Alternative derivation of the correspondence between Rindler and Minkowski particles

We develop an alternative derivation of Unruh and Wald’s seminal result that the absorption of a Rindler particle by a detector as described by uniformly accelerated observers corresponds to the emission of a Minkowski particle as described by inertial observers. Actually, we present it in an inverted version, namely, that the emission of a Minkowski particle corresponds in general to either the emission or the absorption of a Rindler particle. 04.62.+v, 04.70.Dy Typeset using REVTEX 1 We present a brute force, but straightforward method to reobtain Unruh and Wald’s seminal result [1] that the absorption of a Rindler particle by a detector as described by uniformly accelerated observers corresponds to the emission of a Minkowski particle as described by inertial observers. Actually, we present it in an inverted version, namely, that the emission of a Minkowski particle in the inertial vacuum will correspond in general to a thermal emission or absorption of a Rindler particle. In the particular case where the source is a uniformly accelerated Unruh–DeWitt detector [2,3], the emission of a Minkowski particle will uniquely correspond to the absorption of a Rindler particle. We believe that our approach may be particularly useful in understanding the behavior of realistic sources following arbitrary worldlines. We assume natural units (h̄ = c = kB = 1), and an ndimensional Minkowski spacetime with signature (+−−...−) for sake of generality. In order to capture only the essential features of a quantum device without making use of some particular detector, let us motivate the use of complex currents. The current describing the excitation of a DeWitt–like detector [3] can be defined as j(τ) = 〈E|m̂(τ)|E0〉, (1) where m̂(τ) is the monopole which represents the detector, τ is its proper time, and ∆E = E − E0 is the energy gap of the detector. In the Heisenberg picture the monopole is time– evolved as m̂(τ) = e0m̂(0)e0 , where H0 is the free Hamiltonian of the detector. Thus, current (1) is clearly complex since j(τ) ∝ e . This is the reason why we will consider here arbitrary complex currents. These currents should be interpreted as describing the transition of a non–necessarily pointlike quantum device following an arbitrary worldline. For sake of simplicity, we couple our complex current to a real massless scalar field through the interaction operator Ŝ = ∫