Holography and $SL(2,\bR)$ symmetry in 2D Rindler spacetime

TL;DR

This paper demonstrates a unitary bulk-boundary correspondence for a massless scalar field in 2D Rindler spacetime, revealing a geometric interpretation of the hidden $SL(2,R)$ symmetry as horizon diffeomorphisms.

Contribution

It establishes a bulk-boundary quantum field theory correspondence on the Rindler horizon and clarifies the geometric origin of the $SL(2,R)$ symmetry.

Findings

Quantum field theory on the horizon is diffeomorphism invariant.

A unitary transformation relates bulk and boundary theories.

The $SL(2,R)$ symmetry corresponds to horizon diffeomorphisms.

Abstract

It is shown that it is possible to define quantum field theory of a massless scalar free field on the Killing horizon of a 2D-Rindler spacetime. Free quantum field theory on the horizon enjoys diffeomorphism invariance and turns out to be unitarily and algebraically equivalent to the analogous theory of a scalar field propagating inside Rindler spacetime, nomatter the value of the mass of the field in the bulk. More precisely, there exists a unitary transformation that realizes the bulk-boundary correspondence under an appropriate choice for Fock representation spaces. Secondly, the found correspondence is a subcase of an analogous algebraic correspondence described by injective *-homomorphisms of the abstract algebras of observables generated by abstract quantum free-field operators. These field operators are smeared with suitable test functions in the bulk and exact 1-forms on the horizon. In this sense the correspondence is independent from the chosen vacua. It is proven that, under that correspondence the ``hidden'' $SL(2,\bR)$ quantum symmetry found in a previous work gets a clear geometric meaning, it being associated with a group of diffeomorphisms of the horizon itself.