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?