# Foundation of path Integral formulation of QFT, derivation and meaning of LSZ formulas

I'm currently studying path integral in quantum field theory. I am comfortable with path integrals, and also path integral formulation of QM, but I was asking if there is a self consistent coherent formulation of QFT that only use path integrals. I know just a few texts that use this approach, like Zee and Ramond, and I have some conceptual problems. As I understood the topic, the conceptual features are more or less this:

1) We define a infinite degree of freedom quantum system, ad example,by taking the continuous limit of a lattice quantum system, in which each dof is labelled by a lattice index. So we have, a field ket (the equivalent of the position basis in QM), a field operator (the equivalent of position operator) and a field function (the equivalent of the position eigenvalues).

2) We define the amplitude of propagation between two distinct configuration of the field at different times, exactly as in QM, as a functional integrals over all possible configuration of the field, with fixed extremes. Then, from Shwinger-Dyson equation, free field (operator) satisfy the classical equation of motion (when they are linear)

3) We can build one particle states by applying the free field operator to the free vacuum, and find the usual creator-destruction operator decomposition (this point isn't very clear because I didn't find some text that do its in detail)

4) We can define a generating functional Z, that is the integral over all possible configuration of the field from -inf time to + inf time with a source in the lagrangian. Some text says that one can show this functional is the vacuum to vacuum amplitude in presence of source: this is due to the imaginary small factor one add to the mass, that "select" the vacuum states. I have clear that in QM but not so clear in QFT. From Z one can easily find closed expressions of vev of fields' T ordered product (Green function).

Here the point that puzzle me the most: T ordered product and S matrix elements are connected by LSZ reduction formulas. If I well understood the topics, in momentum space this reduction states that to obtain S matrix element we have to multiply the Fourier transform of Green function by inverse of free propagator of final and initial states (e.g. in the scalar case, the factor p2-m2), and by some squared Z and E factors. In the Faynmann diagram, the consequence is that we have to "cut" external legs, and substitute the bare mass with the renormalized one.

However I found only a messy and rather obscure demonstration of this result, involving creator operator ecc ecc and I was wandering if ther is anhoter more direct demonstration that use path integral: I mean, instinctively, green function in momentum space are the amplitude of the process in which I crate a particle in x, and destroy a particle in y, and so its transform its obviously connected to S matrix elements, apart form delta factor. Some tests, as Ramond only say this, but never actually compute cross sections. The inverse free propagator factors are, intuitively, connected with the necessity of "throw away" the free propagation, before and after scattering, we are not interested in. I can understand even the origin of Z factors, that are connected to self interaction of field that we can't never "turn off" but all texts I checked do not explain this factor, or do this in rather obscure ways (e.g. using Lehman decomposition). Apart form self interaction and renormalization problems, are LSZ formulas "obvious"? And how can we validate these Z factors without having before afforded renormalization problems?

I am very confused by the fact every QFT text treat the subject in a completely different manner and I found that nearly no one gives a systematic treatise of path integrals. Do you know such a text?

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• Too long question, try to be more direct. – Nogueira Nov 7 '16 at 20:09
• The standard textbook by L. H. Ryder exploits all and more about definition of QFT using path integral only. Moreover: "I was asking if there is a self consistent coherent formulation of QFT that only use path integrals" the only self consistent formulation of QFT is exactly the one making use of the path integral (the examples with creations and annihilations operators only hold true if the fields equations are linear, which they are not, in general). – gented Nov 7 '16 at 20:19
• Hi and welcome to phyiscs.SE! We want our questions to be understandable and useful to a broad readership. Currently, it is not entirely clear to me what you are asking for - a book recommendation? Someone proving that the path integral formalism can recover the LSZ formulae? Someone showing how renormalization works in the path integral formalism? Something else? – ACuriousMind Nov 7 '16 at 20:19
• Yes @GennaroTedesco I know it's the only real consistent formulation: that's the reason I am very interested in the subject, but I can't find a real self consistent treatment! – Guido Giachetti Nov 7 '16 at 20:25
• Thank you @ACuriousMind . More directly, I was looking for (in general) a good text on the subject, and (in particular), some path integral proof of LSZ, closer to the physics meaning. I have many doubts because the text are so different, and I not sure the pieces are fit each other in my head! – Guido Giachetti Nov 7 '16 at 20:27