7. ROLE OF SYMMETRY IN PHYSICS

Symmetry, often used to describe the balance and proportions in objects or works of art, plays a particularly important role in particle physics. Symmetry simplifies the description of physical phenomena. Without it there would be no clear understanding of the relationships between elementary particles.Steven Weinberg defines symmetry as “a symmetry is a principle of invariance. That is, it tells us that something does not change its appearance when we make certain changes in our point of view—for instance, by rotating it or moving it.”

A human face is symmetric between the right and left of the face. If we put everything that is on the left side to the right, the human face will look the same. A square has a special kind of symmetry—it looks exactly the same if it is turned around by 90 degrees. Figure 7-1: REGULAR POLYHEDRA. TAKEN FROM STEVEN WIENBERG’S ARTICLE: ‘SYMMETRY: A ‘KEY TO NATURE’S SECRETS’.

Consider the five regular polyhedra shown in Figure 7. Steven Weinberg writes that ‘they satisfy the symmetry requirement that every face, every edge, and every corner should be precisely the same as every other face, edge, or corner…. Plato argued in Timaeus that these were the shapes of the bodies making up the elements: earth consists of little cubes, while fire, air, and water are made of polyhedra with four, eight, and twenty identical faces, respectively. The fifth regular polyhedron, with twelve identical faces, was supposed by Plato to symbolize the cosmos.’

However, the concept of symmetry in physics extends beyond its geometric meaning.

When we talk about the role of symmetry in physics, it is not the symmetries of objects we allude to, but the symmetries of the laws of physics. For example, Newton’s laws have built-in symmetry. Newton’s laws of motion and gravitation do not change their form with time. Nor do they change if we alter the point from which distances are measured, or even if we rotate it. In Einstein’s Special Relativity, motion causes objects to shrink and clocks to slow down in such a way that the speed of light is a constant, irrespective of the speed of the observer. But making observations from a moving laboratory does not change the form of the observed laws of nature, although the effect of this motion on measured distances and times is a brand new concept in Special Relativity.

There are various examples of symmetry in all branches of physics. But why is symmetry so important? To understand that, we must digress slightly and talk about the conservation laws.

Encyclopedia Britannica defines conservation laws in physics as:

“…several principles that state that certain physical properties (i.e., measurable quantities) do not change in the course of time within an isolated physical system. In classical physics, laws of this type govern energy, momentum, angular momentum, mass, and electric charge. In particle physics, other conservation laws apply to properties of subatomic particles that are invariant during interactions. An important function of conservation laws is that they make it possible to predict the macroscopic behavior of a system without having to consider the microscopic details of the course of a physical process or chemical reaction.

Conservations laws are taught in basic level physics. The most important ones are:

• Conservation of energy or mass implies that energy (or matter) can be neither created nor destroyed.
• Conservation of linear momentum implies that a body or system of bodies in motion retains its total momentum which is the product of mass and vector velocity, unless an external force is applied to it.
• Conservation of angular momentum (which is a vector quantity) implies that a body or system that is rotating continues to rotate at the same rate unless a torque is applied to it.
• Conservation of charge states that the total amount of electric charge in a system does not change with time. For example, charged particles are created in pairs so that the total amount of charge is always constant (with equal positive and negative charge).

These laws are all derived from classical mechanics. Nevertheless, all remain true in quantum mechanics and relativistic mechanics as well. What’s interesting is, there exists a relationship between symmetries and conservation laws which is known as Noether’s theorem. Some examples of this in classical physics are:

• Invariance under change of time → Conservation of energy.
• Invariance under translation in space → Conservation of momentum
• Invariance under rotation → Conservation of angular momentum

Noether’s theorem is a remarkable result that leads to conserved quantities from symmetries of the laws of nature. This result was proven in 1915 by Emmy Noether soon after she first arrived in Gottingen. Einstein wrote shortly after her death that “Noether was the most significant creative mathematical genius thus far produced since the higher education of women began.”

Eugene Wigner and Hermann Weyl were among the first to recognize the great relevance of symmetry in quantum mechanics and the first to reflect on the meaning of symmetry principles. Group Theory is the mathematical method for exploiting the consequences of symmetries in a system. In fact, Group Theoretical notions are relevant in all areas of theoretical physics and particularly important in quantum mechanics.

7.1 THE EIGHT-FOLD WAY

How can it be that writing down a few simple and elegant formulae, like short poems governed by strict rules such as those of the sonnet or the waka, can predict universal regularities of Nature?

— Murray Gell-Mann’s speech at the Nobel Banquet in Stockholm, December 10, 1969

I have already mentioned that during the 1950s and 1960s, new accelerators and experimental equipment helped to unearth many new elementary particles. Murry Gell-Mann was able to classify these particles (and their interactions) by exploiting certain symmetries intrinsic to baryons and mesons. A symmetry group called SU(3) represented patterns that he was looking for . In 1961 based on his groupings, Gell-Man predicted the existence of the η particle which was needed to complete a pattern. The η particle was discovered a few months later.

To understand Gell-Mann’s groupings, we first need to understand how certain conservation laws in particle physics allow common interactions (and decays) to occur while forbidding certain others. These conservation laws are in addition to the classical conservation laws such as conservation of energy, charge, etc., which are still valid in particle interactions (see Table 4). Figure 7-2: THE EIGHTFOLD WAY.