[Paper Review] Fermion Doubling in Loop Quantum Gravity

Jacob L. Barnett, Lee Smolin|arXiv (Cornell University)|Jul 5, 2015

Noncommutative and Quantum Gravity Theories34 references3 citations

TL;DR

This paper demonstrates that loop quantum gravity (LQG) exhibits a fermion doubling problem due to its discrete spatial structure, using a Born-Oppenheimer approximation around a lattice-based quantum gravity background. By mapping the low-energy fermionic spectrum to a lattice gauge theory, the authors apply the Nielsen-Ninomiya no-go theorem to prove that left- and right-handed fermions must appear in equal numbers, implying chiral symmetry is lost unless fermions double—a fundamental obstacle to realizing chiral fermions in LQG.

ABSTRACT

In this paper, we show that the Hamiltonian approach to loop quantum gravity has a fermion doubling problem. To obtain this result, we couple loop quantum gravity to a free massless scalar and a chiral fermion field, gauge fixing the many fingered time gauge invariance by interpreting the scalar field as a physical clock. We expand around a quantum gravity state based on a regular lattice and consider the limit where the bare cosmological constant is large but the fermonic excitations have energies low in Planck units. We then make the case for identifying the energy spectrum in this approximation with that of a model of lattice fermion theory which is known to double.

Motivation & Objective

• To investigate whether loop quantum gravity (LQG) suffers from a fermion doubling problem due to its discrete spatial structure.
• To examine whether chiral fermions can be consistently realized in LQG, given that chiral anomalies are canceled only if fermions double.
• To establish a mapping between the low-energy spectrum of LQG and a lattice fermion theory, enabling application of known theorems on fermion doubling.
• To demonstrate that the absence of chiral anomalies in LQG implies the necessity of fermion doubling, using a background-independent, Hamiltonian formulation.

Proposed method

• Use the Hamiltonian formulation of loop quantum gravity (LQG) coupled to a scalar field and a chiral fermion field, with the scalar field serving as a physical clock to gauge-fix the many-fingered time symmetry.
• Construct a background quantum gravity state based on a regular lattice Γ, corresponding to a flat spatial metric under coarse-graining.
• Apply a Born-Oppenheimer approximation, expanding the Hamiltonian around the gravitational background and focusing on low-energy fermionic excitations.
• Introduce a degravitating map and dressing map to relate the effective matter Hamiltonian in LQG to a standard lattice fermion theory on Γ.
• Verify that the effective Hamiltonian on Γ satisfies the conditions of the Nielsen-Ninomiya no-go theorem, which forbids chiral fermions on regular lattices.
• Use momentum-space analysis and dispersion relation curves in 3+1D to show that degeneracy points (Weyl nodes) must appear in pairs with opposite helicity, proving fermion doubling.

Experimental results

Research questions

• RQ1 Does loop quantum gravity exhibit a fermion doubling problem due to its discrete spatial structure?
• RQ2 Can chiral fermions be consistently coupled to LQG without violating the Nielsen-Ninomiya no-go theorem?
• RQ3 Is the cancellation of chiral anomalies in LQG's anomaly-free constraint algebra sufficient to allow chiral fermions, or does it necessitate fermion doubling?
• RQ4 Can the low-energy effective theory of LQG be mapped to a lattice fermion model where fermion doubling is known to occur?
• RQ5 What is the topological structure of the energy-momentum dispersion relation in LQG's effective fermionic spectrum, and how does it relate to Weyl nodes and helicity?

Key findings

• The low-energy spectrum of loop quantum gravity, when expanded around a lattice-based background, matches that of a lattice fermion theory on a regular graph Γ.
• The effective Hamiltonian for fermions in LQG satisfies the conditions of the Nielsen-Ninomiya no-go theorem, which forbids the consistent coupling of chiral fermions to a local Hamiltonian on a regular lattice.
• As a consequence, the spectrum of fermionic excitations in LQG must double, with equal numbers of left- and right-handed components, even when the original matter theory is chiral.
• Degeneracy points in the dispersion relation (Weyl nodes) appear in pairs with opposite helicity, and closed curves in momentum space passing through these nodes must cross equally in both directions, enforcing a balance between left- and right-handed states.
• The orientation of these curves, determined by the phase of the wavefunction around degeneracy points, confirms that the number of left-handed and right-handed Weyl fermions must be equal, thus proving fermion doubling in the low-energy limit.
• The result implies that LQG cannot support a chiral fermion spectrum without doubling, unless the underlying lattice structure or dynamics break the symmetries assumed in the no-go theorem.