# What does field operators operate on?

I was reading quantum field theory and came across second quantization. There the wave function was quantized to field operators. The field operator is expressed as a sum of the product of creation/annihilation operator(that depends on time) and wave function satisfying Schrodinger's equation. In the beginning of quantum mechanics we quantized observables to operators and operated them on the wave function that contains all the information about the system under consideration. In QFT, upon what do operate the field operator?

• A Hilbert space. – Slereah Sep 19 '17 at 8:30
• 1. Related: physics.stackexchange.com/q/285710/50583 2. If you think that QM inherently deals with a "wave function" rather than the more general notion of a quantum state as a vector/ray in Hilbert space, you have a very narrow view of QM, e.g. no qubit or other quantum information system is described by a wavefunction since there are no position operators for these. – ACuriousMind Sep 19 '17 at 8:30
• A hilbert space which consists of multi-particle states (starting from the vacuum=zero particles, one-particle, two-particles, three-particles and so on) which is called Fock space. The 2nd quantisation is the application of QM on fields, so it's actually Quantum field theory. The field operators are the dynamical quantities on that space, i.e. no longer position and momentum of a single particles. For more the literature has to be consulted. – Frederic Thomas Sep 19 '17 at 10:54
• @FredericThomas from what I understand, Fock space is an abstract mathematical concept. I cannot seem to understand how this generalized mathematical concept can yield the physical reality. In QM, we chose wave functions based on our specific problem, e.g. LHO, H-atom, Square well potential etc. But here things are way too general. How do I apply this concept of field operators and Fock space to real problems? How do I get real life physical solutions from such an abstract concept? – Samapan Bhadury Sep 21 '17 at 7:16