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. 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.

In quantum mechanics, any particular particle (with a given momentum distribution, location distribution, spin state, etc.) is written as a vector (or "ket") in a Hilbert space H. To help understand what types of particles can exist, it is important to classify the possibilities for H, and their properties. The particle is more precisely characterized by the associated projective Hilbert space PH, since two vectors that differ by a scalar factor (or in physics terminology, two "kets" that differ by a "phase factor") correspond to the same physical quantum state.

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 the simultaneous translation of all particles and fields forward in time by five seconds.) Starting with a particular particle in the state ket ${\displaystyle |p_{0}\rangle }$, and a symmetry transformation g in G, it is possible to apply the symmetry transformation to the particle to get a new state ket ${\displaystyle |p_{g}\rangle =g|p_{0}\rangle }$. For this picture to be consistent, it is necessary that PH holds a projective group representation of G. (For example, this condition guarantees that applying a symmetry transformation, then applying its inverse transformation, will restore the original quantum state.)

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