Boundary Quantum Field Theory on the Interior of the Lorentz Hyperboloid

TL;DR

This paper constructs and analyzes boundary quantum field theory nets on the Lorentz hyperboloid, revealing connections to conformal nets, thermal states, and symmetric inner functions, with implications for different quantum states.

Contribution

It introduces a canonical construction of boundary QFT nets on the Lorentz hyperboloid and explores their relation to symmetric inner functions and various quantum states.

Findings

Constructed boundary QFT nets on the Lorentz hyperboloid.

Established a link between nets and symmetric inner functions.

Demonstrated nets in thermal and ground states.

Abstract

We construct local, boost covariant boundary QFT nets of von Neumann algebras on the interior of the Lorentz hyperboloid LH = {x^2 - t^2 > R^2, x>0}, in the two-dimensional Minkowski spacetime. Our first construction is canonical, starting with a local conformal net on the real line, and is analogous to our previous construction of local boundary CFT nets on the Minkowski half-space. This net is in a thermal state at Hawking temperature. Then, inspired by a recent construction by E. Witten and one of us, we consider a unitary semigroup that we use to build up infinitely many nets. Surprisingly, the one-particle semigroup is again isomorphic to the semigroup of symmetric inner functions of the disk. In particular, by considering the U(1)-current net, we can associate with any given symmetric inner function a local, boundary QFT net on LH. By considering different states, we shall also have nets in a ground state, rather than in a KMS state.