25.4 Examples in d=2 and 3

25.4 Examples in d=2 and 3

25.4.1 Two degrees of freedom and SU(2)

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In the case $d=2$ , the action of the group $U(2)\subset Sp(4,\mathbf{R})$ discussed in section 25.3 commutes with the standard harmonic oscillator Hamiltonian and thus acts as symmetries on the quantum harmonic oscillator state space, preserving energy eigenspaces. Restricting to the subgroup $SU(2)\subset U(2)$ , we’ll see that we can recover our earlier (see section 8.2) construction of $SU(2)$ representations in terms of homogeneous polynomials, in a new context. This use of the energy eigenstates of a two dimensional harmonic oscillator appears in the physics literature as the “Schwinger boson method” for studying representations of $SU(2)$

The state space for the $d$ = 2 Bargmann-Fock representation, restricting to finite linear combinations of energy eigenstates, is

$$\mathcal{H}={\mathcal{F}}_2^{fin}=\mathbf{C}[z_1,z_2]$$

the polynomials in two complex variables $z_1,z_2$ . Recall from our $SU(2)$ discussion that it was useful to organize these polynomials into finite dimensional sets of homogeneous polynomials of degree $n$ for $n=0,1,2,...$

$$\mathcal{H}={\mathcal{H}}^0\oplus {\mathcal{H}}^1\oplus {\mathcal{H}}^2\oplus \dots$$

There are four annihilation or creation operators

$$a_1^{†}=z_1,a_2^{†}=z_2,a_1=\frac{\partial }{\partial z_1},a_2=\frac{\partial }{\partial z_2}$$

acting on $H$. These are the quantizations of complexified phase space coordinates $z_1,z_2,{\overline{z}}_1,{\overline{z}}_2$ , with quantization the Bargmann-Fock construction of the representation ${\Gamma }_{BF}^{\prime }$ of ${\mathfrak{h}}_{2d+1}={\mathfrak{h}}_5$

$${\Gamma }_{BF}^{\prime }(1)=-i1,{\Gamma }_{BF}^{\prime }(z_j)=-i{a}_j^{†},{\Gamma }_{BF}^{\prime }({\overline{z}}_j)=-i{a}_j$$

Quadratic combinations of the creation and annihilation operators give representations on $H$ of three subalgebras of the complexification $\mathfrak{sp}(4,\mathbf{C})$ of sp(4, $R$):

• A three dimensional commutative Lie sub-algebra spanned by ${\overline{z}}_1{\overline{z}}_2,{\overline{z}}_1^2,{\overline{z}}_2^2$ 2 with quantization

$${\Gamma }_{BF}^{\prime }({\overline{z}}_1{\overline{z}}_2)=-i{a}_1{a}_2,{\Gamma }_{BF}^{\prime }({\overline{z}}_1^2)=-i{a}_1^2,{\Gamma }_{BF}^{\prime }({\overline{z}}_2^2)=-i{a}_2^2$$

• A three dimensional commutative Lie sub-algebra spanned by $z_1{z}_2,z_1^2,z_2^2$ 2 with quantization

$${\Gamma }_{BF}^{\prime }(z_1{z}_2)=-i{a}_1^{†}{a}_2^{†},{\Gamma }_{BF}^{\prime }(z_1^2)=-i(a_1^{†}{)}^2,{\Gamma }_{BF}^{\prime }(z_2^2)=-i(a_2^{†}{)}^2$$

• A four dimensional Lie subalgebra isomorphic to gl(2, $C$) with basis

$$z_1{\overline{z}}_1,z_2{\overline{z}}_2,z_2{\overline{z}}_1,z_1{\overline{z}}_2$$

and quantization

$$\begin{array}{rl}{\Gamma }_{BF}^{\prime }(z_1{\overline{z}}_1)& =-\frac{i}{2}(a_1^{†}{a}_1+a_1{a}_1^{†}),{\Gamma }_{BF}^{\prime }(z_2{\overline{z}}_2)=-\frac{i}{2}(a_2^{†}{a}_2+a_2{a}_2^{†})\\ {\Gamma }_{BF}^{\prime }(z_2{\overline{z}}_1)& =-i{a}_2^{†}{a}_1,{\Gamma }_{BF}^{\prime }(z_1{\overline{z}}_2)=-i{a}_1^{†}{a}_2\end{array}$$

Real linear combinations of

$$z_1{\overline{z}}_1,z_2{\overline{z}}_2,z_1{\overline{z}}_2+z_2{\overline{z}}_1,i(z_1{\overline{z}}_2-z_2{\overline{z}}_1)$$

span the Lie algebra $\mathfrak{u}(2)\subset \mathfrak{sp}(4,\mathbf{R})$ , and ${\Gamma }_{BF}^{\prime }$ applied to these $\mathrm{gi}$ ves a unitary Lie algebra representation by skew-adjoint operators.

Inside this last subalgebra, there is a distinguished element $h=z_1{\overline{z}}_1+z_2{\overline{z}}_2$ that Poisson-commutes with the rest of the subalgebra (but not with elements in the first two subalgebras). Quantization of h gives the Hamiltonian operator

$$H=\frac{1}{2}(a_1{a}_1^{†}+a_1^{†}{a}_1+a_2{a}_2^{†}+a_2^{†}{a}_2)=N_1+\frac{1}{2}+N_2+\frac{1}{2}=z_1\frac{\partial }{\partial z_1}+z_2\frac{\partial }{\partial z_2}+1$$

This operator will multiply a homogeneous polynomial by its degree plus one, so it acts by multiplication by $n+1$ on ${\mathcal{H}}^n$ . Exponentiating this operator (multiplied by −$i$) one gets a representation of a $U(1)$ subgroup of the metaplectic cover $Mp(4,\mathbf{R})$ . Taking instead the normal ordered version

$$:H:=a_1^{†}{a}_1+a_2^{†}{a}_2=N_1+N_2=z_1\frac{\partial }{\partial z_1}+z_2\frac{\partial }{\partial z_2}$$

one gets a representation of a $U(1)$ subgroup of $Sp(4,\mathbf{R})$ . Neither H nor $:H:$ commutes with operators coming from quantization of the first two subalgebras.

These will be linear combinations of pairs of either creation or annihilation operators, so will change the eigenvalue of H or :$H$: by ±2, mapping

$${\mathcal{H}}^n\to {\mathcal{H}}^{n\pm 2}$$

and in particular taking |0⟩ to either 0 or a state in ${\mathcal{H}}^2$

$h$ is a basis element for the $\mathfrak{u}(1)$ in $\mathfrak{u}(2)=\mathfrak{u}(1)\oplus \mathfrak{su}(2)$ . For the $\mathfrak{su}(2)$ part, on basis elements $\begin{array}{r}{X}_j=-i\frac{{\sigma }_j}{2}\end{array}$ the moment map 25.3 gives the following quadratic polynomials

$${\mu }_{X_1}=\frac{1}{2}(z_1{\overline{z}}_2+z_2{\overline{z}}_1),{\mu }_{X_2}=\frac{i}{2}(z_2{\overline{z}}_1-z_1{\overline{z}}_2),{\mu }_{X_3}=\frac{1}{2}(z_1{\overline{z}}_1-z_2{\overline{z}}_2)$$

This relates two diferent but isomorphic ways of describing $\mathfrak{su}(2)$ : as 2 by 2 matrices with Lie bracket the commutator, or as quadratic polynomials, with Lie bracket the Poisson bracket.

Quantizing using the Bargmann-Fock representation give a representation of su(2) on $H$

$$\begin{array}{rl}{\Gamma }_{BF}^{\prime }(X_1)& =-\frac{i}{2}(a_1^{†}{a}_2+a_2^{†}{a}_1),{\Gamma }_{BF}^{\prime }(X_2)=\frac{1}{2}(a_2^{†}{a}_1-a_1^{†}{a}_2)\\ {\Gamma }_{BF}^{\prime }(X_3)& =-\frac{i}{2}(a_1^{†}{a}_1-a_2^{†}{a}_2)\end{array}$$

Comparing this to the representation ${\pi }^{\prime }$ of $\mathfrak{su}(2)$ on homogeneous polynomials discussed in chapter ${}^{8,}$ one finds that ${\Gamma }_{BF}^{\prime }$ and ${\pi }^{\prime }$ are the same representation. The inner product that makes the representation unitary is the one of equation 8.2. The Bargmann-Fock representation extends this $SU(2)$ representation as a unitary representation to a much larger group $(H_5⋊Mp(4,\mathbf{R}))$ , with all polynomials in $z_1,z_2$ now making up a single irreducible representation of $H_5$

The fact that we have an $SU(2)$ group acting on the state space of the $d=2$ harmonic oscillator and commuting with the action of the Hamiltonian H means that energy eigenstates can be organized as irreducible representations of $SU(2)$ In particular, one sees that the space ${\mathcal{H}}^n$ of energy eigenstates of energy $n+1$ will be a single irreducible $SU(2)$ representation, the spin $\frac{n}{2}$ representation of dimension $n+1$ (so $n$ + 1 will be the multiplicity of energy eigenstates of that energy).

Another physically interesting subgroup here is the $SO(2)\subset SU(2)\subset$ $Sp(4,\mathbf{R})$ consisting of simultaneous rotations in the position and momentum planes, which was studied in detail using the coordinates $q_1,q_2,p_1,p_2$ in section 20.3.1. There we found that the moment map was given by

$${\mu }_L=l=q_1{p}_2-q_2{p}_1$$

and quantization by the Schr¨odinger representation gave a representation of the Lie algebra so(2) with

$$U_L^{\prime }=-i(Q_1{P}_2-Q_2{P}_1)$$

Note that this is a diferent $SO(2)$ action than the one with moment map the Hamiltonian, it acts separately on positions and momenta rather than mixing them.

To see what happens if one instead uses the Bargmann-Fock representation, using

$$q_j=\frac{1}{\sqrt{2}}(z_j+{\overline{z}}_j),p_j=i\frac{1}{\sqrt{2}}(z_j-{\overline{z}}_j)$$

the moment map is

$$\begin{array}{rl}{\mu }_L& =\frac{i}{2}((z_1+{\overline{z}}_1)(z_2-{\overline{z}}_2)-(z_2+{\overline{z}}_2)(z_1-{\overline{z}}_1))\\ & =i(z_2{\overline{z}}_1-z_1{\overline{z}}_2)\end{array}$$

Quantizing, the operator

$$U_L^{\prime }=a_2^{†}{a}_1-a_1^{†}{a}_2={\Gamma }^{\prime }(2X_2)$$

gives a unitary representation of $\mathfrak{so}(2)$ . The factor of two here reflects the fact that exponentiation gives a representation of $SO(2)\subset Sp(4,\mathbf{R})$ , with no need for a double cover.

25.4.2 Three degrees of freedom and SO(3)

Concept links · terms present in this machine draft; source roles are unverified: eigenvalue · Lie algebra · irreducible representation

The case $d=3$ corresponds physically to the so-called isotropic quantum harmonic oscillator system, and it is an example of the sort of central potential problem we studied in chapter 21 (since the potential just depends on $r^2=q_1+q_2^2+q_3^2)$ . For such problems, we saw that since the classical Hamiltonian is rotationally invariant, the quantum Hamiltonian will commute with the action of $SO(3)$ on wavefunctions, and energy eigenstates can be decomposed into irreducible representations of $SO(3)$ ).

Here the Bargmann-Fock representation gives an action of $H_7⋊Mp(6,\mathbf{R})$ on the state space, with a $U(3)$ subgroup commuting with the Hamiltonian (more precisely one has a double cover of $U(3)$ , but by normal ordering one can get an actual $U(3))$ . The eigenvalue of the $U(1)$ corresponding to the Hamiltonian gives the energy of a state, and states of a given energy will be sums of irreducible representations of $SU(3)$ ). This works much like in the $d=2$ case, although here our irreducible representations are on the spaces ${\mathcal{H}}^n$ of homogeneous polynomials of degree $n$ in three variables rather than two. These spaces have dimension $\frac{1}{2}(n+1)(n+2)$ ). A diference with the $SU(2)$ case is that one does not get all irreducible representations of $SU(3)$ this way.

The rotation group $SO(3)$ will be a subgroup of this $U(3)$ and one can ask how the $SU(3)$ irreducible ${\mathcal{H}}^n$ decomposes into a sum of irreducibles of the subgroup (which will be characterized by an integral spin $l=0,1,2,\cdots )$ . One can show that for even n one gets all even values of l from 0 to $n,$ and for odd n one gets all odd values of l from 1 to n. A derivation can be found in some quantum mechanics textbooks, see for example pages 456-460 of [60].

To construct the angular momentum operators in the Bargmann-Fock representation, recall that in the Schr¨odinger representation these were

$$L_1=Q_2{P}_3-Q_3{P}_2,L_2=Q_3{P}_1-Q_1{P}_3,L_3=Q_1{P}_2-Q_2{P}_1$$

and these operators can be rewritten in terms of annihilation and creation operators. Alternatively, theorem 25.2 can be used, for Lie algebra basis elements $l_j\in \mathfrak{so}(3)\subset \mathfrak{u}(3)\subset \mathfrak{gl}(3,\mathbf{C})$ which are (see chapter 6)

$$l_1=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& -1\\ 0& 1& 0\end{array}\right),l_2=\left(\begin{array}{ccc}0& 0& 1\\ 0& 0& 0\\ -1& 0& 0\end{array}\right),l_3=\left(\begin{array}{ccc}0& -1& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right)$$

to calculate

$$-i{L}_j=U_{l_j}^{\prime }=\sum _{m,n=1}^3{a}_m^{†}(l_j{)}_{mn}{a}_n$$

This gives

$$U_{l_1}^{\prime }=a_3^{†}{a}_2-a_2^{†}{a}_3,U_{l_2}^{\prime }=a_1^{†}{a}_3-a_3^{†}{a}_1,U_{l_3}^{\prime }=a_2^{†}{a}_1-a_1^{†}{a}_2$$

Exponentiating these operators gives a representation of the rotation group $SO(3)$ on the state space ${\mathcal{F}}_3.$ , commuting with the Hamiltonian, so acting on energy eigenspaces (which will be the homogeneous polynomials of fixed degree).

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