# Zwiebach, pages 175,176

1. ### Jimmy Snyder

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

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

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 $\vec{p} = -\vec{p}$, which does not effect $E_p$. Then (10.58) is supposed to be the sum of (10.57) over all values of $\vec{p}$. 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. ### ehrenfest

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
3. ### Jimmy Snyder

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.

4. ### nrqed

3,083
But in each term of equation 10.57, all the components of $\vec{p}$ 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?

5. ### Jimmy Snyder

Is that implied by something in the text? Just above equation (10.58) he says:
phi includes contributions from all values of $\vec{p}$