Differential geometry in quantum mechanics - conserved quantities

[quasar_4]

Hi everyone. This is kind of a geometry/quantum mechanics question (hope this is the right place to post this).

So, in quantum mechanics we consider operators that reside in an infinite dimensional Hilbert space (to speak rather informally). We even have the cool commutator relation, which is remarkably like a Lie derivative between vector fields. I recall from my differential geometry course that given a vector field, if I take its Lie derivative with respect to some vector field and get zero, then I've discovered the Killing vector field that is the infinitesimal generator for some isometry. Of course, our class covered just R^n.

So if I have two quantum operators that commute, like momentum with the Hamiltonian, for example, then I can say that momentum is conserved... what I'd like to think is that mathematically, this means I have found a Killing vector field, which gives us the isometry that is physically manifested as conservation (is that right?).

What I'm not sure of is whether I'm allowed to talk about Killing vector fields in a context outside of R^n - our definition in math said that we had to have a Riemannian metric, and I'm not even sure what metric to talk about for quantum mechanics. Is all this comparison right, to think of it this way? And what metric are we dealing with for quantum mechanics?

 

[John_Phillips]

What you are aiming at here is called Noether's theorem. It shows an equivalence between symmetries and conservation laws. Work sometimes called Noether's second theorem extends this idea to infinite dimensional Lie algebras and systems of differential equations.

John