Quantum loop representation for fermions coupled to an Einstein-Maxwell field

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. The gravitational field is treated in the framework of the Ashtekar formalism; fermions are described by two Grassmann-valued fields. We define a ${\mathit{C}}^{\mathrm{*}}$ 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 ${\mathit{C}}^{\mathrm{*}}$ 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 way which is compatible with their orientation, while in the case of pure gravity this action is more complicated. © 1996 The American Physical Society.