HAAG'S THEOREM IN RENORMALISED QUANTUM FIELD THEORIES LUTZ KLACZYNSKI Abstract. We review a package of no-go results in axiomatic quantum field theory with Haag's theorem at its centre. Since the concept of operator-valued distributions in this framework comes very close to what we believe canonical quantum fields are about, these results are of consequence to quantum field theory: they suggest the seeming absurdity that this highly victorious theory is incapable of describing interactions. We single out unitarity of the interaction picture's intertwiner as the most salient provision of Haag's theorem and critique canonical perturbation theory to argue that renormalisation bypasses Haag's theorem by violating this very assumption. Contents 1. Introduction 2 Part 1. Historical account 5 2. The representation issue & Haag's theorem 5 3. Other versions of Haag's theorem 13 4. Superrenormalisable theories evade Haag's theorem 16 5. What to do about Haag's theorem: reactions 18 Part 2. Axiomatic quantum field theory & Haag's theorem 23 6. The interaction picture in Fock space 23 7. Canonical (anti)commutation relations and no-interaction theorems 27 8. Wave-function renormalisation constant 32 9. Canonical quantum fields: too singular to be nontrivial 36 10. Wightman axioms and reconstruction theorem 40 11. Proof of Haag's theorem 45 12. Haag's theorem for fermion and gauge fields 50 Part 3. Renormalisation wrecks unitary equivalence 56 13. The theorem of Gell-Mann and Low 56 arXiv:1602.00662v1 [hep-th] 1 Feb 2016 14. The CCR question 58 15. Divergencies of the interaction picture 59 16. The renormalisation narrative 63 17. Renormalisation circumvents Haag's theorem 68 18. Conclusion: renormalisation staves off triviality 71 Acknowledgements 71 Part 4. Appendices 71 Appendix A. Baumann's theorem 71 Appendix B. Wightman's reconstruction theorem 73 Appendix C. Jost-Schroer theorem 75 References 77 Date: February 2, 2016. 1 2 LUTZ KLACZYNSKI 1. Introduction Quantum field theory (QFT) is undoubtedly one of the most successful physical theories. Besides the often cited extraordinary precision with which the anomalous magnetic moment of the electron had been computed in quantum electrodynamics (QED), this framework enabled theorists to predict the existence of hitherto unknown particles. As Dirac was trying to make sense of the negative energy solutions of the equation which was later named after him, he proposed the existence of a positively charged electron [Schw94]. This particle, nowadays known as the positron, presents an early example of a so-called antiparticle. It is fair to say that it was the formalism he was playing with that led him to think of such entities. And here we have theoretical physics at its best: the formulae under investigation only make sense provided an entity so-and-so exists. Here are Dirac's words: "A hole, if there were one, would be a new kind of particle, unknown to experi- mental physics, having the same mass and opposite charge to the electron. We may call such a particle an anti-electron. We should not expect to find any of them in nature, on account of their rapid rate of recombination with electrons, but if they could be produced experimentally in high vacuum, they would be quite stable and amenable to observations"[Dir31]. Of course, the positron was not the only particle to be predicted by quantum field theory. W and Z bosons, ie the carrier particles of the weak force, both bottom and top quark and probably also the Higgs particle are all examples of matter particles whose existence was in some sense necessitated by theory prior to their discovery. Yet canonical QFT presents itself as a stupendous and intricate jigsaw puzzle. While some massive chunks are for themselves coherent, we shall see that some connecting pieces are still only tenuously locked, though simply taken for granted by many practising physicists, both of phenomenological and of theoretical creed. 1.1. Constructive and axiomatic quantum field theory. In the light of this success, it seems ironic that so far physically realistic quantum field theories like the standard model (SM) and its subtheories quantum electrodynamics (QED) and quantum chromodynamics (QCD) all defy a mathematically rigorous description [Su12]. Take QED. While gauge transformations are classically well-understood as representations of a unitary group acting on sections of a principle bundle [Blee81], it is not entirely clear what becomes of them once the theory is quantised [StroWi74, Stro13]. However, Wightman and G˚ardinghave shown that the quantisation of the free electromagnetic field due to Gupta and Bleuler is mathematically consistent in the context of Krein spaces (see [Stro13] and references there, p.156). Drawing on the review article [Su12], we make the following observations as to what the state of affairs broadly speaking currently is. First, all approaches to construct quantum field models in a way seen as mathematically • sound and rigorous employ methods from operator theory and stochastic analysis, the latter only in the Euclidean case. This is certainly natural given the corresponding heuristically very successful notions used in Lagrangian quantum field theory and the formalism of functional integrals. These endeavours are widely known under the label constructive quantum field theory, where a common objective of those approaches was to obtain a theory of quantum fields with some reasonable properties. Axiomatic quantum field theory refined these properties further to a system of axioms. Several more or less equivalent such axiomatic systems have been proposed, the most prominent of which are: (1) the so-called Wightman axioms [StreatWi00, Streat75], HAAG'S THEOREM IN RENORMALISED QUANTUM FIELD THEORIES 3 (2) their Euclidean counterparts Osterwalder-Schrader axioms [OSchra73, Stro13] and (3) a system of axioms due to Araki, Haag and Kastler [HaKa64, Ha96, Stro13]. These axioms were enunciated in an attempt to clarify and discern what a quantum field theory should or could reasonably be. In contrast to this, the proponents of the somewhat idiosyncratic school of axiomatic S-matrix theory tried to discard the notion of quantum fields all together by setting axioms for the S- matrix [Sta62]. However, it lost traction when it was trumped by QCD in describing the strong interaction and later merged into the toolshed of string theory [Ri14]. Second, efforts were made in two directions. In the constructive approach, models were • built and then proven to conform with these axioms [GliJaf68, GliJaf70], whereas on the axiomatic side, the general properties of quantum fields defined in such a way were investigated under the proviso that they exist. Among the achievements of the axiomatic community are rigorous proofs of the PCT and also the spin-statistics theorem [StreatWi00]. Third, within the constructive framework, the first attempts started with superrenor- • malisable QFTs to stay clear of ultraviolet (UV) divergences. The emerging problems with these models had been resolved immediately: the infinite volume divergences encountered there were cured by a finite number of subtractions, once the appropri- ate counterterms had been identified [GliJaf68, GliJaf70]. Fourth, however, these problems exacerbated to serious and to this day unsurmountable • obstructions as soon as the realm of renormalisable theories was entered. 4 In the case of (φ )d, the critical dimension turned out to be d = 4, that is, rigorous results were attained only for the cases d = 2; 3. The issue there is that UV divergences cannot be defeated by a finite number of substractions. To our mind, it is their ’prolific’ nature which lets these divergences preclude any nonperturbative treatment in the spirit of the constructive approach. For this introduction, suffice it to assert that a nonperturbative definition of renormalisation for renormalisable fields is clearly beyond constructive methods of the above type. In particular, the fact that the regularised renormalisation Z factors can only be expected to have asymptotic perturbation series is obviously not conducive to their rigorous treatment. Although formally appearing in nonperturbative treatments as factors, they can a priori only be defined in terms of their perturbation series [Ost86]. However, for completeness, we mention [Schra76] in which a possible path towards the (in 4 some sense implicit) construction of (φ )4 in the context of the lattice approach was discussed. As one might expect, the remaining problem was to prove the existence of the renormalised limit to the continuum theory. 1.2. Haag's theorem and other triviality results. Around the beginning of the 1950s, soon after QED had been successfully laid out and heuristically shown to be renormalisable by its founding fathers [Dys49b], there was a small group of mathematical physicists who detected inconsistencies in its formulation. Their prime concern turned out to be the interaction picture of a quantum field theory [vHo52, Ha55]. In particular, Haag concluded that it cannot exist unless it is trivial, ie only describing a free theory. Rigorous proofs for these suspicions could at the time not be given for a simple reason: in order to prove that a mathematical object does not exist or that it can only have certain characteristics, one has to say and clarifiy what kind of mathematical thing it actually is or what it is supposed to be. But the situation changed when QFT was put on an axiomatic footing by Wightman and collaborators who made a number of reasonable assumptions and proved that the arguments put forward earlier against the interaction picture and Dyson's matrix were well-founded [WiHa57]. 4 LUTZ KLACZYNSKI This result was then called Haag('s) theorem. It entails in particular that if a quantum field purports to be unitarily equivalent to a free field, it must be free itself. Other important issues were the canonical (anti)commuation relations and the ill-definedness of quantum fields at sharp spacetime points. The ensuing decade brought to light a number of triviality results of the form "If X is a QFT with properties so-and-so, then it is trivial", where 'trivial' comes in 3 types, with increasing strength: the quantum fields are free fields, identity operators or vanishing.