[Paper Review] Coherent state, local excitation in 2D conformal field theory

Wu-zhong Guo|arXiv (Cornell University)|Oct 24, 2015

Black Holes and Theoretical Physics28 references3 citations

TL;DR

This paper establishes that local excitations of primary operators in 2D conformal field theory (CFT) can be understood as coherent states of the global conformal group $SL(2,\mathbb{C})$, providing a group-theoretic framework for such states. It derives the entanglement entropy between holomorphic and anti-holomorphic sectors as proportional to the quantum dimension of the primary operator, and demonstrates violation of Bell's inequality in entangled coherent states constructed via deformed local excitations.

ABSTRACT

In this paper we discuss the topics concerning the local excitation and coherent state in 2D CFT. It is shown that the local excitation of primary operator can be taken as a coherent state of the global conformal group. We also discuss the entanglement property of such state. For rational CFT the entanglement entropy between the holomorphic and anti-holomorphic sector of the local excitation of some primary operator is related to the quantum dimension of the operator, consistent with previous approach, but by a differentmethod. We comment on the possible application of so-defined group coherent state in the holographic view. We also study the coherent state in the free massless boson field, their time evolution and entanglement property. We introduce the deformed local excitation and the entangled state constructed by them. It is shown the violation of Bell inequality for such entangled state.

Motivation & Objective

• To establish a group-theoretic interpretation of local excitations in 2D CFT as coherent states of the global conformal group $SL(2,\mathbb{C})$.
• To re-derive the entanglement entropy between holomorphic and anti-holomorphic sectors using fusion rules and quantum dimensions, independent of the replica trick.
• To explore the physical behavior and time evolution of Glauber-type coherent states in the free massless boson CFT.
• To define deformed local excitations that create finite-region excitations and construct entangled coherent states.
• To investigate the holographic implications of group coherent states and their potential dual description in classical gravity.

Proposed method

• Constructs coherent states via the coset space $H_4/G_0$ for the Heisenberg-Weyl group, generalizing Glauber's definition to field theories.
• Applies the group coherent state formalism to 2D CFT by mapping local primary operator excitations to $SL(2,\mathbb{C})$ coherent states via the global conformal algebra generators $L_{-1}, L_0, L_1$ and $\bar{L}_{-1}, \bar{L}_0, \bar{L}_1$.
• Uses radial quantization and operator product expansion (OPE) to relate local excitation to coherent state evolution and entanglement structure.
• Introduces deformed local excitations as non-local, finite-region excitations to model realistic physical perturbations.
• Analyzes entanglement entropy via fusion rules and quantum dimensions, showing independence from coordinate choice.
• Applies Bell inequality tests to entangled coherent states formed from deformed excitations, demonstrating non-classical correlations.

Experimental results

Research questions

• RQ1 Can local excitations of primary operators in 2D CFT be systematically described as coherent states of the global conformal group?
• RQ2 How is the entanglement entropy between holomorphic and anti-holomorphic sectors related to the quantum dimension of the primary operator?
• RQ3 What is the time evolution of Glauber-type coherent states in the free massless boson CFT under time-dependent Hamiltonians?
• RQ4 Can deformed local excitations generate entangled coherent states that violate Bell's inequality?
• RQ5 What are the potential holographic implications of defining local excitations as group coherent states?

Key findings

• Local excitations of primary operators in 2D CFT are equivalent to coherent states of the global conformal group $SL(2,\mathbb{C})$.
• The entanglement entropy between holomorphic and anti-holomorphic sectors is independent of the spatial coordinate and equals the logarithm of the quantum dimension of the primary operator.
• In the free massless boson CFT, the Glauber coherent state evolves unitarily under both free and time-dependent Hamiltonians, preserving its coherent nature.
• Deformed local excitations—excitations over a finite spatial region—can be used to construct entangled coherent states with non-trivial entanglement structure.
• The entangled coherent states constructed from deformed excitations violate Bell's inequality, demonstrating non-classical correlations beyond bipartite systems.
• The group coherent state framework provides a natural bridge to holography, suggesting that such boundary coherent states may have a direct classical gravity dual in AdS.