Mathematical Physics and Geometry

Physics and Geometry

November 2, 2004

Mathematical physics, one of whose pioneers was the mathematician Jean-Marie Souriau, proceeds through geometry. Throughout this approach, physical quantities such as energy, mass, momentum, spin, and electric charge become purely geometric entities, thanks to one key tool: the theory of groups. What is required to venture into this universe—or this way of perceiving the universe? Not much: the ability to manipulate matrices. If these objects are unfamiliar to you, make the effort to become acquainted with them—the reward is well worth it. If you've encountered this before, dust off your knowledge—it could take you far and help answer questions such as:

• What is the true nature of particle spin?

• What is antimatter?

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Coadjoint Action of the Poincaré Group
on Its Momentum Space

Warning: intended only for readers strongly oriented toward science. This is not scientific popularization.

October 24, 2004

Physics has always been closely linked to geometry. The mathematician Jean-Marie Souriau is one of the founders of mathematical physics, which proceeds through a highly elegant geometrization of physics. Everything is based on groups with real coefficients, such as the Lorentz group and the Poincaré group, which here are represented by real-coefficient matrices. What follows begins with a single matrix G, related to the metric of Minkowski space—the space of special relativity. Using this matrix, we define a first group L, represented by 4×4 matrices. This group acts on spacetime, composed of event points. From these matrices and a "spacetime translation vector" C, we construct a second group, represented by 5×5 matrices, which also acts on spacetime. In this spacetime, we consider "motions." The concept of trajectory is insufficient. The motion of a particle must be associated with quantities such as its energy E and momentum p. For a theoretical physicist, a particle—viewed as a "material point"—should also possess spin. But what is such an object? Can a material point "rotate about itself"?

Souriau introduced these quantities geometrically, starting solely from group theory. I concede that all this is quite difficult. A group "acts." So everything begins with the concept of action. The group acts on motion in the sense that an element of the Poincaré group transforms one motion into another, which belongs to the space of motions—the spacetime. A group "transports." The Euclidean group, for example, contains translations and rotations in 3D space. It allows us to transport points or sets of points. This idea is quite intuitive. When dealing with spacetime, we "transport" "motions." Consider two identical ashtrays located at different positions in 3D space. There always exists an element of the Euclidean group that, via a translation and a rotation, can move the first ashtray onto the second. Thanks to the group, if we know the description of an ashtray somewhere in space, we can construct "all possible ashtrays," in all locations and orientations.

In spacetime, the object is a "motion." The motions handled by the Poincaré group correspond to those of a "relativistic material point." Similarly, thanks to the group, if we know one such motion, we know them all. But a particle is a particular motion of a material point. We might summarize this way of seeing things using the expression:

Tell me how you move, and I’ll tell you what you are.

Souriau showed that the space of motions must be associated with a second space—the one he called the "moment space." By "moment," Souriau meant the parameters associated with a given particle. When this particle is observed in a certain way—i.e., described in an appropriate coordinate system—three quantities emerge:

E, p, s

Energy E, momentum p, and this mysterious object known as spin s. These quantities then appear as purely geometric entities through the coadjoint action of the group on its momentum space.

Currently, astrophysicists are working with an object they call "dark energy," the only new cosmological ingredient they consider capable of explaining the phenomenon of cosmic reacceleration, linked to repulsive forces. This "dark energy" is... negative. We will see that the approach presented here also leads to the existence of material points with negative energy, as a simple consequence of the properties of the Poincaré group, which can generate motions of this type. Before proceeding further, it would be essential for the scientific reader to read this document and internalize it. Technically, this reading requires nothing more than the ability to manipulate matrices. Fifteen years ago, this was at the level of a French terminal S (pre-university) student, but it seems matrices are no longer taught at that level. A pity—this is an essential tool—but this omission likely reflects a "modernization of curricula."

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Negative Energy Particles

October 25, 2004

In contemporary astrophysics, theorists are increasingly focusing on what they call "dark energy"—negative energy believed to be responsible for cosmic reacceleration, as inferred from observations of distant supernovae.

The theory of dynamical groups in physics (the Poincaré group) provides clarification on this complex topic. Once again, this concerns elements accessible only to scientists or readers strongly oriented toward science.

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Electric Charge: A Geometric Object

November 9, 2004

By employing an invention of the mathematician Jean-Marie Souriau—the coadjoint action of a group on its momentum space—we recalled how he derived energy, momentum, and spin as purely geometric objects. In what follows, we revisit his method to show how electric charge also emerges as a purely geometric object. He adds a fifth dimension to the four-dimensional spacetime. This five-dimensional space is then governed by a new eleven-dimensional dynamical group, a nontrivial extension of the Poincaré group. The increase in the number of group dimensions corresponds to an increase in the components of the momentum, with this eleventh dimension being identified with the electric charge q.

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