Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Modular evolutions and causality in two-dimensional conformal field theory

Dobrica Jovanovic, Mihail Mintchev, Erik Tonni

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.

Modular evolutions and causality in two-dimensional conformal field theory

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.
Paper Structure (31 sections, 265 equations, 18 figures)

This paper contains 31 sections, 265 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Interval in the line, vacuum state
  3. Chiral modular evolution and modular conjugation
  4. Chiral modular correlators
  5. Commutator of the currents
  6. Chiral distance along the modular evolutions
  7. Modular trajectories
  8. Modular properties of the $K$-evolution
  9. Spacetime distance along the modular trajectories
  10. Distance along a modular trajectories in Euclidean signature
  11. Interval in the line, thermal state
  12. Chiral distance along the modular evolutions
  13. Modular trajectories and spacetime distance between their points
  14. Dirac field in the vacuum: Two disjoint intervals in the line
  15. Modular Hamiltonian
  16. ...and 16 more sections

Figures (18)

  • Figure 1: Modular evolutions along the chiral direction in the plane $(\xi, \tau)$ given by (\ref{['xi-map-fund']}), whose initial points correspond to the dots. Any horizontal dashed line is obtained from (\ref{['tauB_def']}) for the initial point in $B$ having the same color.
  • Figure 2: The chiral distance (\ref{['xi12-tau-chiral']}) between the points (red and blue squares) belonging to two distinct chiral modular evolutions whose initial points (red and blue dots) are in $A$ (see also (\ref{['xi12-tau-chiral-2']})), for either $\tau_1 = \tau_2$ (left panel) or $\tau_1 \neq \tau_2$ (right panel).
  • Figure 3: Modular trajectories (solid curves) in the diamond $\mathcal{D}_A$ (light grey region), whose initial points correspond to the black dot and the black square, and the corresponding modular trajectories (dashed curves) in $\mathcal{B}_A$ (light blue region), obtained through the geometric action of the modular conjugation, which relates dashed and solid arcs having the same colour. The union of each modular trajectory and of the corresponding modular trajectory obtained through the modular conjugation map (see (\ref{['mod-conj-traj-tau-line-x']})-(\ref{['mod-conj-traj-tau-line-t']})) gives the hyperbolae (\ref{['mod-hyper']})-(\ref{['mod-hyper-parameters']}).
  • Figure 4: Spacetime distances along the modular evolution either for two events in $\mathcal{D}_A$ (left panel) or for one event in $\mathcal{D}_A$ and the other one in $\mathcal{B}_A$ (right panel).
  • Figure 5: Spacetime distances along the modular evolution for two events in $\mathcal{B}_A$.
  • ...and 13 more figures