the following is well-known (e. g. Ashtekar/Schilling, Brody/Hughston): A bounded self-adjoint operator $A$ on a Hilbert space $H$ induces a globally defined vector field $X$ on the projective Hilbert space $PH$, which is a symmetry of the Fubini-Study metric. $X$ is the Hamilton-vector field for the function $\langle \psi, A \psi \rangle$. Furthermore one recognizes that each Killing vector field on $PH$ can be constructed in this way.

I am interested in the case of an unbounded, densely defined operator $A$. By Chernoff/Marsden "Properties of infinite dimensional Hamiltonian systems" I know that the Hamilton vector field $X$ for $\langle \psi, A \psi \rangle, \psi \in D(A)$ is only defined densely, but generates a global flow. It should also leave the metric invariant. Is the converse also analogous to the bounded case, i.e. are all densely defined Killing vector fields on $PH$ associated to an unbounded self-adjoint operator on $H$?

Thanks and a happy new year! Tobias


1 Answer 1


Unless I am confused (the probability of which is around 75%) a counterexample can be found here, , page 439, Example 4 (for posterity, the paper is:

Geometric POV-Measures, Pseudo-Kahlerian Functions and Time 433 M. Skulimowski

While the book is: Topics in mathematical physics, general relativity and cosmology in honor of Jerzy Plebanski.


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.