Conformal quantum mechanics of causal diamonds

Abstract

It is shown that a general radial conformal Killing vector in Minkowski space-time can be associated to a generator of time evolution in conformal quantum mechanics. Among these conformal Killing vectors there is a class which maps causal diamonds in Minkowski space-time into themselves. The flow of such Killing vectors describes worldlines of accelerated observers with a finite lifetime within a causal diamond. Time evolution of static diamond observers is equivalent to time evolution in conformal quantum mechanics governed by a hyperbolic Hamiltonian and covering only a segment of the time axis. This indicates that the Unruh temperature perceived by static diamond observers in the vacuum state of inertial observers in Minkowski space-time can be obtained from the behaviour of the two-point functions of conformal quantum mechanics. The results presented suggest a group theoretical description of the recently proposed light-cone temperature associated to null surfaces defined by light fronts in Minkowski space-time.