The geometry of particle physics

Alan Kuhn's non-commutative geometry suggests an alternative to string theory. In fact, since it can be tested directly, it may be superior to string theory

If there is a mathematician eagerly awaiting the start-up of the Large Hadron Collider (LHC) near Geneva next year, it is Alain Connes of the Collège de France in Paris. Kuhn, like many physicists, hopes the Higgs boson will show up in detectors. The Higgs is the missing jewel in the crown of the "Standard Model": the theoretical framework that describes subatomic particles and their interactions. For Kuhn, the discovery of the Higgs, which, according to the hypothesis, gives mass to the other particles, has a decisive meaning: its existence, and even its mass, derive from equations known only to those who know Han, and belong to a new mathematical branch known as "non-commutative geometry", of which Kuhn is the main inventor.

Kuhn's idea was to extend the connection between the geometric space on his commutative Cartesian coordinate algebra, such as longitudes and latitudes, and geometry based on non-commutative algebra. In commutative algebra, the product does not depend on the order of the factors: 3X5 = 5X3 but there are some non-commutative operations.

Take for example a stunt plane that is able to roll (rotate on its longitudinal axis) and perform a roll (rotate around an axis parallel to the wings). Imagine a pilot picking up a radio transmitter instructing him to roll 90 degrees and then make a 90 degree yaw towards the belly of the plane. Everything will be fine if the pilot follows the instructions properly. But if the order is reversed, the plane will dive down. Operations in Cartesian coordinates in space are commutative, but rotations in three dimensions are not commutative.

To get a clearer picture of what is happening in nature, physicists occasionally turn to the "phase space". Such a space serves as an alternative to Cartesian coordinates - a researcher can determine that its axes represent the electron's position and momentum, instead of simply specifying its x and y coordinates. Because of Heisenberg's uncertainty principle, it is impossible to measure these two properties at the same time. As a result, the product of the position times the momentum is not equal to the product of the momentum times the position. Therefore the quantum phase space is not commutative. Moreover, introducing such non-commutativity into the regular space - for example, by making the coordinates x and y non-commutative - creates a space with a non-commutative geometry.

Following this type of analysis, Kuhn discovered the strange properties of his new geometry, properties that obey the principles of quantum theory. He spent thirty years perfecting his thoughts, and although he laid the foundations in a book he wrote back in 1994, scholars still flock to hear his lectures. On a windy and rainy day typical of the month of March, about 60 of the most senior French researchers of mathematics gather in hall number 5 of the Collège de France. 59-year-old Cohn paces back and forth between two spotlights hanging above him, talking rapidly, constantly switching slides loaded with equations. Outside, police sirens wail in front of student protesters trying to break into the nearby Sorbonne University in response to the new employment law proposed by the French government.

Kuhn didn't notice the commotion

Kuhn seems oblivious to the commotion - even afterwards, as he crosses rue Saint-Jacques and passes the blue police cars and riot police, he continues to describe how his research led him to new insights into physics. As an example, Kuhn gives the way in which particle physics developed: the concept of space-time is derived from electrodynamics, but electrodynamics is only a small part of the standard model. New particles were added as needed, and confirmation of their existence came when those particles, which were a prediction, appeared in accelerators.

But the space-time used in general relativity, which was also based on electrodynamics, remained unchanged. Kuhn proposed something completely different: instead of new particles, he developed a more sophisticated geometry, and the subtleties in which produce these new particles. In fact, he succeeded in creating a non-commutative space that contains all the abstract algebras (known as "symmetry bundles") that describe the properties of the elementary particles in the standard model.

The picture that emerges from the standard model is therefore a picture of space-time as a non-commutative space that can be seen as consisting of two layers of continuity, like the two sides of a sheet of paper. The space between the two sides of the paper is an additional space, discrete (discontinuous) and non-commutative. The discrete part forms the Higgs, while the continuous parts form the calibration particles, such as the W and Z particles, through which the weak force is transmitted.

Kuhn was convinced that not only did the physical calculations reflect reality, but also that behind their seemingly complex appearance hidden mathematical gems. All that is required is a tool that can peer into the complexity, in the same way that an electron microscope reveals molecular structure. Kuhn says that his "electron microscope" is the non-commutative geometry. "What really interests me are the complicated calculations that physicists perform and that are tested experimentally," he declares. "These calculations are checked up to nine digits after the point, so you can be confident that we will come across a gem, that is, something that requires clarification."

One of the gems had infinite hidden within it. Although the standard model turned out to be a resounding success, he quickly encountered a setback: infinite values ​​appeared in many of his calculations. Physicists, including Gerard T'Hoft and Martinus Flatman from the University of Utrecht in the Netherlands, solved the problem by introducing a new mathematical method known as "renormalization". By adjusting certain values ​​in the model, the physicists were able to avoid these infinities and calculate particle properties that are consistent with reality.

Degree of cheating?

Although some researchers saw this technique as a certain amount of cheating, for Kuhn the renormalization became another opportunity to explore the space in which physics resides. But it was not easy. "I spent 20 years trying to understand renormalization. It's not that I didn't understand what the physicists did, but that I didn't understand what the mathematics behind it meant," says Kuhn. He and the physicist Dirk Cramer from the Institute for Higher Science Studies near Paris soon realized that the recipe for canceling infinities was actually related to one of the 23 great problems in mathematics, formulated by David Hilbert in 1900, a problem that was already on its way to being solved. The link gave normalization a rigorous mathematical foundation - it is no longer "cheating".

The connection between renormalization and non-commutative geometry serves as a starting point for the unification of relativity with quantum mechanics, and therefore for a complete description of gravity. "We must now move to the next step - we must try to understand how a space with broken dimensions," which appears in non-commutative geometry, "communicates with gravity," Kohn claims. He had already shown, in collaboration with the physicist Carlo Rubelli of the University of Marseille, that the concept of time could arise naturally from the non-commutativity of the observed magnitudes of gravity. Time can be compared to a property such as temperature, which needs atoms to exist, Robelli explains.

And what about string theory? Doesn't it unify gravity and the quantum world? Kuhn claims that his approach, which seeks the mathematics behind the physical phenomenon, is fundamentally different from the approach of the string theorists. Because string theory cannot be tested directly - it deals with energies that cannot be produced in a laboratory - whereas non-commutative geometry yields predictions that can be put to the test, such as the Higgs mass (160 billion electron volts), and according to him, even the renormalization can be verified.

The Large Hadron Collider will not only test Kuhn's mathematics, but also generate data for him that will allow him to extend his work to smaller scales. "Non-commutative geometry now equips us with a model of space-time that goes down to 10-16 centimeters," Kuhn says. That's not even halfway. But to Kuhn, there is no doubt that the glass seems half full.