Differential geometry in quantum mechanics - conserved quantities

In summary: Killing vector fields and Noether's theorem. He wonders if this comparison is accurate and if it is applicable outside of R^n. He also questions what metric would be used in quantum mechanics. Noether's theorem shows a connection between symmetries and conservation laws, and there is also an extension of this idea to infinite dimensional Lie algebras and systems of differential equations. In summary, John is curious about the relationship between quantum mechanics and differential geometry, specifically in terms of Killing vector fields and Noether's theorem, and wonders if this comparison is accurate and applicable outside of R^n, as well as what metric would be used in quantum mechanics.
  • #1
quasar_4
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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?
 
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  • #2
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
 

1. What is differential geometry in the context of quantum mechanics?

Differential geometry is a branch of mathematics that studies smooth curved surfaces and spaces. In the context of quantum mechanics, it is used to analyze and describe the geometric properties of quantum systems, such as their curvature and topology.

2. How does differential geometry relate to conserved quantities in quantum mechanics?

In quantum mechanics, conserved quantities are physical quantities that remain constant over time. Differential geometry plays a crucial role in determining these quantities by using geometric techniques to analyze the symmetries and invariances of a quantum system.

3. Can you give an example of how differential geometry is applied in quantum mechanics to determine conserved quantities?

One example is the application of Noether's theorem, which uses differential geometry to study the symmetries of a system and their corresponding conserved quantities. For instance, the rotational symmetry of a quantum system can lead to the conservation of angular momentum.

4. What are some other important concepts in differential geometry that are relevant to quantum mechanics?

Other important concepts in differential geometry that are relevant to quantum mechanics include Riemannian and symplectic geometry, which are used to study the curvature and symplectic structure of quantum systems, respectively. Differential forms and Lie groups are also commonly used in the analysis of conserved quantities in quantum mechanics.

5. How does an understanding of differential geometry in quantum mechanics contribute to our overall understanding of the universe?

Differential geometry has played a significant role in advancing our understanding of the fundamental laws of the universe, including quantum mechanics. By providing a geometric framework for analyzing and describing quantum systems, it has helped us gain a deeper understanding of the underlying structure and symmetries of the universe.

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