Thermal time of noncommutative Minkowski spacetime

TL;DR

This work extends the thermal time hypothesis to noncommutative Minkowski spacetimes by deriving a modular flow from the momentum-space modular function. It demonstrates that in $\kappa$-Minkowski the thermal time corresponds to a global time translation, while in unimodular $\rho$-Minkowski the thermal time is absent due to discrete time, linking noncommutativity to thermality. The analysis employs GNS construction, Tomita–Takesaki theory, and Connes’ cocycle to relate modular structure to dynamics, and discusses inner automorphism freedom and the commutative limit. The results open a route to quantum-gravity phenomenology via thermal field theory on noncommutative spacetimes, and highlight several open problems for extending the framework to gauge theories and beyond.

Abstract

In this paper, we study the thermal time hypothesis of arXiv:gr-qc/9406019 in the context of noncommutative deformations of Minkowski. We show that a natural modular group arises from the modular function of the momentum space. In the specific case of $κ$-Minkowski, we show that this thermal time flow corresponds to the globally defined time coordinate translation. On the other hand, the absence of thermal time for $ρ$-Minkowski is directly related to the discreteness of its global time. The impact of inner automorphism transformation on the physics and the treatment of unimodular case (unthermalised spacetimes) are discussed. Moreover, a reflection on the use of thermal field theory for quantum gravity phenomenology is put forward, as we just bridged thermal spacetimes with $κ$-Minkowski, often considered a "flat limit" of a quantum gravity candidate theory.