- #1

spaghetti3451

- 1,344

- 33

The quantum Klein-Gordon field ##\phi({\bf{x}})## and its momentum density ##\pi({\bf{x}})## are given in Fourier space by

##\phi({\bf{x}}) = \int \frac{d^{3}p}{(2\pi)^{3}} \frac{1}{\sqrt{2 \omega_{{\bf{p}}}}} \big( a_{{\bf{p}}} e^{i{\bf{p}} \cdot {\bf{x}}} + a^{\dagger}_{{\bf{p}}} e^{-i{\bf{p}} \cdot {\bf{x}}} \big)## and

##\pi({\bf{x}}) = \int \frac{d^{3}p}{(2\pi)^{3}} (-i) \sqrt{\frac{ \omega_{{\bf{p}}}}{2}} \big( a_{{\bf{p}}} e^{i{\bf{p}} \cdot {\bf{x}}} - a^{\dagger}_{{\bf{p}}} e^{-i{\bf{p}} \cdot {\bf{x}}} \big)##.

[These are equations (2.25) and (2.26) from the Peskin and Schroeder.]

Now, I used the substitution ##{\bf{p}} \rightarrow {\bf{-p}}## in the expression for ##\phi({\bf{x}})## and obtained

##a_{{\bf{p}}} = a^{\dagger}_{-{\bf{p}}}## and ##a_{-{\bf{p}}} = a^{\dagger}_{{\bf{p}}}##.

On the other hand, I used the same substitution ##{\bf{p}} \rightarrow {\bf{-p}}## in the expression for ##\phi({\bf{x}})## and obtained

##a_{{\bf{p}}} = - a^{\dagger}_{-{\bf{p}}}## and ##a_{-{\bf{p}}} = - a^{\dagger}_{{\bf{p}}}##.

Can someone explain what's going on?

##\phi({\bf{x}}) = \int \frac{d^{3}p}{(2\pi)^{3}} \frac{1}{\sqrt{2 \omega_{{\bf{p}}}}} \big( a_{{\bf{p}}} e^{i{\bf{p}} \cdot {\bf{x}}} + a^{\dagger}_{{\bf{p}}} e^{-i{\bf{p}} \cdot {\bf{x}}} \big)## and

##\pi({\bf{x}}) = \int \frac{d^{3}p}{(2\pi)^{3}} (-i) \sqrt{\frac{ \omega_{{\bf{p}}}}{2}} \big( a_{{\bf{p}}} e^{i{\bf{p}} \cdot {\bf{x}}} - a^{\dagger}_{{\bf{p}}} e^{-i{\bf{p}} \cdot {\bf{x}}} \big)##.

[These are equations (2.25) and (2.26) from the Peskin and Schroeder.]

Now, I used the substitution ##{\bf{p}} \rightarrow {\bf{-p}}## in the expression for ##\phi({\bf{x}})## and obtained

##a_{{\bf{p}}} = a^{\dagger}_{-{\bf{p}}}## and ##a_{-{\bf{p}}} = a^{\dagger}_{{\bf{p}}}##.

On the other hand, I used the same substitution ##{\bf{p}} \rightarrow {\bf{-p}}## in the expression for ##\phi({\bf{x}})## and obtained

##a_{{\bf{p}}} = - a^{\dagger}_{-{\bf{p}}}## and ##a_{-{\bf{p}}} = - a^{\dagger}_{{\bf{p}}}##.

Can someone explain what's going on?

Last edited: