Convergence of vacuum state of Klein Gordon field in a box

1. Jun 3, 2009

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

I get it so far...

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.

2. Jun 3, 2009

DarMM

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.

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.

3. Jun 3, 2009

maverick280857

Can you please elaborate on that?

4. Jun 4, 2009

DarMM

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.

5. Jun 4, 2009

maverick280857

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

6. Jun 4, 2009

Fredrik

Staff Emeritus
http://home.uchicago.edu/~seifert/geroch.notes/ [Broken].

Last edited by a moderator: May 4, 2017
7. Jun 4, 2009

maverick280857

Thanks, I'll take a look at these.