Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Virtual particle propagators in QFT

  1. Jul 15, 2011 #1
    I am reading a nice book (Quarks and Leptons, by Halzen and Martin) about particle physics. It states that the general form of the propagator of a virtual particle is:
    [tex]
    \dfrac{i\sum_{\text{spins}}}{p^2 - m^2}
    [/tex]

    I see that this is the case for the Dirac propagator:

    [tex]
    \dfrac{i(\displaystyle{\not}{p} + m)}{p^2 - m^2} = \dfrac{i\sum_{s}u_s(p)\bar{u}_s(p)}{p^2 - m^2}
    [/tex]

    but how can I prove that always holds?
     
  2. jcsd
  3. Jul 15, 2011 #2

    Bill_K

    User Avatar
    Science Advisor

    Matthaeus , I agree, Halzen and Martin is a very well written book, and highly useful. Although it does tend to be a bit informal, and you are not going to catch them "proving" anything. So in the same spirit, here is not a proof but an argument.

    Let's say you have an operator L, which is generally a differential operator, and in particular will be the Klein-Gordon operator. You want to find the field |φ> at one point produced by a source |S> at another. The equation is L|φ> = |S>. The solution can be written formally as |φ> = L-1|S>. Expand L in terms of its eigenstates, namely L = ∑|n>Ln<n| where |n> are the eigenstates and Ln the eigenvalues. Then L-1 = ∑|n>Ln-1<n|. Thus |φ> = G|S> where G is the Green's function or propagator, G = ∑|n>Ln-1<n|.

    For the Klein-Gordon operator the eigenstates are the plane wave solutions, Ln = k2 + m2, and the numerator ∑|n><n| is the sum over spins. Halzen and Martin go on to write this out in more detail for the only cases that are physically realistic: spin 0, 1/2 and 1.
     
  4. Jul 16, 2011 #3
    Yes, I was also thinking about something along these lines.
    Thanks.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Virtual particle propagators in QFT
  1. Virtual Particles (Replies: 10)

  2. Virtual Particles? (Replies: 3)

  3. Virtual particle (Replies: 3)

  4. Virtual Particles (Replies: 5)

Loading...