Dismiss Notice
Join Physics Forums Today!
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

  1. Jun 3, 2009 #1
    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" [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...

    I have a few questions here:

    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.
     
  2. jcsd
  3. Jun 3, 2009 #2

    DarMM

    User Avatar
    Science Advisor

    No, the point is that [itex]\Phi(t,{\bf{x}})[/itex] 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.
     
  4. Jun 3, 2009 #3
    Can you please elaborate on that?
     
  5. Jun 4, 2009 #4

    DarMM

    User Avatar
    Science Advisor

    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.
     
  6. Jun 4, 2009 #5
    Ok, so its a distribution. Where can I read more about the connection between quantum fields and distributions? More like an expository article/book..
     
  7. Jun 4, 2009 #6

    Fredrik

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    http://home.uchicago.edu/~seifert/geroch.notes/ [Broken].
     
    Last edited by a moderator: May 4, 2017
  8. Jun 4, 2009 #7
    Thanks, I'll take a look at these.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Convergence of vacuum state of Klein Gordon field in a box
  1. Klein Gordon Field (Replies: 4)

Loading...