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There is a natural connection between particle physics and representation theory, as first noted in the 1930s by Eugene Wigner. [1] It links the properties of elementary particles to the structure of Lie groups and Lie algebras.
It is fundamental in theoretical physics. In a physical theory having Minkowski space as the underlying spacetime, the space of physical states is typically a representation of the Poincaré group. (More generally, it may be a projective representation, which amounts to a representation of the double cover of the group.)
It was introduced by Eugene Wigner, to classify particles and fields in physics—see the article particle physics and representation theory. It relies on the stabilizer subgroups of that group, dubbed the Wigner little groups of various mass states.
The representation with = (i.e., =) is the 3 representation, the adjoint representation. It describes 3-d rotations , the standard representation of SO(3), so real numbers are sufficient for it. Physicists use it for the description of massive spin-1 particles, such as vector mesons , but its importance for spin theory is much higher because it ...
The representation theory of SU(3) is well-understood. [12] Descriptions of these representations, from the point of view of its complexified Lie algebra s l ( 3 ; C ) {\displaystyle {\mathfrak {sl}}(3;\mathbb {C} )} , may be found in the articles on Lie algebra representations or the Clebsch–Gordan coefficients for SU(3) .
E.P. Wigner (1902–1995), ForMemRS, first proved the theorem bearing his name.It was a key step towards the modern classification scheme of particle types, according to which particle types are partly characterized by which representation of the Lorentz group under which it transforms.
[7] [8] Furthermore, representation theory is important in physics because it can describe how the symmetry group of a physical system affects the solutions of equations describing that system. [9] Representation theory is pervasive across fields of mathematics. The applications of representation theory are diverse. [10]
A Feynman diagram is a representation of quantum field theory processes in terms of particle interactions. The particles are represented by the diagram lines. The lines can be squiggly or straight, with an arrow or without, depending on the type of particle.