Modular evolutions and causality in two-dimensional conformal field theory

TL;DR

This paper investigates the structure of modular flows and causality in two-dimensional conformal field theories, analyzing how modular trajectories affect spacetime distances, locality, and causality in various states and configurations.

Contribution

It provides new insights into the modular evolution and causality preservation in 2D CFTs, especially for disjoint intervals and different states, including finite temperature cases.

Findings

Modular flows preserve relativistic causality in 2D CFTs.

Bilocal modular Hamiltonians lead to delta-function contributions in commutators.

Modular trajectories can connect spacelike separated points via bilocal terms.

Abstract

In two-dimensional conformal field theories (CFT) in Minkowski spacetime, we study the spacetime distance between two events along two distinct modular trajectories. When the spatial line is bipartite by a single interval, we consider both the ground state and the state at finite different temperatures for the left and right moving excitations. For the free massless Dirac field in the ground state, the bipartition of the line given by the union of two disjoint intervals is also investigated. The modular flows corresponding to connected subsystems preserve relativistic causality. Locality along the modular flows of some fields is explored by evaluating their (anti-)commutators. In particular, the bilocal nature of the modular Hamiltonian of two disjoint intervals for the massless Dirac field provide multiple trajectories leading to Dirac delta contributions in the (anti-)commutators even when the initial points belong to different intervals, thus being spacelike separated.