3
$\begingroup$

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.

$\endgroup$
-1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.