Lie groups

Group theory is the mathematical study of symmetry. A group is simply a mathematical set of objects that are linked by a set of mathematical rules. Group theory began with the work of Evariste Galois, a French mathematician. Although Galois’ ideas were originally met with rejection, group theory has now been shown to have tremendous power and answering mathematical questions that cannot be answered by any other means.

Sophus Lie, a Norwegian mathematician, is going to develop group theory into a more serious branch of mathematics. Lie was able to catalogue all groups into about 7 varieties. These are now known as Lie groups. These 7 kinds of Lie groups are called: A, B, C, D, E, F and G. 

A, B, C and D - are labeled by an integer: “n”. This integer can be arbitrarily large. That being said, there are an infinite amount of these kinds of groups. These are the groups that have historically been the most useful for building models that involve quarks and leptons. 

In today’s notation, we can refer to these groups as:

A(n)=SU(n+1)

B(n)=SO(2n+1)

C(n)=SP(2n)

D(n)=SO(2n)

Where: S = special, O = orthogonal, U = unitary and SP = sympletic.

There have been thousands of papers written using these groups, with the goal of describing elementary particles. However, “n” remains arbitrary, and its value cannot be determined.

E, F and G - however, allow for only a definite number of quarks. Thus, these groups are of interest to physicists. These groups come only in the following sets:

G(2), F(4), E(6), E(7) and E(8)

Superstring theory, it should be noted, has the symmetry E(8) x E(8). This is enough to explain all of the known particles and then some. However, when this superstring symmetry is broken, it is suspected to break down to E(6), which then will break down to SU(3) x SU(2) x U(1).

Indeed, these SU(N) symmetries were discovered by Lie and rotate complex numbers:

U(1) - This is the simplest of the SU(N) symmetries. This is the symmetry that underlies Maxwell’s equations. The “1” represents the fact that there is only one photon.

SU(2) - This is the next simplest set of SU(N) symmetries. The SU(2) symmetry can rotate two particles: both the proton and the neutron. Werner Heisenberg showed in 1932, that the Schrodinger equation for these particles (which are very much alike apart from their electric charge) can be written so as to remain invariant even if the particles are shuffled. Protons could turn into neutrons and vice versa, all the while, the equations stayed the same. We also see this in the Weinberg-Salam theory. The theory remains the same if an electron and neutrino are rotated into each other. That being said, this electroweak symmetry, also includes the symmetry of Maxwell, and can therefore be read as: SU(2) x U(1).

SU(3) - Shoichi Sakata, showed that the SU(3) symmetry group can represent the strong interaction. This is because it rotates the three subnuclear particles that make up the strongly interacting hadrons. If we wanted, we could make “N” represent the number of quarks and we could raise the SU(N) symmetry to as high as we desired. 

SU(5) - The SU(5) symmetry group would be an example of a grand unified theory that can shuffle five components: an electron, a neutrino and three quarks.

The E(N) symmetries are mysterious groups. These are abstract algebraic manipulations that have nothing to do with the natural or physical world at all. The highest value “N” can take, for some complicated mathematical reasons, is 8. It should be noted that the E(8) symmetry appears in superstring theory.