# Superposition of Fock states

Fock states form the most convenient basis of a Fock space. Does it mean that any state of a quantum field (any element of Fock space) can be expressed as a superposition of Fock states? Does this also apply to Fock states?

• For perspective: QFT can be expressed in terms of operators on a Hilbert space, just like any other quantum theory can. Different ways of constructing the Hilbert space may be more convenient in different contexts, but physics doesn't care how we construct it or what basis we use. Those things are purely matters of convenience. "Fock space" can be a convenient way to construct the Hilbert space for some purposes, but ultimately what matters is the map from physical observables to operators on the Hilbert space, regardless of how we constructed the Hilbert space. Nov 21, 2021 at 18:29

There are two different questions here:

1.) Can any element of Fock space be expressed as a superposition of Fock states? and

2.) Can any state of a quantum field be expressed as a superposition of Fock states?

The answer to 1 is yes, but the answer to 2 is (in general) no.

(1) is tautological. If Fock states form a basis for Fock space, then by definition any state in that space can be expressed as a linear combination of them. That's just what the definition of the word "basis" means, for any vector space.

However, there's a false assumption in the question, as written, that any quantum field is a member of a Fock space. I remember there was a quantum mechanics textbook I used to use (I think it was the one by Robert Shenkar?). In graduate school, I distinctly remember many of us giggling about an italicized disclaimer which appears on one page:

WARNING: Fock space is only valid for non-interacting quantum field theories!

We found it entertaining to imagine how big of a head scratcher it would be for most people if they walked into a restaurant or a store and found a sign saying this, or if they found it printed on a clothing label or on the back of a cleaning product.

Anyway, what this means is that the Hilbert space for any quantum field theory containing interaction terms in the Lagrangian (any terms other than kinetic or mass terms, stuff like $$\lambda\phi^4$$, $$ie\bar{\psi} \gamma_{\mu}\psi A^{\mu}$$, $$\lambda\bar{\psi} \psi \phi$$ etc.) is not a Fock space. If the interactions are small, then it looks approximately similar, but unlike a Fock space it cannot be cleanly decomposed into a direct sum of tensor powers of single-particle Hilbert spaces.

For example, there is a state in the Hilbert space for QED corresponding to an electron and a positron which start infinitely far away and move towards each other, until they annihilate with each other, momentarily becoming a virtual photon, then transforming back into an electron-positron pair and moving away in opposite directions along a different axis. This is not in the Fock space at all, so cannot be expressed as a superposition of merely 1-particle or 2-particle states.