Quantum fields for unitary representations of Thompson's groups F and T

TL;DR

This paper introduces a method to define quantum field-like observables within the framework of Thompson's groups F and T, revealing conformal data from correlation functions, with applications to quantum spin systems and anyon chains.

Contribution

It presents a novel approach to constructing quantum fields for Thompson's groups and extracting conformal data, bridging group representations and quantum field theory.

Findings

Correlation functions yield conformal data such as primary fields and scaling dimensions.

Examples from quantum spin systems and anyon chains demonstrate the method.

The approach connects Thompson's groups with quantum field theoretical concepts.

Abstract

We describe how to define observables analogous to quantum fields for the semicontinuous limit recently introduced by Jones in the study of unitary representations of Thompson's groups $F$ and $T$. We find that, in terms of correlation functions of these fields, one can deduce quantities resembling the conformal data, i.e., primary fields, scaling dimensions, and the operator product expansion. Examples coming from quantum spin systems and anyon chains built on the trivalent category $\mathit{SO}(3)_q$ are studied.