- #1
Jimmy Snyder
- 1,127
- 21
Homework Statement
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]
Homework 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]
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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