What is the missing factor in equations (10.58) and (10.60)?

[Jimmy Snyder]

Homework Statement

Equation (10.58) is:
$$\phi(t, \vec{x}) = \frac{1}{\sqrt{V}}\Sigma_{\vec{p}}\frac{1}{\sqrt{2E_p}}(a_{p}e^{-iE_pt + i\vec{p}\cdot\vec{x}} + a_p^{\dagger}e^{iE_pt - i\vec{p}\cdot\vec{x}})$$

Homework Equations

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

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.

 

[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?

 

[Jimmy Snyder]

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?

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.

 

[nrqed]

Homework Statement

Equation (10.58) is:
$$\phi(t, \vec{x}) = \frac{1}{\sqrt{V}}\Sigma_{\vec{p}}\frac{1}{\sqrt{2E_p}}(a_{p}e^{-iE_pt + i\vec{p}\cdot\vec{x}} + a_p^{\dagger}e^{iE_pt - i\vec{p}\cdot\vec{x}})$$

Homework Equations

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

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.

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?

 

[Jimmy Snyder]

But in each term of equation 10.57, all the components of $\vec{p}$ are supposed to be positive.

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