In general I find in books that the path integral approach is an equivalent alternative of the hamiltonian approach for QFT (and for QT in general, but my concern is with QFT). There I usually find that this method is usually developed in a formal way and used to derive Feynmann rules, gauge identities and so on.(adsbygoogle = window.adsbygoogle || []).push({});

However, I did not find lots of explicit applications of this method to QFT (I found the typical example of the path integral representation of the 2 slit experiment but that is non relativistic one particle QT and not QFT).

In this sense, my doubts are related with how amplitudes of QFT should be computed with this method.

Lets put an example: Klein Gordon non interacting theory. I want to calculate the amplitude of finishing with a final 2-particle state with given energy and momentum (lets say (E2a;p2a) y (E2b;p2b)) given that we started with an initial 1 particle state with given energy and momentum (lets say (E1;p1)). (I know that the answer should be 0 because when there is no interaction, we can not go from a 1-particle state to a 2 particle state, but I would like to arrive to this result through path integral formalism.

My questions are:

1) What are the limits of the integral?

2) Are they “fi(0,x)=exp(E1*t+p1*x) and fi(T,x)=exp(E2a*t+p2a*x)+exp(E2b*t+p2b*x)”?

3) If my guess is correct, what would the final limit “fi(T,x)=exp(E2a*t+p2a*x)+0,5*exp(E2b*t+p2b*x)” mean? That we end we half a particle?

4) If that is what that mean, this amplitude should give 0 as a result, isn´t it? Is this the result that we arrive when we compute this amplitude

PS1: Thanks in advance for everything you can do?

PS2: Sorry for my English, I speak Spanish

PS3: Sorry for my physics, I am an actuary!

PS4: I know that there is no point in calculating this amplitude through this method. I just want to know how this amplitude should be calculated, if I had a computer where I can put 1 millon “for i=1 to n”, in order to understand properly what this method exactly mean.

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# Path Integral example, analysis of integral limits for K-G theory

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