Peskin and Schroeder - Derivation of equation (2.45)

[spaghetti3451]

I'm having trouble deriving equation (2.45) on page 25. In particular, in the derivation of

##i\frac{\partial}{\partial t}\pi({\bf{x}},t) = -i(-\nabla^{2}+m^{2}) \phi({\bf{x}},t)##,

I need to show that

##\frac{1}{2}\pi({\bf{x}},t) \phi({\bf{x'}},t)(-\nabla^{2}+m^{2}) \phi({\bf{x'}},t) - \frac{1}{2} \phi({\bf{x'}},t)(-\nabla^{2}+m^{2})\phi({\bf{x'}},t)\pi({\bf{x}},t) = -i \delta^{(3)}({\bf{x}}-{\bf{x'}})(-\nabla^{2}+m^{2})\phi({\bf{x'}},t)##.

Now, this is what I've done so far:

##\frac{1}{2}\pi({\bf{x}},t) \phi({\bf{x'}},t)(-\nabla^{2}+m^{2}) \phi({\bf{x'}},t) - \frac{1}{2} \phi({\bf{x'}},t)(-\nabla^{2}+m^{2})\phi({\bf{x'}},t)\pi({\bf{x}},t)##

##=\frac{1}{2}\pi({\bf{x}},t) \phi({\bf{x'}},t)(-\nabla^{2}+m^{2}) \phi({\bf{x'}},t)-\frac{1}{2}\phi({\bf{x'}},t)\pi({\bf{x}},t) (-\nabla^{2}+m^{2}) \phi({\bf{x'}},t) +\frac{1}{2}\phi({\bf{x'}},t)\pi({\bf{x}},t) (-\nabla^{2}+m^{2}) \phi({\bf{x'}},t)- \frac{1}{2} \phi({\bf{x'}},t)(-\nabla^{2}+m^{2})\phi({\bf{x'}},t)\pi({\bf{x}},t)##

##=\frac{1}{2}[\pi({{\bf{x}},t}), \phi({{\bf{x'}},t})](-\nabla^{2}+m^{2}) \phi({\bf{x'}},t) + \frac{1}{2}\phi({\bf{x'}},t)[\pi({\bf{x}},t), (-\nabla^{2}+m^{2}) \phi({\bf{x'}},t)]##

##=\frac{1}{2}(-i)\delta^{(3)}({\bf{x'}}-{\bf{x}})(-\nabla^{2}+m^{2}) \phi({\bf{x'}},t) + ??##.

How do I evaluate the second commutator?

 

[spaghetti3451]

bumpp!

 

[bapowell]

Did you try breaking the second commutator up as [itex][\pi,m^2\phi] - [\pi,\nabla^2\phi][/itex] and convincing yourself that both the [itex]m^2[/itex] and [itex]\nabla^2[/itex] can be pulled out of their respective commutators?