Peskin-Schroeder - Eqn 2.45 derivation

[giant_bog]

I'm having problems with the equations leading up to eqn 2.45 on page 25. The hamiltonian has a $(\nabla\phi)^2 + m^2 \phi^2$ term in the $\phi(x)$ commutator and in the $\pi(x)$ commutator it's $\phi(-\nabla^2 + m^2) \phi$.

I'm aware of a vector calculus identity that makes $(\nabla\phi)^2 = 1/2 (\nabla^2[\phi^2]) - \phi \nabla^2 \phi$.

That's almost what we have here, but the $\frac{1}{2}(\nabla^2[\phi^2])$ term is missing in the $\pi(x)$ commutator.

Did anybody see where it went?

 

[waht]

That's the n-1 boundary term which is taken be zero.

 

[The_Duck]

You can do this kind of thing using integration by parts. E.g., in one dimension:

$$\int_{-\infty}^{+\infty} \left(\frac{df}{dx}\right)^2 dx = \left[f(x) \frac{df}{dx}\right]_{-\infty}^{+\infty} - \int_{-\infty}^{\infty}f(x)\frac{d^2 f}{dx^2}dx$$

And if f(x) goes to zero at infinity then the first term on the right drops out.

 

[giant_bog]

I see it now. Thanks, folks.