Reflection positivity and its relation to disc, half plane and the strip

TL;DR

This paper explores reflection positivity on the strip by drawing analogies with the disc and upper half plane, linking it to KMS conditions and algebraic quantum field theory through a novel perspective.

Contribution

It introduces a new framework connecting reflection positivity on the strip with classical domains and quantum field theory concepts, expanding the understanding of these mathematical structures.

Findings

Reflection positivity on the strip relates to KMS conditions.

Analogies between disc, half plane, and strip domains are established.

Connections to standard pairs in Algebraic Quantum Field Theory are demonstrated.

Abstract

We develop a novel perspective on reflection positivity (RP) on the strip by systematically developing the analogies with the unit disc and the upper half plane in the complex plane. These domains correspond to the three conjugacy classes of one-parameter groups in the M\"obius group (elliptic for the disc, parabolic for the upper half plane and hyperbolic for the strip). In all cases, reflection positive functions correspond to positive functionals on $H^\infty$ for a suitable involution. For the strip, reflection positivity naturally connects with Kubo--Martin--Schwinger (KMS) conditions on the real line and further to standard pairs, as they appear in Algebraic Quantum Field Theory. We also exhibit a curious connection between Hilbert spaces on the strip and the upper half plane, based on a periodization process.