Representation theory of SU(2)

In the study of the representation theory of Lie groups, the study of representations of SU(2) is fundamental to the study of representations of semisimple Lie groups. It is the first case of a Lie group that is both a compact group and a non-abelian group. The first condition implies the representation theory is discrete: representations are direct sums of a collection of basic irreducible representations (governed by the Peter–Weyl theorem). The second means that there will be irreducible representations in dimensions greater than 1.

SU(2) is the universal covering group of SO(3), and so its representation theory includes that of the latter, by dint of a surjective homomorphism to it. This underlies the significance of SU(2) for the description of non-relativistic spin in theoretical physics; see below for other physical and historical context.

As shown below, the finite-dimensional irreducible representations of SU(2) are indexed by an integer or half-integer $λ \geq 0$ , and have dimension $2 λ + 1$ .

Lie algebra representations

The representations of the group are found by considering representations of ${\mathfrak {su}}(2)$ , the Lie algebra of SU(2). In principle this is the 'infinitesimal version' of SU(2); Lie algebras consist of infinitesimal transformations, and their Lie groups consist of 'integrated' transformations. Since the group SU(2) is simply connected, every representation of its Lie algebra can be integrated to a group representation.^[1] The group representations can be realized on spaces of polynomials in two complex variables.^[2] In what follows, we shall consider the complex Lie algebra (i.e. the complexification of the Lie algebra), which doesn't affect the representation theory. A reference for this material is Section 4.6 of (Hall 2015).

The Lie algebra is spanned by three elements $e$ , $f$ and $h$ with the Lie brackets

{\begin{aligned}{[}h,e]&=e\\[3pt][h,f]&=-f\\[3pt][e,f]&=h\end{aligned}}

(These elements may be expressed in terms of matrices $I 1$ , $I 2$ and $I 3$ which are related to the Pauli matrices by multiplication by a factor of $- i$ . $e = I 1 + i I 2$ , $f = I 1 - i I 2$ , and $h = I 3$ .)

Since ${\mathfrak {su}}(2)$ is semisimple, the representation $ρ (h)$ is always diagonalizable (for complex number scalars). Its eigenvalues are called the weights. Its eigenvectors can be taken as a basis for the vector space the group acts upon. The dimension of the representation can be determined by counting the number of these eigenvectors.

Suppose $x$ is an eigenvector of weight $α$ . Then,

{\begin{aligned}h[x]&=\alpha x\\[3pt]h[e[x]]&=(\alpha +1)e[x]\\[3pt]h[f[x]]&=(\alpha -1)f[x]\end{aligned}}

In other words, $e$ raises the weight by one and $f$ reduces the weight by one. $e$ and $f$ are referred to as ladder operators, taking us between eigenvectors or to 0. A consequence is that

h^{2}+ef+fe

is a Casimir invariant and commutes with the generators of the algebra. By Schur's lemma, its action is proportional to the identity map, for irreducible representations. It is convenient to write the constant of proportionality as $λ (λ + 1)$ . (The expression $h^{2}+ef+fe$ is equal to $I^{2}$ defined as $I_{1}^{2}+I_{2}^{2}+I_{3}^{2}$ , which is related to the magnitude of angular momentum operator in quantum physics.)

Weights

Finite-dimensional representations only have finitely many weights, and have a greatest and least weight. (They are both highest weight representations and lowest weight representations.)

Let $α 1$ be a weight which is greater than all the other weights. Let $x$ be an $h$ -eigenvector of eigenvalue $\alpha _{1}$ . Then $e (x) = 0$ . If the representation is irreducible, using the commutation relations we can calculate that $(h^{2}+ef+fe)x=(\alpha _{1}^{2}+\alpha _{1})x=\lambda (\lambda +1)x$ . Since $x$ is nonzero, $\alpha _{1}$ is either $λ$ or $- λ - 1$ .

Likewise, let $α 2$ be a weight which is lower than all the other weights. Let $x$ be an eigenvector of $α 2$ , so $f (x) = 0$ . If the representation is irreducible, using the commutation relations $(\alpha _{2}^{2}-\alpha _{2})x=\lambda (\lambda +1)x$ , and so $\alpha _{2}$ is either $λ + 1$ or $- λ$ .

For an irreducible finite-dimensional representation, the highest weight can't be less than the lowest weight. In addition, the difference between them has to be an integer because if the difference isn't an integer, there will always be a weight which is one more or one less than any given weight, contradicting the assumption of finite-dimensionality.

Since $λ < λ + 1$ and $- λ - 1 < - λ$ , without any loss of generality we can assume the highest weight is $λ$ (if it's $- λ - 1$ , just redefine a new $λ'$ as $- λ - 1$ ) and the lowest weight would then have to be $- λ$ . This means λ has to be an integer or half-integer. Every weight is a number between $λ$ and $- λ$ which differs from them by an integer.

Furthermore, each weight has multiplicity one. If this were not the case, we could define a proper subrepresentation generated by an eigenvector of $λ$ and $f$ applied to it any number of times, contradicting the assumption of irreducibility.

This construction also shows for any given nonnegative integer multiple of half $λ$ , all finite-dimensional irreps with $λ$ as its highest weight are equivalent (just make an identification of a highest weight eigenvector of one with one of the other).

Another approach

See under the example for Borel–Weil–Bott theorem.

Properties

All irreps for a half-integer (not integer) $λ$ are faithful. All irreps for an integer $λ$ have the kernel $\pm 1$ and are virtually representations of SO(3), faithful ones for $λ \geq 1$ .

Most important irreducible representations and their applications

As stated above, representations of SU(2) describe non-relativistic spin, due to being a double covering of the rotation group of Euclidean 3-space. Relativistic spin is described by the representation theory of SL₂(C), a supergroup of SU(2), which in a similar way covers SO⁺(1;3), the relativistic version of the rotation group. SU(2) symmetry also supports concepts of isobaric spin and weak isospin, collectively known as isospin.

$λ =$ ⁠1/2⁠ gives the 2 representation, the fundamental representation of SU(2). When an element of SU(2) is written as a complex $2 \times 2$ matrix, it is simply a multiplication of column 2-vectors. It is known in physics as the spin-½ and, historically, as the multiplication of quaternions (more precisely, multiplication by a unit quaternion).

$λ = 1$ gives the 3 representation, the adjoint representation. It describes 3-d rotations, the standard representation of SO(3), so real numbers are sufficient for it. Physicists use it for the description of massive spin-1 particles, such as vector mesons, but its importance for spin theory is much higher because it anchors spin states to the geometry of the physical 3-space. This representation emerged simultaneously with the 2 when William Rowan Hamilton introduced versors, his term for elements of SU(2). Note that Hamilton did not use standard group theory terminology since his work preceded Lie group developments.

The $λ =$ ⁠3/2⁠ representation is used in particle physics for certain baryons, such as the Δ.

References

^ Hall 2015 Theorem 5.6
^ Hall 2015 Section 4.2

Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer
Gerard 't Hooft (2007), Lie groups in Physics, Chapter 5 "Ladder operators"
Iachello, Francesco (2006), Lie Algebras and Applications, Lecture Notes in Physics, vol. 708, Springer, ISBN 3540362363

[1] Hall 2015 Theorem 5.6

[2] Hall 2015 Section 4.2

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