for thermal capacity of solids and indistinguishability of the oscillators Albert Einstein's theory of thermal capacity of a solids makes the assumption that a crystal is made up from oscillators which of course oscillate, in all three directions. Thus, for N atoms of the ... 1answer 418 views ### Expectation value of a_i^\dagger a_i for thermal density matrix Suppose we have some heat bath with Hamiltonian,$$H=\sum_n \left(a^{\dagger}_na_n+\frac{1}{2}\right)\hbar\omega_n and a density matrix $\rho=Z\exp(-\beta H)$ for some normalisation $Z$. ...
This is a problem (Problem 3.16) from the book Statistical Mechanics 2nd Ed. by Pathria. In the problem I have to calculate the partition function of an ultra-relativistic 1D gas ($E_i=cp_i$) ...