State space of QFT, CCR and quantization, and the spectrum of a field operator?

In the canonical quantization of fields, CCR is postulated as (for scalar boson field ): $$[\phi(x),\pi(y)]=i\delta(x-y)\qquad\qquad(1)$$ in analogy with the ordinary QM commutation relation: $$[x_i,p_j]=i\delta_{ij}\qquad\qquad(2)$$ However, using (2) we could demo the continuum feature of the spectrum of $x_i$, while (1) raises the issue of $\delta(0)$ for the searching of spectrum of $\phi(x)$, so what would be its spectrum?

I guess that the configure space in QFT is the set of all functions of $x$ in $R^3$, so the QFT version of $\langle x'|x\rangle=\delta(x'-x)$ would be $\langle f(x)|g(x)\rangle=\delta[f(x),g(x)]$,
but what does $\delta[f(x),g(x)]$ mean?

If you will say it means $\int Dg(x) F[g(x)]\delta[f(x),g(x)]=F[f(x)]$, then how is the measure $Dg(x)$ defined?

And what is the cardinality of the set $\{g(x)\}$? Is the state space of QFT a separable Hilbert sapce also? Then are field operators well defined on this space?

Actually if you choose to quantize in an $L^3$ box, many issues will not emerge, but many symmetries cannot be studied in this approximation, such as translation and rotation, so that would not be the standard route,
so I wonder how the rigor is preserved in the formalism in the whole space rather than in a box or cylinder model?

I'm now beginning learning QFT, and know little about the mathematical formulation of QFT, so please help me with these conceptual issues.

Answer

The objects such as $\hat \phi(x,y,z,t)$ in a QFT are strictly speaking "operator distributions". They differ from "ordinary operators" in the same way how distributions differ from functions. Only if you integrate such operator distributions over some region with some weight $\rho$, $$\int d^3 x\,\hat\phi(x,y,z,t) \rho(x,y,z,t)=\hat O,$$ you obtain something that is a genuine "operator".

In a free QFT, the state vectors may be built as combinations of states in the Fock space – an infinite-dimensional harmonic oscillator. But you may also represent them via "wave functional". Much like the wave function in non-relativistic quantum mechanics $\psi(x,y,z)$ depends on 3 spatial coordinates, a wave functional depends on a whole function, $\Psi[\phi(x,y,z)]$. For each allowed configuration of $\phi(x,y,z)$, there is a complex number.

Yes, one may also integrate over all classical functions $\phi(x,y,z)$. There also exists a Dirac delta-like object, the Dirac "delta-functional", and it is usually denoted $\Delta$, $$\int {\mathcal D}\phi(x,y,z) F[\phi(x,y,z)] \Delta[\phi(x,y,z)] = F[0(x,y,z)]$$ I wrote the zero as a function of $x,y,z$ to stress that the argument of $F$ is still a function.

The functional integration is a sort of infinite-dimensional integration and the delta-functional is an infinite-dimensional delta-function. One must be careful about these objects, especially if we integrate amplitudes that may have amplitudes and especially if we integrate over curved infinite-dimensional objects such as infinite-dimensional gauge groups etc. – there may be subtleties such as anomalies.

Yes, the Hilbert space of a free QFT is still isomorphic to the usual Hilbert space: there is a countable basis. But we're talking about the finite-energy excitations only. There are lots of "highly excited states" that aren't elements of the Fock space – one would need infinite occupation numbers for all one-particle states. Physically, such states are inaccessible because the energy can't be infinite. However, when one is changing the energy from one Hamiltonian to another (e.g. by simple operations such as adding the interaction Hamiltonian), finite-energy states of the former $H_1$ may be infinite-energy states of the latter $H_2$ and vice versa.

So one must be careful: the physically relevant finite-energy Hilbert space may be obtained from some infinite-occupation-number states in a different, e.g. approximate, Hamiltonian. It's still true that the relevant Hilbert space is as large as a Fock space and it has a countable basis. The "totally inaccessible" states that are too strong deformations have an important example or name – they're "different superselection sectors".

Rigor is a strong word. People tried to define a QFT rigorously – by AQFT, the Algebraic/Axiomatic Quantum Field Theory. These attempts have largely failed. It doesn't mean that there isn't any "totally set of rules" that QFT obeys. Instead, it means that it's not helpful to be a nitpicker when it comes to the new issues that arise in QFT relatively to more ordinary models of quantum mechanics; it's neither fully appropriate to think that a QFT is "exactly just like a simpler QM model" but it's equally inappropriate to forget that it's formally an object of the same kind. Formally, many things proceed exactly in the same way and there are also new issues (unexpected surprises that contradict a "formal treatment") that have some physical explanation and one should understand this explanation. Some of these new subtleties are "IR", connected with long distances, some of them are "UV", connected with ever shorter distances.

The fact that a QFT has infinitely many degrees of freedom is both an IR and UV issue. So even if you put a QFT into a box, you won't change the fact that you need wave functionals, delta-functionals, and that there are superselection sectors and states inaccessible from the Fock space. By the box, you only regulate the IR subtleties but there are still the UV subtleties (momenta, even in a box, may be arbitrarily large). Those may be regulated by putting the QFT on a lattice. This has some advantages but some limitations, too.