Dirac Operators on Configuration Spaces: Fermions with Half-integer Spin, Real Structure, and Yang-Mills Quantum Field Theory

In this paper we continue the development of a spectral triple-like construction on a configuration space of gauge connections. We have previously shown that key elements of bosonic and fermionic quantum field theory emerge from such a geometrical framework. In this paper we solve a central problem concerning the inclusion of fermions with half-integer spin into this framework. We map the tangent space of the configuration space into a similar space based on spinors and use this map to construct a Dirac operator on the configuration space. We also construct a real structure acting in a Hilbert space over the configuration space. Finally, we show that the self-dual and anti-self-dual sectors of the Hamiltonian of a non-perturbative quantum Yang-Mills theory emerge from a unitary transformation of a Dirac equation on a configuration space of gauge fields. The dual and anti-dual sectors emerge in a two-by-two matrix structure.