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A generalization of the concept of a multiplet. A multiplet is a set of quantum-mechanical states, each of which has the same value of some fundamental quantum number and differs from the other members of the set by another quantum number which takes values from a range of numbers dictated by the fundamental quantum number. The number of states in the set is called the multiplicity or dimension of the multiplet. The concept was originally introduced to describe the set of states in a nonrelativistic quantum-mechanical system with the same value of the orbital angular momentum, L, and different values of the projection of the angular momentum on an axis, M. The values that M can take are the integers between -L and L, 2L + 1 in all. This is the dimension of the multiplet. If the hamiltonian operator describing the system is rotationally invariant, all states of the multiplet have the same energy. A supermultiplet is a generalization of the concept of multiplet to the case when there are several quantum numbers that describe the quantum-mechanical states. See Angular momentum, Symmetry laws (physics)
Both concepts, multiplet and supermultiplet, acquire a precise mathematical meaning by the use of the theory of group transformations. A multiplet is an irreducible representation of a group, G. The quantum number called fundamental described above labels the representation of the group. The other quantum number labels the representation of a subgroup G′ of G. For angular momentum, the group G is the rotation group, called special orthogonal group in three dimensions, SO(3), and its subgroup G′ is the group of special orthogonal transformations in two dimensions, SO(2). A supermultiplet is a generalization to the case in which the group G is not a group of rank one but has larger rank. A group of rank one has only one quantum number to label its representations. The concept of a multiplet or supermultiplet is particularly useful in the classification of states of physical systems. See Quantum mechanics, Quantum numbers
The term supermultiplet was first used by E. P. Wigner in 1932 in order to classify the quantum-mechanical states of light atomic nuclei. The constituents of these are protons, p, and neutrons, n. Each proton and neutron has an intrinsic spin, S, of ½ in units of ℏ, which is Planck's constant divided by 2&pgr;. The projection of the intrinsic spin on an axis, Sz, is then Sz = ½ or -½ (spin up or down). In addition to having the same spin, the proton and neutron have essentially the same mass but differ in that the proton is charged whereas the neutron is not. They can thus be regarded as different charge states of the same particle, a nucleon. The distinction can be made formal by introducing a quantum number called isotopic spin, T, which has the value ½. The two charge orientations, Tz, are taken to be ½ for the proton and -½ for the neutron. There are thus four constituents of nuclei, protons and neutrons with spin up and down, that is, p ↑, p ↓, n ↑, and n ↓. The set of transformations among these constituents forms a group called SU(4), the special unitary group in four dimensions. This is the group G for Wigner's theory. The representations of SU(4), that is, Wigner supermultiplets, are characterized by three quantum numbers (λ1, λ2, λ3) with λ1 ≥ λ2 ≥ λ3. See I-spin, Nuclear structure