Spin Chronicles Part 46 - Closing the GNS construction door

 

"When one door closes another door opens" - how very true. But first this one door needs to be tightly closed. Today we will close this door - the GNS construction door.

When one door closes another door opens

The following theorem is an adaptation of Theorem 1.6.6. from William Arveson, "An Invitation to C*-Algebras", Springer 1976, p. 30. Originally I thought that I will follow Naimark (cf. Part 45), but the proof in Naimark's monograph is, as I have realized today, incomplete. So, here is the Theorem, as it is stated, in Arveson, together with half of the proof (in this note we will follow Arveson's notation, which is a good gymnastic for the mind). First the Theorem:

Then the first part of the proof

This first part of the proof should be clear. Since we are in finite dimension, we can just skip the  " π(A)ξ is dense" parts. We see that if f is a pure state, then there are no non-trivial projections in the commutant. But if there are no non-trivial projections, there are no non-trivial operators at all. (Why?). That means, by Shur's lemma, that the representation is irreducible.

If you have any questions - ask, and I will do my best to answer.

The second part, the converse, we can prove easier than it is done in Arveson, using the theorem from Part 45. Let us assume that f is a mixture of states, f=tf₁ + (1-t)f₂, where t is in (0,1). Then f₁ is f-dominated (Why?). Therefore, using the theorem from Part 45, there exists a positive operator B in the commutant π(A)' such that

f₁(a) = (ξ,Bπ(a)ξ).

Supposing that π is irreducible, B would have to be a scalar B=rI. From f₁(1)=1, it would then follows that B=I. But then f₁=tf, f₂=(1-t)f, contrary to the assumption that f is a nontrivial mixture. Thus π must be reducible. QED.

And so we ready now to move to the next issue: the "baby" version of the Tomita-Takesaki theory of "modular automorphisms". Here I will make use of the paper by Roberto Longo "A simple proof of the existence of modular automorphisms ...".

Spin Chronicles Part 45: positive domination

For some reasons I consider the concept of state on a *-algebra as important. I wrote a whole post on the subject, with a Theorem, its detailed proof, and with exercises for the Reader. I also wrote a long introduction. Then I have made one wrong click, and all was gone. Well, not all, the introduction has been preserved, since I have already managed to copy it to the blog. But introduction is just words. All the hard stuff, all formulas, are gone. So I decided to ask the I Ching oracle for advice. I Ching told me:

"Thunder and wind symbolize Duration,

The superior man stands firm

Without changing direction."

Fine. I stand firm. Will not change direction. But will provide a theorem, and skip the proof. The proof is lengthy, but not difficult. There will be no harm in skipping it. Maybe it will even be good. No distraction. So, first the beginning of the Introduction. The rest of the introduction will be at the end.

Introduction Part 1

 In this post, we will delve deeper into the concept of states on *-algebras. States are fundamental in physics, serving as a crucial link between the abstract formalism of algebra and the concrete realm of empirical measurement. They transform abstract algebraic elements into numbers, which we then interpret as experimental values. In this way, states act as a bridge between the invisible, abstract structures of theoretical physics and the tangible, measurable phenomena of the observable world. But this raises a profound and provocative question: Are states themselves "visible"? Or, more precisely, do they belong to the realm of the measurable, or do they reside in a liminal space between the abstract and the concrete?

This question is not merely technical; it is deeply philosophical. It forces us to confront the nature of representation, measurement, and the limits of human knowledge. States, in their role as mediators, challenge the boundary between the intelligible and the sensible, between what can be thought and what can be experienced. To ask whether states are "visible" is to grapple with the very nature of reality and our capacity to comprehend it. It is a question that mingles different levels of description—mathematical, physical, and metaphysical—in a way that is both illuminating and perilous. Such questions are logically treacherous, reminiscent of the paradoxical barber who shaves only those who do not shave themselves. They expose the fragility of our conceptual frameworks and the limitations of language itself.

The barber paradox

Let A be a finite-dimensional *-algebra with unit 1. Let f be a state on A, and let H be the Hilbert space of the GNS construction with cyclic vector Ω for a representation π (which previously we have denoted by ρ).

Definition. If f₁ is a positive functional on A, we say that f₁ is f-dominated if there exists a constant λ>0 such that

f₁(a*a) ≤ λ f(a*a) for all a in A.

We say that f₁ is f-absolutely continuous if f(a*a)=0 implies f₁(a*a) = 0.

It is evident that every f-dominated functional is f-absolutely continuous. (can you see it?) The converse is not immediately evident. But it will be seen from the following theorem that I adapt from the monograph by Naimark.

Here is the original theorem:

Let H be the Hilbert space of the GNS construction with cyclic vector Ω for a representation π (which previously we have denoted by ρ).

Theorem.  Let f₁ be a positive functional on A that is f-absolutely continuous. Then there exists a unique positive operator B in the commutant π(A)' of π(A) such that

f₁(a) = (Ω,π(a)BΩ).         (1)

Conversely, every such B determines, by (1) a positive functional  f₁ that is f-dominated.

Proof. Gone with the wind.

In a future post we will use this theorem to prove that a GNS representation is irreducible if and only if f is a pure state.

Introduction Part 2

Yet, it is precisely these kinds of questions that I find most compelling. They reveal the boundaries of our understanding and force us to confront the inadequacy of communicable language when grappling with the ineffable. By asking such questions, we quickly come to realize how constrained our knowledge is when confined to the tools of language and logic. Language, as a system of symbols, is inherently limited in its ability to capture the full depth of abstract thought and the complexity of reality. It is a filter through which we attempt to convey the inexpressible, but it inevitably falls short.

This tension between the communicable and the ineffable is at the heart of both science and philosophy. States on *-algebras, as abstract entities that give rise to measurable quantities, embody this tension. They are both a product of human thought and a reflection of an external reality that exists independently of our conceptual frameworks. In this sense, they invite us to consider the nature of existence itself: Is reality fundamentally mathematical, as some physicists and philosophers suggest? Or is mathematics merely a tool, a language we have invented to describe a reality that ultimately transcends it?

These questions are not merely academic; they have profound implications for how we understand the universe and our place within it. They challenge us to think beyond the limits of our current knowledge and to embrace the uncertainty and ambiguity that come with exploring the unknown. By engaging with such questions, we not only deepen our understanding of states on *-algebras but also confront the very nature of knowledge, reality, and the human condition. And in doing so, we may come to appreciate the beauty and mystery of a universe that is far more complex and enigmatic than our language and logic can ever fully capture.

Spin Chronicles Part 44: Archimedes Principle

 This is a continuation of "Spin Chronicles Part 43: Feeding GNS doll", where we have treated the GNS construction as if it was a lovely sweet doll. We played with it, and we have discussed three different ways of looking at it. The third method, called there Feed 3 is the most satisfactory, and in the following we will use it for the discussion of "time" emerging from space a'la Connes-Rovelli and Heller-Sasin. Do not worry now, we will get there soon, and all will be clear.

What if there is always a little bit of imaginary space and imaginary time?
Like a boat that floats on the surface, but is always embedded in water.
Perhaps there is a kind of  "Archimedes Principle" here? 

Why do I connect GNS to spinors? To answer this question let us first take a look how spinors are defined when we want to connect them to geometry, so that, for instance, they can be used also in curved spacetimes. There are several standard ways of doing so. There is no problem with getting the Clifford algebra and spin group inside it. In General Relativity it will be the Clifford algebra of the tangent space at a given point. But then what? One way is: take an irreducible representation of this algebra. Call vectors of this representation "spinors". Problem solved. Yes and no. There is unanswered question: where do I get this representation from? Which one? To ease this uneasiness we give the representation space a name "Clifford module". We are dealing now with named objects, we feel better.

The second approach is: introduce "spinor structure". We postulate its existence together with the covariant 2:1 map from spin frames to orthonormal frames. But where does it come from?  The answer is: "somehow, does it really matter?". And then it is added: for some spacetimes there may exist several inequivalent spin structures! 

The third approach is: take a minimal left ideal in the Clifford algebra, perhaps take two or four of them - you will fit different spinors to different Fermions. Isn't it nice? Even better, the right action will shuffle the ideals. Left action corresponds to spacetime rotations, right action corresponds to "internal symmetry operations". It can even fit the Standard Model. For all practical purposes it can even work, but the question remains: which ideal and why this and not another one? I am not completely happy with this. We are onto something, but what is it, this "something"?

GNS construction associates representations to "states". If we follow this philosophy, spacetime (or just "space") at a given point may be in some "state". It fits my intuition. Once We have "state", we have the associated representation. Pure states lead to irreducible representations. Mixed states lead to reducible ones (we will discuss this issue in the next post). This opens my memory bank. Pure states are "extreme points" of the convex set of all states. They are at the boundary. It is hard to get dynamically exactly to the boundary. Rather, when we experimentally attempt to get exactly pure state, we will obtain "almost pure" state, never "exactly pure". Pure states is an extreme idealization. No state is in the Nature is exactly pure. This opens a whole new perspective with possible physical consequences. I like it. It leads to another idea: we think that our space is "real", time is "real". Imaginary space and imaginary time are simply mathematical tricks. But are they? What if "impurities" ("defects")  are important? What if there is always a little bit of imaginary space and imaginary time? Like a boat that floats on the surface, but is always embedded in water. Perhaps there is a kind of  "Archimedes Principle" here?  In recent years Peter Woit develops a similar vision.

That is just "talk". Next post will be just "math". We will discuss "states dominated by other states", representations, cyclic vectors and irreducibility. 

Spin Chronicles Part 43: Feeding GNS doll

 Grothendieck, the great and strange mathematician, complained about the nature of mathematics:

"I think there's an inherent flaw in the mathematical way of thinking and I have the impression that it's also reflected in this Manichaean vision of human nature. On the one hand, there's the good, and on the other, the bad, and in the best case scenario, we see both living side by side."

Grothendieck, disappointed,  has left mathematics and went into seclude in Pyrenees. I do not understand it. Personally I do not see anything wrong with mathematics, and with yes-no Aristotelian logic:

There is good and there is evil and there is the specific situation that determines which is which.  What's wrong with this?

From The Guardian article about Alexander Grothedieck of Aug 31, 2024: "He was in mystic delirium’: was this hermit mathematician a forgotten genius whose ideas could transform AI – or a lonely madman?"

"I open a first page at random. The writing is spidery; there are occasional multicoloured topological diagrams, namechecks of past thinkers, often physicists – Maxwell, Planck, Einstein – and recurrent references to Satan and “this cursed world”

Alexander Grothendieck - forgotten genius
whose ideas could transform AI

So I stay with the devil of the algebra and the angel of geometry, and with their happy marriage - geometric algebra. Which geometric algebra? This is a good question. Certainly time is not the same as space. Einstein and Minkowski merged space with time, but their reasoning was highly questionable. Maxwell equations were taken as a starting point. 10-parameters Poincare group came out rather easily as an invariance group, and it was set on a pedestal. It was given the absolute power over all other possible interactions, for no good reason. Then it was discovered that Maxwell equations are invariant under the 15-parameters conformal group, containing the Poincare group as a subgroup leaving the "conformal infinity" (Dupin cyclide)  as the unmoved "absolute", but it didn't help us with understanding the nature of time. Is time "real" or "imaginary"? Or both? There is a lot that needs to be done in physics before jumping ahead into the future of fancy mathematics with no "yes-no"  dichotomy. Some physicists promote the view that the universe is a quantum computer - like it would explain anything. It does not explain anything at all, it just sweeps problems under the carpet, since we do not understand what quantum theory is about. Quantum theory does not explain consciousness. We need to understand what "consciousness", "information", and "measurements" are first. Then, perhaps, we will be able to get some idea about quantum theory, the theory of "measurements".

Let's go back to algebra, geometric algebra. The mystery of spin is hidden there. We do not need to mix space and time from the start (see Peter Woit blog "The Mystery of Spin"). If time comes out of space and spin - that would be a nice surprise. Geometric algebra A = Cl(V) of space happens to be isomorphic to complex quaternions that can be faithfully represented by 2×2 complex matrices Mat(2,C). Unexpected meeting of geometry with the formalism of quantum mechanics. Coincidence? Or there is some deep meaning hidden in this fact? Can we read/decode the message?  

Hamilton invented quaternions when trying to make space into an algebra. He was forced to add the fourth dimension. What is the meaning of this fourth dimension? Is it just pure mathematics that has nothing to do with reality? Certainly the fourth dimension of Hamilton is not the "real time" of special relativity theory. So what is it? For a while we have a toy that we are playing with, discovering new functionalities. I talked to Laura about all these mysteries, and she told me that as a little girl she owned a doll that could even pee. Of course you had first to fill the doll with water. What can our Cl(V) produce if properly filled with  proper stuff? Shall we get spinors?

Feed 1.

First we have fed A with an orthogonal  projector p = (1+n)/2,  p = p* = p². It produced the right eigensubspace

{u: up=u} = Ap,

which is a non-trivial left ideal of A. It carried an irreducible representation of A,  where "spinors" live. We also got its orthogonal complement A(1-p). Spinors can reside also there.

Feed 2.

Then we have chosen a different approach, through the Gelfand-Neumark-Segal (GNS) construction. This time we fed A with a "state" - a positive linear functional f on A. It produced a left ideal I_f:

I_f = {u: f(u*u)=0}

which we didn't like (as it consists of norm zero vectors), so we have taken the quotient Hilbert space H_f = A/I_f, build as the set of equivalence classes [a], and we got a representation  ρ_f of A on H_f with cyclic vector Ω_f = [1] such that

f(a) = (Ω_f,ρ_f(a)Ω_f). 

If f is a pure state, then the representation ρ_f is irreducible (we did not show it yet). So elements of H_f are also candidates for spinors. The relation to Feed 1 is still waiting for an explanation.

Feed 3.

Here again we start with a state f, but this time we add some salt - we use the fact that our A is not only a *-algebra, but also a Hilbert space on its own. This allows us to represent f through an algebra element, denoted by the same letter, so that

f(a) = <f*, a>_{.                             (1)}

Taking the positive square root f^½  of f  we get

f(a) = <f^½, L(a)f^½ >,

where L is the left regular representation of A on A.

Connecting Feed 2 to Feed 3.

We can now happily use Proposition 4.5.3 from the GNS Promenade to connect the two constructions. For this to take place, Feed 3 should result in a cyclic representation. As it stands, the representation is not necessarily cyclic in A. Let us understand this problem better. Let us look at Eq. (1), but from the right to the left. If f, on the right,  is a positive element of the algebra A, then (1) defines a positive functional. What are the simplest possible positive elements of A? The answer is: they are orthogonal projections p=p*=p². So, what happens if we take f = p = (1+n)/2? Since p = p* = p², it follows that f^½ = p. In that case L(A)f^½ = Ap is our left ideal of Feed 1. It is certainly not the whole A - for this choice of f. Thus, in this case
f^½  is not cyclic in A. But it is cyclic in the left ideal Af^½ ! Define H'_f = L(A)f^½,
Ω'_f = f^½, ρ'_f = L restricted to H'_f. Then, according to Proposition 4.5.3,  there exists an isometry U from H_f to H'_f, that maps  Ω_f  into
f^½, and U intertwines the two representations:

Uρ_f(a) = L(a)U   on H_f.

The GNS construction (Feed 2) is not really needed in our case. The point is that GNS construction is quite general, it works with any C*-algebra. Here we have a particular case, when our algebra is already a Hilbert space (perhaps too big), so we can apply Feed 3.

We have already made a connection to Feed 1 by taking f = p, but this needs to be made clear. We will do it in the next post.

As for spinors - they are lurking, always nearby, but still not as main characters.

P.S. 04-02-25 From my correspondence: "... Allais' effect only worked when his pendulum was triggered by a man. Allais' effect was officially recognized by the French academy of sciences."

Interesting. Quoting from Wikipedia

https://en.wikipedia.org/wiki/Allais_effect

Allais' paraconical pendulum

Maurice Allais states that the eclipse effect is related to a gravitational anomaly that is inexplicable in the framework of the currently admitted theory of gravitation, without giving any explanation of his own.^[30] Allais's explanation for another anomaly (the lunisolar periodicity in variations of the azimuth of a pendulum) is that space evinces certain anisotropic characteristics, which he ascribes to motion through an aether which is partially entrained by planetary bodies.

His hypothesis leads to a speed of light dependent on the moving direction with respect to a terrestrial observer, since the Earth moves within the aether but the rotation of the Moon induces a "wind" of about 8 km/s. Thus Allais rejects Einstein's interpretation of the Michelson–Morley experiment and the subsequent verification experiments of Dayton Miller.^[31][32]

Spin Chronicles Part 42: GNS Promenade

 Promenade: /ˌprɑm·əˈneɪd, -ˈnɑd/ to walk slowly along a street or path, usually where you can be seen by many people, for relaxation and pleasure. (Definition of promenade from the Cambridge Academic Content Dictionary.

In this post I am taking my Readers for a walk along the GNS construction that we have discussed in a number of previous posts Part 36-Part 41.

It will be a lazy promenade. Walking is important for philosophers. From Five philosophers on the joys of walking

"For Rousseau, the great benefit of walking is that you can move at your own time, doing as much or as little as you choose. You can see the country you’re traveling through, turn off to the right or left if you fancy, examine anything which interests you."

"Santayana points out that plants cannot move, whilst animals can.(...) Moving around allows animals to experience more of the world, to imagine how it might be elsewhere."

You can see the country you’re traveling through,
turn off to the right or left if you fancy,
examine anything which interests you

While walking we may recognize shapes and forms that are somewhat familiar to us, but there is always something new that enriches our knowledge and understanding of the reality. During the promenade we may notice other people, who look and dress differently. We were discussing the GNS construction. So let's see how it is being treated by mathematicians in an advanced textbook. The particular textbook I am inviting you to is "Fundamentals of the Theory of Operator Algebras, Volume 1", by Richard V. Kadison and John R. Ringrose, Academic Press 1983. It is a very nice book.

Operator algebras on promenade

It starts with Linear Spaces in Chapter 1, then, in Chapter 2, we have Basics of Hilbert Space and Linear Operators. Chapter 3 treats Banach Algebras. Chapter 4 discusses Elementary C*-Algebra Theory. In Ch. 4.5, p. 275,  we finally find the landscape that we are looking for "States and representations". On p. 277 we find, in Proposition 4.51,  something that looks familiar:

So, let's take a look.

Here the letter ρ is used for state. We used letter f. Our algebra A is denoted by a symbol like 𝒜, that i don't know how to reproduce using available fonts. Then we have a left ideal L_ρ, that we have denoted I_f in Part 40.

So, it is just a different notation, but nothing new for us. Of course the book has an adult C*-algebra, that is it works also for infinite-dimensional algebras, while we were playing with a finite-dimensional toy.

Let us walk a little bit further, on the next page we find Theorem 4.5.2 - the GNS construction. 

We see the term "cyclic representation". That means *-representation with a cyclic vector. We know what a cyclic vector is. We checked that "our" representation is cyclic in  Part 40. The book denotes the representation π_ρ, we called it ρ_f. The book calls the Hilbert space ℋ_ρ, we called it H_f. The book calls the cyclic vector xρ, we called it  Ω_f . Not a big deal. Finally, the last formula on the theorem is similar to our

(Ω, ρ(a)Ω) = f(a),

but the book has a different order under scalar product. That is because in the book the scalar product is assumed to be linear in the first argument and anti-linear in the second (the convention used by mathematicians), while our scalar product is anti-linear in the first argument and linear in the second (the convention used by physicists). Easy to remember.

So, thus far, nothing really new, just a different dress. Let us therefore move to the next picture at the exhibition. On p. 279 we meet Proposition 4.5.3:

This is something new. We need to stop here and examine it. So let us rewrite it using our notation. A is our finite-dimensional *-algebra. f is a state on A, ρ is a GNS cyclic representation of A on a Hilbert space H, with cyclic vector Ω, such that

f(a) = (Ω,ρ(a)Ω).                            (1)

Then, suppose we have some ρ',H',Ω' with the same properties. Then, the theorem states:

There is an isomorphism of Hilbert spaces U, from H to H', such that

UΩ = Ω',                                    (2)

and

ρ'(a) = Uρ(a)U^-1                        (3)

for all a in A.

The book uses U* instead of U^-1, but it has the same meaning.

This is a new result for us, and it needs a proof. It says that the GNS construction is essentially unique, up to a unitary equivalence. So, let us see about the proof. I will follow the reasoning in the book, with comments.

We start with establishing the equality that will be used to construct U. The equality is

||ρ(a)Ω|| = ||ρ'(a)Ω'||   for all a in A.        (4)

On the left we have the norm in H, on the right we have the norm in H'. To see that the equality indeed holds we use (1), which by assumption holds for ρ,H,Ω and for ρ',H',Ω':

||ρ(a)Ω||² = (ρ(a)Ω,ρ(a)Ω) = (Ω,ρ(a)*ρ(a)Ω) = (Ω,ρ(a*)ρ(a)Ω) = (Ω,ρ(a*a)Ω) = f(a*a),
||ρ'(a)Ω'||² = (ρ'(a)Ω',ρ'(a)Ω') = (Ω',ρ'(a)*ρ'(a)Ω') = (Ω',ρ'(a*)ρ'(a)Ω') = (Ω',ρ'(a*a)Ω') = f(a*a).

So, we have (4). We will now define U, a map from H to H'. We choose a vector in H. Since Ω is cyclic for ρ in H, there is an element a in A such that our chosen vector is of the form  ρ(a)Ω. In general there will be more than one such an "a", so we will need to be careful. We define U acting on ρ(a)Ω to be, by definition ρ'(a)Ω'.

Uρ(a)Ω = ρ'(a)Ω'.

Now, does it define a map from H to H'? What if ρ(a)Ω=ρ(b)Ω? Well, then we have
0=ρ(a)Ω-ρ(b)Ω=ρ(a-b)Ω.

But  then, using (4), 0= ||ρ(a-b)Ω||=||ρ'(a-b)Ω'||=||ρ'(a)Ω'-ρ'(b)Ω'||, so that
ρ'(a)Ω'=ρ'(b)Ω'.

Therefore U is well defined, and it evidently maps H onto H' (Why?). The map U is evidently linear (Why?), and it is an isometry (Why?). Moreover, we have UΩ = Ω' (Why?), as required. Finally for any a,b in A we have

Uρ(a)ρ(b)Ω = Uρ(ab)Ω = ρ'(ab)Ω' = ρ'(a)ρ'(b)Ω' =ρ'(a)Uρ(b)Ω,

and since vectors ρ(b)Ω span the whole H, we have

Uρ(a) = ρ'(a)U, or

Uρ(a)U^-1 = ρ'(a),

QED.

We are done. We will use this result in the next post, where we will compare the GNS construction with that of Part 41. We can have a real promenade now.

  
Pictures at an Exhibition is a piano suite in ten movements, plus a recurring and varied Promenade theme, written in 1874 by Russian composer Modest Mussorgsky. 1874 My dear généralissime, Hartmann is boiling as Boris boiled—sounds and ideas hung in the air, I am gulping and overeating, and can barely manage to scribble them on paper. I am writing the 4th No.—the transitions are good (on the 'promenade'). I want to work more quickly and steadily. My physiognomy can be seen in the interludes. So far I think it's well turned ..