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 equilibrium condition in quantum statistical mechanics (Haag, Hugenholtz, Winnink), thus modular time is related to temperature. Indeed, positivity of temperature fixes a quantum-thermodynamical arrow of time. We shall sketch some aspects of our recent work extending the modular evolution to a quantum operation (completely positive map) level and how this gives a mathematically rigorous understanding of entropy bounds in physics and information theory. A key point is the relation with Jones’ index of subfactors. In the last part, we outline further recent entropy computations in relativistic quantum field theory models by operator algebraic methods, that can be read also within classical information theory. The information contained in a classical wave packet is defined by the modular theory of standard subspaces and related to the quantum null energy inequality.

传统上，人们可以想象一个完全静止的空间，因此没有时间。众所周知，由于真空波动，该图像在量子物理学中是不可想象的。这两个框架之间的根本区别在于，经典物理学是可交换的（可观测的同时），而量子物理学本质上是不可交换的（海森堡不确定性关系）。从这个意义上讲，我们可以说时间是由非交换性产生的。如果这个陈述是正确的，我们应该能够从非交换空间中得出时间。我们知道冯·诺依曼代数是一个非交换空间。大约50年前，富田-崎崎模块化理论揭示了与冯·诺依曼代数的任何给定（忠实，正常）状态有关的内在演化，因此非交换空间本质上是动态的。这种演化的特征是量子统计力学（Haag，Hugenholtz，Winnink）中的Kubo-Martin-Schwinger热平衡条件，因此模块化时间与温度有关。确实，温度的正向固定了时间的量子热力学箭头。我们将概述我们最近工作的一些方面，这些方面将模块化演化扩展到量子运算（完全正图）级别，以及这如何在数学和信息论中给出对数学界的数学严格理解。关键是与琼斯子因子指数的关系。在最后一部分中，我们概述了通过算符代数方法在相对论量子场理论模型中进行的最新熵计算，这在经典信息论中也可以理解。