A new Hamiltonian for the Topological BF phase with spinor networks

TL;DR

This paper introduces a novel scalar Hamiltonian for the SU(2) BF topological phase, formulated with spinor networks, and explores its quantum properties and differences from existing models.

Contribution

It presents a new Hamiltonian based on spinor variables, differing from the traditional plaquette operator, and analyzes its quantum behavior in various representations.

Findings

The Hamiltonian leads to a difference equation involving Wigner 6j-symbols.

It provides a new perspective on the gauge-theoretical structure of the topological phase.

Comparison with other models highlights unique features of the proposed Hamiltonian.

Abstract

We describe fundamental equations which define the topological ground states in the lattice realization of the SU(2) BF phase. We introduce a new scalar Hamiltonian, based on recent works in quantum gravity and topological models, which is different from the plaquette operator. Its gauge-theoretical content at the classical level is formulated in terms of spinors. The quantization is performed with Schwinger's bosonic operators on the links of the lattice. In the spin network basis, the quantum Hamiltonian yields a difference equation based on the spin 1/2. In the simplest case, it is identified as a recursion on Wigner 6j-symbols. We also study it in different coherent states representations, and compare with other equations which capture some aspects of this topological phase.