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.

Spin Chronicles Part 26: Imagine Clifford Algebra

 The song is in C major. Even if I like C minor better, I like the song.

"Imagine all the people

Livin' life in peace

You"

Can you imagine it? I can, even though it is not in my memory banks. Imagination is a powerful mechanism. I am reading "The Story of the Tower of Babel: Human Aspirations and Dreams".

There:

"Human imagination is a unique and powerful ability that allows us to conceive of things that do not exist, create new concepts, explore possibilities, and even transcend the current physical limitations in spirit. This ability is indeed rare in the natural world, and, at least based on our current knowledge, humans are unique in their use of complex symbolic thinking and creative imagination."

Imagination and creativity are close friends. I am imagining what I will write today. Then I start writing. The result will probably be not quite as I have imagined it.

"You may say I'm a dreamer

But I'm not the only one

I hope someday you'll join us

And the world will be as one"

You may say I'm a dreamer....

Yes, you may say I'm a dreamer. But I'm not the only one. Last night I dreamed about Hilbert Algebra. The dream had its roots in my past memory. My PhD thesis was partly about equilibrium states in quantum thermodynamics, the algebraic approach, with KMS (Kubo-Martin_Schwinger)  states. Tomita and Takesaki, the two Japanese mathematicians, developed the theory of Hilbert algebras and modular automorphisms. Much later, in 1994,  Alain Connes and Carlo Rovelli  applied it  to the "flow of time":

"We consider the cluster of problems raised by the relation between the notion of time, gravitational theory, quantum theory and thermodynamics; in particular, we address the problem of relating the “timelessness” of the hypothetical fundamental general covariant quantum field theory with the “evidence” of the flow of time. "

But now is now, and we are dealing with the Ur atom of space - the geometric Clifford algebra of 3D space Cl(V). We have an algebra. Buit where is "Hilbert"? Searching for "Hilbert Algebra" leads us to Wolfram MathWorld. There we find conditions 1, 2, 3, 4, 5 that need to be satisfied.

In our case only condition 3 counts. All other are automatically satisfied. So, we should have an algebra with involution (we have one), which is a Hilbert space (and Cl(V) is a Hilbert space, as we will see soon),  with a scalar product <a,b> satisfying

<ba,c> = <b,ca*>.                    (1)

We need to define Hilbert space scalar product on Cl(V). We have already done it in Part 11,

where we have defined:

"... But then we have also the second natural "bilinear" form:

B_τ(u,v) =t_n(τ(u)v),

In fact, this form is, when considered as a form on a complex vectors space,  Hermitian: it is complex linear in v, but complex anti-linear in u."

In Part 12 we further wrote:

B_τ(u,u) = |p₀|² + |p|².

This is our Hilbert space scalar product - the scalar part of τ(u)v: we set

<u,v> = the scalar part of the product τ(u)v. If u = (p₀,p), v = (q₀,q), then

<u,v> = p_0^*q₀ + p^*· q.

Then we need an involution in our algebra, we need a *-algebra. We take τ for our involution. We just set u* = τ(u). There may arise some confusion now. Because in my blog I often used the star to denote the complex conjugation. And now I am using the same symbol to denote the conjugation in the algebra! Well, the meaning will be defined by the context, and when confusion can arise, it will have to be explicitly addressed.

Let us recall the action of τ:

If u = (p₀,p), then τ(u) = (p₀*,p*), thus u* =  (p₀*,p*). In other words:

(p₀,p)* = (p₀*,p*).

Not too bad! I suppose we can live with that.

Now comes the crutial point:

Exercise 1. Prove (1).

Hint: there are at least two ways of proving it. Hard way, using definitions, and not-so-hard, if you are creative.