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The aim of this work is to firstly demonstrate the efficacy of the recentlyproposed Orlicz space formalism for Quantum theory \cite{ML}, and secondly toshow how noncommutative differential structures may naturally be incorporatedinto this framework. To start off with we specifically propose regularityconditions which in the context of local algebras corresponding to Minkowskispace, ensure good behaviour of field operators as observables, and then showthat fields obtained by the Osterwalder-Schrader reconstruction theorem areregular in this sense. The pair of Orlicz spaces we explicitly use for thispurpose, are respectively built on the exponential function and on an entropictype function. This formalism has been shown to be well suited to a descriptionof quantum statistical mechanics, and in the present work we show that it isalso a very useful and elegant tool for Quantum Field Theory. We then introducethe class of tangentially conditioned algebras, which is a large class of localalgebras corresponding to globally hyperbolic Lorentzian manifolds that locally``look like'' the local algebras of Minkowski space. On the one hand thisensures that at a local level, the Orlicz space formalism discussed above isalso relevant for a much more general class of local algebras. On the otherhand, the structure of this class of algebras, allows for the development of anon-commutative differential geometric structure along the lines of the duBois-Violette approach to such a theory. In this way we obtain a completedepiction: integrable structures based on local algebras provide a staticsetting for an analysis of Quantum Field Theory and an effective tool fordescribing regular behaviour of field operators, whereas differentiablestructures posit indispensable tools for a description of equations of motion.

Integral and differential structures for quantum field theory