Mathematics of light

Summary: Now understood as both a particle and a wave, light is a recurring subject of interest in physics.

Light, a form of electromagnetic energy, mediates the electrostatic interactions between particles. Under some experimental conditions, it acts as a particle, and under others, as a wave. Attempts by physicists to reconcile this dual nature and to otherwise exploit this duality have been the impetus for the development of large areas of mathematics.

Particle or Wave?

Isaac Newton advocated the particle nature of light, initiating the study of geometric, or ray, optics. This form of optics treats light as rays that travel in straight lines, though capable of bending near objects. It is based on two laws. The law of reflection states that when light is reflected from a surface, the angle of incidence equals the angle of reflection. The law of refraction says that light will bend when it passes from one medium to another according to Snell’s Law, named for mathematician Willebrord Snell, a relation between the angles of incidence and refraction and light’s speed in the two media.

At about the same time, Christiaan Huygens discovered polarized light and explained it with a wave theory. From this beginning, Thomas Young and Augustin-Jean Fresnel developed physical optics. The resulting mathematics allowed engineers to construct extremely faithful lenses; its close cousin, wave acoustics, helped architects design performance halls. Scientists pursued these optics to ever-finer scales. Eventually, they developed the electron microscope, which permits biologists to see individual DNA molecules. Physical biochemists use a related technique called “crystallography.” When X-rays are shot through crystals of protein molecules, they form a diffraction pattern, which when transformed by a technique called Fourier analysis (named for mathematician and physicist Joseph Fourier) allows the precise determination of the protein’s atomic structure. Many owe their Nobel Prizes to this transformation.

The wave theory of light provides the most natural explanation for the spectrum of visible light. What the physicist calls “light” varies from about 1023 cycles per second, corresponding to gamma rays, down to roughly 1000 cycles per second for the electron waves in plasma. What humans can see is but a small part of this, varying from purple at a wavelength of 380 nanometers (nm) or 7.8 × 10¹⁴ cycles per second, to red at about 780 nm, or 3.8 × 10¹⁴ cycles per second.

Light Speed

In 1861, James Clerk Maxwell wrote down his famous equations describing the interactions between electric and magnetic fields in terms of their sources. Four years later, he derived from them an electromagnetic wave equation, which physicists soon understood to be a description of light waves. In 1907, Edward Rosa and Noah Dorsey used these equations to calculate the speed of light at 299,784 km/sec. The accuracy of this calculation was not matched by experiment until 1926, when Albert Michelson obtained a value of 299,796 km/sec. In 1983, the 17th Conférence Général des Poids et Mesures established a new standard for the length of the meter by fixing the speed of light at 299,792,458 meters/second.

In the 1890s, Hendrik Lorentz, George Fitzgerald, and Joseph Larmor noticed that Maxwell’s equations did not change under a certain type of transformation. Henri Poincaré called these “Lorentz transformations” and noticed that they formed a group of symmetries on four-dimensional space-time. Albert Einstein incorporated this symmetry into his theory of special relativity. One of the key postulates is that light travels at the universe’s speed limit and so nothing can travel faster. Hermann Minkowski developed from these theories a four-dimensional geometry called “Minkowski space,” in which Einstein’s famous theory is understood as geometric properties of the space.

Quantum Phenomena

Light held yet further mysteries. Nineteenth-century physics predicted that heated bodies should radiate infinite amounts of energy and that an atomic electron should plunge into the nucleus. Max Planck eliminated the first problem by postulating the quantization of light. Einstein used this idea to explain properties of the photoelectric effect, the phenomenon behind solar panels. Niels Bohr expanded these ideas into an explanation of why electrons in atoms do not continuously radiate light until they collapse into the nucleus. All three won Nobel Prizes for their work and quantum physics was born.

John von Neumann developed a mathematical description of these quantum phenomena involving Hilbert spaces and operator algebras. As a result, research into Operator Algebras became a major research focus of the last half of the twentieth century. To further explain quantum behavior, von Neumann and Garrett Birkoff developed quantum logic, a subject pursued not only by mathematicians but also by many philosophers. In a high point of this endeavor, John Bell developed the Bell inequalities in 1966. Sixteen years later, Alain Aspect confirmed that quantum systems do violate these inequalities, and provided strong evidence that the mysterious results of quantum mechanics are not solely because of our difficulties in measuring systems on such a fine scale but are because of the very nature of these small-scale systems. These experiments exploited a quantum property called “entanglement.” Richard Feynman hypothesized this entanglement might be exploitable as a computational resource. In recent decades, Peter Schor, Lov Grover, and others have developed algorithms based on Feynman’s idea and created the field of quantum computing.

Quantum mechanics has, in the last half century, developed into quantum field theory (QFT). QFT attempts to explain all particles and forces by equations that are modeled on Maxwell’s. In developing their models, mathematical physicists rely on physical properties to perform manipulations mathematicians find objectionable because of their lack of rigor. Many great mathematicians have taken up the challenge of developing a rigorous axiomatic basis for QFT. Lying at the intersection of philosophy, mathematics, and physics, many mathematicians see this as one of the great challenges of the twenty-first century.

Bibliography

Baierlein, Ralph. Newton to Einstein: The Trail of Light. Cambridge, England: Cambridge University Press, 1992.

Farndon, John. From Newton’s Rainbow to Frozen Light: Discovering Light. Chicago: Heinemann Library, 2007.

Smith, Francis Graham, and John Hunter Thomson. Optics. Hoboken, NJ: Wiley, 1971.

Sobel, Michael. Light. Chicago: University of Chicago Press, 1987.

Zeilinger, Anton. Dance of the Photons: From Einstein to Quantum Teleportation. New York: Farrar, Straus & Giroux, 2010.