Integral and differential structures for quantum field theory

The aim of this work is to firstly demonstrate the efficacy of the recently proposed Orlicz space formalism for Quantum theory [44], and secondly to show how noncommutative differential structures may naturally be incorporated into this framework. To start off with we specifically propose regularity conditions which in the context of local algebras corresponding to Minkowski space, ensure good behaviour of field operators as observables, and then show that fields obtained by the Osterwalder–Schrader reconstruction theorem are regular in this sense. This complements earlier work by Buchholz, Driessler, Summers and Wichman, etc, on generalized $H$-bounds. The pair of Orlicz spaces we explicitly use for this purpose, are respectively built on the exponential function (for the description of regular field operators) and on an entropic type function (for the description of the corresponding states). This formalism has been shown to be well suited to a description of quantum statistical mechanics, and in the present work we show that it is also a very useful and elegant tool for Quantum Field Theory. We then introduce the class of tangentially conditioned algebras, which is a large class of local algebras corresponding to globally hyperbolic Lorentzian manifolds that locally “look like” the local algebras of Minkowski space. On the one hand this ensures that at a local level, the Orlicz space formalism discussed above is also relevant for a much more general class of local algebras. On the other hand, the structure of this class of algebras, allows for the development of a non-commutative differential geometric structure along the lines of the du Bois–Violette approach to such a theory. In this way we obtain a complete depiction: integrable structures based on local algebras provide a static setting for an analysis of Quantum Field Theory and an effective tool for describing regular behaviour of field operators, whereas differentiable structures posit indispensable tools for a description of equations of motion.

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The contribution of L. E. Labuschagne is based on research partially supported by the National Research Foundation (IPRR Grant 96128).

Any opinion, findings and conclusions or recommendations expressed in this material, are those of the author, and therefore the NRF do not accept any liability in regard thereto.

Published 30 June 2023