# Holography and $SL(2,\mathbb{R})$ symmetry in 2D Rindler spacetime

Valter Moretti, Nicola Pinamonti
April 11, 2003
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'' $SL(2,\mathbb{R})$ quantum symmetry found in a previous work gets a clear geometric meaning, it being associated with a group of diffeomorphisms of the horizon itself.