Particle physics and representation theory
There is a natural connection between particle physics and representation theory, as first noted in the 1930s by Eugene Wigner.
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, corresponding to "approximate symmetries" of the universe.
In quantum mechanics, any particular one-particle state is represented as a vector in a Hilbert space
To help understand what types of particles can exist, it is important to classify the possibilities for
The particle state is more precisely characterized by the associated projective Hilbert space
, also called ray space, since two vectors that differ by a nonzero scalar factor correspond to the same physical quantum state represented by a ray in Hilbert space, which is an equivalence class in
By definition of a symmetry of a quantum system, there is a group action on
is some symmetry of the system (say, rotation about the x-axis by 12°), then the corresponding transformation
An electron, for example, is a spin-one-half particle; its Hilbert space consists of wave functions on
, Bargmann's theorem tells us that every projective unitary representation of
is infinite dimensional, then to obtain the desired conclusion, some algebraic assumptions must be made on
[4] Fortunately, in the crucial case of the Poincaré group, Bargmann's theorem applies.
[5] (See Wigner's classification of the representations of the universal cover of the Poincaré group.)
Thus, in favorable cases, the quantum system will carry a unitary representation of the universal cover
An example in which Bargmann's theorem does not apply comes from a quantum particle moving in
This failure to commute reflects the failure of the position and momentum operators—which are the infinitesimal generators of translations in momentum space and position space, respectively—to commute.
In this case, to obtain an ordinary representation, one has to pass to the Heisenberg group, which is a nontrivial one-dimensional central extension of
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.
(There are in fact other possible representations, such as tachyons, infraparticles, etc., which in some cases do not have quantized spin or fixed mass.)
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.
[6] In the finite-dimensional case—and the infinite-dimensional case, provided that Bargmann's theorem applies—irreducible projective representations of the original group correspond to ordinary unitary representations of the universal cover.
This is the case, notably, for studying the irreducible projective representations of the rotation group SO(3).
These are in one-to-one correspondence with the ordinary representations of the universal cover SU(2) of SO(3).
As an example of what an approximate symmetry means, suppose an experimentalist lived inside an infinite ferromagnet, with magnetization in some particular direction.
The 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.
Some examples help clarify the possible effects of these transformations: In general, particles form isospin multiplets, which correspond to irreducible representations of the Lie algebra SU(2).
[7] This is, again, an approximate symmetry, violated by quark mass differences and electroweak interactions—in fact, it is a poorer approximation than isospin, because of the strange quark's noticeably higher mass.
Nevertheless, particles can indeed be neatly divided into groups that form irreducible representations of the Lie algebra SU(3), as first noted by Murray Gell-Mann and independently by Yuval Ne'eman.
