Differential geometry in quantum mechanics - conserved quantities

  • Thread starter quasar_4
  • Start date
290
0
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?
 
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
 

Related Threads for: Differential geometry in quantum mechanics - conserved quantities

Replies
2
Views
4K
  • Posted
Replies
4
Views
2K
Replies
1
Views
874
Replies
13
Views
4K
Replies
28
Views
8K
Replies
1
Views
555
  • Posted
Replies
5
Views
2K
Replies
1
Views
6K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top