(adsbygoogle = window.adsbygoogle || []).push({}); 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.

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# Zwiebach, pages 175,176

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