The study of the symmetries that define the properties of a system. Invariance under symmetry operations enables much about a system to be deduced without knowing explicitly the solutions to the equations of motion. Newton’s law of gravitation, for instance, exhibits spherical symmetry. The force of gravity due to the attraction of a planet to a star is the same for all positions that are equidistant from the centre of mass of the star. However, the possible trajectories of the planet include nonsymmetric elliptical orbits. These elliptical orbits are solutions to the Newtonian equations. However, one discovers on solving them that the planet does not move at a constant speed around the ellipse: it speeds up when it approaches perihelion and slows down approaching aphelion, which is consistent with what one might expect from a spherically symmetric force law. This behaviour, first formulated as one of Kepler’s laws of planetary motion, is now accepted as a result of the conservation of angular momentum.
This association of a dynamical symmetry with a conservation law was first suggested in a general way by Emmy Noether in 1918 (see Noether’s theorem). For example, the laws of physics are invariant under translations in time; they are the same today as they were yesterday. Noether’s theorem relates this invariance to the conservation of energy. If a system is invariant under translations in space, linear momentum is conserved.
The symmetry operations on any physical system must possess the following properties:
These are the defining properties of a group in group theory. Group elements need not commute; i.e. Ri Rj ≠ Rj Ri, in general; if all the elements do commute then the group is said to be Abelian. Though translations in space and time are Abelian, it is easily verified that rotations about axes in three-dimensional space are not. Groups can be finite (for example, the group of rotations of an equilateral triangle) or infinite (for example, the set of all integers, with addition used to combine the members). Groups can also be classed as continuous or discrete. An example of a continuous group is the group of all continuous rotations about the centre of a sphere. Discrete groups have elements that may be labelled by an index that takes on only integer values. All finite groups and some infinite groups, such as the group of integers described above, are discrete. Discrete groups have been used extensively in the 21st century to discuss family symmetry.
Group elements are conveniently represented in matrix form. In elementary-particle physics, the most common groups are unitary groups, denoted U(n), the group of n × n unitary matrices, i.e. matrices U with complex elements such that the product of U and its complex conjugate transpose matrix is always the unit matrix. If the unitary matrices are further restricted to have a determinant of 1, the group is called a special unitary group, denoted SU(n). The analogous matrices to unitary matrices in which all the elements are real are called orthogonal matrices, with orthogonal groups, denoted O(n), being the set of n × n orthogonal matrices. Finally, the group of real, orthogonal, n × n matrices of determinant 1 is SO(n). SO(n) may be thought of as the group of all rotations in n-dimensional space. Thus SO(3) is the name of the group of all rotations in three-dimensional space. This symmetry is related to the conservation of angular momentum by Noether’s theorem. Group theory is particularly important in quantum mechanics because most quantum numbers are associated with the symmetry group of the system.
In discussing the particles of high-energy physics and the special unitary groups SU(2) and SU(3), it is helpful to consider an analogous example with SO(3) symmetry, such as the hydrogen atom. The hydrogen atom is composed of an electron trapped within the spherically symmetric potential of the atomic nucleus (a proton). The quantum-mechanical treatment of this problem leads to a description of the electron states in terms of standing waves on a spherical surface. These standing waves are labelled by three integers or quantum numbers n, l, and m (see atom). The quantum numbers are associated with the nodes and antinodes on the spherical surface undergoing standing-wave oscillations. The energy associated with these standing waves, however, is (2l+1)-fold degenerate; that is, for a given standing wave characterized by the quantum numbers n and l, there are (2l+1) values of m characterizing states of the same energy. The reason for this degeneracy lies in the fact that the potential is spherically symmetric and independent of the angular position of the electron with respect to the nucleus. As a consequence of the spherical symmetry of the potential, the angular momentum L is conserved. The quantum number n is associated with a ‘hidden’ symmetry of the inverse-square law. See Fock degeneracy.
The degeneracy of the hydrogen atom may be eliminated by the application of a magnetic field. The directional nature of this field destroys the spherical symmetry of the problem and leads to the Zeeman effect. The degeneracy is lifted by this symmetry-breaking term.
In the 1930s, Heisenberg proposed a model for nuclear forces that was analogous to the group theory of atoms. The only known nuclear particles at the time were the proton and the neutron, which were very similar in mass. Heisenberg proposed that nuclear interactions were independent of electric charge and that the proton and neutron were the same particle in the absence of electromagnetic interactions. This meant that the proton and neutron were members of a degenerate state and that the electromagnetic interaction broke the symmetry of nuclear interactions, lifting this degeneracy. Just as L is the conserved quantity for the hydrogen atom analogy, the conserved quantity for nuclear interactions between protons and neutrons is a quantity labelled I that was later called isospin (see isotopic spin).
In the absence of the electromagnetic interaction, isospin is conserved (the proton and neutron have the same mass) and there is a two-fold degeneracy, i.e. the proton and neutron are regarded as a doublet. Equivalently, the nuclear interaction (strong interaction) must be invariant under the group that has matrices that are described by isospin matrices. This group is SU(2).
By the 1960s, many more particles had been discovered and the ideas of isospin were applied to them. It turns out that it is convenient to describe these particles by characteristic quantum numbers, I for isospin and Y for hypercharge. The particles may be grouped into multiplets. Experiments involving these particles show that hypercharge Y and isospin I are conserved under strong interactions.
In 1961 Gell-Mann, and independently Yuval Ne’eman, suggested that the strong interaction should be invariant under SU(3); that is, should have SU(3) symmetry. Gell-Mann called this scheme the eight-fold way. In the absence of any other interaction, the strong interaction sees all particles in a multiplet as the same particle, i.e. the same state. It is the effects of the weak and electromagnetic interactions that break the SU(3) symmetry and lead to the lifting of the degeneracy and the splitting of the masses of these particles. Thus, the particles in a multiplet are only approximately degenerate, just as multiplets in atomic spectra are split up. SU(3) is a more approximate symmetry than isospin.