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:

• ${\displaystyle {\text{U}}(1)}$: Part of the gauge group of the Standard model, and the gauge group for theories of electromagnetism. Irreps are all 1 dimensional and are indexed by integers ${\displaystyle \mathbb {Z} }$, given explicitly by ${\displaystyle \rho _{n}:{\text{U}}(1)\rightarrow {\text{GL}}(\mathbb {C} );e^{i\theta }\mapsto e^{in\theta }}$. The index can be understood as the winding number of the map.
• ${\displaystyle {\text{SU}}(2)\cong {\text{Spin}}(3)}$: Part of the gauge group of the Standard model. Irreps are indexed by non-negative integers in ${\displaystyle n\in \mathbb {N} _{\geq 0}}$, with ${\displaystyle n}$ describing the dimension of the representation, or, with appropriate normalisation, the highest weight of the representation. In physics it is common convention to label these by half-integers instead. See Representation theory of SU(2).
• ${\displaystyle {\text{SO}}(3)}$: The group of rotations of 3D space. Irreps are the odd-dimensional irreps of ${\displaystyle {\text{SU}}(2)}$
• ${\displaystyle {\text{SU}}(3)}$: Part of the gauge group of the Standard model. Irreps are indexed pairs of non-negative integers ${\displaystyle (m,n)}$, describing the highest weight of the representation. See Clebsch-Gordan coefficients for SU(3).
• ${\displaystyle {\text{SO}}(1,3)}$: The Lorentz group, the linear symmetries of flat spacetime. All representations arise as representations of its corresponding spin group. See Representation theory of the Lorentz group.
• ${\displaystyle {\text{SL}}(2,\mathbb {C} )\cong {\text{Spin}}(1,3)}$: The spin group of ${\displaystyle {\text{SO}}(1,3)}$. Irreps are indexed by pairs of non-negative integers ${\displaystyle (\mu ,\nu )}$, indexing the dimension of the representation.
• ${\displaystyle {\text{E}}(1,3)\cong \mathbb {R} ^{1,3}\rtimes {\text{SO}}(1,3)}$: The Poincaré group of isometries of flat spacetime. This can be understood in terms of the representation theory of the groups above. See Wigner's classification.

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:

• Conformal symmetry: For pseudo-Euclidean space, symmetries are described by the conformal group ${\displaystyle {\text{Conf}}(p,q)\cong O(p,q)/\mathbb {Z} _{2}}$.
• Supersymmetry: Symmetry described by a supergroup.
• Grand unified theories: Gauge groups which contain the Standard model gauge group as a subgroup. Proposed candidates include ${\displaystyle {\text{SU}}(5),{\text{SO}}(10)}$ and ${\displaystyle {\text{E}}_{6}}$.

Quantum field theory

In quantum physics, the mathematical notion is usually applied to representations of the gauge group. For example, an ${\displaystyle {\text{SU}}(2)}$ gauge theory will have multiplets which are fields whose representation of ${\displaystyle {\text{SU}}(2)}$ is determined by the single half-integer number ${\displaystyle s=:n/2}$, the isospin. Since irreducible ${\displaystyle {\text{SU}}(2)}$ representations are isomorphic to the ${\displaystyle n}$th symmetric power of the fundamental representation, every field has ${\displaystyle n}$ symmetrized internal indices.

Fields also transform under representations of the Lorentz group ${\displaystyle {\text{SO}}(1,3)}$, or more generally its spin group ${\displaystyle {\text{Spin}}(1,3)}$ which can be identified with ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$ due to an exceptional isomorphism. Examples include scalar fields, commonly denoted ${\displaystyle \phi }$, which transform in the trivial representation, vector fields ${\displaystyle A_{\mu }}$ (strictly, this might be more accurately labelled a covector field), which transforms as a 4-vector, and spinor fields ${\displaystyle \psi _{\alpha }}$ such as Dirac or Weyl spinors which transform in representations of ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$. A right-handed Weyl spinor transforms in the fundamental representation, ${\displaystyle \mathbb {C} ^{2}}$, of ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$.

Beware that besides the Lorentz group, a field can transform under the action of a gauge group. For example, a scalar field ${\displaystyle \phi (x)}$, where ${\displaystyle x}$ is a spacetime point, might have an isospin state taking values in the fundamental representation ${\displaystyle \mathbb {C} ^{2}}$ of ${\displaystyle {\text{SU}}(2)}$. Then ${\displaystyle \phi (x)}$ 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.

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.