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The term representation of a group is also used in a more general sense to mean any "description" of a group as a group of transformations of some mathematical object. More formally, a "representation" means a homomorphism from the group to the automorphism group of an object. If the object is a vector space we have a linear representation.
The two-dimensional "spin 1/2" representation of the Lie algebra so(3), for example, does not correspond to an ordinary (single-valued) representation of the group SO(3). (This fact is the origin of statements to the effect that "if you rotate the wave function of an electron by 360 degrees, you get the negative of the original wave function.")
Representation theory is used in many parts of mathematics, as well as in quantum chemistry and physics. Among other things it is used in algebra to examine the structure of groups. There are also applications in harmonic analysis and number theory. For example, representation theory is used in the modern approach to gain new results about ...
In mathematics and theoretical physics, Wigner's classification is a classification of the nonnegative ( ) energy irreducible unitary representations of the Poincaré group which have either finite or zero mass eigenvalues.
In mathematics and theoretical physics, a representation of a Lie group is a linear action of a Lie group on a vector space. Equivalently, a representation is a smooth homomorphism of the group into the group of invertible operators on the vector space. Representations play an important role in the study of continuous symmetry.
The column "representation" indicates under which representations of the gauge groups that each field transforms, in the order (SU(3), SU(2), U(1)) and for the U(1) group, the value of the weak hypercharge is listed. There are twice as many left-handed lepton field components as right-handed lepton field components in each generation, but an ...
Many of the representations, both finite-dimensional and infinite-dimensional, are important in theoretical physics. Representations appear in the description of fields in classical field theory, most importantly the electromagnetic field, and of particles in relativistic quantum mechanics, as well as of both particles and quantum fields in quantum field theory and of various objects in string ...
In mathematics, the representation theory of the Poincaré group is an example of the representation theory of a Lie group that is neither a compact group nor a semisimple group. It is fundamental in theoretical physics .