1a. Classical geometries

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We would like to develop some noncommutative geometry theory, that can be of help in quantum mechanics, a bit as classical geometry is of help in classical mechanics. So, this will be a book about mathematical physics. With mathematical physics meaning, as usual, mathematics developed with physics motivations in mind.

Before anything, let us recommend some reading. Physics and quantum mechanics in particular can be learned from the books of Feynman , or Griffiths , , or Weinberg , . For quantum mechanics, a look at the old books of Dirac , von Neumann and Weyl can be instructive too. Also, never forget that physics is a whole, and do not hesitate to complete your electrodynamics and quantum mechanics knowledge with some solid classical mechanics, and some thermodynamics too.

Back to our goals, noncommutative geometry, a look at all this physics does not help that much. There are certainly a few things to be learned, as for instance the fact that noncommutative geometry should have something to do with the linear operators [math]T:H\to H[/math] over a complex Hilbert space [math]H[/math]. But passed that, we are a bit in the dark. As an example, the Hilbert space [math]H[/math] used to be something quite abstract, [math]H=l^2(\mathbb N)[/math], for Heisenberg, then something more concrete, [math]H=L^2(\mathbb R^3)[/math], for Schrödinger, and then Pauli and others added a copy of [math]K=\mathbb C^2[/math], in order to take into account the spin of the electron. And this was only what happened in the 1920s, and there is no telling of what happened afterwards, up to the present days. It is probably safe to say that no one really knows what [math]H[/math] is. And even worse, no one really knows if there is one such [math]H[/math] at all.

The same story goes with the linear operators [math]T:H\to H[/math]. These used to be densely defined and unbounded, during the good old days of Heisenberg, Schrödinger and Dirac. But then, with quantum mechanics evolving into quantum electrodynamics, then into the Standard Model, and then into more technical versions of quantum field theory, such operators often became everywhere defined, and bounded, [math]T\in B(H)[/math]. And there is even worse, because at the truly advanced level you often manage to find a way to deal with usual complex matrices, [math]T\in M_N(\mathbb C)[/math], or sometimes with random matrices.

Summarizing, all this looks complicated, and my proposal would be to leave physics for later, although please have a look at the above-mentioned physics books, because what we will be doing here, and I insist, is mathematical physics, and not pure mathematics, and you won't understand otherwise, and have some inspiration from pure mathematics instead. We have for instance the following question, that we can try to solve: \begin{question} What are the noncommutative analogues of the geometry of [math]\mathbb R^N[/math], and of the geometry of [math]\mathbb C^N[/math]? \end{question} Here by “noncommutative” we mean with the standard coordinates not commuting, [math]x_ix_j\neq x_jx_i[/math]. But this is something a bit vague, because shall we look here for some weakenings of the commutation relations [math]x_ix_j=x_jx_i[/math], and as we will soon see, there are plently of interesting choices here, or shall we just look, for simplifying and to start with, for “free” geometries, where [math]x_i,x_j[/math] are not subject to any kind of relation.

As for “geometry”, things are quite vague here as well. We have indeed algebraic geometry, differential geometry, symplectic geometry, Riemannian geometry, and many more. Also, when talking [math]\mathbb C^N[/math], our manifolds can be taken to be real, or complex. And also, all these geometries usually come in two flavors, affine and projective.

In short, we have an idea with our Question 1.1, but everything is still too vague. So here we are back to physics, and quantum mechanics, for inspiration, and based on some knowledge there, let us formulate the following fact: \begin{fact} The spaces [math]\mathbb R^N[/math] and [math]\mathbb C^N[/math] have no interesting free analogues, and this due to the fact that these spaces are not compact. \end{fact} Obviously, this is something subjective. My point comes from the fact that, while you can certainly talk about the free algebra generated by variables [math]x_1,\ldots,x_N[/math], with some care of course in relation with conjugation and the complex structure, there is no way of putting a reasonable norm on this algebra, due to the fact that [math]x_1,\ldots,x_N[/math] are unbounded. And so this algebra is just some abstract, pure mathematical beast, totally unrelated to analysis, physics, probability, quantum mechanics, you name it.

In short, you'll have to trust me here. And we will talk about this more in detail later in this chapter, when learning about operator algebras. By the way, let me point out that Question 1.1 was something controversial too, because we assumed somehow by definition that the interesting fields are [math]F=\mathbb R,\mathbb C[/math]. Which is far from being something accepted, with many mathematical physicists, myself included, agreeing that, at a more advanced level, other fields than [math]F=\mathbb R,\mathbb C[/math] are interesting too. But this is another story.

Going ahead with our study, Fact 1.2 suggests replacing [math]F^N=\mathbb R^N,\mathbb C^N[/math] with a suitably chosen compact manifold [math]X\subset F^N[/math]. And here, we have several choices. The first thought goes to the unit sphere [math]S\subset F^N[/math]. But then, why not looking instead at the unit torus [math]T\subset F^N[/math], which in addition is a group. And then, talking now groups, why not being a bit advanced, and looking instead at the unitary group [math]U\subset\mathcal L(F^N)[/math]. And then why not, for being even more advanced, looking at the reflection group [math]K\subset\mathcal L(F^N)[/math].

These choices are all reasonable, with the mathematics of each of [math]S,T,U,K[/math] being well-known to be interesting, and describing a bit of the mathematics of [math]F^N[/math] itself. So, why not putting all these objects [math]S,T,U,K[/math] together, as to have a complete, robust object replacing [math]F^N[/math]. We are led in this way to the following answer to Question 1.1: \begin{answer} An affine noncommutative geometry should come from a diagram

[[math]] \xymatrix@R=50pt@C=50pt{ S\ar[r]\ar[d]\ar[dr]&T\ar[l]\ar[d]\ar[dl]\\ U\ar[u]\ar[ur]\ar[r]&K\ar[l]\ar[ul]\ar[u] } [[/math]]

consisting of a sphere [math]S[/math], a torus [math]T[/math], a unitary group [math]U[/math], and a reflection group [math]K[/math]. \end{answer} So, this will be our starting point, for the considerations in this book, and our guiding philosophy, in what follows, until the end. However, obviously, this is something quite advanced, and before starting our study, a few comments, and some advice:

(1) As already said, and above everything, this is something advanced. The above answer emerged in the late 10s, based on substantial work, a few dozen research papers, written all over the 90s, 00s, 10s. So, modesty, and it will take us a few 100 pages in this book, or perhaps the whole book, to understand what Answer 1.3 really says.

(2) Also importantly, Answer 1.3 talks about “noncommutative geometries” in general, and not about manifolds constructed inside such geometries. We will first try to understand these noncommutative geometries themselves, via some axiomatization and classification work. And the study of the corresponding manifolds will come after.

(3) Looking now at Answer 1.3 as it is, that sounds like some kind of pure mathematics, for the most involving algebraic geometry and Lie groups. Although not really necessary for reading this book, some knowledge here would be welcome. You can learn these for instance from Shafarevich and Fulton-Harris , respectively.

(4) In fact, we will see in a moment that what is really needed for understanding Answer 1.3 is rather basic analysis. So, getting now to the true prerequisites for the present book, these will be Rudin . With perhaps a bit of familiarity with algebra too, say from Lang , and a bit of functional analysis too, say from Lax .

Getting started now, let us first discuss the case of the usual geometry, in [math]\mathbb R^N[/math]. We must construct here the corresponding quadruplet [math](S,T,U,K)[/math], as per the requirements of Answer 1.3, and the definitions here, all very familiar, are as follows:

Definition

The real sphere, torus, unitary group and reflection group are:

[[math]] \begin{eqnarray*} S^{N-1}_\mathbb R&=&\left\{x\in\mathbb R^N\Big|\sum_ix_i^2=1\right\}\\ T_N&=&\left\{x\in\mathbb R^N\Big|x_i=\pm\frac{1}{\sqrt{N}}\right\}\\ O_N&=&\left\{U\in M_N(\mathbb R)\Big|U^t=U^{-1}\right\}\\ H_N&=&\left\{U\in M_N(-1,0,1)\Big|U^t=U^{-1}\right\} \end{eqnarray*} [[/math]]

These are the usual sphere, cube, orthogonal group, and hyperoctahedral group.

To be more precise here, all the objects on the right are certainly familiar, but the notations and terminology for them are perhaps not, and here are the details:

(1) The sphere [math]S^{N-1}_\mathbb R[/math] is the unit sphere of [math]\mathbb R^N[/math] as we know it, and we will often say sphere instead of unit sphere. As for the superscript [math]N-1[/math], which is very standard, this stands for the real dimension as a manifold, which is indeed [math]N-1[/math].

(2) Regarding [math]T_N[/math], this is the standard cube in [math]\mathbb R^N[/math], with the [math]1/\sqrt{N}[/math] normalization being there in order to have an embedding [math]T_N\subset S^{N-1}_\mathbb R[/math], which will be useful for us. We also call [math]T_N[/math] torus, standing for “discrete torus”, and more on this later.

(3) Regarding now [math]O_N[/math], this is the orthogonal group as we know it. We also call it unitary group of [math]\mathbb R^N[/math], because, a bit as for the cube/torus before, we are using here in this book a hybrid real/complex terminology for everything. More on this later.

(4) Finally, regarding [math]H_N[/math], this is the hyperoctahedral group, which is by definition the symmetry group of the hypercube in [math]\mathbb R^N[/math], which means our cube/torus [math]T_N[/math]. The formula for [math]H_N[/math] in the statement is something elementary, coming from definitions.

Regarding now the correspondences between our objects, there are many ways of establishing them, depending on knowledge and taste. Assuming that you followed my advice, and that you are a bit familiar with basic algebraic geometry and Lie theory, you would probably say that our objects [math](S,T,U,K)[/math] are trivially in correspondence with each other, QED. On the opposite, assuming that you are not familiar with all this, and that your mathematical background is basically the officially needed Rudin , along with a bit of Lang and Lax , you might have a bit of troubles with all this.

Well, good news, establishing the correspondences for [math]\mathbb R^N[/math] is actually not crucial for us, at this point. And with this coming from the fact that, no matter what things can be said about [math]\mathbb R^N[/math], we will be doing noncommutative geometry in this book, and in the noncommutative setting things are far more rigid, and so the correspondences between [math](S,T,U,K)[/math], even for [math]\mathbb R^N[/math], are to be discussed later, once we know what we're doing.

So, in the hope that you got my point. We are just having some preliminary fun, and we need to know, for getting started, that we are on the right track, and that our objects [math](S,T,U,K)[/math] from Definition 1.4 are indeed in correspondence, be that a bit informal. So here is the statement, formulated informally, and coming with an informal proof, and with this, being informal, being, and I insist, the right thing to do, right now:

Theorem

We have a full set of correspondences, as follows,

[[math]] \xymatrix@R=50pt@C=50pt{ S^{N-1}_\mathbb R\ar[r]\ar[d]\ar[dr]&T_N\ar[l]\ar[d]\ar[dl]\\ O_N\ar[u]\ar[ur]\ar[r]&H_N\ar[l]\ar[ul]\ar[u] } [[/math]]

obtained via various results from basic geometry and group theory.

Show Proof

As already mentioned, there are several possible solutions to the problem, and all this is not crucial for us. Here is a way of constructing these correspondences:

(1) [math]S^{N-1}_\mathbb R\leftrightarrow T_N[/math]. Here [math]T_N[/math] comes from [math]S^{N-1}_\mathbb R[/math] via [math]|x_1|=\ldots=|x_N|[/math], while [math]S^{N-1}_\mathbb R[/math] appears from [math]T_N\subset\mathbb R^N[/math] by “deleting” this relation, while still keeping [math]\sum_ix_i^2=1[/math].

(2) [math]S^{N-1}_\mathbb R\leftrightarrow O_N[/math]. This comes from the fact that [math]O_N[/math] is the isometry group of [math]S^{N-1}_\mathbb R[/math], and that, conversely, [math]S^{N-1}_\mathbb R[/math] appears as [math]\{Ux|U\in O_N\}[/math], where [math]x=(1,0,\ldots,0)[/math].

(3) [math]S^{N-1}_\mathbb R\leftrightarrow H_N[/math]. This is something trickier, but the passage can definitely be obtained, for instance via [math]T_N[/math], by using the constructions in (1) above and (5) below.

(4) [math]T_N\leftrightarrow O_N[/math]. Here [math]T_N\simeq\mathbb Z_2^N[/math] is a maximal torus of [math]O_N[/math], and the group [math]O_N[/math] itself can be reconstructed from this maximal torus, by using various methods.

(5) [math]T_N\leftrightarrow H_N[/math]. Here, similarly, [math]T_N\simeq\mathbb Z_2^N[/math] is a maximal torus of [math]H_N[/math], and the group [math]H_N[/math] itself can be reconstructed from this torus as a wreath product, [math]H_N=T_N\wr S_N[/math].

(6) [math]O_N\leftrightarrow H_N[/math]. This is once again something trickier, but the passage can definitely be obtained, for instance via [math]T_N[/math], by using the constructions in (4) and (5) above.

■

The above result is of course something quite non-trivial, and having it understood properly would take some time. However, as already said, we will technically not need all this. Our purpose for the moment is just to explain our [math](S,T,U,K)[/math] philosophy.

As a second basic example of geometry, we have the usual geometry of [math]\mathbb C^N[/math]. Here the corresponding quadruplet [math](S,T,U,K)[/math] can be constructed as follows:

Definition

The complex sphere, torus, unitary group and reflection group are:

[[math]] \begin{eqnarray*} S^{N-1}_\mathbb C&=&\left\{x\in\mathbb C^N\Big|\sum_i|x_i|^2=1\right\}\\ \mathbb T_N&=&\left\{x\in\mathbb C^N\Big|\ |x_i|=\frac{1}{\sqrt{N}}\right\}\\ U_N&=&\left\{U\in M_N(\mathbb C)\Big|U^*=U^{-1}\right\}\\ K_N&=&\left\{U\in M_N(\mathbb T\cup\{0\})\Big|U^*=U^{-1}\right\} \end{eqnarray*} [[/math]]

These are the usual complex sphere, torus, unitary group, and complex reflection group.

As before, the superscript [math]N-1[/math] for the sphere does not fit with the rest, but is quite standard, somewhat coming from dimension considerations. We will use it as such. Also, the [math]1/\sqrt{N}[/math] factor is there in order to have an embedding [math]\mathbb T_N\subset S^{N-1}_\mathbb C[/math].

Also as before, in what regards the correspondences between our objects, there are many ways of establishing them, will all this being not crucial for us. In analogy with Theorem 1.5, let us formulate a second informal statement, as follows:

Theorem

We have a full set of correspondences, as follows,

[[math]] \xymatrix@R=50pt@C=50pt{ S^{N-1}_\mathbb C\ar[r]\ar[d]\ar[dr]&\mathbb T_N\ar[l]\ar[d]\ar[dl]\\ U_N\ar[u]\ar[ur]\ar[r]&K_N\ar[l]\ar[ul]\ar[u] } [[/math]]

obtained via various results from basic geometry and group theory.

Show Proof

We follow the proof in the real case, by making adjustments where needed, and with of course the reiterated comment that all this is not crucial for us:

(1) [math]S^{N-1}_\mathbb C\leftrightarrow\mathbb T_N[/math]. Same proof as before, using [math]|x_1|=\ldots=|x_N|[/math].

(2) [math]S^{N-1}_\mathbb C\leftrightarrow U_N[/math]. Here “isometry” must be taken in an affine complex sense.

(3) [math]S^{N-1}_\mathbb C\leftrightarrow K_N[/math]. Trickier as before, best viewed by passing via [math]\mathbb T_N[/math].

(4) [math]\mathbb T_N\leftrightarrow U_N[/math]. Coming from the fact that [math]\mathbb T_N\simeq\mathbb T^N[/math] is a maximal torus of [math]U_N[/math].

(5) [math]\mathbb T_N\leftrightarrow K_N[/math]. Once again, maximal torus argument, and [math]K_N=\mathbb T_N\wr S_N[/math].

(6) [math]U_N\leftrightarrow K_N[/math]. Trickier as before, best viewed by passing via [math]\mathbb T_N[/math].

■

As a conclusion, our [math](S,T,U,K)[/math] philosophy seems to work, in the sense that these 4 objects, and the relations between them, encode interesting facts about [math]\mathbb R^N,\mathbb C^N[/math]. Our plan in what follows will be that of leaving aside the complete understanding of what has been said above, and going directly for the noncommutative case. We will see that in the noncommutative setting things are more rigid, and therefore, simpler. And then, with this simpler theory axiomatized, we will come back of course to [math]\mathbb R^N,\mathbb C^N[/math], with full details about the correspondences between [math](S,T,U,K)[/math] here, don't worry about that.

General references

Banica, Teo (2024). "Affine noncommutative geometry". arXiv:2012.10973 [math.QA].