# 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.