
TL;DR
This paper explores how time may emerge from noncommutative quantum spaces, using operator algebra theory to connect modular evolution, entropy bounds, and the quantum null energy condition.
Contribution
It extends modular evolution to quantum operations and relates it to entropy bounds and the Jones' index, providing a rigorous mathematical framework for understanding time emergence.
Findings
Modular theory relates quantum time to temperature and entropy.
Extension of modular evolution to quantum operations clarifies entropy bounds.
Operator algebra methods compute entropy in relativistic quantum field models.
Abstract
Classically, one could imagine a completely static space, thus without time. As is known, this picture is unconceivable in quantum physics due to vacuum fluctuations. The fundamental difference between the two frameworks is that classical physics is commutative (simultaneous observables) while quantum physics is intrinsically noncommutative (Heisenberg uncertainty relations). In this sense, we may say that time is generated by noncommutativity; if this statement is correct, we should be able to derive time out of a noncommutative space. We know that a von Neumann algebra is a noncommutative space. About 50 years ago the Tomita-Takesaki modular theory revealed an intrinsic evolution associated with any given (faithful, normal) state of a von Neumann algebra, so a noncommutative space is intrinsically dynamical. This evolution is characterised by the Kubo-Martin-Schwinger thermal…
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