Fock Space in QFT vs SFT

Question

In quantum field theory, the Fock Space decomposition of the state space is a central part of the mathematical formalism. In statistical field theories (SFT), (edit: for example, the Ginzburg-Landau theory or the XY model) the Fock Space formalism is almost never used, at least not at an introductory level in the way it is in QFT. Yet at a cursory glance, the mathematical settings of QFT and SFT are very similar– both take as the fundamental object a density of some sort over field configurations, the difference being the usual difference between the complex-valued wavefunctions of quantum mechanics and the purely real probability densities of statistical mechanics.

My question is: why are Fock Spaces preferred for one theory and not the other, despite their apparent mathematical similarities?

Answer 1

A Fock space is only meaningful in free theories; there is no notion of an $n$-particle state in an interacting theory.

In QFT a Fock space is still used because people do perturbation theory of the interacting theory around the free theory, so they use the $n$-particle states of the free theory as approximate states of the (weakly) interacting theory. This is so, because historically QFT (from the high-energy point of view) was developed to describe particles, as excitations of the various fields.

On the other hand, SFT tries to describe the collective behaviour of things, at low energies, and thus the notion of a particle, and hence an $n$-particle state is not important. For example, the XY model, which you mentioned, is, crucially, an interacting theory. As such, it does not make much sense to treat it using a Fock space approximation for the Hilbert space of the theory.

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For the record, Jean Zinn-Justin's book "Quantum Field Theory & Critical Phenomena", which (despite the title) has mainly a statistical physics point of view, discusses Fock space at length.

Answer 2

I think that the Fock space is also a fundamental concept of statistical physics. But you are right in saying that most of the introductory curses skip the presentation of Fock states. I guess that the main reason is that in equilibrium statistical mechanics you don't have to make much use of the algebra of vectors in a Fock space when presenting the main notions of the field.

For example, when working in the grand-canonical ensemble, the energy ($E$) and the number of particles ($N$) are random variables following the probability distribution: $$P(E,N)=\Xi^{-1}e^{\mu N-\beta E}.$$ You can compute many useful quantities using the above expression (e.g. averages and fluctuations). In order to see the Fock space "hidden" in this statement, one only has to redefine in terms of occupancy numbers: say that $n_i$ is the number of particles in energetic state $i$ with energy $\epsilon_i$, then: $$N=\sum_i n_i,$$ $$E=\sum_i \epsilon_i n_i,$$ $$P(\{n_i\})=\Xi^{-1} \prod_i e^{\mu n_i-\beta \epsilon_i n_i}.$$ Where the last distribution weights the states of a Fock space $|n_1,n_2,...\rangle$.

In non-equilibrium statistical mechanics, the role of the Fock space is more relevant at an introductory level and can be found in many texts (e.g. Haye Hinrichsen's Non-equilibrium critical phenomena Book).

P.S. In your question, you were speaking about fields, but I preferred to refer to the second quantization framework because I am more used to working with it. In any case, one can extend these ideas to the field representation.