# Formalism Of Quantum Field Theory vs Quantum Mechanics

How far can we extend the formalisms on quantum mechanics (QM) to quantum field theory (QFT)? In particular,

1. How is a Fock space $\mathcal{F}$ different from a Hilbert space $\mathcal{H}$? Can a general state in Fock space $\mathcal{F}$ be written as the superposition of number operator eigenstates? If yes, are the number operator eigenstates the only basis states for the Fock space states?

2. Are all the postulates of quantum mechanics hold in QFT as well? What is the interpretation of norm of one or many particle states in QFT? Is there a concept of position basis or momentum basis in Fock states?

• – Qmechanic Jan 27 '14 at 19:30

The position basis and momentum basis (or representation) are particular bases (or representations) for non-relativistic quantum mechanics. They're composed of the continuous eigenstates of the operators $x_i$ and $p_i$, respectively. But these are not well-defined operators in quantum field theory – after all, even the number of particles is variable in QFTs so there can't be any "fixed number of particles' positions or momenta". Quantum field theories have other operators (observables). The existence of the "momentum basis" or "position basis" is a particular property of a class of (non-relativistic) models of quantum mechanics; this existence is not belonging among the general postulates of quantum mechanics and these theories (with a fixed number of particles with positions or momenta) don't describe our Universe accurately.