Convergence of vacuum state of Klein Gordon field in a box

[maverick280857]

Hi

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" \mathbb{B}: \left[-\frac{1}{2}L,\frac{1}{2}L\right]^3 in \mathbb{R}^3. Here \bf{p} lies in the lattice:

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

the field being given by

\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})

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

\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

(with the 1 in the p^{th} place).

The author goes on to say that

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

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

I get it so far...

That is, \Phi is the linear map that assigns to each compactly supported C^{\infty} function \chi_{1} on \mathbb{R} and each C^{\infty} \Lambda-periodic function \chi_{2} on \mathbb{R}^3 the operator

\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]

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

I have a few questions here:

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

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

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

Thanks.

 

[DarMM]

I have a few questions here:

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

No, the point is that \Phi(t,{\bf{x}}) is not an operator. The only way to get an operator from it is to "smear" it by integrating it against a function. This results in a different operator for every choice of function. Here the author is just using two different functions to smear in time and space.

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

Fock space. Basically get the Hilbert space for a Klein-Gordon particle, attach the Hilbert space for two particles and so on, until you have a Hilbert space which can deal with any particle number. It's all the Hilbert spaces for different particle numbers put together.

 

[maverick280857]

No, the point is that \Phi(t,{\bf{x}}) is not an operator. The only way to get an operator from it is to "smear" it by integrating it against a function. This results in a different operator for every choice of function. Here the author is just using two different functions to smear in time and space.

Can you please elaborate on that?

 

[DarMM]

Can you please elaborate on that?

The quantum field operator isn't an operator. It simply gives you operators when you integrate it against a function. Just like the dirac delta isn't a function, but gives you a number when integrated against a function.

 

[maverick280857]

The quantum field operator isn't an operator. It simply gives you operators when you integrate it against a function. Just like the dirac delta isn't a function, but gives you a number when integrated against a function.

Ok, so its a distribution. Where can I read more about the connection between quantum fields and distributions? More like an expository article/book..

 

[Fredrik]

http://home.uchicago.edu/~seifert/geroch.notes/ .

 

[maverick280857]

Thanks, I'll take a look at these.