Quantum Loop Representation for Fermions coupled to Einstein-Maxwell field

TL;DR

This paper develops a unified quantum theory for gravitational, fermionic, and electromagnetic fields using a loop representation, extending Ashtekar formalism to include fermions with a novel algebraic structure.

Contribution

It introduces a loop-based quantum representation for Einstein-Maxwell-fermion systems, incorporating fermionic degrees of freedom with a simplified loop operator action.

Findings

Constructed a $C^{*}$-algebra of configurational variables with loops and curves.

Defined a quantum representation space as cylindrical functionals on the algebra spectrum.

Described creation and annihilation operators as loop and curve operators in the basis.

Abstract

Quantization of the system comprising gravitational, fermionic and electromagnetic fields is developed in the loop representation. As a result we obtain a natural unified quantum theory. Gravitational field is treated in the framework of Ashtekar formalism; fermions are described by two Grassmann-valued fields. We define a $C^{*}$-algebra of configurational variables whose generators are associated with oriented loops and curves; ``open'' states -- curves -- are necessary to embrace the fermionic degrees of freedom. Quantum representation space is constructed as a space of cylindrical functionals on the spectrum of this $C^{*}$-algebra. Choosing the basis of ``loop'' states we describe the representation space as the space of oriented loops and curves; then configurational and momentum loop variables become in this basis the operators of creation and annihilation of loops and curves. The important difference of the representation constructed from the loop representation of pure gravity is that the momentum loop operators act in our case simply by joining loops in the only compatible with their orientaiton way, while in the case of pure gravity this action is more complicated.