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Mandl & Shaw QFT problem 14.1

  1. Aug 2, 2013 #1
    The problem is on pages 323 and 324 of the second edition.

    1. The problem statement, all variables and given/known data
    Given the lagrangian
    [tex]\mathcal{L} = -\frac{1}{4}F_{\mu\nu}(x)F^{\mu\nu}(x) - \frac{1}{2\alpha}(\partial_{\mu}A^{\mu})^2[/tex]
    show that the momentum space photon propoagator is given by
    [tex]D_F^{\mu\nu}(k) = \frac{-g^{\mu\nu} + \delta k^{\mu}k^{\nu}/k^2}{k^2 + i\epsilon}[/tex]

    2. Relevant equations
    [tex]\delta = 1 - \alpha^{-1}[/tex]

    3. The attempt at a solution
    I can solve this problem if I set
    [tex]\delta = 1 - \alpha[/tex]
    but not with the delta stated in the book.

    My question is this:

    Should the book say [itex]\delta = 1 - \alpha[/itex] and not [itex]\delta = 1 - \alpha^{-1}[/itex]?

    This question and this question only. The meat of the answer will be one word.
     
  2. jcsd
  3. Sep 3, 2013 #2
    Yes, the text is a typo. It should say [itex]\delta = 1 - \alpha[/itex]. This can be seen by referencing eqn (8.40) on page 154 in the second edition of the book "A First Book of Quantum Theory", by Lahiri & Pal. Eqn (8.40) is
    [tex]D_{\mu\nu}(k) = -\frac{1}{k^2 + i\epsilon}[g_{\mu\nu} - (1 - \xi)\frac{k_{\mu}k_{\nu}}{k^2}][/tex]
    and is the photon propagator when the lagrangian is given by eqn (8.11) on page 148 together with eqn (8.26) on page 152 to get
    [tex]\mathcal{L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} - \frac{1}{2\xi}(\partial_{\mu}A^{\mu})^2[/tex]
    which is the same as the eqn at the bottom of page 323 in Mandl & Shaw.
     
    Last edited: Sep 3, 2013
  4. Sep 11, 2013 #3
    I suppose you should give thanks to yourself!
     
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