Non‐perturbative Quantum Field Theory and the Geometry of Functional Spaces.

In this paper we construct a non‐commutative geometry over a configuration space of gauge connections and show that it gives rise to a candidate for an interacting, non‐perturbative quantum gauge theory coupled to a fermionic field on a curved background. The non‐commutative geometry is given by an infinite‐dimensional Bott‐Dirac type operator, whose square gives the Hamilton operator, and which interacts with an algebra generated by holonomy‐diffeomorphisms. The Bott‐Dirac operator and the associated Hilbert space relies on a metric on the configuration space of connections, which effectively works as a covariant ultra‐violet regulator. We show that the construction coincides with perturbative quantum field theory in a local limit. Questions concerning Lorentz invariance and the fermionic sector as well as the issue of existence are left open. In this paper a non‐commutative geometry over a configuration space of gauge connections is constructed and it is shown that it gives rise to a candidate for an interacting, non‐perturbative quantum gauge theory coupled to a fermionic field on a curved background. The noncommutative geometry is given by an infinite‐dimensional Bott‐Dirac type operator, whose square gives the Hamilton operator, and which interacts with an algebra generated by holonomy‐diffeomorphisms. The Bott‐Dirac operator and the associated Hilbert space relies on a metric on the configuration space of connections, which effectively works as a covariant ultra‐violet regulator. It is shown that the construction coincides with perturbative quantum field theory in a local limit. Questions concerning Lorentz invariance and the fermionic sector as well as the issue of existence are left open. [ABSTRACT FROM AUTHOR]