The connection between the Wightman formalism and the standard formalism has been given in a few papers. In two dimensions: For example the fact that the taylor expansion of the Wightman functions in the non-Fock space agree with renormalized perturbation theory can be found in: Dimock J.
This is proven in: Osterwalder K. Also Borel summation of the renormalized perturbation theory gives the Wightman functions on the non-Fock space. This is proven in: Sokal A. Proof found in: Magnen J.
SCATTERING IN QUANTUM FIELD THEORIES, THE AXIOMATIC AND CONSTRUCTIVE APPROACHES
Phys, 56 , This is also true of the S-matrix: Constantinescu F. Physics, , The simplest example of explicit n-point functions is usually found in work on the Thirring model. For example: D.
Stoyanov and L. Khadzhiivanov, "Theory of Wightman functions in the thirring model" , Theoretical and Mathematical Physics 46 , 3, This is because the Thirring model has a rigorously defined transformation to a free scalar field, so you obtain the free scalar correlation functions and then transform back to the Thirring model. If anybody would like more difficult 2d models please say so. DarMM said:. The simplest example of explicit n-point functions is usually found in work on the Thirring model.
Scattering in Quantum Field Theories : The Axiomatic and Constructive Approaches
Do you mean that they can be written as the same limit, if one uses the right cutoff-dependent counterterms? The start apparently is to assume having the right inner product? But this is not really known And another question: Where would one find all this information if one would have to search for oneself? I found the literature on these topics extremely difficult to access. How did you learn it? The expectations and existence of objects like U f and their products is basically related to verifying the Haag-Kastler axioms.
It was easier to prove the existence, causality, etc. It is extremely difficult to access. I essentially read Glimm and Jaffe and PCT, Spin and statistics and kept reading references based on questions they generated. I basically wrote notepad after notepad of notes, cross-referencing papers and every time a question occured to me I searched everywhere I knew to find the answer.
This often involved making notes on what authors said their references demonstrated so that I could quickly find the right paper. Basically I climbed the ladder with a sequence of questions for every rung, crossing it off when I had read the appropriate paper. There is no easy way to do it because there is no good monograph on the subject for an outsider I hope to write one. Last edited by a moderator: May 5, Why should that be important? These states are nonobservable, anyway! I think that they are observable, at least, in principle. Nevertheless, such states exist in the Fock-space theory, and given enough instrumental resolution we should be able to distinguish such states.
So, in principle, experiments can tells us whether Fock-space or non-Fock-space is realized in nature. I wonder if somebody has analyzed the experimental differences between the two approaches? You argue that ''it is important that one can always form states in which the number of particles is determined exactly'' in order to justify your preference for Fock space, and ''such states exist in the Fock-space theory, [so] we should be able to distinguish such states''.
This is called circular reasoning. Let me be clear.
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The relevant experiment should be able to tell exactly how many particles there are in a given state. In practice this is not possible. A theory which used Fock spaces for relativistic interactions would not be a field theory. So your questions concerns the difference between field theory and some other formalism. They would not be different approaches to field thoery, but totally different formalisms. One is the field-based approach.
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