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I've been reading through the book "Quantum Field Theory: A Tourist Guide for Mathematicians" by George B. Folland. On page 101, he describes the construction of a scalar field "in a box" [itex]\mathbb{B}: \left[-\frac{1}{2}L,\frac{1}{2}L\right]^3[/itex] in [itex]\mathbb{R}^3[/itex]. Here [itex]\bf{p}[/itex] lies in the lattice:

[tex]\Lambda = \left[\frac{2\pi}{L}\mathbb{Z}\right]^3[/tex]

the field being given by

[tex]\Phi(t,{\bf{x}) = \sum_{\Lambda}\frac{1}{\sqrt{2\omega_{p}L^3}}(e^{ip\cdot x - i\omega_{p}t}A_{p} + e^{-ip\cdot x + i\omega_{p}t}A_{p}^{\dagger})[/tex]

Suppose we apply the operator [itex]\Phi(t,{\bf{x}})[/itex] to the vacuum state:

[tex]\Phi(t,{\bf{x}})|0,0,\ldots\rangle = \sum_{\Lambda}\frac{1}{\sqrt{2\omega_{p}L^3}}e^{-ip\cdot x + i\omega_{p}t}|0,0,\ldots,0,1,0,\ldots\rangle[/tex]

(with the 1 in the [itex]p^{th}[/itex] place).

The author goes on to say that

I get it so far... The square of the norm of this alleged vector is [itex]\sum_{\Lambda}1/2(|{\bf{p}}|^2+m^2)[/itex], which is infinite (by comparison to [itex]\int_{\mathbb{R}^3}d{\bf{p}}/(|{\bf{p}}|^2+m^2) = 4\pi\int_{0}^{\infty} r^{2}dr/(r^2 + m^2)[/itex])...

The way out is to interpret [itex]\Phi[/itex] as a operator-valued distribution rather than an operator-valued function.

I have a few questions here: That is, [itex]\Phi[/itex] is the linear map that assigns to each compactly supported [itex]C^{\infty}[/itex] function [itex]\chi_{1}[/itex] on [itex]\mathbb{R}[/itex] and each [itex]C^{\infty}[/itex] [itex]\Lambda[/itex]-periodic function [itex]\chi_{2}[/itex] on [itex]\mathbb{R}^3[/itex] the operator

[tex]\int_{\mathbb{B}}\int_{\mathbb{R}}\Phi(t,{\bf{x}})\chi_{1}(t)\chi_{2}({\bf{x}})dt d{\bf{x}} = \sum_{\Lambda}\frac{1}{\sqrt{2\omega_{p}L^3}}\left[\hat{\chi}_{1}(\omega_{p})\hat{\chi}_{2}(-p)A_{p} + \hat{\chi}_{1}(-\omega_{p})\hat{\chi}_{2}(p)A_{p}^{\dagger}\right][/tex]

with the obvious interpretation of the Fourier coefficients [itex]\hat{\chi}_{1}[/itex] and [itex]\hat{\chi}_{2}[/itex]. The rapid decay of these coefficients as [itex]|{\bf{p}}|\rightarrow\infty[/itex] guarantees that this series converges nicely as an operator on the finite-particle space [itex]\mathcal{F}_{s}^{0}(\mathcal{H})[/itex].

1. Have we introduced the two functions [itex]\chi_{1}[/itex] and [itex]\chi_{2}[/itex] as a means to redefine the notion of convergence of the operator [itex]\Phi(t,{\bf{x}})[/itex]?

2. If yes, what is the motivation for such a definition?

3. What is [itex]\mathcal{F}_{s}^{0}(\mathcal{H})[/itex] really?

Thanks.

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# Convergence of vacuum state of Klein Gordon field in a box

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