When dealing with non-relativistic particle dynamics in quantum physics, we can formulate the theory on the space of quantum states, i.e. the projective Hilbert space.

Can we do the same thing with quantum field theory?

The biggest problem is, I think, with the normalization of states. Again, in particle dynamics, we can fix this problem by taking the space of functions $L^2(\mathbb{R}^3)$ as our Hilbert space.

Is the same procedure applicable in the field theory? If so, how do we define expectation values of fields and observables?

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    $\begingroup$ QFT is done on a Fock Space. $\endgroup$
    – user73352
    Jan 17 '17 at 18:54
  • $\begingroup$ I'm not sure what the exact question is. As a quantum mechanical theory (albeit one with infinitely many degrees of freedom), QFT has a Hilbert space and therefore a projective Hilbert space. You see this e.g. in the asymptotic Fock spaces, or in the "wavefunctionals" of the QFT Schrödinger picture. What exactly do you want to know about that? $\endgroup$
    – ACuriousMind
    Jan 17 '17 at 19:25
  • $\begingroup$ I would like to know how the Hermitian inner-product on the Hilbert space is related to the inner-product on the projective Hilbert space, and then calculate the expectation value of a quantum observable in a normalized state. $\endgroup$ Jan 19 '17 at 20:49

The space of quantum states has a much more natural "algebraic" definition as the convex hull of positive and norm one elements of the continuous dual of the C* algebra of quantum observables.

It is a natural definition, and better than the one you give as elements of a projective Hilbert space, since it is not possible to represent all algebraic quantum states at once in the same (projective) Hilbert space.

In addition, all separable infinite dimensional Hilbert spaces are isomorphic, but they can be "inequivalent" in the algebraic sense, for they may represent inequivalently the given algebra of quantum observables.

Given that, states in qft are defined in the same way as in non-relativistic quantum mechanics, and share the same properties. In particular, given any state there is a representation on which it acts as an element of some (projective) Hilbert space. This result takes the name of Gel'fand-Naimark-Segal construction.


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