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I'm studying statistical mechanics in particular correlation function:

https://en.wikipedia.org/wiki/Correlation_function_(statistical_mechanics)

and I have understood it. Now searching on internet I found this:

https://en.wikipedia.org/wiki/Correlation_function_(quantum_field_theory)

I don't know quantum field theory and I was wondering which was the link between the two function? For example why the correlation function in quantum field theory is defined as $ \langle \phi_1,\phi_2,\dots,\phi_n \rangle $ and not $ \langle \phi_1,\phi_2,\dots,\phi_n \rangle - \langle \phi_1 \rangle \cdots \langle \phi_n \rangle $ as in statistical mechanics.

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The "link" comes from the path integral formulation of quantum mechanics.

There's a certain dictionary that maps quantities from the canonical formulation to path integrals which closely resemble correlation functions from statistical mechanics. Specifically, suppose that $\varphi_1, \dots, \varphi_n$ are $n$ values of certain physical observables which correspond to quantities measured at times $t_1 > \dots > t_n$.

A quantum transition amplitude is given by

$$ \left< 0 \right| \hat{\varphi}_1 \dots \hat{\varphi}_n \left| 0 \right>, $$

where $\left| 0 \right>$ is the vacuum state of the quantum system, and quantities with "hats" represent quantizations of physical observables (linear operators acting on the Hilbert space).

It encodes a certain probabilistic property of quantum systems. For example, for $n = 2$, its absolute value squared encodes the probability density of a transition between two quantum states.

On the other side of the correspondence is the path integral

$$ \int Dx e^{i \hbar^{-1} S[x]} \varphi_1[x] \dots \varphi_n[x], $$

where all quantities are just numbers. The expression

$$ \rho[x] = e^{i \hbar^{-1} S[x]} $$

can be thought of as the probability density functional defined on the space of all trajectories. However, the similarity is only formal: unlike probability densities, it is complex-valued, and generally ill-defined without delicate procedures called renormalizations.

This link can be made precise for Wightman QFT and statistical mechanics with Osterwalder-Schrader axioms. However, the absolute majority of realistic QFT models are based on the gauge theory, for which there's no known axiomatization, so the link remains just a vague conjecture.

Actually, making this precise for gauge theories is related to one of the millennium prize problems.

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