mathematical physics - Constructive vs Algebraic Quantum Field Theory

I am interested to know how the (non)existence theorems of constructive QFT and algebraic QFT are related (or not). I have only a weak grasp of either, so I'm looking for something like a quick overview. Here's how I understand things:

Constructive QFT has shown that quantum fields can be well-defined in $d$ < 4, by showing that the distributions and interactions exist in that case. Apparently there are specific examples, but this is an existence result for 1) techniques coming from analysis, 2) $d$ < 4, and 3) interactions are included.

Algebraic QFT comes from looking at the canonical commutation relations on spacetime, and has shown that a single Hilbert space cannot represent both the interacting and non-interaction picture (I think this is one statement of Haag's theorem). So this is a non-existence result 1) based on the algebra of CCR and 2) includes interactions (apparently a way out is to assume periodic boundary conditions - I'm not overly interested in that part).

So several specific questions - can I think of these as a Lagrangian vs Hamiltonian picture? What do these two results mean to each other - are they even related at all since they are essentially QFT in different dimensions?

References with reviews of these theories would be nice - I've checked out Haag's book enough to know that if I spent time I could probably understand it, but I'd rather good review articles if anyone knows of any.

Answer

The no-go results from Algebraic and Constructive QFT you mention deal with related but slightly different matters.

(Edit: the previous version of the following paragraph was slightly misleading - Haag's theorem is actually stronger than I stated before; see below for details)

• Haag's theorem (which actually slightly predates the inception of Algebraic QFT) tells us that we cannot write interaction picture dynamics within Hilbert spaces which are free field representations of the CCR's. This is not the same as to say that interacting dynamics does not exist at all - it simply says that we cannot implement it as unitary operators in the interaction picture. This is done by showing that the possibility to do so in some Hilbert space does imply that we are dealing with a free field representation of the CCR's. The argument is closed by a "soft triviality" result by Jost, Schroer and Pohlmeyer arguing that the latter implies that all truncated $n$-point functions vanish for $n>2$, hence the field is really free - in particular, the "interaction Hamiltonian" is zero.

This has consequences for both scattering theory and attempts to rigorously construct field theoretical models starting from free fields. In the first case, Haag's theorem is circumvented by either the LSZ of Haag-Ruelle scattering formalisms, which obtain the S-matrix by respectively taking infinite time limits in the weak (matrix elements) and strong (Hilbert space vectors) sense. Recall that both setups require the assumption of a mass gap in the joint energy-momentum spectrum (i.e. an isolated, non-zero mass shell), otherwise we run into the notorious "infrared catastrophe", which is dealt with using "non-recoil" (i.e. Bloch-Nordsieck) approximation methods in formal perturbation theory but remains a challenge in a more rigorous setting, save in some non-relativistic models. In the second case, one is led to consider representations of the CCR's which are inequivalent to free field ones. Since field theories living in the whole space-time have infinite degrees of freedom, the Stone-von Neumann uniqueness theorem no longer holds (actually, Haag's theorem can be seen as a manifestation of this particular failure mechanism), and hence such representations should exist in abundance. Motivated by these results, Algebraic QFT was devised with a focus on structural (i.e. "model-independent") aspects of QFT in a way that does not depend on a particular representation; on other front, one may also try to explore this abundance of representations to construct models rigorously, which brings us to the realm of Constructive QFT.

• The "existence" (a.k.a "non-triviality") and "non-existence" (a.k.a. "triviality") results in Constructive QFT tell us which interactions survive after non-perturbative renormalization. More precisely, you construct field theoretical models in a mathematically rigorous way by first considering "truncated" interacting theories (i.e. with UV and IR cutoffs), and then carefully removing the cutoffs in a sequence of controlled operations. The resulting model may be interacting (i.e. "non-trivial") or not (i.e. "trivial"), in the sense that its truncated $n$-point correlation functions for $n>2$ may be respectively non-vanishing or not. In the first case, any representation of the CCR's in the Hilbert space where the interacting vacuum state vector lives is necessarily inequivalent to a free field one - in particular, one cannot write the interacting dynamics as unitary operators in the interaction picture, in accordance with Haag's theorem. In the second case, you really obtain a free field representation of the CCR's, but here because renormalization has completely killed the interaction.

Finally, it is important to notice that triviality of a model may stem from reasons unrelated to the underlying mechanism of Haag's theorem. The latter, once more, is a consequence of having an infinite number of degrees of freedom in infinite volumes (this theorem does not hold "in a box", for instance), whereas the former usually derives from an interaction which has too singular a short-distance behavior, as argued in the previous paragraph. This can be intuitively be understood by the (local) singularity and (global) integrability of the free field's Green functions: the lower the space-time dimension, the better the singular (UV) behaviour and the worse the integrability (IR) behaviour, and vice-versa. That's the underlying reason why $\lambda\phi^4$ scalar models are super-renormalizable in 2 and 3 dimensions (having only tadpole Feynman graphs as divergent in 2 dimensions) and non-perturbatively trivial in $>4$ dimensions.

Ah, I've almost forgotten about the references: in my opinion, the best discussion of triviality results in QFT from a rigorous viewpoint is the book by R. Fernández, J. Fröhlich and A. D. Sokal, "Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory" (Springer-Verlag, 1992), specially Chapter 13. There both the above "hard triviality" results for $\lambda\phi^4$ models and "soft triviality results" such as the Jost-Schroer-Pohlmeyer theorem (which underlies Haag's theorem, as mentioned at the beginning of my answer) are discussed. The book is not exactly for the faint of the heart, but the first sections of this Chapter provide a good discussion of the statements of the theorems, before proceeding to the proofs of the above "hard triviality" results. For a detailed discussion of Jost-Schroer-Pohlmeyer's and Haag's theorems, as well as their proofs, I recommend the book of J. T. Lopuszanski, "An Introduction to Symmetry and Supersymmetry in Quantum Field Theory" (World Scientific, 1991). The classic book of R. F. Streater and A. S. Wightman, "PCT, Spin and Statistics, and All That" (Princeton Univ. Press) also discusses these two results.