A primer on the Universe

Galileo Galilei was a key precursor to Isaac Newton and modern science. In Pisa, he famously conducted (or at least proposed) experiments by dropping objects of different masses from the Leaning Tower to demonstrate that their fall speed is independent of mass, directly challenging Aristotle’s longstanding views on gravity. This foundational work paved the way for Newton’s later formulation of the laws of motion and universal gravitation.

Galileo Galilei was a key precursor to Isaac Newton and modern science.

Isaac Newton created the modern view of the universe by conceiving space as a vast, three-dimensional arena and time as a uniform, linear progression—an independent backdrop against which all physical events unfold. He used mathematics to give precise form to the law of universal gravitation, expressing it as a rigorous equation: every mass attracts every other mass with a force proportional to their masses and inversely proportional to the square of the distance between them. In this way, Newton unified terrestrial and celestial mechanics under a single mathematical framework.

Albert Einstein, together with Hermann Minkowski, transformed the Newtonian paradigm by fusing space and time into a single, four-dimensional continuum called spacetime. In this new setting, the geometry of events is governed by Lorentz transformations, revealing that space and time are interdependent and relative to the observer. Their work showed that simultaneity, length, and the passage of time are not absolute but depend on the observer's motion. Physical laws remain invariant under transformations that mix space and time coordinates—a cornerstone of special relativity. For example, two events that appear simultaneous in one reference frame may not be simultaneous in another moving at a different velocity.

Mainstream physics rapidly adopted this paradigm, and today the standard picture still rests on these principles—even though quantum theory and recent astronomical discoveries highlight persistent mismatches with the classical framework. Increasingly, there is a sense that space and time may not be fundamental entities. This shift motivates newer programs such as “process physics” and “relational physics,” in which the basic constituents of reality are not extended in space or time, but are instead conceived as Leibniz-like monads, processes, or relations. In such approaches, our experience of space and time emerges statistically from the collective behavior of countless underlying primary entities, and from the web of connections among them. As a concrete example, some relational models postulate a discrete network of interactions—rather than a smooth spacetime continuum—out of which familiar geometry only appears at large scales.

For further reading on the relational approach, you may refer to the review "Relational Paradigm in Asks and Answers" by M. I. Suslova and A. A. Sidorova-Biryukova here, and "On Explanations of Magnetic Fields of Astrophysical Objects in the Geometric and Relational Approaches" by Y.S. Vladimirov, S.V. Bolokhov, and I.A. Babenko here.

Perhaps physics requires a complete paradigm shift—perhaps it even needs entirely new mathematics, with “circular arrows” as invoked in some relational approaches. Personally, however, I favor a middle path, one based on algebra and geometry. I believe that algebra and geometry have not yet been exploited to their full potential.

Scientists usually believe that there exist fundamental laws or principles, and that their task is to discover and study them. There are at least two possible strategies for this pursuit: we may either collect data, analyze it, and try to deduce the laws by making educated guesses, or we may rely on our intuition or flashes of insight, boldly postulating fundamental principles (such as “duality” or “triality”), and then deducing more complex laws from these "prime principles." In practice, it is usually best to blend these approaches—combining empiricism and intuition—if we are to make significant progress within any reasonable time frame. I witness such a healthy attitude among participants in Yu.S. Vladimirov’s weekly seminar “Foundations of Fundamental Physics” at Peoples’ Friendship University of Russia, which I have been attending regularly for several months. The Vladimirov school is developing a mathematics of complex binary relations and, on this basis, constructing a worldview that incorporates the ideas of Leibniz and Mach, including the key notion of action at a distance.

“Foundations of Fundamental Physics” seminar 2025-11-06.

Y.S. Vladimirov on the far right.

Personally, I am a syncretist, blending elements from different traditions. Building the universe out of primitive yes-no alternatives, even with circular arrows and complex numbers, is a formidable task and may take an extremely long time—even following clever shortcuts. Instead of focusing on elementary monads or "atoms of existence," I prefer the attitude of a chemist, who works with molecules rather than elementary particles. Thus, I take algebras as already given, and geometry—which naturally follows from these algebras—as sufficiently stable to serve as the building blocks of our knowledge. These structures are the prototypes I use for modeling and describing our reality, both material and immaterial.

In my previous post, I mentioned a particular paper, written by five mathematicians, that describes the synthetic geometry of a family of such models. The title of the paper is "A Primer on the (2 + 1) Einstein Universe," and it is intended to introduce the “synthetic geometry of the Einstein universe.” Synthetic geometry is, in principle, based on axioms—like those found in Euclidean geometry. Fortunately, the paper does not pursue this formal path. Personally, I do not care for axiom systems, since they can often be replaced by others that express the opposite or yield incompatible structures. I prefer “constructions.” But constructions, in turn, often come with cumbersome formulas and calculations, making it easy to lose sight of the underlying essence. The paper I refer to, fortunately, takes a balanced approach: it introduces the main constructions, but then focuses on the “objects” involved and the web of relations (mostly incidence relations) between these objects—this is the part of synthetic geometry that I truly appreciate! Let us now delve into the paper’s details.

• The math part will follow in the next post
• A Substack version of this post is here:

Afternotes:

12-11-25 12:02 A passing Reader inquired in a comment to the previous post if I will discuss the Penrose diagram and twistors. Yes, this is related to the content of "A Primer", and we will talk about these concepts. There will be a separate post dealing with the Penrose diagram. BTW Y.S Vladimirov likes to quote the Penrose program and expands it into new areas using new methods,

12-11-25 18:47 Somewhat related new post  on Substack by Laura:
Mind, Matter, and the Epistemic Asymmetry: A Close Reading of Kastrup’s Argument.

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 40: GNS construction

Inspiring thought in the biography of Walter Russell "The Man who Tapped the Secrets of the Universe' by Glenn Clark:

""You say that the thought which flows through you," I interrupted, "is itself never created; the thought belongs to the universe; it is only the form of the thought that is created?"
"Yes," he replied. "I can go back to the answer which Rodin gave to Lillian Russell  when  she  asked him  if  it would be very difficult to learn to be a great sculptor. 'No, Madam,' he replied, 'it is not difficult. It is very simple. All you have to do is to buy a block of marble and knock off what you do not want.'"

All you have to do is to buy a block of marble
and knock off what you do not want.

So simple: just knock off what you do not want! I will try to follow this advice.

This is a continuation of Part 39: Inside a state. We have a finite-dimensional complex *-algebra A, with unit 1, and we have a state f.  Thus f is a linear functional on A, satisfying f(a*a) ≥ 0 for all a in A, and f(1)=1. We have seen that then f has the Hermitian property f(a*) = f(a)*, and it satisfies the Cauchy inequality  (we use essentially only the Cauchy part of Cauchy-Bunyakovsky–Schwarz):

|f(a*b)|² ≤ f(a*a) f(b*b).  

What can we do with such an f?

Well, here we can also ask another, even more relevant, question: how do we know that the set of states is non-empty? Why should we analyze consequences of some assumptions, if the set of mathematical objects satisfying these assumptions is either empty or trivial, non-interesting?

In mathematics there are two main approaches to such an existential problems. We can try using the axiom of choice. Sometimes it works, sometimes not. It is a rather unsatisfactory solution. That is how we prove existence of functions that are not Borel-measurable. After that we know they exist, but we can't give even one concrete example. Not so useful in applications. The second method is to "construct". Can we "construct" a state? Perhaps we can "construct" every state? By "construction" I mean using bricks that already are at our disposal. Fortunately we can do it. We will use the constructive way later on, after we are done. So, have a faith - we are dealing with objects that exist in abundance.

The Cauchy inequality for states is quite similar in form to a well known inequality for Hilbert space scalar product:

|(x,y)|² ≤ (x,x) (y,y).

This suggests that we can try to use f to define a scalar product (a,b)_f on A:

(a,b)_f ≝ f(a*b).

But in a Hilbert space we should have the property that  (x,x) = 0 implies x=0. With f we can have f(a*a) = 0 even if a≠0. It would be nice to see examples of f with this property, but with examples we have to wait until we know how to construct states.

In Part 39 we have seen that

{a: f(a*a)=0} = {a:f(a*b)=0 for all b},

which implies the the set {a: f(a*a)=0} is a linear subspace of A. Thus "bad" vectors, norm zero vectors, they form a linear subspace. We want all these vectors to become "zero" vectors. In linear algebra there is a standard way of performing such a task - take the quotient of a vector space by an unwanted subspace. So, we define the "unwanted" subspace

I_f = {a∈A: f(a*a)=0} = {a∈A: f(a*b) = 0, ∀ b∈A}.

We define H_f = A/I_f.

Which means that we introduce an equivalence relation in A: a~b if (a-b)∈I_f, and we define
H_f  as the set of equivalence classes: [a] ≝ a+I_f. Then H_f becomes automatically a vector space: [a]+[b] ≝ [a+b],  λ[a] ≝ [λa]. In particular [0] = I_f. One easily checks that these are correct definitions (if you never did it before - do it, check it!). The whole subspace I_f becomes a zero vector of the quotient space. On H_f we define now the scalar product:

([a],[b])_f ≝ f(a*b).

Notice that this is a good definition: if [a']=[a], [b']=[b], then a'=a+u, b'=b+v with u,v in I_f. Then

f(a'*b')=f((a+u)*(b+v))=f(a*b)+f(a*v)+f(u*b)+f(u*v)

Now f(u*b) and f(u*v) are zero because u is in I_f. What to do with f(a*v)? We use the  Hermitian property of f: f(a*v)=f((a*v)*)*=f(v*a)*, which is zero because v is in I_f. So f(a'*b')=f(a,b) - the product ([a],[b])_f is well defined, it does not depend on the choice of representatives of equivalence classes. It is a matter of writing a couple of lines to check that ([a],[b])_f is linear in the second argument and anti-linear in the first - as it should be.

In a Hilbert space we should have the property that the only vector orthogonal to all vectors is the zero vector. Do we have it here? Suppose ([a],[b])_f=0 for all b. That means f(a*b)=0 for all b. That means a is in I_f. That means [a]=[0].

What about zero norm vectors? Suppose ([a],[a])_f=0. That means f(a*a)=0. That means a∈I_f. That means [a]=[0] - the zero vector of H_f.

So far so good. We have constructed a Hilbert space (finite-dimensional Hilbert spaces are also called "unitary" spaces. "Hilbert" is usually used for infinite-dimensional spaces). But there is more. I used the symbol I_f for a reason: I_f is a left ideal in A!

Indeed, suppose a∈I_f, and u∈A is arbitrary. Does it follow that ua∈I_f? We check:

f((ua)*b)= f((a*u*)b)= f(a*(u*b))=0.

So it works.  I_f  is a left ideal! In previous posts we used left ideals to construct a representation of A. Here we do something that looks as completely opposite: we have a nice left ideal, and we are getting rid of it! What a shame! Yet there is a method in this madness. The fact that  I_f  is a left ideal will now let us to construct a *-representation of A on H_f. Let us see how it works, and only after doing that we will be able to understand what is going on here.

So we define a representation by an almost evident formula:

ρ_f(a)[b]≝[ab].

Is it well defined? Is it a representation? Is it a *-representation? Let's check. Suppose [b]=[b']. Is it then true that [ab]=[ab']? If [b]=[b'] then b'=b+u, u∈I_f. Then ab'=ab+au. But  I_f  is a left ideal, therefore au∈I_f. Therefore [ab]=[ab'], and so ρ_f(a) is well defined. Checking that ρ_f(ab)=ρ_f(a)ρ_f(b) for all a,b in A is then a matter of using associativity - not a big deal. What about ρ_f(a*)=ρ_f(a)*? Here we need to use the scalar product of H_f.

Exercise 1. Do it.

We have prepared the scene. It is time to introduce the main actor. Our algebra has a distinguished element - the unit 1. We set

Ω_f ≝ [1].

It is a vector in H_f.

Calculate the norm squared of Ω_f :

(Ω_f ,Ω_f)_f = ([1],[1])_f  = f(1*1)= f(1) = 1.

So Ω_f is a unit vector. Moreover, Ω_f  is a cyclic vector for ρ_f. Let us verify it. Take any vector [a] in H_f. Then [a]=[a1]=ρ_f(a)[1]=ρ_f(a)Ω_f . So every vector of H_f can be obtained by acting with a representation operator ρ_f(a) on Ω_f .

But there is more.

To make it more transparent, we will skip in the following the subscript f. We have

(Ω,ρ(a)Ω)=([1],ρ(a)[1])=([1],[a])=f(1*a)=f(a).

Thus the values of our positive functional f (a) is recovered as an expectation value of the representing operator ρ(a) in the (vector) state Ω. We ended up with a Hilbert space, a *-representation, and a distinguished cyclic vector that realizes the functional as a quantum mechanical expectation value. A nice reward for the construction work.

So this is the Gelfand-Neumark-Segal construction in its finite-dimensional baby version.

There are still unanswered questions: How that relates to our previous constructions with left ideals? How to construct states? And can we cut off what we do not want? For instance I do not want the positivity prison. What will happen if we do not want to use the positivity restriction? There are people who would like to cut even more. For instance some do not like real numbers, they prefer finite fields. Some other would go beyond finite fields, but to non-Archimedean fields, like p-adic numbers.... Well, if you sculpt, you must be careful. If you cut too much, you may cut off the nose part, and your sculpture will get dysfunctional. So, step by step, carefully.

We will come back to the hanging questions in the next post.