Commutators, Lie groups, and quantum systems

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Discussion Overview

The discussion revolves around the controllability of quantum systems through the analysis of Lie groups and commutators related to Hamiltonians. Participants explore the implications of these mathematical structures for understanding symmetries and physical interpretations within quantum mechanics.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant proposes a method to determine the controllability of a quantum system based on the Lie group generated by the commutator of Hamiltonians.
  • There is a question about whether to use "is" or "is isomorphic to" when discussing the relationship between the dynamical Lie group and U(N).
  • Another participant clarifies that using "is" is acceptable, as isomorphism pertains to notation rather than structural differences.
  • A participant suggests that the discussion should focus on the Lie algebra generated rather than the Lie group, emphasizing the role of the Lie bracket in defining the operator algebra.
  • The original poster acknowledges the correction regarding the Lie algebra and expresses a desire for further insights into the physical interpretation of the symmetries involved.

Areas of Agreement / Disagreement

Participants generally agree on the terminology regarding Lie groups and Lie algebras, but the discussion remains unresolved regarding the physical implications of the symmetries and the interpretation of controllability in quantum systems.

Contextual Notes

Participants have not fully explored the implications of the commutator on the controllability of operators, and there are unresolved questions about the physical meaning of the symmetries derived from the mathematical structures discussed.

Einstein Mcfly
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Hi folks. I've come across a method to determine the controllability of a quantum system that depends on the Lie group generated by the commutator of the skew-Hermetian versions of the field free and interaction (dipole) Hamiltonians. If, for an N dimensional system the dimension of the group generated is N^2, then the dynamical Lie group is U(N) (is it correct to say “is” or should I say “is isomorphic to”?) and “every unitary operator can be dynamically generated”. My questions are: What symmetry are we exploring here with this group? What does the commutator tell us about the symmetry of the operators that determines if they’re fully controllable? What does this mean physically?

I have little experience with this sort of argument (that is, deriving physical information from the structure of a group with the same symmetry), so any insight is much appreciated. Thanks.
 
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I'll just answer this little part of your question:

EinsteinMcfly said:
is it correct to say “is” or should I say “is isomorphic to”?

"Is" is fine. If two groups are isomorphic then the difference between them is just notation. Every element matches up and all products and inverses precisely match too.
 
selfAdjoint said:
I'll just answer this little part of your question:



"Is" is fine. If two groups are isomorphic then the difference between them is just notation. Every element matches up and all products and inverses precisely match too.

Thanks. That's what I thought.


Anyone else? I know there are a lot of very smart folks on here with a lot more experience than me. Is anyone good at interpreting these symmetries physically?
 
Do you mean to say Lie algebra generated as opposed to Lie group. The Lie bracket defines the operator algebra.
 
Epicurus said:
Do you mean to say Lie algebra generated as opposed to Lie group. The Lie bracket defines the operator algebra.


Yes, indeed I did. Thanks.
 
I just thought I'd bump this and see if anyone had any ideas.
 

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