The integral does not converge....

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In summary, we are asked to compute the commutator ##[\phi(x), \phi^\dagger(y)]## for a non-relativistic field with a spacelike vector. This leads to an integral that does not converge, but by shifting a quantity ##z_0## and taking the limit, we are able to make the integral convergent and obtain the desired result. This technique is justified by Distribution theory.
  • #1
Jufa
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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.
 
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What does it mean when the integral does not converge?

When the integral does not converge, it means that the area under the curve of the function being integrated does not have a finite value. This could be due to the function having infinite values or oscillating infinitely.

Why is it important to determine if an integral converges or not?

Determining if an integral converges or not is important because it helps us understand the behavior of the function being integrated. It also allows us to properly evaluate the integral and make accurate calculations.

What are some common methods used to determine if an integral converges?

Some common methods used to determine if an integral converges include the comparison test, the limit comparison test, and the ratio test. These tests compare the given integral to a known convergent or divergent integral to determine its behavior.

Can an integral diverge even if the function being integrated is continuous?

Yes, an integral can still diverge even if the function being integrated is continuous. This is because the continuity of a function only guarantees its behavior at individual points, but not necessarily its behavior over a larger interval.

What are some real-world applications of understanding the convergence of integrals?

Understanding the convergence of integrals is important in various fields of science and engineering. For example, in physics, it is used to calculate the work done by a force, and in economics, it is used to calculate the area under a demand curve to determine consumer surplus. It is also used in probability and statistics to calculate the probability of certain events occurring.

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