Before I get to the point, let me quickly describe the context and what level of understanding I’m trying to achieve, if possible. I’d like to get some intuition on more rigorous approaches of QFTs, how they relate to “standard textbook perturbative approaches” (for lack of a better name) and how they deal with some of the issues and important concepts that arose in these standard approaches. At that point, I do not necessarily want to get the concrete proofs behind rigorous QFT but rather understand the main statements: in some sense my main goal would be to build a picture for myself where I have a physical and practical understanding of QFT, not necessarily fully rigorous, but which I’d know to be in line with the existing rigorous results. In that spirit, even some intuition coming from simple lower dimensional examples sounds good to me. Most of the resources I have looked at were rather technical and a bit difficult to get into for someone who only wants to get the big picture, so I’d be very happy if specialists could help me navigate this domain.
Being more of an algebraist at heart, I’d be happiest with an approach that roughly include the following notions:
- a non-commutative algebra of observables $A$
- a Hilbert space $H$ on which elements of $A$ act as operators
- a notion of fields that can essentially be seen as “space-time dependent quantum operators”
- an hamiltonian in $A$ built as a local combination of these fields (possibly part of a full Poincaré representation)
Of course this is very vague and would need to be supplemented with more precise features (domain of definition for operators, well-behaved definition for fields for instance as distributions like in Wightmann axioms, notions of locality, …) but these would be the big requirements. From this framework, one could then define various physical quantities to be measured (probability amplitudes, energy spectrum, …). I'll refer to any formulation including these ingredients as an "algebraic formulation" of a QFT, which seems to broadly include approaches such as Wightman, Haag-Kastler, ...
In my current view, textbook perturbative QFT tries to start with such a setup using a “naive” Fock space quantisation and the interaction picture but fails and hits various problems (Haag’s theorem, divergences, ...). The processes of regularisation and renormalisation then allow to give a sensible meaning for some physical quantities such as probability amplitudes, at least in some cases. But as far as I can see, in doing this we lost the connection with the initial algebraic picture: the latter was merely a symbolic guide for the starting point. In the end, the objects which are well-defined after renormalisation are not the operators, the fields or the states, but just some physical observables like amplitudes. I see how this could be enough from the point of view of physics, but I would like to know if one can go further and also reach a proper well-defined algebraic formulation of the finite theory. As far as I can understand, this would for instance be the expected framework for solving the Yang-Mills millenium problem, at least in its initial formulation (not that I aim to solve it, but it would be nice to at least understand its expectations).
My main question is then the following. Can we construct a finite well-defined algebraic formulation of a renormalised QFT in the above sense and if yes, what are the tools used to implement renormalisation on the fields, operators and states of the Hilbert space?
From what I can understand, works of Stueekelberg-Petermann, Bogoliubov-Parasiuk-Hepp, Epstein-Glaser, etc could be relevant in that direction but it’s a bit hard for me to really extract the overall picture from those.
I get that this post is long so thanks a lot for reading until here. Any answer, comment or bibliographic suggestion, even partial, is most welcomed.
