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Wikipedia / 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.

Mathematical formulation

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).

where

  • is a vector space over a field

  • is a Lie group. This is often a compact Lie group.

  • .

, where

  • is as before.

  • .

  • .

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.

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.

-plet.

. The group of 3D rotations SO(3). Since the dimension of this representation space is 3, this is known as the triplet.

Application to theoretical physics

For applications to theoretical physics, we can restrict our attention to the representation theory of a handful of physically important groups. Many of these have well understood representation theory:

These groups all appear in the theory of the Standard model. For theories which extend these symmetries, the representation theory of some other groups might be considered:

  • .

  • Supersymmetry: Symmetry described by a supergroup.

  • .

Physics

Quantum field theory

In quantum physics, the mathematical notion is usually applied to representations of the gauge group gauge theory will have multiplets which are fields symmetrized internal indices.

due to an exceptional isomorphism. Examples include scalar fields such as Dirac or Weyl.

is a vector valued function of spacetime, but is still referred to as a scalar field, as it transforms trivially under Lorentz transformations.

In quantum field theory different particles correspond one to one with gauged fields transforming in irreducible representations of the internal and Lorentz group. Thus, a multiplet has also come to describe a collection of subatomic particles described by these representations.

Examples

The best known example is a spin multiplet, which describes symmetries of a group representation of an SU(2) subgroup of the Lorentz algebra, which is used to define spin quantization. A spin singlet is a trivial representation, a spin doublet is a fundamental representation and a spin triplet is in the vector representation or adjoint representation.

In QCD, quarks are in a multiplet of SU(3), specifically the three-dimensional fundamental representation.

Other uses

Spectroscopy

In spectroscopy, particularly Gamma spectroscopy and X-ray spectroscopy, a multiplet is a group of related or unresolvable spectral lines. Where the number of unresolved lines is small, these are often referred to specifically as doublet or triplet peaks, while multiplet is used to describe groups of peaks in any number.

References

  • Georgi, H. (1999). Lie Algebras in Particle Physics: From Isospin to Unified Theories (1st ed.). CRC Press. https://doi.org/10.1201/9780429499210

See also