(adsbygoogle = window.adsbygoogle || []).push({}); State space of QFT,CCR and quantization,spectrum of a field operator?

In the canonical quantization of fields, CCR is postulated as (for scalar boson field ):

[ϕ(x),π(y)]=iδ(x−y) ------ (1)

in analogy with the ordinary QM commutation relation:

[xi,pj]=iδij ------ (2)

However, using (2) we could demo the continuum feature of the spectrum of xi, while (1) raises the issue of δ(0) for the searching of spectrum of ϕ(x), so what would be its spectrum?

I guess that the configure space in QFT is the set of all functions of x in [itex]R^3[/itex], so the QFT version of ⟨x′|x⟩=δ(x′−x) would be ⟨f(x)|g(x)⟩=δ[f(x),g(x)],

but what does δ[f(x),g(x)] mean?

If you will say it means ∫Dg(x)F[g(x)]δ[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 [itex]L^3[/itex] 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.

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

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