Rigorous Quantum Field Theory.

[zetafunction]

from the book of Zeidler http://www.flipkart.com/book/quantum-electrodynamics-eberhard-zeidler-bridge/3540853766 i heard that all the 'divergent' quantities were encoded in the linear combination of dirac delta funciton

$$\sum_{n\ge 0}c_{n}\delta ^{n} (x)$$ so when taken x=0 the expression was divergent. As far as i know Epstein-Glasser method allowed you to recover the Scattering S-matrix perturbatively plus a distributional contribution involivng dirac derivatives, also the fact that '2 distributions can not be multiplied' avoided us from getting finite result

could anyone give a lazyman intro to Epstein-Glasser theory ??

 

[DrFaustus]

DarMM, or anyone else for that matter, I'm trying to figure out the rigorous construction of the $$\varphi_2^4$$ and I'm reading Glimm and Jaffe, "Quantum field theory and Statistical mechanics - Expositions". Problem I find it a rather hard nut to crack. Tons of technicalities and I'm also failing to grasp the big picture, i.e. how are all the technicalities supposed to fit together. So question is, do yo know of any "pedagogical" account on the rigorous construction of $$\varphi_2^4$$ in a Minkowski setting? That is, no Haag-Kastler nor Osterwalder-Schrader. Would really appreciate any refrences. DarMM, you mentioned you found your notes... I'm guessing they're not in electronic format, are they?

zetafunction -> I did not use the Epstein-Glaser approach, so this is just the idea of how it works. Essentially, if you know the time ordered product of one Wick monomial, then by causality you know the TOP of 2 Wick monomials. And if you know the TOP of 2 WM, then you know the TOP of 3 WM. And so on. Here, when I say you know I mean "you can construct". For instance, ion the case of the usual $$\varphi^4$$ theory, causality will allow you to construct the following chain of TOP
$$T[:\varphi^4:] \longrightarrow T[:\varphi^4::\varphi^4:] \longrightarrow T[:\varphi^4::\varphi^4::\varphi^4:] \longrightarrow \dots$$.
Double dots denote normal ordering and the fields are free fields. Now, the problem with the above chain is that you have products of distributions which are generally ill defined for coinciding points. The extension of the TOP of 2 or more WM to the diagonal, i.e. to coinciding points, then amounts to renormalization. And the extension is also not unique, which corresponds to the usual renormalization ambiguities. Note that there are no divergencies here, everything's finite.