Particle physics and representation theory

There is a natural connection, first discovered by Eugene Wigner, between the properties of particles, the representation theory of Lie groups and Lie algebras, and the symmetries of the universe. According to it, the different states of an elementary particle furnishes an irreducible representation of the Poincare group. Moreover, the spectrum of different particles, and their properties, can be related to representations of Lie algebras which correspond to "approximate symmetries" of the universe.

General picture

In quantum mechanics, any particular particle (with a given momentum distribution, location distribution, spin state, etc.) is treated as a vector, (or "ket") in a Hilbert space. Let "G" be the "symmetry group of the universe" -- that is, the set of symmetries under which the laws of physics are invariant. (For example, one element of "G" is translation forward in time by five seconds.) Starting with a particular particle in the state ket $|p\_0\; angle$, there must be some well-defined (up to a phase factor) ket $|p\_g\; angle=g|p\_0\; angle$ that results from applying the symmetry transformation "g" to the particle, for any "g" in "G". For this picture to be consistent, it needs to be the case that applying two transformations consecutively is equivalent to applying the combined transformation -- i.e. $g\_1(g\_2|p\_0\; angle)\; =\; (g\_1g\_2)|p\_0\; angle$ (again, up to a phase factor, but we'll ignore this detail; see Weinberg, Ch. 2 Appendix B for a proof that the phase factors can be chosen to be 1 in most cases). We recognize this as the definition of a group representation; therefore, any given particle is associated with a unique representation of "G" on the vector space spanned by $\{g|p\_0\; angle|gin\; G\}$ (We say the particle "lies in", or "transforms as" the representation). One can prove that this representation is irreducible; and in fact, Wigner's Theorem proves that it is also a unitary representation, or possibly anti-unitary (see Weinberg, Ch. 2 Appendix A for a proof of this.)

So we conclude that each type of particle corresponds to an irreducible representation of "G"; and if we can classify the representations of "G", we will have much more information about what types of particles can exist.

Poincaré group

The group of translations and Lorentz transformations form the Poincaré group, and this group is certainly a subgroup of "G". Hence, any representation of "G" will in particular be a representation of the Poincaré group. The representations of the Poincaré group are characterized by a nonnegative mass and a half-integer spin; this can be thought of as the reason that spin is quantized.

Other symmetries

While the spacetime symmetries in the Poincaré group are particularly easy to visualize and believe, there are also other types of symmetries, called internal symmetries. One example is color SU(3), an exact symmetry corresponding to continuous interchange of the three quark colors.

Approximate symmetries

Although the above symmetries are believed to be exact, other symmetries are only approximate.

Hypothetical example

As an example of what an approximate symmetry means, suppose we lived inside an infinite ferromagnet, with magnetization in some particular direction. An experimentalist in this situation would find not one but two distinct types of electrons: one with spin along the direction of the magnetization, with a slightly lower energy (and consequently, a lower mass), and one with spin anti-aligned, with a higher mass. Our usual SU(2) rotational symmetry, which ordinarily connects the spin-up electron with the spin-down electron, has in this hypothetical case become only an "approximate" symmetry, relating "different types of particles" to each other.

Lie algebras versus Lie groups

Many (but not all) symmetries or approximate symmetries, for example the ones above, form Lie groups. Then representations of the group are closely related to representations of its Lie algebra; since the latter is usually simpler to compute, that is the way it is usually done.

General definition

In general, an approximate symmetry arises when there are very strong interactions that obey that symmetry, and weaker interactions that do not. In the electron example above, the two "types" of electrons would behave identically under the strong force and weak force, but differently under the electromagnetic force.

Example: flavour symmetry

An example from the real world is flavour symmetry, an SU(3) group corresponding to varying quark flavour. This is an approximate symmetry, believed to be exact under strong interactions, but violated by electroweak interactions. Indeed, we see experimentally that particles can be neatly divided into groups that form irreducible representations of the Lie algebra "SU"(3), as first noted by Murray Gell-Mann (see the eightfold way).

References

*Sternberg, Shlomo (1994). "Group Theory and Physics". Cambridge University Press. ISBN 0-521-24870-1. Especially pp. 148-150.
*cite book | author=Weinberg, Steven | title=The Quantum Theory of Fields, Volume 1: Foundations | publisher=Cambridge Univ. Press | year=1995 | id=ISBN 0-521-55001-7 See especially appendices A and B to Chapter 2.
*Coleman, Sidney (1985). "Aspects of Symmetry: Selected Erice Lectures of Sidney Coleman". Cambridge University Press. ISBN 0-521-26706-4.
*Georgi, Howard (1999). "Lie Algebras in Particle Physics". Reading, MA: Perseus Books. ISBN 0-7382-0233-9.
* Hall, Brian C. "Lie Groups, Lie Algebras, and Representations: An Elementary Introduction", 1st edition, Springer, 2006. ISBN 0-387-40122-9.