Quantum Virasoro algebra with central charge c=1 on the horizon of a 2D-Rindler spacetime

TL;DR

This paper demonstrates that the hidden SL(2,R) symmetry in a scalar quantum field on Rindler spacetime extends to a Virasoro algebra with central charge c=1, revealing a geometric interpretation and thermal properties related to the horizon.

Contribution

It establishes a Virasoro algebra structure with c=1 on the Rindler horizon, linking holography, conformal symmetry, and thermal states in a novel way.

Findings

Virasoro algebra with c=1 is realized on the horizon.

The Rindler Hamiltonian corresponds to a Virasoro generator.

Thermal states emerge at specific ground energies, matching Unruh temperature.

Abstract

Using the holographic machinery built up in a previous work, we show that the hidden SL(2,R) symmetry of a scalar quantum field propagating in a Rindler spacetime admits an enlargement in terms of a unitary positive-energy representation of Virasoro algebra, with central charge c=1, defined in the Fock representation. The Virasoro algebra of operators gets a manifest geometrical meaning if referring to the holographically associated QFT on the horizon: It is nothing but a representation of the algebra of vector fields defined on the horizon equipped with a point at infinity. All that happens provided the Virasoro ground energy h vanishes and, in that case, the Rindler Hamiltonian is associated with a certain Virasoro generator. If a suitable regularization procedure is employed, for h=1/2, the ground state of that generator corresponds to thermal states when examined in the Rindler wedge, taking the expectation value with respect to Rindler time. This state has inverse temperature 1/(2beta), where beta is the parameter used to define the initial SL(2,R) unitary representation. (As a consequence the restriction of Minkowski vacuum to Rindler wedge is obtained by fixing h=1/2 and 2beta=beta_U, the latter being Unruh's inverse temperature). Finally, under Wick rotation in Rindler time, the pair of QF theories which are built up on the future and past horizon defines a proper two-dimensional conformal quantum field theory on a cylinder.