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.