Group theory and quantum optics

Question

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.

This post imported from StackExchange Physics at 2015-11-02 22:08 (UTC), posted by SE-user Cosmo Lupo

Comments

Worth noting that apparently "commutant" is another word for "centralizer," for those who were confused like me. Maybe this is a regional thing?

This post imported from StackExchange Physics at 2015-11-02 22:08 (UTC), posted by SE-user Chris White

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commutant is used in functional analysis, centralizer in group theory.