8.2 Representations of SU(2) : construction

Concept links · terms present in this machine draft; source roles are unverified: vector space · complex inner product · eigenvalue · Lie algebra · Lie algebra representation · irreducible representation · Lie group

The argument of the previous section only tells us what properties possible finite dimensional irreducible representations of $S U (2)$ must have. It shows how to construct such representations given a highest weight vector, but does not provide any way to construct such highest weight vectors. We would like to find some method to explicitly construct an irreducible $(π_{n}, V^{n})$ for each highest weight $n .$ There are several possible constructions, but perhaps the simplest one is the following, which gives a representation of highest weight $n$ by looking at polynomials in two complex variables, homogeneous of degree $n$ . This construction will produce representations not just of $S U (2)$ , but of the larger group $G L (2, C)$

Recall from equation 1.3 that if one has an action of a group on a space $M$ , one can get a representation on functions $f$ on $M$ by taking

(π (g) f) (x) = f (g^{- 1} \cdot x)

For the group $G L (2, C)$ , we have by definition an action on $M = C^{2}$ , and we look at a specific class of functions on this space, the polynomials. We can break up the infinite dimensional space of polynomials on $C^{2}$ into finite dimensional subspaces as follows:

Definition (Homogeneous polynomials)

The complex vector space of homogeneous polynomials of degree $n$ in two complex variables $z_{1}, z_{2}$ is the space of functions on $C^{2}$ of the form
$f (z_{1}, z_{2}) = a_{0} z_{1}^{n} + a_{1} z_{1}^{n - 1} z_{2} + \dots + a_{n - 1} z_{1} z_{2}^{n - 1} + a_{n} z_{2}^{n}$

The space of such functions is a complex vector space of dimension $n + 1$

This space of functions is exactly the representation space $V^{n}$ that we need to get the spin $\frac{n}{2}$ irreducible representation of $S U (2) \subset G L (2, C)$

If we choose a basis $e_{1}, e_{2}$ of $C^{2}$ , then we can write $g$ as the matrix

g = (α γ β δ) \in G L (2, C)

The coordinates $z_{1}, z_{2}$ will be the dual basis of the linear functions on $C^{2}$ and (see the discussion at the end of sections 4.1 and 4.2) g will act on them by

(z _{2} z _{1}) \to g^{- 1} (z _{2} z _{1})

The representation $π_{n} (g)$ on homogeneous polynomial functions will be given by this action on the $z_{1}, z_{2}$ in the expression for the polynomial.

Taking the derivative, the Lie algebra representation is given by

π_{n}^{'} (X) f = \frac{d}{d t} π_{n} (e^{tX}) f_{∣ t = 0} = \frac{d}{d t} f (e^{- tX} (z _{2} z _{1}))_{∣ t = 0}

where $X \in gl (2, C)$ is any 2 by 2 complex matrix. By the chain rule, for

(z _{2} ( t ) z _{1} ( t )) = e^{- tX} (z _{2} z _{1})

this is

π_{n}^{'} (X) f = (\frac{\partial f}{\partial z _{1}}, \frac{\partial f}{\partial z _{2}}) (\frac{d}{d t} e^{- tX} (z _{2} z _{1}))_{∣ t = 0} = - \frac{\partial f}{\partial z _{1}} (X_{11} z_{1} + X_{12} z_{2}) - \frac{\partial f}{\partial z _{2}} (X_{21} z_{1} + X_{22} z_{2})

where the $X_{j k}$ are the components of the matrix $X$ .

Computing what happens for $X_{j} = - i \frac{σ _{j}}{2}$ (a basis of su(2), we get

(π_{n}^{'} (X_{3}) f) (z_{1}, z_{2}) = \frac{i}{2} (\frac{\partial f}{\partial z _{1}} z_{1} - \frac{\partial f}{\partial z _{2}} z_{2})

π_{n}^{'} (X_{3}) = \frac{i}{2} (z_{1} \frac{\partial}{\partial z _{1}} - z_{2} \frac{\partial}{\partial z _{2}})

and similarly

π_{n}^{'} (X_{1}) = \frac{i}{2} (z_{1} \frac{\partial}{\partial z _{2}} + z_{2} \frac{\partial}{\partial z _{1}}), π_{n}^{'} (X_{2}) = \frac{1}{2} (z_{2} \frac{\partial}{\partial z _{1}} - z_{1} \frac{\partial}{\partial z _{2}})

The $z_{1}^{k} z_{2}^{n - k}$ for $k = 0, \dots n$ are eigenvectors for $S_{3} = i X_{3}$ with eigenvalue $\frac{1}{2} (n - 2 k)$ since

π_{n}^{'} (S_{3}) z_{1}^{k} z_{2}^{n - k} = \frac{1}{2} (- k z_{1}^{k} z_{2}^{n - k} + (n - k) z_{1}^{k} z_{2}^{n - k}) = \frac{1}{2} (n - 2 k) z_{1}^{k} z_{2}^{n - k}

$z_{2}^{n}$ will be an explicit highest weight vector for the representation $(π_{n}, V^{n})$

An important thing to note here is that the formulas we have found for $π_{n}^{'}$ are not in terms of matrices. Instead we have seen that when we construct our representations using functions on $C^{2}$ , for any $X \in gl (2, C)), π_{n}^{'} (X)$ is given by a diferential operator. These diferential operators are independent of $n,$ with the same operator $π^{'} (X)$ on all the $V^{n}$ . This is because the original definition of the representation

(π (g) f) (x) = f (g^{- 1} \cdot x)

is on the full infinite dimensional space of polynomials on $C^{2}$ . While this space is infinite dimensional, issues of analysis don’t really come into play here, since polynomial functions are essentially an algebraic construction.

Restricting the diferential operators $π^{'} (X)$ to $V^{n}$ , the homogeneous polynomials of degree $n,$ they become linear operators on a finite dimensional space. We now have an explicit highest weight vector, and an explicit construction of the corresponding irreducible representation. If one chooses a basis of $V^{n}$ then the linear operator $π^{'} (X)$ will be given by a $n$ + 1 by $n + 1$ matrix. Clearly though, the expression as a simple first-order diferential operator is much easier to work with. In the examples we will be studying in later chapters, the representations under consideration will often be on function spaces, with Lie algebra representations appearing as diferential operators. Instead of using linear algebra techniques $to$ find eigenvalues and eigenvectors, the eigenvector equation will be a partial diferential equation, with our focus on using Lie groups and their representation theory to solve such equations.

One issue we haven’t addressed yet is that of unitarity of the representation. We need Hermitian inner products on the spaces $V^{n}$ , inner products that will be preserved by the action of $S U (2)$ that we have defined on these spaces. A standard way to define a Hermitian inner product on functions on a space $M$ is to define them using an integral: for $f, g$ complex-valued functions on $M .$ , take their inner product to be

⟨ f, g ⟩ = \int_{M} \overline{f} g

While for $M = C^{2}$ this gives an $S U (2)$ invariant inner product on functions (one that is not invariant for the full group $G L (2, C))$ , it is useless for $f, g$ polynomial, since such integrals diverge. In this case an inner product on polynomial functions on $C^{2}$ can be defined by

⟨ f, g ⟩ = \frac{1}{π ^{2}} \int_{C^{2}} \overline{f (z_{1}, z_{2})} g (z_{1}, z_{2}) e^{- (∣ z_{1} ∣^{2} + ∣ z_{2} ∣^{2})} d x_{1} d y_{1} d x_{2} d y_{2} (8.2)

Here $z_{1} = x_{1} + i y_{1}, z_{2} = x_{2} + i y_{2}$ . Integrals of this kind can be done fairly easily since they factorize into separate integrals over $z_{1}$ and $z_{2}$ , each of which can be treated using polar coordinates and standard calculus methods. One can check by explicit computation that the polynomials

\frac{z _{1}^{j} z _{2}^{k}}{j ! k !}

will be an orthonormal basis of the space of polynomial functions with respect to this inner product, and the operators $π^{'} (X), X \in su (2)$ will be skew-adjoint.

Working out what happens for the first few examples of irreducible $S U (2)$ representations, one finds orthonormal bases for the representation spaces $V^{n}$ of homogeneous polynomials as follows

∙ For n = s = 0

∙ For n = 1, s = \frac{1}{2}

z_{1}, z_{2}

• For $n$ = 2, $s$ = 1

\frac{1}{2} z_{1}^{2}, z_{1} z_{2}, \frac{1}{2} z_{2}^{2}

• For $n$ = 3, $s$ = ³₂

\frac{1}{6} z_{1}^{3}, \frac{1}{2} z_{1}^{2} z_{2}, \frac{1}{2} z_{1} z_{2}^{2}, \frac{1}{6} z_{2}^{3}

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8.2 Representations of SU(2) : construction

8.2 Representations of SU(2) : construction

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