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

Abstract

In this paper, the development of a spectral triple-like construction on a configuration space of gauge connections is continued. It has previously been shown that key elements of bosonic and fermionic quantum field theory emerge from such a geometrical framework. In this paper, a central problem concerning the inclusion of fermions with half-integer spin into this framework is solved. The tangent space of the configuration space is mapped into a similar space based on spinors, and this map is used to construct a Dirac operator on the configuration space. A real structure acting in a Hilbert space over the configuration space is also constructed. Finally, it is shown that the self-dual and anti-self-dual sectors of the Hamiltonian of a nonperturbative 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 are shown to emerge in a two-by-two matrix structure.