Do cyclic product vectors generatating irreducible representation of a Lie group come from a unique orbit?

Question

Consider a Hilbert space $\mathcal{H}$ which is a carrier space of a unitary, irreducible and strongly continuous representation $\Pi$ of a Lie group $G$. Let $\Pi\otimes \Pi$ denote the corresponding representation of $G$ in $\mathcal{H}\otimes \mathcal{H}$.

Assume that $\phi\in\mathcal{H}$ is such that $\phi\otimes \phi$ is a cyclic vector of some irreducible representation $\tilde{\Pi}$ (with a carrier space $\tilde{\mathcal{H}}\subset \mathcal{H}\otimes\mathcal{H}$) of the group $G$. Let $\psi\in\mathcal{H}$ be another vector with this property.

Question: Are vectors $\phi$ and $\psi$ related by the action of $G$? More precisely, do we have

$\Pi(g)\mathbb{P}_\phi\Pi(g)^\dagger=\mathbb{P}_\psi$,

where $g\in G$ and $\mathbb{P}_\phi$, $\mathbb{P}_\psi$ are projectors onto one dimensional subspaces spanned by $\phi$ and $\psi$ respectively. (In order to disregard the trivial problem of the "global phase" I passed to the projective space).

Motivation I am interested in this problem because it gives the possibility of characterising (in some cases) particular orbits of $G$ in the projective space via single homogenous polynomial condition. In the compact case the situation is easy. I want to generalise what I have to the non-compact cases. In particular I am interested in groups like the Heisenberg Group, SU(1,1) or the Metaplectic group.

Remark: In the case of compact connected group $G$ the answer is yes by the structural theory of representations of semisimple Lie groups.

Answer 1

I wouldn't use the projective space, although it can be useful to diagonalise the Hermitian matrix which defines the unitary operator.

I have results for SU(2), and my co-worker had some similar results from which I worked off, which are in the following link http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.96.060503 . The principal idea is developing a unitary transformation which maps the system from its initial to final state. Maybe his result on SU(2) could be extended to SU(1,1). I already have done the SU(3) and SU(4) calculations to find all the unitary transformations and it seems to work fine.

As far as I can decipher your question, you are asking whether the conjoined system defined by the state vector ϕ⊗ϕ is mapped under the action of the tensor product of the Hamiltonians. It's not necessarily the case. For example, if you take a 2x2 matrix and a 3x3 matrix, which is the simplest form of a non-identical tensor product, then there are two different Hamiltonian operators which can operate on the conjoined product (3+2 vs 2+3). It's not even certain whether, given a time optimal unitary operator for each of the component subspaces, that the unitary for the conjoined system would be the tensor product of the subsystems.

Computationally, I would say that the homogenous polynomial condition generally tends to be the Cayley Hamilton equation, in that it enables a quick and dirty way of calculating exponents of these types of time dependent matrices.

Hopefully that helps more than it hinders.