# Is the usage of the Fock space a postulate in QFT?

In this question, when I write Fock space, I mean "the direct sum of the symmetric or antisymmetric tensors in the tensor powers of a single-particle Hilbert space H", as it is described by Wikipedia.

When I began with many-particle-QM in my college-course, we constructed the many-particle space as a tensorproduct of single-particle-states, and then we constructed operators that act on this many-particle-space. From that point of view, it was natural to use the fock space as the space that constains the many-particle-states.

As I understand Quantum Field Theory, everything I do is, I take a classical field theory, and try to quantize it, which means I replace the field variables (like $\phi(x,t)$) with an Operator, then I set the Operator's commutation-relations in a way that makes sense. Afterwards I write the Hamiltonian down, and express it as a sum of some operators, which I afterwards interpret as "Creation and Annihilation"-operators, the create excitation of the quantum field. We then call this excitation "one particle".

My Question: Is the space that the operators act uppon a Fock-state by definition? Or is it just "any Hilbertspace"? If it's the latter: Can we interpret the Hilbertspace to be a Fock-space, after we work with the annihilation and creation operators? For example, If I found some strange EM-Field-space in QED, can I express it as the sum of tensorproducts of "single-particle-photon"-states?

More generaly (and that is the core of my question): Is there something like a "single-particle" in QFT? While in many-particle-QM, the single-particle-QM is the basis, QFT doesn't rely on this concept, that's why I'm asking.

• The Fock space formalism only works for very specific theories (mostly free theories). Interacting theories should not, if exactly solved, have a Fock space for their Hilbert space. – Slereah Jun 23 '16 at 12:16

Let me expand the ideas a bit more, related to the last part of your question. When we write the canonical commutation relations for quantum fields, it is customary to start from a suitable "one-particle" space $\mathfrak{h}$ (at least it was called like that by Irving Segal). Such space is e.g. the Hilbert space on which you build the Fock space $\Gamma_{s/a}(\mathfrak{h})$ (as direct sum of tensor products), and it encodes roughly speaking the properties of the related classical field (scalar/vector/spinor field,...). However, contrarily to what happens in quantum mechanics, there are infinitely many unitarily inequivalent irreducible representations of the canonical commutation relations $\mathrm{C}^*$-algebra built on such one-particle space $\mathfrak{h}$. One of those representations is the Fock one, the others still encode the commutation relations of the creation/annihilation operators $\{a^{\#}(h),h\in\mathfrak{h}\}$, but such operators may not behave as in the Fock representation: in particular, the intersection of their domains of definition $\bigcap_{h\in\mathfrak{h}}D(a(h))$ may fail to be a dense subset of the Hilbert space.
• @Quantumwhisp By interacting theory I mean any theory representing a field (or many fields) in interaction. A simple example is the scalar $\varphi^4$ model, given by the lagrangian density $\mathcal{L}(\varphi)= \partial^\mu\varphi\partial_\mu\varphi - m^2\varphi^2-\lambda \varphi^4$. Also any theory of fermion fields interacting with a scalar or vector field are, well, interacting of course ;-) – yuggib Jun 23 '16 at 14:33