Preface
Classical geometry has its origins in classical mechanics, with some of its most fundamental objects, such as the conics, coming from the trajectories of planets and other celestial objects around the Sun. Similarly, quantum mechanics has inspired several theories of quantum geometry, more commonly called “noncommutative geometry”.
The idea of noncommutative geometry goes back to Heisenberg. Back in the 1920s, the main problem in physics was that of understanding the mechanics of the hydrogen atom, and it was known since Bohr that the Maxwell equations do not work. Heisenberg came with a clever idea for solving the problem, namely looking for some sort of “quantum trajectory” for the electron, instead of a classical, honest trajectory, and with his mathematics involving the algebra [math]M_\infty(\mathbb C)[/math] of the complex infinite matrices.
A few years later Schrödinger came with something better, namely a PDE for the wave function of the electron. This improved Heisenberg's findings, with [math]M_\infty(\mathbb C)[/math] being now understood to correspond to the algebra of operators on the Hilbert space [math]H=L^2(\mathbb R^3)[/math] of such wave functions. Later, Pauli and others added a copy of [math]\mathbb C^2[/math] to this space, as to account for the electron spin, and in this form, that of the late 1920s, quantum mechanics was powerful enough for solving many questions, such as the structure of all atoms. Making Bohr's dream, who was the initiator of the whole program, come true.
Einstein disagreed with all this, saying that such things, probability, noncommutative geometry, you name it, while certainly great, should be regarded as being temporary, and so homework for us for going towards determinism, meaning true, honest geometry.
Generally speaking, modern physics is about making Einstein's dream come true. Quantum mechanics has evolved several times since the 1920s, with the fine structure of hydrogen, then with quantum electrodynamics, then with the discovery, at smaller scales, of quantum chromodynamics, then with the Standard Model, then with all sorts of efforts in general quantum field theory, and then with string theory, which is something geometric. Slowly but surely, we are going towards Einstein's determinism.
This being said, I don't know about you, but personally I'm still waiting for nuclear-powered cars, first for the unlimited horsepower, and then for not having to refuel. And also, why not for nuclear watches too, because every time the battery of my Casio gives up, a whole pain with replacing it. And shall we trust modern physics with coming up soon with concrete answers, to these very concrete life questions that we have.
Shall we perhaps downgrade a bit our dreams in physics? Noncommutative geometry, in its modern formulations, is a bit about this. Forget about Einstein's determinism, or rather leave that for later, and more modestly, try instead to have some sort of noncommutative geometry theory working, improving what Heisenberg was saying, and of course, with the whole thing being as close as possible to the modern advances in physics.
The credit for such ideas goes to Connes, who created in the 80s a noncommutative geometry theory which is definitely simple, beautiful, and modern too. Connes looked at the noncommutative manifolds [math]X[/math], with this meaning that [math]A=C(X)[/math] is an operator algebra, [math]A\subset B(H)[/math], which are smooth and Riemannian, in a certain technical sense, with remarkable results in connection with physics, obtained all over the 90s and 00s.
Our aim here is to talk about noncommutative geometry too, from a point of view very close to the one of Connes, but with slightly different motivations in mind. We will keep from Connes his two main principles, namely that the noncommutative manifolds [math]X[/math] should appear from operator algebras, and also, that they should be Riemannian. However, based on our belief that at very small scales, smaller than those of the Standard Model, there is no room for smoothness, we will ditch the assumption that [math]X[/math] should be smooth, and so Riemannian for us will rather mean that [math]X[/math] is real algebraic, a bit a la Nash, and coming with an integration functional [math]\int:C(X)\to\mathbb C[/math], that we will be eager to compute explicitly, using techniques of Jones, Voiculescu and Woronowicz.
This book will be purely mathematical. The applications to physics, involving some more mathematics, such as PDE over our free manifolds, will be hopefully discussed in a series of further books. As for nuclear cars and watches, I am currently trying to build some in my garage, for my personal usage, but things difficult here. More later.
This book is partly based on a number of recent papers on quantum groups and noncommutative geometry, and I am particularly grateful to Julien Bichon, for his heavy involvement in the subject. Many thanks go as well to my cats. Their timeless views and opinions, on everyone and everything, have always been of great help.
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Cergy, August 2024
Teo Banica \baselineskip=15.95pt \tableofcontents \baselineskip=14pt
General references
Banica, Teo (2024). "Affine noncommutative geometry". arXiv:2012.10973 [math.QA].