Multiplet

In physics and particularly in particle physics, a multiplet is the state space for 'internal' degrees of freedom of a particle, that is, degrees of freedom associated to a particle itself, as opposed to 'external' degrees of freedom such as the particle's position in space. Examples of such degrees of freedom are the spin state of a particle in quantum mechanics, or the color, isospin and hypercharge state of particles in the Standard model of particle physics. Formally, we describe this state space by a vector space which carries the action of a group of continuous symmetries. Mathematically, multiplets are described via representations of a Lie group or its corresponding Lie algebra, and is usually used to refer to irreducible representations (irreps, for short). At the group level, this is a triplet where is a vector space over a field (in the algebra sense) , generally taken to be or is a Lie group. This is often a compact Lie group. is a group homomorphism , that is, a map from the group to the space of invertible linear maps on . This map must preserve the group structure: for we have . At the algebra level, this is a triplet , where is as before. is a Lie algebra. It is often a finite-dimensional Lie algebra over or . is an Lie algebra homomorphism . This is a linear map which preserves the Lie bracket: for we have . The symbol is used for both Lie algebras and Lie groups as, at least in finite dimension, there is a well understood correspondence between Lie groups and Lie algebras. In mathematics, it is common to refer to the homomorphism as the representation, for example in the sentence 'consider a representation ', and the vector space is referred to as the 'representation space'. In physics sometimes the vector space is referred to as the representation, for example in the sentence 'we model the particle as transforming in the singlet representation', or even to refer to a quantum field which takes values in such a representation, and the physical particles which are modelled by such a quantum field.