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Zwiebach, pages 175,176 
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#1
Sep2707, 06:01 PM

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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. 


#2
Oct707, 12:23 AM

P: 1,996

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? 


#3
Oct707, 05:13 AM

P: 2,179




#4
Oct707, 08:33 AM

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P: 3,017

Zwiebach, pages 175,176
Does that make sense? 


#5
Oct707, 08:01 PM

P: 2,179

phi includes contributions from all values of [itex]\vec{p}[/itex] 


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