Thermality of the Rindler horizon: A simple derivation from the structure of the inertial propagator

Abstract: The Feynman propagator encodes all the physics contained in a free field and transforms as a covariant bi-scalar. Therefore, we should be able to discover the thermality of the Rindler horizon, just by probing the structure of the propagator, expressed in the Rindler coordinates. I show that the thermal nature of the Rindler horizon is indeed contained --- though hidden --- in the standard, inertial, Feynman propagator. The probability $P(E)$ for a particle to propagate between two events, with energy $E$, can be related to the temporal Fourier transform of the propagator. A strikingly simple computation reveals that: (i) $P(E)$ is equal to $P(-E)$ if the propagation is between two events in the same Rindler wedge while (ii) they are related by a Boltzmann factor with temperature $T=g/2\pi$, if the two events are separated by a horizon. A more detailed computation reveals that the propagator itself can be expressed as a sum of two terms, governing absorption and emission, weighted correctly by the factors $(1+n_\nu)$ and $n_\nu$ where $n_\nu$ is a Planck distribution at the temperature $T=g/2\pi$. In fact, one can discover the Rindler vacuum and the alternative (Rindler) quantization, just by probing the structure of the inertial propagator. These results can be extended to local Rindler horizons around any event in a curved spacetime. The implications are discussed.