Mathematics and the study of elementary particles

Summary: Various branches of mathematics are employed to study elementary particles, the smallest particles in the universe.

Particle physics is a branch of physics that seeks to describe and explain the universe on the smallest scales. The particles thought to be the fundamental building blocks of matter and force are called “elementary particles.” Like all branches of physics, the study of elementary particles relies heavily upon many branches of mathematics, including calculus, geometry, group theory, algebra, and statistics. Particle physics also contributes to mathematical research by posing questions that give rise to new mathematical theories.

History

For thousands of years, scientists and philosophers have been asking the questions, “What is the universe made of?” and “Are there fundamental units that make up space, matter, energy, and time, or are these infinitely divisible?” As early as the fifth century b.c.e., Greek philosopher Democritus (c. 460-370 b.c.e.) hypothesized that all matter is made of indivisible, fundamental units called “atoms.” Despite these early hypotheses, there was very little progress in this field until the dawn of the twentieth century.

The twentieth century saw the emergence of several new branches of physics. Among these was particle physics, a field that seeks to explore the universe on the smallest scales. Particle physicists try to identify the particles that form matter and force, describe their properties, and understand how these particles relate to each other. Some of these particles are not composed of any other particles and are therefore called “elementary particles.” These elementary particles form the basic building blocks of the universe .

The understanding of particle physics at the beginning of the twenty-first century is embodied in the Standard Model of Particle Physics, an elaborate yet still incomplete model that attempts to list and describe all existing particles. Jokingly referred to as “The Particle Zoo,” the Standard Model lists dozens of particles and includes elementary particles with exotic names such as “gluon,” “muon,” and “quark.” Many of the particles in the Standard Model have yet to be detected experimentally, and their existence is conjectured based on theoretical work.

Mathematics Used in the Study of Particle Physics

Like all physical theories, particle physics relies heavily upon mathematics, which provides the theoretical framework physicists use to explain and describe physical phenomena. Mathematics also enables physicists to make predictions that can later be tested using modern tools, such as particle accelerators.

One of the most useful branches of mathematics is calculus, a field that has applications in practically all branches of the natural sciences, as well as in engineering and even in the social sciences. It is therefore not surprising that calculus occupies a central role in the theory of elementary particles. Differential calculus may be used to describe properties of particles at an instant, while integral calculus is used to describe cumulative effects of a particle or a system of particles over time and space.

Calculus is but one branch of the mathematical field of analysis that is useful in particle physics. Other branches of analysis, such as partial differential equations, complex analysis, and functional analysis, play important roles as well.

Geometry has traditionally been used to describe the universe on the grandest scales, those of galaxies, galaxy clusters, and the universe as a whole. Recently, geometry has found a place in elementary particle research as well. French mathematician Alain Connes (1947-) has described a theoretical model for particle physics that is based on noncommutative geometry, which is a geometrical representation of noncommutative algebras systems in which the order of factors in an operation determines the value of the operation. For example, if a and b are real numbers, then it is always true that a×b=b×a, as multiplication is commutative for real numbers. However, if A and B are matrices, then generally A×B≠B×A. Matrix multiplication is therefore noncommutative.

Symmetry, Group Theory, and Quantum Mechanics

One of the most fundamental mathematical concepts in elementary particles is symmetry. In mathematics, symmetry is defined as an operation on an object that leaves some of the object’s properties unchanged. As an example, consider a square drawn in the plane and an axis of rotation that passes through the square’s center, perpendicular to the plane. If the square is rotated by 90 degrees around that axis, the square will appear unchanged. Rotation by 90 degrees is thus called a “symmetry” of the square. The set of all symmetries of an object forms a mathematical construct called a group (a set with an operation that obeys several axioms). Group theory, a branch of algebra, plays an important role in particle physics, as properties of many elementary particles can be explained and described by the use of symmetry.

The chief group-theoretic structure in particle physics is the Lie (pronounced “Lee”) group, named after Norwegian mathematician Sophus Lie (1842-1899). Lie groups are groups that posses the properties of geometric constructs known as “differentiable manifolds.” Lie groups thus provide yet another connection between geometry and elementary particles.

One of the most important physical theories of the twentieth century is quantum mechanics, a theory that holds that, at the atomic and subatomic levels, behavior of particles is a statistical rather than a deterministic phenomenon. Since elementary particles obey quantum-mechanical laws, statistics and probability are invariably major components of the mathematical framework of elementary particles.

While physicists use mathematics as a tool for exploring the universe, the relationship between particle physics and mathematics is not one-directional. Research in particle physics drives the emergence of new mathematical theories, just as mechanics drove the emergence of calculus in the seventeenth century. In 1990, American theoretical physicist Edward Witten (1951-) won the Fields Medal, the highest honor in mathematics, for his many contributions to mathematics. He is the only non-mathematician ever to win the prestigious award. As both mathematicians and physicists continue to explore new horizons, the cross-fertilization of ideas will benefit both fields in decades to come.

Bibliography

Griffiths, David. Introduction to Elementary Particles. Weinheim, Germany: Wiley-VCH, 2008.

Hellemans, Alexander. “The Geometer of Particle Physics.” Scientific American 295, no. 2 (2006).

Mann, Robert. An Introduction to Particle Physics and the Standard Model. Boca Raton, FL: CRC Press, 2010.