Spin Chronicles Part 47: Emergence of time I

 It took me a while to gather my thoughts and finally sit down to write about the "emergence of time," as I had promised. But today, the moment felt right—a kairos, as the ancient Greeks might say. And so, here we are.

Let me start with a little detour into history. The ancient Greeks had two words for time: chronos and kairos. While chronos refers to chronological, sequential time—the kind we measure with clocks—kairos is something more elusive. It means "the right or critical moment," a qualitative, almost magical sense of timing. In modern Greek, kairos even extends to mean "weather," hinting at its connection to the unpredictable and the timely.

Kairos emerging from pleroma

As Wikipedia eloquently puts it:

"Kairos has classically been defined as a concept that focused on 'the uniquely timely, the spontaneous, the radically particular.' Ancient Pythagoreans considered kairos one of the most fundamental laws of the universe. It was said to weave together the dualistic nature of existence. Empedocles, the philosopher, linked kairos to the principle of opposites and harmony, making it a cornerstone of conflict and resolution—a concept that even found its way into rhetoric."

Kairos as portrayed in a 16th-century fresco by Francesco Salviati

What is he doing there? Drilling with an electric drill?

But philosophy, as fascinating as it is, often feels like an attempt to impose logical order on a tangled web of ideas. These ideas, of course, spring from our interactions with reality—both conscious and unconscious. For me, though, the real magic lies in mathematical structures that can model both the real and the unreal. And when it comes to making sense of "time," I believe algebra is the key.

It was likely John von Neumann who first pioneered the algebraic approach to quantum theory. His seminal work, Mathematical Foundations of Quantum Mechanics, played a pivotal role in my own intellectual journey. From there, I fell head over heels for the perspectives of Araki, Haag, and Kastler. When I first began studying their works, I had no idea that one day I’d be discussing the intricacies of quantum theory with them in their homes. Most of these conversations were with Rudolf Haag, but it was Daniel Kastler with whom I co-authored three of my papers:

1. A. Jadczyk and D. Kastler, “Graded Lie Cartan Pairs”, Rep. Math. Phys., 25 (1988), 1–51 pdf 
2. A. Jadczyk and D. Kastler, “Graded Lie Cartan Pairs. 2. The Fermionic Differential Calculus”, Ann. Phys., 179 (1987), 169–200 pdf  
   
3. R. Coquereaux, A. Jadczyk, D. Kastler, “Differential and Integral Geometry of Grassmann Algebras”, Rev. Math. Phys., 3 (1991), 63–99 pdf  

These collaborations were a masterclass in honing my abstract algebraic skills—far removed from any immediate real-world applications. At the time, Kastler was deeply invested in promoting the ideas of noncommutative geometry, a field being developed by Alain Connes. Kastler believed this would be the future of physics, and in 1992, he and Thomas Schücker published a paper in the Russian journal Theoretical and Mathematical Physics titled “Remarks on Alain Connes' Approach to the Standard Model in Non-Commutative Geometry,” dedicated to M.C. Polivanov.

Then, in 1994, something remarkable happened. Alain Connes, a mathematician, joined forces with theoretical physicist Carlo Rovelli. Together, they wrote a groundbreaking paper linking noncommutative geometry to the "emergence of time" [1]. This collaboration laid the foundation for a new way of thinking about time—one that transcends the classical, linear view.

Later, Carlo Rovelli, renowned for his work on quantum gravity, distilled these ideas into his popular book The Order of Time [2]. In Part III, titled “The Sources of Time – Quantum Time,” he writes:

"Connes has provided a refined mathematical version of this idea: he has shown that a kind of temporal flow is implicitly defined by the noncommutativity of the physical variables. Due to this noncommutativity, the set of physical variables in a system defines a mathematical structure called a 'noncommutative von Neumann algebra,' and Connes has shown that these structures contain an implicitly defined flow. Surprisingly, there is an extremely close relation between Alain Connes’s flow for quantum systems and the thermal time I have discussed above. Connes has shown that, in a quantum system, the thermal flows determined by different macroscopic states are equivalent, up to certain internal symmetries, and that, together, they form precisely the Connes flow. Put simply: the time determined by macroscopic states and the time determined by quantum noncommutativity are aspects of the same phenomenon."

 

This, I believe, is the essence of what we call "time" in our universe—a variable that doesn’t exist at the fundamental level but emerges from the interplay of quantum and thermal dynamics.

The notes in Rovelli’s book elaborate further:

"[85]. The theorem of Tomita-Takesaki shows that a state on a von Neumann algebra defines a flow (a one-parameter family of modular automorphisms). Connes has shown that the flows defined by different states are equivalent up to internal automorphisms, and therefore define an abstract flow determined only by the noncommutative structure of the algebra.

[86]. The internal automorphisms of the algebra referred to in the above note.

[87]. In a von Neumann algebra, the thermal time of a state is exactly the same as Tomita’s flow! The state is KMS with respect to this flow."

 

On the same subject, M. Heller and W. Sasin published a paper titled Emergence of Time [3]. In Section 6, "Interpretation," they write:

"To define the modular group at, Connes and Rovelli have distinguished the state on A of the form ω(a) = Tr[aω] for every a∈A (which, in the language used by physicists, is a density matrix). Owing to this choice, they were able to argue that the time flow has a statistical (thermodynamic) origin. They emphasize that it is not only the arrow of time that emerges in this way, but also the time flow itself." 

 

Similarly, R. Longo’s paper Emergence of Time states in its abstract:

"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 characterized 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."

In the following posts, we’ll explore Tomita’s flow of “thermal time” through the lens of a simple geometric Clifford algebra, A ≃ Mat(2,C). We’ll identify A with Mat(2,C) and begin with the concept of a "state," which we’ve already touched on.

References:

[1] A. Connes, C. Rovelli, "Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories", Class. Quantum Grav. 11 (1994) 2899 .

[2] Carlo Rovelli, "The Order of Time", Penguin Books 2018, ISBN 9780735216105.

[3] M. M. Heller and W. Sasin, "Emergence of Time", Phys. Lett. A 250 (1998) 48-54.

[4] R. Longo, "Emergence of Time", Expo. Math. 38 (2020) 240-258.