The Noncommutative Geometry of Configuration Spaces and the Problem of Reconciling General Relativity With Quantum Field Theory

In this paper a candidate for a spectral triple on a quotient space of gauge connections modulo gauge transformations is construted and it is shown that it is related to a Kasparov type bi-module over two canonical algebras: the $\mathrm{HD}$-algebra, which is a non-commutative $C^{\ast }$-algebra generated by parallel transports along flows of vectorfields, and an exterior algebra on a space of gauge transformations. The latter algebra is related to the ghost sector in a BRST quantisation scheme. It is Previously shown that the key elements of bosonic and fermionic quantum field theory on a curved background emerge from a spectral triple of this type. In this paper it is shown that a dynamical metric on the underlying manifold also emerges from the construction. We first rigourously construct a Dirac type operator on the a quotient space of gauge connections modulo gauge transformations, and the commutator between this Dirac type operator and the $\mathrm{HD}$-algebra is discussed. To do this a gauge-covariant metric on the configuration space is first constructed and it is used to construct the triple. The key step is that the volume of the quotient space to required to be finite, which amounts to an ultra-violet regularisation. Since the metric on the configuration space is dynamical with respect to the time-evolution generated by the Dirac type operator, The regularisation is interpreted as a physical feature (as opposed to static regularisations, which are always computational artefacts). Finally, a Bott–Dirac operator that connects our construction with quantum Yang-Mills theory is constructed.