Higher Dimensional Oscillators, Revisited

We sharpen our focus now to the relatively simple case of $ \text{SU}(3) $, since there is a neat trick that allows us to generate all representations $ (n,m) $.

Recall that we constructed the representation $ (N,0) $ using three oscillators. The fact that the algebra of these oscillator creation and annihilation operators is

\[\begin{align} \left[ a _ {i}, a ^{\dagger j} \right] = \delta _ {i}{}^{j} \ , \end{align}\]

means that the states we constructed earlier correspond to totally symmetric tensors. Note that in the above equation we have subtly changed notation: we are now careful to place indices for creation and annihilation operators in such a way that they are consistent with the tensor considerations in the previous section. Thus, not only do these states just correspond to symmetric tensors; they correspond to symmetric tensors \emph{with upper indices}.

This business with indices is going to get confusing unless we keep our wits about us, so let’s be particularly careful going forward. There are in fact four related representation spaces: the fundamental $ V $, the dual of the fundamental $ V ^{\star } $, the conjugate of the fundamental $ \overline{V} $, and the conjugate-dual of the fundamental $ \overline{V} ^{\star } $. We place indices on elements of these spaces as follows:

\[\begin{align} v ^{i} \in V \quad & \quad v _ {i} \in V ^{\star } \ , \\ v ^{\bar{\imath} } \in \overline{V} \quad & \quad v _ {\bar{\imath} } \in \overline{V} ^{\star } \ . \end{align}\]

Under the action of an unitary operator $ U $, each of these transforms differently:

\[\begin{align} v ^{i} \mapsto U ^{i}{} _ {j} v ^{j} \quad & \quad v _ {i} \mapsto v _ {j} \Big( U ^{\dagger } \Big) ^{j}{} _ {i} \ , \\ v ^{\bar{\imath} } \mapsto \overline{U} ^{\bar{\imath } }{} _ {\bar{\jmath } } v ^{\bar{\jmath } } \quad & \quad v _ {\bar{\imath} } \mapsto v _ {\bar{\jmath } } \Big( \overline{U} ^{\dagger } \Big) ^{\bar{\jmath } }{} _ {\bar{\imath } } \ , \end{align}\]

where $ \overline{U} $ denotes the complex conjugate of $ U $.

Let us introduce two independent pairs of creation and annihilation operators satisfying the non-trivial commutation relations

\[\begin{align} \left[ a _ {i}, a ^{\dagger j} \right] = \delta _ {i}{} ^{j} \quad \text{and} \quad \left[ b _ {\bar{\imath } }, b ^{\dagger \bar{\jmath } } \right] = \delta _ {\bar{\imath } }{} ^{\bar{\jmath } } \ . \end{align}\]

The vacuum is defined as the state satisfying both $ a _ {i} \left\vert 0 \right\rangle = 0 $ and $ b _ {\bar{\imath } } \left\vert 0 \right\rangle = 0 $. The placecment of indices on these operators is intentional: $ a ^{\dagger i} $ and $ b ^{\dagger \bar{\imath } } $ create $ 1 $-particle states that transform in the fundamental $ \left( \mathbf{3} \right) $ and antifundamental $ \left( \overline{\mathbf{3} } \right) $ representations respectively. The corresponding annihilation operators “eat” fundamental and antifundamental indices and therefore transform in the corresponding dual representations. Finally, as a consequence of the above commutation relations, a general state in this Fock space looks like

\[\begin{align} T ^{(i _ {1} \cdots i _ {N})(\bar{\jmath } _ {1} \cdots \bar{\jmath } _ {M})} = \left[ \prod _ {k=1} ^{N} \frac{1}{\sqrt{N _ {k}!} } \left( a ^{\dagger i _ {k}} \right)^{N _ {k}} \right] \left[ \prod _ {\ell =1} ^{M} \frac{1}{\sqrt{M _ {\ell }!} } \left( b ^{\dagger \bar{\jmath } _ {\ell }} \right)^{M _ {\ell }} \right] \left\vert 0 \right\rangle \ , \end{align}\]

and manifestly transforms as an element of the representation $ (N,M) $.

How do we represent unitaries of the form $ U = e^{i \theta ^{a}t _ {a}} $, where $ t _ {a} = \frac{1}{2} \lambda _ {a} $ are given in terms of the Gell-Mann matrices, on this Fock space? From the infinitesimal form of the transformation rules

\[\begin{align} \mathcal{U}a ^{\dagger i} \mathcal{U}^{\dagger } = U ^{i}{} _ {j} a ^{\dagger j} \quad \text{and} \quad \mathcal{U}b ^{\dagger \bar{\jmath } } \mathcal{U}^{\dagger } = \overline{U} ^{\bar{\imath } }{} _ {\bar{\jmath } } b ^{\dagger \bar{\jmath } } \ , \end{align}\]

and assuming $ \mathcal{U} = e^{i \theta ^{a}T _ {a}} $ for some judiciously chosen $ T _ {a} $, we must have

\[\begin{align} \left[ T _ {a}, a ^{\dagger i} \right] = \left( t _ {a} \right) ^{i}{} _ {j} a ^{\dagger j} \quad \text{and} \quad \left[ T _ {a}, b ^{\dagger \bar{\imath } } \right] = -\left( \overline{t} _ {a} \right) ^{\bar{\imath } }{} _ {\bar{\jmath } } b ^{\dagger \bar{\jmath } } \ . \end{align}\]

Note that this is as it should be: the antifundamental is a conjugate representation, and so it is generated by $ \overline{U} = e^{- i \theta ^{a} \bar{t} _ {a}} $. We may conclude, therefore, that

\[\begin{align} T _ {a} = a ^{\dagger j} \left( t _ {a} \right) ^{i}{} _ {j} a _ {i} - b ^{\dagger \bar{\jmath } } \left( \overline{t} _ {a} \right) ^{\bar{\imath } }{} _ {\bar{\jmath } } b _ {\bar{\imath } } \ , \end{align}\]

does the job and satisfies the Lie algebra of $ \mathfrak{su}(3) $. Notice that the operators $ T _ {a} $ commute with the number operators $ N _ {a} = a ^{\dagger i}a _ {i} $ and $ N _ {b} = b ^{\dagger \bar{\imath } }b _ {\bar{\imath } } $, decomposing our Hilbert space into invariant sectors, each corresponding to a representation $ (N,M) $. It now remains for us to ensure that our representation is irreducible.

Consider a map $ h : V \times \overline{V} \rightarrow \mathbb{C} $ such that for $ v ^{i} \in V $ and $ w ^{\bar{\imath } }\in \overline{V} $, we get

\[\begin{align} h: \left( v ^{i},w ^{\bar{\jmath } } \right) \mapsto h _ {i \bar{\jmath } } v ^{i} w ^{\bar{\jmath } } \ . \end{align}\]

We might want to set $ h _ {i \bar{\jmath } } = \delta _ {i \bar{\jmath } } $, but we will keep the dependence on $ h $ arbitrary for what follows in the interest of generality. Now, we want this pairing to be invariant under unitaries, which requires

\[\begin{align} h _ {k \bar{\ell } } = h _ {i \bar{\jmath } } U ^{i}{} _ {k} \overline{U} ^{\bar{\jmath } }{} _ {\bar{\ell } } \ , \end{align}\]

which infinitesimally means

\[\begin{align} h _ {i \bar{\ell } } \left( t _ {a} \right) ^{i}{} _ {k} - h _ {k \bar{\jmath } } \left( \overline{t} _ {a} \right) ^{\bar{\jmath } }{} _ {\bar{\ell } } = 0 \ . \label{eq:h-t-condition} \end{align}\]

Define the operators

\[\begin{align} K _ {+} = a ^{\dagger ^{i}} h _ {i \bar{\jmath } } b ^{\dagger \bar{\jmath } } \quad \text{and} \quad K _ {-} = a _ {i} h ^{i \bar{\jmath } } b _ {\bar{\jmath } } \ , \end{align}\]

which send

\[\begin{align} K _ {\pm }: (N,M) \mapsto (N \pm 1, M \pm 1) \ . \end{align}\]

Using \eqref{eq:h-t-condition}, we find that $ K _ {\pm } $ are $ \text{SU}(3) $-invariant operators, i.e. \begin{equation} \label{eq:kpm-ta} \left[ T ^{a}, K _ {\pm } \right] = 0 \ . \end{equation} Now, the action of $ K _ {-} $ on states in a representation have the effect of contracting one fundamental (unbarred) and one antifundamental (barred) index using the Hermitian metric $ h $. Further, recall that in order to achieve an irreducible representation $ (N,M) $, we must consider traceless representations. Therefore, the irreducible representations live in the kernel of the $ K _ {-} $ map: \begin{equation} \label{eq:su3-irrep-kernel-km} (N,M) = \operatorname{ker} K _ {-} \ . \end{equation} That is, unitary irreducible representations of $ \text{SU}(3) $ are constructed in the oscillator formalism provided one projects onto the kernel of the $ K _ {-} $ operator. This completes the oscillator construction for the group $ \text{SU}(3) $.