Archives - Mathematics: Page 15
Author: paul carson (Tue Aug 08, 2006 1:46 pm)
Title: The Geometer of Particle Physics
This article came from sciam.com and was posted here because of its importance to the fields of mathematics and physics.
The geometer of particle physics
Alain Connes’s noncommutative geometry offers an alternative to string theory. In fact, being directly testable, it may be better than string theory By ALEXANDER HELLEMANS
If there is a mathematician eagerly waiting for the Large Hadron Collider near Geneva to start up next year, it is Alain Connes of the College de France in Paris. Like many physicists, Connes hopes that the Higgs particle will show up in detectors. The Higgs is the still missing crowning piece of the so-called Standard Model-the theoretical framework that describes subatomic particles and their interactions. For Connes, the discovery of the Higgs, which supposedly endows the other particles with mass, is crucial: its existence, and even its mass, emerges from the arcane equations of a new form of mathematics called noncommutative geometry, of which he is the chief inventor.
Connes’s idea was to extend the relation between geometric space and its communicative algebra of Cartesian coordinates, such as latitude and longitude, to a geometry based on noncommutative algebra. In commutative algebra, the products is independent of the order of the factors: 3x5=5x3. but some operations are noncommutative. Take, for example, a stunt plane that can aggressively roll (rotate over the longitudinal axis) and pitch (rotate over an axis parallel to the wings). Assume a pilot receives radio instructions to roll over 90 degrees and then to pitch over 90 degrees toward the underside of the plane. Everything will be fine if the pilot follows the commands in that order. But if the order is inverted, the plane will take a nosedive. Operations with Cartesian coordinates in space are commutative, but rotations over three dimensions are not.
To gain a clearer vision of what goes on in nature, physicists sometimes resort to “phase space”. Such a space is an alternative to Cartesian coordinates – a researcher can plot the position of an electron against its momentum rather than simply its x and y locations. Because of the Heisenberg uncertainty principle, one cannot measure both quantities simultaneously. As a consequence, position times momentum does not equal momentum times position. Hence, the quantum phase space is noncommutative. Moreover, introducing such noncommutativity into an ordinary space – say, by making the x and the y coordinates noncommutative-produces a space that has noncommutative geometry.
Through such analyses, Connes discovered the peculiar properties of his new geometry, properties that corresponded to the principles of quantum theory. He has spent three decades refining his thinking, and even though he laid down the basics in a 1994 book, researches beat a path to listen to him. On a day plagued by typical March showers and wind, about 60 of the crème de la crème of French mathematicians fill Salle 5 at the Collége de France. Like a caged lion, the 59-year-old Connes walks quickly back and forth between two overhead projectors, talking rapidly continually replacing transparencies filled with equations. Outside, police sirens scream amid student protestors trying to occupy the Sorbonne next door in response to the French government’s proposed new employment law.
Cones seems oblivious to the commotion – even afterward, while crossing the rue Saint-Jacques past blue police vans and officers in riot gear, he keeps talking about how his research has led him to new insights into physics. As an example, Connes refers to the way particle physics has grown: The concept of spacetime was derived from electrodynamics, but electrodynamics is only a small part of the Standard Model. New particles were added when required, and confirmation came when these predicted particles emerged in accelerators.
But the spacetime used in general relativity, also based on electrodynamics, was left unchanged. Cones proposed something quite different: “Instead of having new particles, we have a geometry that is more subtle, and the refinements of this geometry generate these new particles.” In fact, he succeeded in creating a noncommutative space that contains all the abstract algebras (known as symmetry groups) that describe the properties of elementary particles in the Standard Model.
The picture that emerges from the Standard Model, then, is that of spacetime as a noncommutative space that can be viewed as consisting of two layers of a continuum, like the two sides of a piece of paper. The space between the two sides of the paper is an extra discrete (noncontinuous), noncommutative space. The discrete part creates the Higgs, whereas the continuum parts generate the gauge bosons, such as the W and Z particles, which mediate the weak force.
Cones has become convinced that physics calculations not only reflect reality but hide mathematical jewels behind their apparent complexity. All that is needed is a tool to peer into the complexity, the way the electron microscope reveals molecular structure, remarks Connes, whose “electron microscope” is noncommutative geometry. “what I’m really interested in are the complicated computations performed by physicists and tested by experiment”, he declares. “these calculations are tested at up to nine decimals, so one is certain to have come across a jewel, something to elucidate”.
One jewel held infinities. Although the Standard Model proved phenomenally successful, it quickly hit an obstacle: infinite values appeared in many computations. Physicists, including Gerard ’t Hooft and Martinus Veltman of the University of Utrecht in the Netherlands, solved this problem by introducing a mathematical technique called renormalization. By tweaking certain values in the models, physicists could avoid these infinities and calculate properties of particles that corresponded to reality.
Although some researchers viewed this techniques as a bit like cheating, for Connes renormalization became another opportunity to explore the space in which physics lives. But it wasn’t easy. “I spent 20 years trying to understand renormalization. Not that I didn’t understand what the physicists were doing, but I didn’t understand what the meaning of the mathematics was that was behind it”, Connes says. He and physicist Dirk Kreimer of the Institute des Hautes Études Scientifiques near Paris soon realized that the recipe to eliminate infinities is in fact linked to one of the 23 great problems in mathematics formulated by David Hilbert in 1900 – one that had been solved. The linkage gave renormalization a mathematically rigorous underpinning – no longer was it “cheating”.
The relation between renormalization and noncommutative geometry serves as a starting point to unite relativity and quantum mechanics and thereby fully describe gravity. “We now have to make a next step – we have to try to understand how space with fractional dimensions,” which occurs in noncommutative geometry, “couples with gravitation,” Connes asserts. He has already shown, with physicist Carlo Rovelli of the University of Marseille, that time can emerge naturally from the noncommutativity of the observable quantities of gravity. Time can be compared with a property such as temperature, which needs atoms to exist, Rovelli explains.
What about string theory? Doesn’t that unify gravitation and the quantum world? Connes contends that his approach, looking for the mathematics behind the physical phenomena, is fundamentally different from that of string theorists. Whereas string theory cannot be tested directly – it deals with energies that cannot be created in the laboratory – Connes points out that noncommutative geometry makes testable predictions, such as the Higgs mass (160 billion electron volts), and he argues that even renormalization can be verified.
The Large Hadron Collider will not only test Connes’s math but will also give him data to extend his work to smaller scale. “noncommutative geometry now supplies us with a model of spacetime that reaches down to 10-16 centimeter, which still is a long way to go to reach the Planck scale, which is 10-33 centimeter,” Connes says. That is not quite halfway. But to Connes, the glass undoubtedly appears half full.
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