# Group theory and quantum optics

This is a question about application of group theory to physics.

The starting point is the group $SU(n)$. I have a representation $R$ of $SU(n)$ that takes values on the unitary group on an infinite dimensional separable Hilbert space $H$. This representation can be written as the direct sum of finite-dimensional irreducible representations. Let me write $R(g) = \oplus_j R_j(g)$ for $g \in SU(n)$. The space $H_j$ of each irreducible representation is finite-dimensional. I denote as $P_j$ the projector on $H_j$. For those who are familiar with the subject, what I have in mind is the representation of $SU(n)$ obtained when applying to a set of $n$ bosonic modes the canonical transformations that are linear and preserve the photon-number operator. In this setting, the subspaces $H_j$ are the subspaces with $j$ photons, with $j=0,1,..,\infty$.

My question is the following:

From the representation $R$ given above we can define another representation: $R \otimes R \, : \, g \to R(g) \otimes R(g)$. What is the commutant (also known as centralizer) of the representation $R \otimes R$?

It is easy to see that the commutant of the representation $R$ is given by the projectors $P_j$'s. It is also easy to check that the following operators belong to the commutant of $R \otimes R$:

• $P_i \otimes P_j$, for $i,j=0,1,...,\infty$

• $S_{jj}$

where $S_{jj}$ is the "swap" operator in the subspace $H_j \otimes H_j$. I wonder if there are other operators in the commutant. How can I check it?

Thanks a lot and please accept my apologies if my notation is not very clear.

• Worth noting that apparently "commutant" is another word for "centralizer," for those who were confused like me. Maybe this is a regional thing?
– user10851
Jan 27 '15 at 0:35

There are necessarily further operators: a von Neumann algebra of operators if, as I am assuming, your representations are made of unitary operators.

All those operators can be constructed out of the orthogonal projectors commuting with the representation as complex linear combinations of them, since a complex von Neumann algebra is the closure of the complex span of its orthogonal projectors and here the closure is not necessary since the dimension is finite.

As a matter of fact you therefore should look for the invariant subspaces of each subrepresentation $R_i \otimes R_j$ in the corresponding product space $\cal{H}_i \otimes {\cal H}_j$ and the span of the orthogonal projectors onto these subspaces gives rise to the full space of operators commuting with $R_i \otimes R_j$. In particular, reversing the procedure, with $S_{jj}$ you can construct the subspace of symmetric and the subspace of antisymmetric tensors in ${\cal H}_j \otimes {\cal H}_j$, the projectors are $(I+S_{jj})/2$ and $(I-S_{jj})/2$, all complex combinations of these operators commute with the representation.

The fact that there are however many subspaces is already evident if looking at the case of $SU(2)$. In this case there is a canonical decomposition of $R_i\otimes R_j$ and $\cal{H}_i \otimes {\cal H}_j$ into orthogonal irreducible representations and the complex span of the corresponding projectors define operators in the commutant of $R_i\otimes R_j$ which actually exhaust this commutant (see below).

The said decomposition of $R_i\otimes R_j$ is nothing but the standard Clebsh-Gordan decomposition. In that case the procedure is relatively easy since the irreducible representations of $SU(2)$ are completely fixed by the eigenvalue of the unique Casimir operator $J^2$ and the overall argument reduces to look for the single eigenvalues of $J^2$ admitted for a product $R^{(j)}\otimes R^{(j')}$ of a pair of irreducible representations of $SU(2)$. Each eigenvalue defines an invariant subspace and the complex linear combinations of the corresponding orthogonal projectors are in the commutant of the representation in ${\cal H}^{(j)}\otimes {\cal H}^{(j)'}$.

Here you find the analogous decomposition for $SU(3)$, for $n>3$... I do not know, sorry :). You could try to have a look at Barut-Raczka's book on representations of groups.

An interesting fact is the following.

If the irreducible representations of a compact group are exactly all common eigenspaces of (mutually commuting) Casimir operators and each set of eigenvalues appears at most once, then the operators which commute with a tensor product of two irreducible representations $R_i \otimes R_j$ (a) are mutually commuting and (b) they are exactly the complex combinations of the orthogonal projectors onto the said common eigenspaces.

In fact: the space $\cal{H}_i \otimes {\cal H}_j$ is direct sum of mutually orthogonal irreducible representations by Peter-Weyl theorem which are eigenspaces of the Casimirs. If an operator $A$ commutes with the product representation $R_i \otimes R_j$ it also commute with its Casimir operators and thus it leaves invariant their common eigenspaces, i.e., the spaces of the irreducible representations that decompose $\cal{H}_i \otimes {\cal H}_j= \sum_k {\cal H}^{(i,j)}_k$. Since every such subspace ${\cal H}^{(i,j)}_k$ is irreducible, Schur's lemma entails that $A$ has the form $a_k I_k$ in every irreducible subspace, where $I_k$ is the identity operator on ${\cal H}^{(i,j)}_k$, and $a_k \in \mathbb C$. In summary the von Neumann algebra of the operators commuting with $R_i\otimes R_j$ is just made of the operators of form $\sum_k a_kI_k$, for some compex numbers $a_k$. This algebra is evidently Abelian.