What are some good references for field theory via functional analysis?

Many of the aspects of QFT are traditionally done in ways incompatible with a rigorous mathematical treatment, calling for a variety of tricks to fix essentially what was caused by unjustified calculations or generalizations. One example for all is treating the Fock space as the state space of a collection of infinitely many harmonic oscillators, leading to infinite zero-point energy and related problems. This is at odds with the mathematical postulates of quantum physics, because

1. it is easily proven that Fock space over any separable single-particle state space is separable itself, which precludes the use of an infinite tensor product of $$L^2(\mathbb{R})$$ which is an inseparable space,
2. although the latter space can in principle be constructed, infinite eigenvalues don't make any kind of sense, so the sum of energies of all of the oscillators would not be assigned any self-adjoint operator on it – it would simply not be an observable.

I'm looking for recommendations on references that follow the Reed & Simon approach and build quantum field theory upon proper Hilbert space operator theory, evading such discrepancies from the beginning. Practicality is secondary, I don't expect the reference to get me to any point where I could actually compute any real world problem.

Disclaimer: I haven't read all parts of R&S, it's possible some answer is also found therein.

• You can try Haag's Local Quantum Physics or Streater and Wightman's PCT, Spin and Statistics, and all that – Slereah May 12 at 7:40

There is a large rigorous mathematical literature on quantum field theory. This literature is not complete, in that it does not cover the QFTs of greatest interest to particle physicists, the 4d Yang-Mills theories. But it does cover a great many other models, both toy examples and statistical models used in condensed matter theory, and it also covers the perturbative aspects of the Standard Model in complete rigor.

What follows is an idiosyncratic view of this literature.

The ur-text in the subject is Streater & Wightman's PCT, Spin-Statistics, and All That. This book doesn't have a lot of examples, but it does cover non-interacting scalar and spinor theories. It should at least answer your questions about Fock spaces.

There's also the algebraic QFT approach, which focuses on the properties of the operator algebra of local observables. The classic book here is Haag's Local Quantum Physics. I confess the algebraic QFT approach is not really to my taste, so I'm giving it short shrift here.

An interesting related read is Robert Wald's QFT in Curved Spacetime and his papers with Stefan Hollands. The language is not explicitly rigorous -- it is written more for physicists than mathematicians -- but it is careful and correct. Worth looking at to see some of the limitations of the flat space QFT axioms.

If you want to get closer to particle physics, there is a considerable literature on rigorous perturbation theory. Urs Schreiber has a nice overview of the subset of this literature that intersects with algbraic QFT. Also very much worth looking at is Kevin Costello's book Renormalization & Effective Field Theory, which cover perturbative QFT from a Wilsonian point of view.

Finally, and not least if functional analysis is what you want, there's the 'constructive quantum field theory' literature, which focuses on the rigorous mathematical construction of interacting Euclidean QFTs.

The standard text here is Glimm & Jaffe's "Quantum Physics: A Functional Integral Point of View". This book describes the Euclidean path integral rigorously and constructs interacting scalar and spinor theories in spacetime dimensions 2 & 3. Also worth looking at are Rivasseau's From Perturbative To Constructive Renormalization, and Battle's Wavelets & Renormalization.

As I alluded to, this program aimed at eventually constructing the relativistic 4d theories we use in particle physics. It has not succeeded. Partly that's because the analysis is quite difficult in 4d. Partly, it's because some of the components of the Standard Model (the $$U(1)$$ and Higgs fields) probably can't be made rigorous, as they seem to become trivial in the continuum limit. I think it's reasonable to hope for a construction of a non-trivial gauge theory in our lifetime, but a lot of the energy in constructive QFT has in recent years gone into studying systems that emerge in statistical physics. A nice jumping off point to this literature is the IAS/Park City lecture collection Statistical Mechanics.