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1 About this webpage

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2 Introductory example

While this appendix does depend on the main document, I will also try to make this as self–contained as possible. To that end, I will provide a well-known motivation coming from quantum harmonic oscillators. Even from this most basic example, we will be able to conclude that orthogonal polynomials occur in the matrix entries of representations. Since vertex weights are entries of the \(R\)–matrix coming from representation theory, we expect that vertex weights are orthogonal polynomials.

In contrast to classical mechanics where position and momentum are either vectors or scalars, in quantum mechanics these are operators. In general, these operators do not commute, and the non–commutativity of the position and momentum operators is related to Heisenberg’s uncertainty principle. If the two operators commuted, then both position and momentum could be measured simultaneously. Furthermore, states in a quantum mechanical system are actually “wavefunctions,” which are modeled as an element of some Hilbert space (on which the operators act).

In one dimension, for example, the position operator \(\hat {x}\) is multiplication by \(x\) while the momentum operator \( \hat {p}\) is \(-i \hbar \frac {\partial }{\partial x}. \) One can verify that if \( f_n \) denotes the function \(x^n\) then \[ \hat {x}\hat {p} f_n = - i \hbar n f_n , \quad \hat {p}\hat {x} = -i \hbar (n+1)f_n, \] and so (using Lie bracket notation) \[ [\hat {p}, \hat {x}]f_n = - i \hbar f_n \neq 0. \]

More generally, the time–dependent Schrödinger’s equation states that for the Hamiltonian operator \(\hat {H}\) and a time–dependent state \( \Psi (t) \) we have \[ i\hbar \frac {d}{dt} \vert \Psi (t) \rangle = \hat {H} \vert \Psi (t) \rangle \] where \(i\) is the imaginary unit and \(\hbar \) is Planck’s constant. In the time–independent version, we consider the case when \( \Psi (t) \) is a stationary state, so \( \Psi (t) = \Psi \) for all times \(t\). Then Schrödinger’s equation tells us that \[ \hat {H} \vert \Psi \rangle = E \vert \Psi \rangle , \] where \(E\) is the energy of the state. Note that this means that the energy can only take discrete, “quantized” values. Generally speaking, the Hamiltonian can be expressed as \[ \hat {H} = \hat {T} + \hat {V} \] where \(\hat {T} \) and \( \hat {V} \) are the kinetic and potential energy.

2.1 Quantum Harmonic Oscillator and Hermite Polynomials

For the harmonic oscillator, we get (in analogy with the classical harmonic oscillator) \[ \frac {\hat {p}^2}{2m} + \frac {1}{2}k \hat {x}^2. \] Plugging in \( \hat {p},\hat {x} \) we get a differential equation which can be explicitly solved. More specifically, the steady states are \[ \psi _n(x) = \frac {1}{\sqrt {2^n\,n!}} \left (\frac {m\omega }{\pi \hbar }\right )^{1/4} e^{- \frac {m\omega x^2}{2 \hbar }} H_n\left (\sqrt {\frac {m\omega }{\hbar }} x \right ), \qquad n = 0,1,2,\ldots . \] where \(H_n\) are the physicists’ Hermite polynomials \[ H_n(z)=(-1)^n e^{z^2}\frac {d^n}{dz^n}\left (e^{-z^2}\right ). \] The corresponding energy levels are: \[ E_n = \hbar \omega \bigl (n + \tfrac {1}{2}\bigr ). \] The Hermite polynomials are orthogonal polynomials in the sense that \[ \int _{-\infty }^{\infty } H_m(x) H_n(x) e^{-x^2}dx = \sqrt {\pi }2^n n! \delta _{nm}. \] They satisfy the generating function \[ e^{2xt - t^2} = \sum _{n=0}^\infty H_n(x) \frac {t^n}{n!} \] and the recurrence relation \[ H_{n+1}(x) = 2xH_n(x) - 2nH_{n-1}(x). \]

2.2 CCR relations and Lie algebras

Now, what does this have to do with Lie algebras? To answer this, we approach the quantum harmonic oscillator using “ladder operators.” Define two abstract algebra elements \( \hat {a} \text { and }\hat {a}^{\dagger }\) with canonical commutation relation \[ [\hat {a}, \hat {a}^{\dagger }]=1. \] There is a representation onto an infinite–dimensional vector space with bases \( \vert n \rangle , n=0,1,2\) given by \[ \begin{align} \hat {a}^\dagger |n\rangle &= \sqrt {n + 1} | n + 1\rangle \\ \hat {a}|n\rangle &= \sqrt {n} | n - 1\rangle . \end{align} \] For this reason, \(\hat {a}\) and \(\hat {a}^{\dagger }\) are called lowering and raising operators, respectively. We also introduce a “number operator” \[ \begin{align} N &= \hat {a}^\dagger \hat {a} \\ N\left | n \right \rangle &= n\left | n \right \rangle . \end{align} \]

In the context of the quantum harmonic oscillator, these algebra elements can be represented as \begin{align} \hat {a} &=\sqrt {m\omega \over 2\hbar } \left (\hat x + {i \over m \omega } \hat p \right ), \\ \hat {a}^\dagger &=\sqrt {m\omega \over 2\hbar } \left (\hat x - {i \over m \omega } \hat p \right ). \end{align}

Then \[ \hat {H} = \hbar \omega \left (N + \frac {1}{2}\right ), \] so the energy levels are \(E_n = n + \tfrac {1}{2},\) as we found before.

To relate the two methods in more mathematical formalism, we define the Hilbert space \(L^2(\mathbb {R},w) \) of square–integrable functions on \(\mathbb {R}\) with respect to the weight \(w(x)=e^{-x^2/2}.\) Then this Hilbert space has an orthogonal basis \( \{H_n: n \geq 0\} \) consisting of the Hermite polynomials. If we let \(\mathcal {A}\) denote the algebra generated by \(\hat {a},\hat {a}^{\dagger }\) then we just defined a representation of \(\mathcal {A}\) on \(L^2(\mathbb {R},w)\) where \(\vert n \rangle = H_n. \) Note that we can define a map from \(\mathcal {A}\) to \(B(L^2(\mathbb {R},w))\). Using the generating function and recurrence relation, we have \[ x H_n(x) = \frac {1}{2}H_{n+1}(x) + n H_{n-1}(x) \] and \[ \frac {d}{dx} H_n(x) = 2n H_{n-1}(x). \] Then the ladder operators act as \[ \hat {a} = \frac {1}{\sqrt {2}}\left (x + \frac {d}{dx}\right ), \quad \hat {a}^\dagger = \frac {1}{\sqrt {2}}\left (x - \frac {d}{dx}\right ). \] Then using the formulas above, these act on the wavefunctions as \[ \hat {a}^\dagger \psi _n = \sqrt {n+1} \, \psi _{n+1} \] \[ \hat {a} \psi _n = \sqrt {n} \, \psi _{n-1} \]

Next, we relate Lie algebras to ladder operators using the Jordan–Schwinger realization. It may be more illustrative to demonstrate this first in the case of \(\mathfrak {sl}_2\). Let \(\mathcal {A}_n\) be generated by \(\hat {a}_i,\hat {a}_i^{\dagger }, i=1,2,\ldots ,n\) with the commutation relations \[ [\hat {a}_i, \hat {a}_j^{\dagger }]=1 \delta _{ij}, \quad [\hat {a}_i,\hat {a}_j] = [\hat {a}^{\dagger }_i, \hat {a}^{\dagger }_j]=0 \text { for } i\neq j. \] The Jordan–Schwinger realization of \(\mathfrak {sl}_2\) is then an injection from \(\mathfrak {sl}_2\) to \(\mathcal {A}_2\). In a typical physics introduction, this example is discussed in the context of angular momentum. In this context, we recall the Pauli spin matrices \[ \sigma ^1 = \left (\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right ), \quad \quad \sigma ^2 = \left ( \begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right ), \quad \quad \sigma ^3 = \left ( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right ). \] which span \(\mathfrak {sl_2}\). The change of basis to \(e,f,h\) is given by \begin{align*} e &= \frac {\sigma _1 + i\sigma _2}{2} & \sigma _1 &= e+f \\ f &= \frac {\sigma _1 - i\sigma _2}{2} & \sigma _2 &= -i(e-f) \\ h &= \sigma _3 & \sigma _3 &= h \end{align*}

The Pauli matrices are then written in terms of the ladder operators as \[ \sigma _1 = \hat {a}_1\hat {a}_2^{\dagger } + \hat {a}_1^{\dagger }\hat {a}_2, \quad \sigma _2 = -i( \hat {a}_1^{\dagger }\hat {a}_2 - \hat {a}_1\hat {a}_2^{\dagger } ), \quad \sigma _3 = \hat {a}_1^{\dagger }\hat {a}_1 - \hat {a}_2^{\dagger }\hat {a}_2 \] By comparing the last formula to the change of basis, it is reasonable to conjecture that the elements \(\hat {a}_1^{\dagger }\hat {a}_2, \hat {a}_1\hat {a}_2^{\dagger }, \hat {a}_1^{\dagger }\hat {a}_1 - \hat {a}_2^{\dagger }\hat {a}_2\) then satisfy the same relations as \(e,f,h\in \mathfrak {sl}_2\), respectively. Indeed this is the case; for example, the relation \([h,e]=2e\) can be seen in \begin{align*} \left ( \hat {a}_1^{\dagger }\hat {a}_1 - \hat {a}_2^{\dagger }\hat {a}_2 \right ) \cdot \left ( \hat {a}_1^{\dagger }\hat {a}_2 \right ) &= \hat {a}_1^{\dagger }\hat {a}_1\hat {a}_1^{\dagger }\hat {a}_2 - \hat {a}_2^{\dagger }\hat {a}_2\hat {a}_1^{\dagger }\hat {a}_2\\ &= \hat {a}_1^{\dagger }( \hat {a}_1^{\dagger }\hat {a}_1 + 1)\hat {a}_2 - \hat {a}_1^{\dagger } (\hat {a}_2\hat {a}_2^{\dagger } - 1) \hat {a}_2 \\ &= \hat {a}_1^{\dagger }\hat {a}_2\hat {a}_1^{\dagger }\hat {a}_1 + \hat {a}_1^{\dagger }\hat {a}_2 - \hat {a}_1^{\dagger }\hat {a}_2\hat {a}_2^{\dagger }\hat {a}_2 + \hat {a}_1^{\dagger }\hat {a}_2 \\ &= \left ( \hat {a}_1^{\dagger }\hat {a}_2 \right )\cdot \left ( \hat {a}_1^{\dagger }\hat {a}_1 - \hat {a}_2^{\dagger }\hat {a}_2 +2\right ). \end{align*}

More generally, there is an injection from any finite–dimensional Lie algbra \(\mathfrak {g}\) to some \(\mathcal {A}_n\). Since there is an injection \(\mathrm {ad}: \mathfrak {g} \rightarrow \mathfrak {gl}(\mathfrak {g})\), without loss of generality we can take \( \mathfrak {g} = \mathfrak {gl}(n)\). The Jordan–Schwinger realization is then an injection of Lie algebras defined by \[ M \rightarrow \sum _{i,j=1}^n \hat {a}^{\dagger }_i M_{ij} \hat {a}_j. \] One can check that this generalizes the case when \(\mathfrak {g}=\mathfrak {sl}_2\).

As a result, if there is a Jordan–Schwinger realization \(\mathfrak {g} \hookrightarrow \mathcal {A}_n\) then any representation of \(\mathcal {A}_n\) pulls back to a representation of \(\mathfrak {g}.\) Since we have already seen the Hermite polynomials appear as an orthonormal basis of a representation of \(\mathcal {A}_1\), it is not unreasonable to expect orthogonal polynomials to appear in the representations of \(\mathfrak {g}.\) Note that if we had instead picked a basis of \(1,x,x^2,\ldots \), then after a change of basis, we would see the Hermite polynomials in the matrix elements of representations of \(\mathcal {A}_1\).

Of course, the Jordan–Schwinger realization does not produce interesting examples of Lie algebras inside \(\mathcal {A}_1\), so we will have to go to higher–dimensional examples. This will result in generalizations of the Hermite polynomials, which leads into the next section.

3 Hierarchy of Orthogonal Polynomials

The Askey–Wilson scheme is a way of organizing orthogonal polynomials of hypergeometric or basic hypergeometric type into a hierarchy. Below is the hierarchy for basic hypergeometric type (courtesy of Koornwinder):

Note that Hermite polynomials are in this flowchart, as a \(q\rightarrow 1\) limit of the various \(q\)–Hermite polynomials. We note that the \(q\)–deformation here is a different quantization than the quantization in the quantum harmonic oscillator; roughly, this corresponds to quantizing the product in one case, and quantizing the co–product in the other case.

3.1 Definitions

Define the \(q\)–Pochhammer symbol by \[ (a;q)_n = (1-a)(1-aq) \cdots (1-aq^{n-1}) \] and \[ (a_1,\ldots ,a_r)_n = (a_1;q)_n \cdots (a_r;q)_n \] The basic hypergeometric series, or \(q\)–hypergeometric series is defined by ([KLSBook]) \[ { }_{r} \phi _{s}\left [\begin{array}{cccc}a_{1} & a_{2} & \ldots & a_{j} \\ b_{1} & b_{2} & \ldots & b_{k}\end{array} ; q, z\right ] :=\sum _{k=0}^{\infty } \frac {\left (a_{1}, \ldots , a_{r} ; q\right )_{k}}{\left (b_{1}, \ldots , b_{s} ; q\right )_{k}}(-1)^{(1+s-r) k} q^{(1+s-r)\binom {k}{2}} \frac {z^{k}}{(q ; q)_{k}} \] The various orthogonal polynomials are defined in terms of the \(q\)–hypergeometric series. Note that if \(a_1=q^{-n}\) then the series terminates, justifiably calling it a polynomial. One example are \(q\)–Racah polynomials, which are defined by \[ R_{n}(\mu (x) ; \alpha , \beta , \gamma , \delta | q) = { }_{4} \phi _{3}\left (\begin{array}{c}q^{-n}, \alpha \beta q^{n+1}, q^{-x}, \gamma \delta q^{x+1} \\ \alpha q, \beta \delta q, \gamma q\end{array} ; q, q\right ), n = 0,1,2,\ldots , N. \] where \[ \mu (x):=q^{-x}+\gamma \delta q^{x+1} \] and \[ \alpha q=q^{-N} \quad \text { or } \quad \beta \delta q=q^{-N} \quad \text { or } \quad \gamma q=q^{-N}. \] Note that \( {}_{4} \phi _{3} \) depends on seven variables whereas the \(q\)–Racah depends on six variables; if the relation \(a_1a_2a_3a_4 = q^{-1}b_1b_2b_3\) holds, and one of \( \{b_1,b_2,b_3\}\) is of the form \(q^{-N}\), then the \( {}_4 \phi _{3} \) function can be written as a \(q\)–Racah.

It turns out that the weights of the \(R\)–matrix of \( \mathcal {U}_q(\widehat {sl_2}) \) are given by the \(q\)–Racah polynomials. Before stating that result, we define some notation. We will let \(m\) denote the maximum number of particles that can enter from the vertical direction, and let \(l\) denote the maximum number of particles that can enter from the horizontal direction. So the stochastic matrix is an operator on \(V_l \otimes V_m\) where \(V_l\) is \(l+1\)–dimensional and \(V_m\) is \(m+1\)–dimensional. We denote a matrix entry by \[ [S(z)]_{j_1,k_1}^{j_2,k_2}, \quad 1 \leq j_1,j_2 \leq l, \quad 1 \leq k_1,k_2 \leq m. \] This corresponds to this weight:

Of course, the weights of the \(R\)–matrix are generally not stochastic. There is a type of transformation that will make it stochastic.

We will state a theorem from [CP16] and [Kua18]; see also [KMMO16] and [BM16] for higher rank generalizations; and the previous papers of [Pov13] and [Man14].

References