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
In mathematics, it is common to refer to the homomorphism
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.
Generally, a group may have multiple non-isomorphic representations of the same dimension, so this does not fully characterize the representation.
For example, consider real three-dimensional space,
This explicit realisation of the rotation group is known as the fundamental representation
For applications to theoretical physics, we can restrict our attention to the representation theory of a handful of physically important groups.
For theories which extend these symmetries, the representation theory of some other groups might be considered: In quantum physics, the mathematical notion is usually applied to representations of the gauge group.
gauge theory will have multiplets which are fields whose representation of
is determined by the single half-integer number
th symmetric power of the fundamental representation, every field has
Fields also transform under representations of the Lorentz group
, or more generally its spin group
Examples include scalar fields, commonly denoted
, which transform in the trivial representation, vector fields
(strictly, this might be more accurately labelled a covector field), which transforms as a 4-vector, and spinor fields
such as Dirac or Weyl spinors which transform in representations of
A right-handed Weyl spinor transforms in the fundamental representation,
Beware that besides the Lorentz group, a field can transform under the action of a gauge group.
For example, a scalar field
is a spacetime point, might have an isospin state taking values in the fundamental representation
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.
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.
In QCD, quarks are in a multiplet of SU(3), specifically the three-dimensional fundamental representation.
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.