Fermions in Loop Quantum Gravity and Resolution of Doubling Problem

TL;DR

This paper derives the fermion propagator in loop quantum gravity, showing that superposing over graphs suppresses fermion doubling, thus resolving a key issue and linking quantum geometry to continuum physics.

Contribution

It introduces a novel approach using graph superposition in loop quantum gravity to resolve the fermion doubling problem.

Findings

Fermion propagator derived from LQG with superposition over graphs.

Fermion doubling problem is resolved through this superposition.

Results suggest a connection between quantum geometry superposition and continuum limit.

Abstract

The fermion propagator is derived in detail from the model of fermion coupled to loop quantum gravity. As an ingredient of the propagator, the vacuum state is defined as the ground state of some effective fermion Hamiltonian under the background geometry given by a coherent state resembling the classical Minkowski spacetime. Moreover, as a critical feature of loop quantum gravity, the superposition over graphs is employed to define the vacuum state. It turns out that the graph superposition leads to the propagator being the average of the propagators of the lattice field theory over various graphs so that all fermion doubler modes are suppressed in the propagator. This resolves the doubling problem in loop quantum gravity. Our result suggests that the superposition nature of quantum geometry should, on the one hand, resolve the tension between fermion and the fundamental discreteness and, on the other hand, relate to the continuum limit of quantum gravity.