- #1

Jufa

- 101

- 15

- Homework Statement
- I encounter a divergent integral when computing a commutator of two fields in quantum field theory homework

- Relevant Equations
- ##\phi = \int \frac{dp^3}{(2\pi)^3}e^{-ipx}\hat{a}(\vec{p})##

I am asked to compute ##[\phi(x), \phi^\dagger(y)]## , with

##\phi = \int \frac{dp^3}{(2\pi)^3}e^{-ipx}\hat{a}(\vec{p})## and with z=x-y a spacelike vector. And show that this commutator does not vanish, which means that for this non-relativsitic field i.e. with ##p^0 = \frac{\vec{p}^2}{2m}## causality is violated.

After two straightforward steps I get to this

##[\phi(x), \phi^\dagger(y)] = \int \frac{dp^3}{(2\pi)^3}e^{-i \frac{\vec{p}^2}{2m}z_0}e^{i\vec{p}\cdot\vec{z}} ##

It is self-evident that this integral does not converge and, therefore, I have been suggested to shift ##z_0## by adding to it a quantity ##-i\epsilon##. This makes the integral convergent and allows one to take the limit ##\epsilon \rightarrow 0## in the end, which works perfectly.

But why are we allowed to do this? To me it seems that the integral I am asked to compute is just (I would end the problem here and say that it is not zero, just what I was demanded to prove) divergent and that this trick allows to compute not the requested integral but a different one.

I need some help on this.

Thanks in advance.

##\phi = \int \frac{dp^3}{(2\pi)^3}e^{-ipx}\hat{a}(\vec{p})## and with z=x-y a spacelike vector. And show that this commutator does not vanish, which means that for this non-relativsitic field i.e. with ##p^0 = \frac{\vec{p}^2}{2m}## causality is violated.

After two straightforward steps I get to this

##[\phi(x), \phi^\dagger(y)] = \int \frac{dp^3}{(2\pi)^3}e^{-i \frac{\vec{p}^2}{2m}z_0}e^{i\vec{p}\cdot\vec{z}} ##

It is self-evident that this integral does not converge and, therefore, I have been suggested to shift ##z_0## by adding to it a quantity ##-i\epsilon##. This makes the integral convergent and allows one to take the limit ##\epsilon \rightarrow 0## in the end, which works perfectly.

But why are we allowed to do this? To me it seems that the integral I am asked to compute is just (I would end the problem here and say that it is not zero, just what I was demanded to prove) divergent and that this trick allows to compute not the requested integral but a different one.

I need some help on this.

Thanks in advance.