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Ryder, QFT 2nd ed, page 127

  1. Aug 23, 2007 #1
    1. The problem statement, all variables and given/known data
    The first expression in equation (4.4) is:
    [tex]\frac{d^4k}{(2\pi)^4}2\pi\delta(k^2 - m^2)\theta(k_0)[/tex]

    2. Relevant equations
    Ryder is most ungenerous on this page. Some concepts, important to understanding the entire chapter are left unexplained. For instance, the reasoning that leads from the assumption that [itex]\phi[/itex] is Hermitian, to the fact that
    [tex]a(k)^{\dagger} = a(-k)[/tex]
    is not displayed or even mentioned. Also, unless I missed it, this is the first use of the Heaviside function in the book, yet it is not defined. Its use in this expression relates to the text where it says "with [itex]k_0 > 0[/itex]". The use of the delta function relates to the text where it says "we have the 'mass-shell' condition". However I do not know where the factor of [itex]2\pi[/itex] comes from.

    3. The attempt at a solution
    I assume that since there is one factor of [itex]2\pi[/itex] in the denominator for each degree of freedom, and the mass-shell condition removes one of those degrees of freedom, the factor is justified. Is there a more direct way of seeing why the factor is there?
    Last edited: Aug 23, 2007
  2. jcsd
  3. Aug 24, 2007 #2


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    That [itex]1/\left(2\pi\right)^{4} [/itex] comes out the convention he used to define the Fourier transform of the classical field [itex] \varphi [/itex].
    I have no idea where that [itex] 2\pi [/itex] in the numerator comes from. In my notes (4.3) appears after a redefinition of the classical complex amplitudes which is simply a [itex] 1/2\pi [/itex] rescaling, so instead of f(p) and [itex] 1/\left(2\pi\right)^{4} [/itex] i have a(p) and [itex] 1/\left(2\pi\right)^{3} [/itex]. And this has nothing to do with any factors of [itex]2\pi [/itex] coming from the mass-shell condition.

    It's annoying to see the face that Ryder doesn't use integrals in his derivation of (4.4). Mathematically speaking his entire calculation is garbage.
    Last edited: Aug 24, 2007
  4. Aug 27, 2007 #3
    I hope to hear from someone who has.
    Actually, except for the factor of 2 pi, I understand everything else in his derivation, so I don't think it's garbage. He does mention at the top of page 128 that there is an implied integration in equation (4.4).

    In my opinion, he should have started with the plain vanilla fourier expansion:
    [tex]\phi(x) = \frac{1}{(2\pi)^4}\int d^4k a(k)e^{-ikx}[/tex]
    as it would have made the rest of the page easier to follow. It would allow him to emphasize the difference between eqn. (4.4) and (4.33) on page 135 where [itex]\phi[/itex] is not assumed to be hermitian.
    Last edited: Aug 27, 2007
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