Clothes for the Standard Model Beggar

WARNING: If algebra and geometry caused you headaches in school, this post could bring back suppressed memories.

According to neuroscience, the left brain is responsible for understanding algebra while geometry is a right brain pursuit. (Image right, via flickr, courtesy of vaXzine.) However, algebra and geometry represent the same math, but disguised in a different languages. Take any line in a plane, for instance: Its points obey a linear equation. Similarly, the intersection of two lines is given by solving two linear equations with two unknowns. Or any half plane can be codified as a linear inequality. And so on and so forth.

At this point, you may well be asking, who cares? I argue that it turns out that this algebraic-geometric duality is at the core of understanding quantum mechanics, the Standard Model, and even this year's FQXi essay question: if nature is digital or analog. (I haven't entered this year's contest, so I am not trying to promote my entry!)

So, if you can suppress the bad memories of algebra and geometry at school, please read on:

Let's start the story with quantum mechanics. Solving Schr枚dinger's equation is a central part of any quantum mechanics introductory class. In there one learns that the wavefunction lives in a so-called Hilbert space. What is not immediately apparent is that the Hilbert space is a very simplistic mathematical object used to codify continuity and whose main characteristic is its dimensionality. Far richer mathematical objects are operator algebras. And in fact, quantum mechanics can be axiomatized in the algebraic framework of the so-called C* algebras. Then one recovers the Hilbert space by a now standard method called the GNS construction.

In technical terms, by a theorem due to Gel'fand, (the G in GNS) any compact space X can be characterized by a C* algebra C(X) of continuous functions on X. This is nothing but the generalization of the mapping between analog properties of geometry/topology and digital properties of algebra.

It is interesting to see how the duality between geometry and algebra holds in the case of stranger geometries where notions of points, lines, and planes do not occur. Wait a minute, you may ask, what geometries are those where points, lines and planes do not make sense?

Here are some examples. I hope the reader is aware of Penrose's non-periodic tiling of the plane. A beautiful example is a Julia set. (Image right, courtesy of Simpsons contributor.)

There are also other non-glamorous examples from quantum mechanics: Alain Connes developed an entire branch of mathematics called non-commutative geometry to solve this algebra-geometry duality mapping in the most general sense, including all pathological cases. And it turns out that the strange cases correspond to non-commutative algebras, or algebras where a*b is not equal with b*a. Therefore the geometry side was aptly named: non-commutative geometry.

Fine then, this is what mathematicians do: they unify separated areas of math in a hopelessly incomprehensible mathematical formalism which maybe our grand-grand kids will understand 300-400 years from now. (That may sound harsh, but we still have problems today with learning calculus developed by Newton and Leibniz quite a while back.)

But this is not only pure math, as it has dramatic physical consequences. So let's go back to quantum mechanics in its C* algebraic formalism. One can further go down the algebraic road by studying physical symmetries, and a C* algebra of operators defined by symmetries is called a von Neumann algebra.聽 They have very useful properties and their complete classification was obtained by mathematicians, including Alain Connes. (This classification contains three basic classes, boringly named factors of Type I, II, and III.) Commutative von Neumann algebras correspond to defining a unique measure space on compact spaces. Calculus works because of this and in turn it allows us to easily compute things which we can then compare with experiments. Loosely speaking, the type I von Neumann algebras correspond to classical measure theory. Type II algebras showcase a new behavior of continuous dimension spanning a continuum of values. Gone are the geometry of lines, planes, and discrete n-dimensional subspaces. Type III is stranger still.

So now let's go back in time a few decades ago and pretend you are Alain Connes. You know quantum mechanics very well and you also have this new non-commutative geometry tool at your disposal. From non-commutative geometry it is clear that there is an entire gamut of options between pure discrete/digital and 100% continuous/analog. Moreover the Standard Model of particle physics, while agreeing with experiments, lacked a coherent mathematical description, but the core symmetries U(1), SU(2), SU(3) have a quantum mechanical origin. What would you do at this point?

In Alain Connes' words, the Standard Model was a "beggar" in search for mathematical "clothes". Why not then try to get a "best fit" of non-commutative geometry objects which can recover all mathematical properties of the Standard Model? This is a mathematical rewrite in a new language. If you have a hammer, everything looks like a nail says an old adage. But it turns out that this was an inspired guess and the new language is a much more economical way of characterizing Standard Model, and the theory appears naturally in this formalism.

But how can we describe nature in non-commutative geometry terms? To characterize a geometry, Connes uses a so-called spectral triple (A, H, D) where A is the algebra, H is a Hilbert space, and D is a self-adjoined operator. The way to understand it, is that the triple (A, H, D) codifies completely the geometrical distance information in an equivalent way. The details are too technical to explain in a short blog post, but the remarkable fact emerging from this work is the natural unification of the three forces and moreover unification with general relativity as well. There is also a striking similarity with string theory (although this is by no means string theory).

For the Standard Model, in the words of Alain Connes from a Fields lecture:

"I will show that the first five axioms I had given in 96 on spectral triples suffice in the commutative case to characterize smooth compact manifolds. I will also define a new invariant in Riemannian geometry, which when combined with the spectrum of the Dirac operator is a complete invariant of the geometry. It is an analogue of the CKM mixing matrix of the Standard model."

Also: "What we find is a geometric space which is neither a continuum nor a discrete space but a mixture of both. This space is the product of the ordinary continuum by a discrete space with only two points, which, for reasons which will become clear later, we shall call L and R. [...] The naive picture that emerges is that of a double space-time, i.e. the product of ordinary space-time by a very tiny discrete two-point space. By construction, purely left-handed particles such as neutrinos live on the left-handed copy XL while electrons involve both XL and XR in X = XL or聽 XR."

You may now be thinking that this sounds too good to be true. At this time is worth noting that this is still a work in progress, as a recent attempt of predicting Higgs' mass was ruled out by experiments. It still requires some "tailoring of the clothes", and it most likely needs to incorporate supersymmetry. However, for all its successes, the non-commutative Standard Model does not have many followers. I think the reason is twofold. First, let me say that even with advanced geometry knowledge, the math there is not for the faint of heart. Second, theoretical physicists tend to view the pursuit of algebraic methods as a useless crossing of the t(s) and dotting of the i(s) exercise, removed from practical questions.

It is very unfortunate that not many physicists understand Alain Connes' results. But this work is of the highest caliber on par with the best mathematical results from history. And the lesson for the nature of reality: analog vs. digital is clear: nature is both, and particle chirality plays a major role in the answer.

Still, there is hope for the rest of us to further contribute here. Non-commutative geometry is an excellent tool for understanding the Standard Model, but the clothes are not yet finished, and the question of why those particular A, H, D are selected by nature is not answered.