Hi(adsbygoogle = window.adsbygoogle || []).push({});

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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

**Physics Forums | Science Articles, Homework Help, Discussion**