Braided Actions of DHR Categories and Reconstruction of Chiral Conformal Field Theories

Abstract

Quantum Field Theory (QFT) is our modern understanding of particles and matter at small scales, where quantum behaviour replaces macroscopical phenomena, which are much closer to intuition, and dynamics is rather driven by the more fundamental interactions between fields. Using quantum fields one can describe particle production, annihilation and scattering processes and they can all together be cast into the Standard Model of particle physics. The latter gives a recipe to predict cross sections of high-energy collisions which fit remarkably well with experimental data. On the other hand our mathematical understanding of the framework, and how to replace diverging series, ad hoc renormalized or truncated to get finite numbers, is still a deep open question. Since the early days it was clear that quantum fields, even when they arise from the classical picture of Lagrangian functionals and actions, are more singular objects than those employed in classical physics. Their values in points of spacetime, i.e., their point-like dependence as operator-valued functions, is easily seen to clash with their realizability as operators on an Hilbert space on one hand, on the other hand it is neither dictated by physics. The structure of spacetime itself, at very small scales, is by now out of our experimental reach. In order to overcome the previous difficulties the notion of field can be relaxed to that of an (unbounded) operator-valued distribution (Wightman axiomatization), elevating the smearing with test functions to an essential feature of a local quantum theory. This generalization introduces more difficult mathematical objects (distributions, compared to functions) but which can be rigorously (without ambiguities) treated, and which are suitable enough to obtain a complete scattering theory, once a Wightman QFT is assigned. In the same spirit, but with different mathematics, QFTs can be dealt with using techniques from the theory of operator algebras. The first main characteristics of the algebraic approach (AQFT) is that one describes local measurements or observable fields and regards them as the primary objects of interest to study matter, particles and fundamental interactions, relegating the non-observable quantities to theoretical tools. Secondly, one treats them by means of bounded operators on Hilbert space (e.g., by considering bounded functions of the fields), advantageous at least for the analysis of the framework. More in details, physically relevant quantities such as observables (and states) of a QFT are described in terms of abstract operator algebras associated to open bounded regions of spacetime (“local algebras”). By abstract we mean independent of any specific Hilbert space realization, and then we regard the choice of different representations of the local algebras as the choice of different states (mathematically speaking via the GNS construction). In particular these objects encode both quantum behaviour, in their intrinsic non-commutativity, and Einstein’s causality principle, in the triviality of commutation relations between local algebras sitting at space-like distances. This second approach is what this thesis is devoted to. The relation between these two formalisms is not completely understood, from distributions to local algebras one has to take care of spectral commutation relations on suitable domains, vice versa one should control the scaling limits of the local algebras in order to exploit the distributional “point-like” generators. In both cases, and (theoretically) in any other mathematically sound description of QFT, consequences become proofs, and different features of models or more general model-independent principles (particle content, covariance, local commutation relations) can be separated and analysed. Moreover, beyond the needs of rigorous description of models, the “axiomatic” approaches to QFT have the advantage of being more independent from classical analogies, like field equations and Lagrangians. Fields themselves are not an essential input to model local measurements obeying the constraints of Einstein’s special relativity and quantum theory. AQFT can be thought of as being divided into two lines, the first aims to the construction of models (either in low or high dimensions, both starting from physical counterparts or using the theory of operator algebras), the second is devoted to the analysis of the assumptions and of the possibly new mathematical structures arising from them. The work presented in this thesis has been developed and expresses its contribution in the second line of research. Our aim is to introduce new invariants for local quantum field theories, more specifically to complete a well established construction (the DHR construction) which associates a certain category of representations (collection of superselection sectors together with their fusion rules, exchange symmetry, statistics) to any local quantum field theory, once the latter is formulated as a local net of algebras.