The Jordan-Schwinger Oscillator Formalism: Part I

The oscillator formalism was developed by Jordan and (independently by) Schwinger to construct irreducible representations of $\text{SU}(2)$ in the following papers:

Der Zusammenhang der symmetrischen und linearen Gruppen und das Mehrkörperproblem
P. Jordan
Zeitschrift für Physik 94 (1935), No. 7, 531–535.

On Angular Momentum
J. Schwinger
Technical Report NYO-3071, Harvard University
and Nuclear Development Associates, Inc. (US), 1952.

Symmetries in Quantum Theory

Before we consider specific examples, it is useful to consider in an abstract, formal sense what happens to the spectrum of a Hamiltonian in the presence of a symmetry. Let us suppose that the dynamics of a quantum system is governed by a Hamiltonian $H$ and that there exists a unitary operator $\mathcal{U} = e^{i a F} $ corresponding to some Hermitian generator $ F $. The Hamiltonian is transformed under the action of this unitary operator as $ H \mapsto H’ = \mathcal{U} H \mathcal{U}^{\dagger }$. When $ a $ is infinitesimal, we may expand as \begin{equation} H’= e^{i a F} H e^{-i a F} \approx (1+i a F) H (1-i a F)=H + i a [F, H] \ , \end{equation} We conclude from this that $\delta H = + i a[F, H]$, and it follows that if $[F, H]=0$, then $ \delta H = 0 $ and $ F $ generates a symmetry. It is easy to see using Heisenberg’s equations of motion that if $ F $ generates a symmetry, it is conserved. This is essentially the quantum equivalent of Noether’s theorem, which we discussed here.

The existence of symmetries implies strong constraints on the spectrum of the Hamiltonian. Consider an eigenstate $\left\vert n \right\rangle $ of the Hamiltonian with eigenvalue $E_ {n}$. We have just seen that unitary operators corresponding to symmetries of the Hamiltonian commute with it, i.e. if $ U $ is a symmetry then $ \left[ \mathcal{U}, H \right] = 0 \Rightarrow H \mathcal{U} = \mathcal{U}H $. From this it follows that \begin{equation} H(\mathcal{U} \left\vert n \right\rangle ) = \mathcal{U} H \left\vert n \right\rangle = E_ {n}(\mathcal{U} \left\vert n \right\rangle ) \ . \end{equation} That is, $\mathcal{U} \left\vert n \right\rangle $ is also an eigenstate of the Hamiltonian with the same energy. More generally, all states related to each other by symmetry transformations must have the same energy. These states are sometimes said to form a multiplet.

Consider starting with a state $ \left\vert n \right\rangle $ and acting with all elements of some group of symmetry transformations $ \mathcal{U} _ {a} $ on $ \left\vert n \right\rangle $ to generate the states $ \left\lbrace \left\vert n _ {a} \right\rangle \right\rbrace $. We have already seen that all these states are degenerate. In addition, the set $ \left\lbrace \left\vert n _ {a} \right\rangle \right\rbrace $ so generated is closed under the action of $ \mathcal{U} $, which in turn implies that the matrix elements of $ \mathcal{U} $ in this degenerate subspace \begin{equation} U _ {ab} = \left\langle n _ {a} \right\vert \mathcal{U} \left\vert n _ {b} \right\rangle \ , \end{equation} furnishes a finite-dimensional irreducible representation of the group of transformations that $ \mathcal{U}_ {a} $ comes from.

In a nutshell, this is one of the reasons why quantum mechanics is interesting to mathematicians: unitary irreducible representations of groups are furnished by quantum systems which exhibit these symmetries. We now proceed to a simple illustrations of this fact.

Jordan-Schwinger Formalism for SU(2) Representations

Let us consider a system of two harmonic oscillators, whose position and momentum operators are labelled by the indices $ i \in \left\lbrace 1,2 \right\rbrace $. Both oscillators are assumed to have the same (unit) angular frequency. The Hamiltonian of this system is then \begin{equation} \label{eq:hamiltonian-2d-oscillator} H = \sum_ {i = 1}^{2} \frac{1}{2} \left( p _ {i}^{2} + q _ {i}^{2} \right) \ . \end{equation} Constructing the raising $ (a ^{\dagger }_ {i}) $ and lowering $ (a _ {i}) $ operators for both these oscillators as we usually do, it is easy to see that the only non-trivial commutators between them are \begin{equation} \label{eq:2d-heisenberg-algebra} \left[a_ {i}, a_ {j}^{\dagger}\right]=\delta_ {i j} \ , \end{equation} and all other commutators are identically zero. In essence, this is two copies of the Heisenberg algebra.

The Hamiltonian in \eqref{eq:hamiltonian-2d-oscillator} in terms of the raising and lowering operators is then \begin{equation} H = \sum_ {i=1}^{2} \left( a ^{\dagger }_ {i}a _ {i} + \frac{1}{2} \right) = N_ {1} + N_ {2} + 1 \ , \end{equation} where $ N _ {i} = a _ {i}^{\dagger }a _ {i} $ is the number operator for the $ i ^{\text{th}} $ oscillator. The ground state is defined by \begin{equation} a _ {i} \left\vert 0 \right\rangle = 0 \quad \text{for} \quad i \in \left\lbrace 1,2 \right\rbrace \ , \end{equation} and has energy $ E _ {0} = 1 $. Excited states can be constructed by acting with either of the creation operators, and since these commute with each other, a general excited state may be labelled and constructed as \begin{equation} \label{eq:states-2d-oscillator} \left\vert N _ {1}, N _ {2} \right\rangle = \frac{1}{\sqrt{N _ {1}! N _ {2}!} } \left( a _ {1}^{\dagger } \right)^{N _ {1}} \left( a _ {2}^{\dagger } \right)^{N _ {2}} \left\vert 0 \right\rangle \ . \end{equation} The degeneracy of these states is easily determined. Every state of the form $ \left\vert k, n - k \right\rangle $ for $ k \in \left\lbrace 0, 1, \cdots , n \right\rbrace $ is an eigenstate of the Hamiltonian with eigenvalue $ E _ {n} = n+1 $ and there are $ n+1 $ such states. Acting with the creation (annihilation) operators, in the usual way, moves to higher (lower) energy eigenstates.

Symmetries in quantum mechanical systems present themselves to us as degeneracies in the spectrum. We might therefore ask: what are the operators are that take us from one degenerate state to another? It is easy to write down two: \begin{equation} \label{eq:kpm-defn} K _ {+} = a _ {1}^{\dagger }a _ {2} \quad \text{and} \quad K _ {-} = a _ {2}^{\dagger }a _ {1} \ , \end{equation} which send \begin{equation} K _ {+} : \left\vert N _ {1}, N _ {2} \right\rangle \mapsto \left\vert N _ {1}+1, N _ {2} - 1 \right\rangle \quad \text{and} \quad K _ {-} : \left\vert N _ {1}, N _ {2} \right\rangle \mapsto \left\vert N _ {1} - 1, N _ {2}+1 \right\rangle \ . \end{equation} These are precisely the kind of operators we are looking for. On defining the operator \begin{equation} \label{eq:k3-defn} K _ {3} = \frac{1}{2} \left( N _ {1} - N _ {2} \right) \ , \end{equation} it is easy to check that the operators $ \left\lbrace K _ {+}, K _ {-}, K _ {3} \right\rbrace $ obey the following commutation relations: \begin{equation} \label{eq:Kpm3-algebra} \left[ K _ {+}, K _ {-} \right] = 2 K _ {3} \quad , \quad \left[ K _ {3}, K _ {+} \right] = + K _ {+} \quad , \quad \left[ K _ {3}, K _ {-} \right] = - K _ {-} \quad . \end{equation} This algebra ought to be familiar: on constituting the combinations \begin{equation} \label{eq:k12-defn} K _ {1} = \frac{1}{2} \left( K _ {+} + K _ {-} \right) \quad \text{and} \quad K _ {2} = \frac{1}{2i} \left( K _ {+} - K _ {-} \right) \ , \end{equation} the algebra in \eqref{eq:Kpm3-algebra} simplifies to \begin{equation} \label{eq:su2-algebra} \left[ K _ {a}, K _ {b} \right] = i \epsilon _ {abc} K _ {c} \ , \end{equation} where the indices $ \left\lbrace a, b, c, \cdots \right\rbrace \in \left\lbrace 1,2,3 \right\rbrace $. This is the familiar $ \mathfrak{su}(2) $ Lie algebra.

Using the states in \eqref{eq:states-2d-oscillator}, we can construct $ \left( n+1 \right)\times \left( n+1 \right) $ matrix representations of the angular momentum operators using the $ n+1 $ degenerate states corresponding to the $ n ^{\text{th}} $ excited energy level. As an example, consider the case of $ n = 1 $, i.e. the first excited energy level which corresponds to two degenerate states $ \left\vert 1 \right\rangle = a _ {1}^{\dagger }\left\vert 0 \right\rangle $ and $ \left\vert 2 \right\rangle = a _ {2}^{\dagger }\left\vert 0 \right\rangle $. The matrix elements of $ K _ {a} $ are easily computed using \eqref{eq:kpm-defn}, \eqref{eq:k3-defn}, and \eqref{eq:k12-defn}, and one finds \begin{equation} K _ {a} = \frac{1}{2} \sigma _ {a} \ . \end{equation} That is, the Pauli matrices furnish a two-dimensional representation of the $ \mathfrak{su}(2) $ Lie algebra. As another example: since the Lie algebras $ \mathfrak{su}(2) $ and $ \mathfrak{so}(3) $ are isomorphic, a similar exercise for the $ n=2 $ case reproduces the three-dimensional representation of $ \mathfrak{so}(3) $ which may be familiar to the reader as the algebra of generators of three-dimensional rotations.

Finally, note that the $ K _ {a} $ are Hermitian matrices, which in turn means the operators of the form $ U = e^{i K _ {a} \theta ^{a}} $ are unitary matrices. Thus, we conclude that the degenerate states of a two-dimensional harmonic oscillator furnish a unitary representation of $ \text{SU}(2) $.

We will discuss how this story generalises to other groups in a later note.