The emergence of time

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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 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.

Introduction

In 1967, Richard Kadison organised the first conference on Operator Algebras at Baton Rouge, an event that set together essentially all the experts in the subject at that time, both from Mathematics and Physics, and put the seed for much of the future developments in the subject.

Due to a remarkable historical accident, two related breakthroughs were announced at that conference. M. Tomita distributed a (never published) preprint on modular theory [57], that set a profound and fundamental change of view in the subject. R. Haag presented in a talk the characterisation of equilibrium states in Quantum Statistical Mechanics at infinite volume [25], extending the Gibbs condition: the KMS (Kubo–Martin–Schwinger) property.

During the next year, fervid discussions on modular theory took place, in particular at Kadison’s seminar on Operator Algebras at Penn. Eventually, Masamichi Takesaki provided a complete, extended, rigorous formulation of modular theory and showed that the modular group is characterised by the KMS condition [56]. Clearly, this was the basis for a new deep interplay between Mathematics and Physics, that is still producing impressive results.

In this article, we briefly outline a path from these general basic results to some present developments where the interplay between Operator Algebras and Quantum Field Theory is playing a key role.

We refer to [32] and to [2], [24] for the basics on Operator Algebras and on Local Quantum Field Theory.

Section snippets

Space and time

We begin with general considerations of interdisciplinary nature about space and time, that explain why we expect Operator Algebras and modular theory to play an even more central role in Science in the future.

From a mathematical point of view, a space is a set X endowed with a structure that put relations among elements (points): for example a distance, a topology, a measure. Time then allows to compare different configurations, in particular it provides a (partial) order.

Now, in Quantum

Araki’s relative entropy

Quantum Information is providing an increasingly important interplay with Quantum Field Theory, that naturally originated in the framework of quantum black hole thermodynamics (see [59]). The first noncommutative entropy notion, von Neumann’s quantum entropy, was originally designed as a Quantum Mechanical version of Shannon’s entropy: if a state ψ is given a density matrix ρψ von Neumann entropy of ψ:Tr(ρψlogρψ).As said, an infinite quantum system is described by a von Neumann algebra M

Landauer’s bound for infinite systems

In short, Landauer’s principle states that “information is physical” [35]. One of its main motivations was to provide a solution to the Maxwell’s demon paradox. More extensively, it can be stated as follows:

Any logically irreversible manipulation of information, such as the erasure of a bit or the merging of two computation paths, must be accompanied by a corresponding entropy increase in non-information bearing degrees of freedom of the information processing apparatus or its environment [4].

Energy conditions in QFT

As is known, positivity of the energy plays an important role in classical physics, general relativity in particular. In QFT, the local energy may have negative density states [21], although energy lower bounds may occur (see [22], [60]).

The Quantum Null Energy Condition (QNEC) was considered in [8] (see [12]). It is associated with a deformation Wλ of a wedge W in the null direction (see Fig. 1 where the boundary deformation is given by the function fλ, with λ the deformation parameter). By

Final comments

The above thermodynamical interpretation of time concerns a system in thermal equilibrium; this is an asymptotic situation, that may apply to our present universe, not soon after the big bang. So far, the understanding of a system out of equilibrium is definitely incomplete and mathematical methods are to be developed, in particular Operator Algebraic ones. However, Operator Algebras have recently provided a model independent framework, with effective computational methods, for non-equilibrium

Acknowledgments

Our discussion on time was initiated to be presented at the workshop “The Origins and Evolution of Spacetime” in the Vatican City at the Pontifical Lateran University, on November 27–28, 2018. We thank the IRAFS Director Gianfranco Basti and vice-Director Flavia Marcacci for the kind invitation.

We acknowledge the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.

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    Supported by the ERC Advanced Grant 669240 QUEST “Quantum Algebraic Structures and Models”, MIUR FARE R16X5RB55W QUEST-NET and GNAMPA-INdAM.

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