I'm studying statistical mechanics in particular correlation function:


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


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.


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.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.