GROUPS OF PARTICLES - MATHEMATICS AND THE MODERN VIEW OF THE WORLD - The Remarkable Role of Evolution in the Making of Mathematics - Mathematics and the Real World

Mathematics and the Real World: The Remarkable Role of Evolution in the Making of Mathematics (2014)

CHAPTER IV. MATHEMATICS AND THE MODERN VIEW OF THE WORLD

31. GROUPS OF PARTICLES

At the beginning of the 1930s the subatomic situation was relatively simple: the atom was made up of a nucleus, inside which were neutrons and protons, and around which revolved things that were waves and also particles, that is, electrons. Light particles were also known, that is, photons. But it soon became apparent that subatomic reality was far more complex. First, following precise experiments that analyzed the frequency of the radiation it was found that there was not just one type of electron. In effect there are two types. The difference between the two types of radiation that they created was explained by the suggestion that the electron, while revolving around the nucleus, also revolves on its own axis, and the direction of this movement, to the left or to the right, is what gives the different types of radiation. The physicists called this turning on its axis “spin.” Again, there is no guarantee that the electron particle, which is also a wave, does actually spin on its own axis. This property, however, would provide a good explanation for the difference between the two types of electrons.

Then the positron was discovered, a particle similar to an electron but with a positive charge. The positron was discovered as a result of a mathematical analysis of Dirac's equation, which is a version of Schrödinger's equation adapted for the electron, yet with an additional solution with a positive charge. This solution indicated the possibility that an “antimatter” particle existed. If this particle meets the “matter” particle, they will both disappear and become energy. Within a short time it was indeed found that the mathematical solution, somewhat exotic we might add, was realized by a real particle in nature, the positron. Within a few years other “matter” (and “antimatter”) particles were discovered, which were then given the collective name of elementary particles. Initially these particles were studied by examining cosmic radiation and the results of the cosmic particles striking the earthly atoms. Cosmic radiation has great energy, but a large part of it is absorbed in the atmosphere. Research into the elementary particles was therefore then carried out by raising photographic plates to a great height and documenting the impact between the cosmic radiation and the earthly particles. Later, other means were developed, such as bubble chambers and later still particle accelerators, that recorded the collision between the accelerated particles and other particles and the changes that the collisions caused, including the creation of new particles. These were characterized by their energy, the frequency of their wave, their mass, and their spin, the level of which increased and now not only reflected direction but was also given values of a half, a third, and so on. The episode of creating a picture of the subatomic world is fascinating but falls outside the scope of this book. Suffice it to say here that a list was formed of the elementary particles, but to understand the order underlying it required mathematics. Schrödinger's equations, other equations that developed from them, Born's interpretation, and what was derived from them were sufficient to describe properties of the particles but did not explain their allocation according to the various properties. For that, a mathematical element that had not been used before in this area was incorporated, namely, groups.

The classification of the various particles by their properties led to their being arranged in tables according to their type. A particular type of particle is called a hadron. Two scientists, Murray Gell-Mann of Caltech in California and the Israeli Yuval Ne'eman (1925–2006), who was then (in 1961) at Imperial College, London, and later at Tel Aviv University, both noticed in that year, independently, that hadrons can be placed in several tables according to the characteristics of their spins so that each table consisted of exactly eight particles. Moreover, the relation between the particles in every table matched what the mathematicians had for a long time called the SU(3) group.

It is not necessary to get involved deeply in group theory to understand its role in describing the physical situation. A group is a collection of mathematical elements and the relations between them, for example, revolving the plane through either 90, 180, 270, or 360 degrees, with the last of those alternatives bringing us back to the starting position. This is a group whose elements are the turns, and the action between every two elements is the turn obtained after two consecutive revolutions. That is to say the relationships are: a turn of 180 degrees followed by a turn of 270 degrees is equal to a turn of 90 degrees, and so on. This is a group with four elements. To describe such turns, there is clearly no need to call them a group and to use sophisticated mathematics. What is special about mathematical terminology is that it enables us to describe more-complex systems. For instance, turning a die through 90 degrees in one direction around one of the three axes will define a more complex group. Mathematics studies even more complex groups and groups in which the relations between the elements are more complicated.

In the nineteenth century a Norwegian mathematician, Sophus Lie (1842–1899), found groups that described symmetries of differential equations. These are today referred to as Lie groups, and one of them is the SU(3) group, which Gell-Mann and Ne’eman suggested constituted the basis of the arrangements of the hadrons.

Yet not all the elements in the group were represented by particles that were known when Gell-Mann and Ne’eman put forward their hypothesis. In particular, in a lecture delivered by Gell-Mann at the European Organization for Nuclear Research (CERN) in Geneva, which currently has the largest particle accelerator, he presented the SU(3) group and its properties and noted that a particle was missing; he called it omega minus. That particle would have completed the group model. Moreover, its existence would have absolutely contradicted another model for classifying hadrons proposed by Japanese researchers. Gell-Mann, however, was apparently not familiar with the literature on experiments then being performed. One of the people who attended Gell-Mann's talk was Luis Alvarez, of the University of California, Berkeley. He was a famous scientist who later received the Nobel Prize in Physics in 1968, and among his many scientific achievements he put forward the hypothesis that dinosaurs became extinct due to a meteorite impacting on the Earth. Alvarez stated that the missing particle, omega minus, had been identified seven years previously by Yehuda Eisenberg, a physicist at the Weizmann Institute of Science in Rehovot, Israel. Eisenberg had made his discovery using the technique of placing photographic plates at the height of the atmosphere, but that was an isolated occurrence not backed by a mathematical explanation, so the experimental discovery of the particle entered the catalogue of particles without attracting any great attention. Following Gell-Mann's lecture, several groups of experimenters started again to look for that same omega minus using the bubble-cell technique until in 1964 a team of physicists led by Nicholas Samios at the Brookhaven Laboratory in New York eventually again succeeded in identifying the missing particle several times and thus corroborated Eisenberg's finding.

Therefore, it can be claimed that in a certain sense, ex post facto, the existence of the particle was predicted by group theory. Enlisting group theory to characterize and classify elementary particles scored its first success. The order in which the elementary particles are recorded moved up from the level of simply a table to a mathematical theory, which enables sophisticated predictions to be suggested and examined.

It should also be noted, however, that there is no fundamental or logical explanation for the fact that group theory and the structure of the elementary particles match each other. From a conceptual viewpoint, that agreement might remind us of the match that Plato found between the world and the four elements of nature that it comprises, and the five perfect solids, or the matching, with the detailed calculations, that Kepler found between the paths of the six celestial bodies and the perfect solids (see sections 10 and 17). Will we in the future assess the role of group theory in describing the elementary particles in the same way as we assess Kepler's model of the perfect solids?

The combination of the existing mathematics, physical principles, and technology of experiments in physics resulted in the discovery, mapping, and understanding of the structure of a larger number of elementary particles and the interaction between them. In addition to the two forces already known, that is, gravity and electromagnetism, two new forces were discovered: a strong nuclear force and a weak nuclear force. Gell-Mann used mathematical principles and put forward the hypothesis that particles exist that are parts of protons and that have a fractional charge. These are the quarks, combinations of which make up the various protons. Although quarks cannot be isolated, their existence was proven beyond all doubt, and this earned Gell-Mann the Nobel Prize in Physics in 1969.

Since then the picture has broadened, and other experimental findings have been added, as well as many mathematical items. Yet the picture is still not complete or final. Right now an experiment is underway, at enormous expense, at the Geneva particle accelerator, intended to reveal the particle known as the Higgs particle (or Higgs boson), named after the British physicist Peter Higgs, who predicted the existence of the particle already in 1964. Initial reports indicate that a new particle has been found in the range of mass and energy in which the Higgs particle is predicted to be. Higgs and his colleague François Englert, who independently made the same prediction, won the Nobel Prize in Physics for 2013. If the initial findings are confirmed, this will corroborate the model known as the standard model. If it transpires that the particle discovered is not the Higgs particle, the physicists will have to rethink the picture of the subatomic world and perhaps will have to adopt a new mathematics to do so.