Entanglement

      Quantum fields can also be described in a self-contained fashion in Rindler space, but a new twist is encountered. Our goal is to describe the usual physics of a quantum field in Minkowski space, but from the viewpoint of the Fidos in Region I, i.e. in Rindler space. To understand the new feature, recall that in the usual vacuum state, the correlation between fields at different spatial points does not vanish. For example, in free massless scalar theory the equal time correlator is given by

 (1.1)

Fig. 1.1. Equal time and proper distance surfaces in Rindler space.

where Δ is the space-like separation between the points (X, Y,Z) and (X', Y ', Z' )

 (1.2)

The two points might both lie within Region I, in which case the correlator in equation (1.1) represents the quantum correlation seen by Fido's in Region I. On the other hand, the two points might lie on opposite sides of the horizon at Z = 0.In that case the correlation is un measurable to the Fidos in Region I. Nevertheless it has significance. When two subsystems (fields in Regions I and III) become correlated, we say that they are quantum entangled, so that neither can be described in terms of pure states. The appropriate description of an entangled subsystem is in terms of a density matrix.