There is a natural connection between particle physics and representation theory, as first noted in the 1930s by Eugene Wigner. It links the properties of elementary particles to the structure 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, corresponding to "approximate symmetries" of the universe.
General picture
Symmetries of a quantum system
In quantum mechanics, any particular one-particle state is represented as a vector in a Hilbert space H {\displaystyle {\mathcal {H}}} . To help understand what types of particles can exist, it is important to classify the possibilities for H {\displaystyle {\mathcal {H}}} allowed by symmetries, and their properties. Let H {\displaystyle {\mathcal {H}}} be a Hilbert space describing a particular quantum system and let G {\displaystyle G} be a group of symmetries of the quantum system. In a relativistic quantum system, for example, G {\displaystyle G} might be the Poincaré group, while for the hydrogen atom, G {\displaystyle G} might be the rotation group SO(3). The particle state is more precisely characterized by the associated projective Hilbert space P H {\displaystyle \mathrm {P} {\mathcal {H}}} , 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 H {\displaystyle {\mathcal {H}}} and, under the natural projection map H → P H {\displaystyle {\mathcal {H}}\rightarrow \mathrm {P} {\mathcal {H}}} , an element of P H {\displaystyle \mathrm {P} {\mathcal {H}}} . By definition of a symmetry of a quantum system, there is a group action on P H {\displaystyle \mathrm {P} {\mathcal {H}}} . For each g ∈ G {\displaystyle g\in G} , there is a corresponding transformation V ( g ) {\displaystyle V(g)} of P H {\displaystyle \mathrm {P} {\mathcal {H}}} . More specifically, if g {\displaystyle g} is some symmetry of the system (say, rotation about the x-axis by 12°), then the corresponding transformation V ( g ) {\displaystyle V(g)} of P H {\displaystyle \mathrm {P} {\mathcal {H}}} is a map on ray space. For example, when rotating a stationary (zero momentum) spin-5 particle about its center, g {\displaystyle g} is a rotation in 3D space (an element of S O ( 3 ) {\displaystyle \mathrm {SO(3)} } ), while V ( g ) {\displaystyle V(g)} is an operator whose domain and range are each the space of possible quantum states of this particle, in this example the projective space P H {\displaystyle \mathrm {P} {\mathcal {H}}} associated with an 11-dimensional complex Hilbert space H {\displaystyle {\mathcal {H}}} . Each map V ( g ) {\displaystyle V(g)} preserves, by definition of symmetry, the ray product on P H {\displaystyle \mathrm {P} {\mathcal {H}}} induced by the inner product on H {\displaystyle {\mathcal {H}}} ; according to Wigner's theorem, this transformation of P H {\displaystyle \mathrm {P} {\mathcal {H}}} comes from a unitary or anti-unitary transformation U ( g ) {\displaystyle U(g)} of H {\displaystyle {\mathcal {H}}} . Note, however, that the U ( g ) {\displaystyle U(g)} associated to a given V ( g ) {\displaystyle V(g)} is not unique, but only unique up to a phase factor. The composition of the operators U ( g ) {\displaystyle U(g)} should, therefore, reflect the composition law in G {\displaystyle G} , but only up to a phase factor:
… excerpt ends here. Continue reading the full article.
