**The Connes' intuition about the emergence of time through the notion of quantum variability**

Here is the first oral presentation by Alain Connes of one of his most personal philosophical ideas "backed-up by an intuition which comes from many years of work" as he says :

At the philosophical level there is something quite satisfactory in the variability of the quantum mechanical observables. Usually when pressed to explain what is the cause of the variability in the external world, the answer that comes naturally to the mind is just : the passing of time.

But precisely the quantum world provides a more subtle answer since the reduction of the wave packet which happens in any quantum measurement is nothing else but the replacement of a “q-number” by an actual number which is chosen among the elements in its spectrum. Thusthere is an intrinsic variability in the quantum world which is so far not reducible to anything classical. The results of observations are intrinsically variable quantities, and this to the point that their values cannot be reproduced from one experiment to the next, but which, when taken altogether, form a q-number.How can time emerge from quantum variability ? As we shall see the study of subsystems as initiated by von Neumann leads to a potential answer...

IHES, Jeudi 9 avril 2015

*(presentation in English)*

Une conférence en français sur le même sujet (filmée elle aussi), plus courte et pour un public plus large a été donnée un peu plus tard à l'académie des sciences :

Je présenterai une réflexion sur la notion de "variabilité", en commençant par la notion de variable réelle" en mathématique puis par la "variabilité" inhérente aux phénomènes quantiques qui prescrit la reproductibilité de certains résultats d'expériences tout en prédisant la probabilité des résultats.J'énonceraiensuiteune conjecture sur l'écoulement du temps comme phénomène émergent de nature thermodynamique gouverné par une équation mathématique ayant pour origine la non-commutativité des observables de la mécanique quantique."

Alain CONNESMardi 19 mai 2015 16:30 - 17:00

**The Rovelli's conjecture to understand the time arrow as perspectival**

In his presentation, Connes explains the connection between his work and the speculations about time from Carlo Rovelli. Here are the most recent developments on the philosophical reflexions of this theoretical physicist:

An imposing aspects of the Cosmos is the mighty daily rotation of Sun, Moon, planets, stars and all galaxies around us. Why does the Cosmos so rotate? Well,it is not really the Cosmos to rotate, it is us. The rotation of the sky is a perspectival phenomenon: we understand it better as due to the peculiarity of our own moving point of view, rather than a global feature of all celestial objects....

The list of conspicuous phenomena that have turned out to be perspectival is long; recognising them has been a persistent aspect of the progress of science.A vivid aspect of reality is the flow of time; more precisely: the fact that the past is different from the future. Most observed phenomena violate time reversal invariance strongly. Could this be a perspectival phenomenon as well? Here I suggest that this is a likely possibility.

Boltzmann’s H-theorem and its modern versions show that for most microstates away from equilibrium, entropy increases in both time directions [1, 2, 3].Whythenwe observe lower entropy in the past? For this to be possible, most microstates around us appear to be very non generic. This is the problem of the arrow of time, or the problem of the source of the second law of thermodynamics [3, 4]. The common solution is to believe that the universe was born in an extremely non-generic microstate[5]...

Here I point out that there is a different possibility: past low entropy might be a perspectival phenomenon, like the rotation of the sky.

This is possible because entropy depends on the system’s microstate but also on the coarse graining under which the system is described. In turn, the relevant coarse graining is determined by the concrete existing interactions with the system. The entropy we assign to the systems around us depends on the way we interact with them —as the apparent motion of the sky depends on our own motion. A subsystem of the universe that happens to couple to the rest of the universe via macroscopic variables determining an entropy that happens to be low in the past, is a system to which the universe appears strongly time oriented. As it appears to us. Past entropy may appear low because of our own perspective on the universe...

Quantum phenomena provide a source of entropy distinct from the classical one generated by coarse graining: entanglement entropy. The state space of any quantum system is described by a Hilbert space H, with a linear structure that plays a major role for physics. If the system can be split into two components, its state space splits into the tensor product of two Hilbert spaces: H=H_{1}⊗H_{2}, each carrying a subset of observables. Because of the linearity, a generic state is not a tensor product of component states; that is, in general ψ≠ ψ_{1}⊗ψ_{2}. This is entanglement.This means: with a sufficient complex algebra of observables and a Hamiltonian which is suitably “ergodic” with respect to it. A quantum system is not determined uniquely by it Hilbert space, Hamiltonian and state. All separable Hilbert space are isomorphic, and the spectrum of the Hamiltonian, which is the only remaining invariant quantity, is not sufficient to characterise the system.Restricting the observables to those of a subsystem, say system 1, determines a quantum entropy over and above classical statistical entropy. This is measured by the von Neumann entropy S=-tr[ρlogρ] of the density matrix ρ=tr_{H2}|ψ><ψ|.Coarse graining is given by the restriction to the observables of a single subsystem.The conjecture presented in this paper can then be extended to the quantum context. Consider a “sufficiently complex” quantum system []. Then:

Conjecture: Given a generic state evolving in time as ψ(t), there exists splits of the system into subsystems such that the von Neumann entropy is low at initial time and increases in time.

This conjecture, in fact, is not hard to prove. A separable Hilbert space admits many discrete bases |n>. Given any ψ∈H, we can always choose a basis |ni where ψ=|1>. Then we can consider two Hilbert spaces, H1 and H2, with bases |ki and |mi, and map their tensor product to H by identifying |k>⊗|m> with the state |n> where (k, m) appear, say, in the n-th position of the Cantor ordering of the (n, m) couples ((1,1),(1,2),(2,1),(1,3),(2,2),(3,1),(1,4)...). Then, ψ = |1>⊗|1> is a tensor state and has vanishing von Neumann entropy. On the other hand,recent results show that entanglement entropy generically evolve towards maximizing entropy of a fixed tensor split(see [9]).

Thereforefor any time evolution ψ(t) there is a split of the system into subsystems such that the initial state has zero entropy and then entropy grows. Growing and decreasing of (entanglement) entropy is an issue about how we split the universe into subsystems, not a feature of the overall state of things(on this, see [10]).Notice that in quantum field theory there is no single natural tensor decomposition of the Fock space.

Finally, let me get to general relativity.In all examples above, I have considered non-relativistic systems where a notion of the single time variable is clearly defined. I have therefore discussed the direction of time, but not the choice of the time variable. In special relativity, there is a different time variable for each Lorentz frame. In general relativity, the notion of time further breaks into related but distinct notions, such as proper time along worldliness, coordinate time, clock time, asymptotic time, cosmological time... Entropy increase becomes a far more subtle notion, especially if we take into account the possibility that thermal energy leaks to the degrees of freedom of the gravitational field and therefore macrostates can includes microstates with different spacetime geometries. In this context, a formulation of the second law of thermodynamics requires to identify not only a direction for the time variable, but also the choice of the time variable itself in terms of which the law can hold [11]. In this context, a spit of the whole system into subsystems is even more essential than in the non-relativistic case, in order to understand thermodynamics [11]. The observation made in this paper therefore apply naturally to the non relativistic case.

The reason for the entropic peculiarity of the past should not be sought in the cosmos at large. The place to look for them is in the split, and therefore in the macroscopic observables that are relevant to us. Time asymmetry, and therefore “time flow”, might be a feature of a subsystem to which we belong, features needed for information gathering creatures like us to exist, not a feature of the universe at large.

(Submitted on 4 May 2015 (v1), last revised 10 May 2015 (this version, v2))

**The Connes and Rovelli joint work**

The hypothesis that we have put forward in this paper is that physical time has a thermodynamical origin. In a quantum generally covariant context, the physical time is determined by the thermal state of the system, as its modular flow...The main indications in support this hypothesis are the following

• Non-relativistic limit.In the regime in which we may disregard the effect of the relativistic gravitational field, and thus the general covariance of the fundamental theory,physics is well described by small excitations of a quantum field theory around a thermal state |ω>. Since |ω> is a KMS state of the conventional hamiltonian time evolution, it follows that the thermodynamical time defined by the modular flow of |ω> is precisely the physical time of non relativistic physics.

• Statistical mechanics of gravity. The statistical mechanics of full general relativity is a surprisingly unexplored area of theoretical physics [4].In reference [4] it is shown that the classical limit of the thermal time hypothesis allows one to define a general covariant statistical theory, and thus a theoretical framework for the statistical mechanics of the gravitational field.

• Classical limit; Gibbs states.The Hamilton equations, and the Gibbs postulate follow immediately from the modular flow relation...

• Classical limit; Cosmology. We refer to [11], whereit was shown that (the classical limit of) the thermodynamical time hypothesis implies that the thermal time defined by the cosmic background radiation is precisely the conventional Friedman-Robertson-Walker time.

• Unruh and Hawking effects. Certain puzzling aspects of the relation between quantum field theory, accelerated coordinates and thermodynamics, as the Unruh and Hawking effects, find a natural justification within the scheme presented here.

• Time–Thermodynamics relation. Finally,the intimate intertwining between the notion of time and thermodynamics has been explored from innumerable points of view [1], and need not be expanded upon in this context.

...We leave a large number of issues open.It is not clear to us, for instance, whether one should consider all the states of a general covariant quantum system on the same ground, or whether some kind of maximal entropy mechanism able to select among states may make sense physically.

//edit 30 June 2015Von Neumann Algebra Automorphisms and Time-Thermodynamics Relation in General Covariant Quantum TheoriesA. Connes, C. Rovelli(Submitted on 14 Jun 1994)

The most synthetic review at the technical level of this idea of passing of time in noncommutative von Neumann algebra can probably be found in the appendix of

*by Alain Connes and Matilde Marcolli.*__Noncommutative Geometry, Quantum Fieldsand Motives__
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