The dual role of (anti-)Hermitian operators in quantum mechanics

Hermitian (or anti-Hermitian) operators are of central importance in quantum mechanics in at least two different incarnations:

• Observables are represented by Hermitian operators on the quantum mechanical state space.


• Transformations of the state space have to preserve the Hilbert space structure: they are unitary. (Anti-)Hermitian operators are the infinitesimal generators of unitary transformations.

It follows that every observable generates a transformation of the state space, and conversely, that to a transformation of the state space corresponds an observable (or at least a Hermitian operator).

My question: is this generally a meaningful correspondence?

I learned elsewhere on this site (I cannot find the question anymore) that not all Hermitian operators correspond to observables (superselection rules), therefore a first part of the question would be if among Hermitian operators the observables correspond to a physically identifiable subset of all unitary operators.

I know of some cases in which the correspondence does seem to be meaningful:

• when the observable is the Hamiltonian, the generated transformation (one-parameter group of transformations) is the time evolution.



• when the transformation is a symmetry of the system, the associated generator is a conserved quantity.