Extra terms in the Action in Zee chapter 1.3?

[looseleaf]

Hi, could someone please explain where the tau and sigma terms come from in this expression:

I see the kinetic energy and quadratic "mattress" potential terms, but the tau and sigma kinda come out of nowhere. Where are they from?

 

[Hans de Vries]

These are just arbitrary coupling constants. They originate from the self interaction energy V of the field $\varphi$ with itself in equation (2). It is a way of expressing the self interaction part of the arbitrary function V in a series of $\varphi$.

 

[looseleaf]

These are just arbitrary coupling constants. They originate from the self interaction energy V of the field $\varphi$ with itself in equation (2). It is a way of expressing the self interaction part of the arbitrary function V in a series of $\varphi$.

Thanks that makes sense.

I was wondering if anyone could address another question I had?
This one is from the Sidney Coleman lectures (2.72)

$e^{i P . x}\alpha(p)e^{-i P . x} = e^{-i p . x}\alpha(p)$

i'm confused why we drop the operator on the right side when we go from 3-momentum to 4-momentum?