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 Coll¿ge 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 commutative algebra of Cartesian coordinates, such as latitude and longitude, to a geometry based on noncommutative algebras. In commutative algebra, the product is independent of the order of the factors: 3 x 5 = 5 x 3. 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.
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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, researchers beat a path to listen to him. On a day plagued by typical March showers and wind, about 60 of the cr¿me de la cr¿me of French mathematicians fill Salle 5 at the Coll¿ge 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.
To Connes, physics calculations not only reflect reality but hide mathematical jewels.
Connes 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. Connes 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.
