On nonperturbative quantum field theory and noncommutative geometry

Abstract

A general framework of non-perturbative quantum field theory on a curved background is proposed. A quantum field theory is in this setting characterised by an embedding of a space of field configurations into a Hilbert space over R∞. This embedding, which is only local up to a scale that we interpret as the Planck scale, coincides in the local and flat limit with the plane wave expansion known from canonical quantisation. We identify a universal Bott–Dirac operator acting in the Hilbert space over R∞ and show that it gives rise to the free Hamiltonian both in the case of a scalar field theory and in the case of a Yang–Mills theory. These theories come with a canonical fermionic sector for which the Bott–Dirac operator also provides the Hamiltonian. We prove that Hilbert space representations of algebras of observables exist non-perturbatively for a real scalar theory and for a gauge theory, both with or without the fermionic sectors, and show that the free theories are given by semi-finite spectral triples over the respective configuration spaces. Finally, we propose a class of quantum field theories whose interactions are generated by inner fluctuations of the Bott–Dirac operator.