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Zwiebach, pages 175,176

  1. Sep 27, 2007 #1
    1. The problem statement, all variables and given/known data
    Equation (10.58) is:
    [tex]\phi(t, \vec{x}) = \frac{1}{\sqrt{V}}\Sigma_{\vec{p}}\frac{1}{\sqrt{2E_p}}(a_{p}e^{-iE_pt + i\vec{p}\cdot\vec{x}} + a_p^{\dagger}e^{iE_pt - i\vec{p}\cdot\vec{x}})[/tex]

    2. Relevant equations
    Here is equation (10.57)
    [tex]\phi_{p}(t, \vec{x}) =\frac{1}{\sqrt{V}}\frac{1}{\sqrt{2E_p}}(a_{p}e^{-iE_pt + i\vec{p}\cdot\vec{x}} + a_p^{\dagger}e^{iE_pt - i\vec{p}\cdot\vec{x}})[/tex]
    [tex]+\frac{1}{\sqrt{V}}\frac{1}{\sqrt{2E_p}}(a_{-p}e^{-iE_pt - i\vec{p}\cdot\vec{x}} + a_{-p}^{\dagger}e^{iE_pt + i\vec{p}\cdot\vec{x}})[/tex]

    3. The attempt at a solution
    The idea is that the second term on the r.h.s. of (10.57) is the same as the first term evaluated for [itex]\vec{p} = -\vec{p}[/itex], which does not effect [itex]E_p[/itex]. Then (10.58) is supposed to be the sum of (10.57) over all values of [itex]\vec{p}[/itex]. My problem is that I think there is a factor of 2 missing on the r.h.s. of (10.58) because each of the terms in (10.57) should appear twice in the sum. What am I missing? The same problem arises on page 176 for equations (10.60) and (10.61) which are sums of (10.55) and (10.56) respectively.
     
    Last edited: Sep 27, 2007
  2. jcsd
  3. Oct 7, 2007 #2
    You're argument makes sense, but I don't understand it well enough to say conclusively. 10.57 could be the equation for both p and -p, but again I am not sure.

    I'll post back if this becomes clear to me.

    I do think 10.63 and 10.64 are wrong if 10.60 and 10.61 are correct, however. Where does the commutator come from if both equations sum over all possible values of a their vector index?
     
    Last edited: Oct 7, 2007
  4. Oct 7, 2007 #3
    No, I think (10.63) and (10.64) follow from (10.60) and (10.61). The product equals the commutator in this case because the annihilator annihilates the vacuum.
     
  5. Oct 7, 2007 #4

    nrqed

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    But in each term of equation 10.57, all the components of [itex] \vec{p} [/itex] are supposed to be positive 9again, this is true for each of the two terms of 10.57). But in 10.58 the components of p are allowed to be negative. So in the sum of 10.58 here is what happens: when the p's are positive, one generates the pieces corresponding to the first term of 10.57. When the p components in 10.58 are negative, one generates the pieces coresponding to the second term of 10.57.

    Does that make sense?
     
  6. Oct 7, 2007 #5
    Is that implied by something in the text? Just above equation (10.58) he says:
    phi includes contributions from all values of [itex]\vec{p}[/itex]
     
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