Particle physics and representation theory
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In physics, the connection between particle physics and representation theory is a natural connection, first noted by Eugene Wigner, between the properties of elementary particles and the representation theory of Lie groups and Lie algebras. According to this connection, the different quantum states of an elementary particle give rise to an irreducible representation of the Poincaré group. Moreover, the properties of the various particles, including their spectra, can be related to representations of Lie algebras, algebras corresponding to "approximate symmetries" of the universe.
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[edit] 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 
, there must be some well-defined (up to a phase factor) ket 
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. 
[1] This defines a group representation; therefore, any given particle is associated with a unique representation of G on the vector space spanned by 
(We say the particle "lies in", or "transforms as" the representation.) One can prove that this representation is irreducible. Wigner's Theorem proves that it is also a unitary representation, or possibly anti-unitary (for a proof, see Weinberg (1995), Appendix A to Ch. 2.)
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.
[edit] Poincaré group
The group of translations and Lorentz transformations form the Poincaré group, and this group is certainly a subgroup of G (neglecting general relativity effects, or in other words, in flat space). Hence, any representation of G will in particular be a representation of the Poincaré group. Representations of the Poincaré group are in many cases characterized by a nonnegative mass and a half-integer spin (see Wigner's classification); this can be thought of as the reason that particles have quantized spin. (Note that there are in fact other possible representations, such as tachyons, infraparticles, etc., which in some cases do not have quantized spin or fixed mass.)
[edit] 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 the continuous interchange of the three quark colors.
[edit] Approximate symmetries
Although the above symmetries are believed to be exact, other symmetries are only approximate.
[edit] 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 SO(3) 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.
[edit] Lie algebras versus Lie groups
Many (but not all) symmetries or approximate symmetries, for example the ones above, form Lie groups. Rather than study the representation theory of these Lie groups, it is often preferable to study the closely related representation theory of the corresponding Lie algebras, which are usually simpler to compute.
[edit] General definition
In general, an approximate symmetry arises when there are very strong interactions that obey that symmetry, along with weaker interactions that do not. In the electron example above, the two "types" of electrons behave identically under the strong and weak forces, but differently under the electromagnetic force.
[edit] 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, violated by quark mass differences and 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).
[edit] See also
[edit] References
- Coleman, Sidney (1985) Aspects of Symmetry: Selected Erice Lectures of Sidney Coleman. Cambridge Univ. 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., (2006) Lie Groups, Lie Algebras, and Representations: An Elementary Introduction. Springer. ISBN 0-387-40122-9.
- Sternberg, Shlomo (1994) Group Theory and Physics. Cambridge Univ. Press. ISBN 0-521-24870-1. Especially pp. 148-150.
- Steven Weinberg (1995). The Quantum Theory of Fields, Volume 1: Foundations. Cambridge Univ. Press. ISBN 0-521-55001-7. Especially appendices A and B to Chapter 2.
- ^ That this holds (again) up to a phase factor will be ignored. See Weinberg (1995), Appendix B to Ch. 2 for a proof that in most cases, the phase factors can be set to 1.
