Commutators, Lie groups, and quantum systems

In summary, the commutator of the skew-Hermetian versions of the field free and interaction (dipole) Hamiltonians tells us the symmetry of the operators that determine if they're fully controllable. This means that the symmetry of the system is explored, and this information can be used to interpret the physical behavior of the system.
  • #1
Einstein Mcfly
162
3
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.
 
Physics news on Phys.org
  • #2
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.
 
  • #3
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?
 
  • #4
Do you mean to say Lie algebra generated as opposed to Lie group. The Lie bracket defines the operator algebra.
 
  • #5
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.
 
  • #6
I just thought I'd bump this and see if anyone had any ideas.
 

1. What is a commutator?

A commutator is a mathematical operation that measures the extent to which two operations do not commute with each other. In other words, it measures how much the order of performing two operations affects the final result. In quantum mechanics, commutators are used to describe the behavior of quantities such as position and momentum.

2. What are Lie groups?

Lie groups are mathematical structures that are used to describe the symmetries of physical systems. They are groups, meaning they have a set of elements and an operation that combines those elements, but they also have the additional property of being continuously differentiable. Lie groups are useful in physics, particularly in quantum mechanics, because they allow us to understand the symmetries of a system and how they relate to the system's behavior.

3. How are commutators related to Lie groups?

Commutators and Lie groups are closely related because commutators are used to define the structure of a Lie group. In fact, the commutator operation is used to define the algebra of a Lie group, which is a set of operations that describe the group's properties. In quantum mechanics, the commutator of two operators corresponds to the Lie bracket, which is a mathematical operation used to define the algebra of a Lie group.

4. What is the significance of Lie groups in quantum mechanics?

Lie groups are significant in quantum mechanics because they are used to describe the symmetries of physical systems. In quantum mechanics, symmetries are important because they correspond to conserved quantities, such as energy and momentum. By understanding the symmetries of a system, we can better understand its behavior and make predictions about its properties.

5. How are quantum systems described using Lie groups?

In quantum mechanics, quantum systems are described using the language of Lie groups and their associated algebras. The operators that describe physical quantities, such as position and momentum, are represented as elements of a Lie group, and their commutators correspond to the Lie brackets of the group's algebra. This allows us to use the powerful tools of Lie group theory to analyze and understand quantum systems.

Similar threads

A Unitary representations of Lie group from Lie algebra
    • Quantum Physics
Replies
5
Views
1K
I Relationship between a Lie group such as So(3) and its Lie algebra
    • Quantum Physics
Replies
6
Views
949
A Symmetries of particle interacting with external fields (Ballentine)
    • Quantum Physics
Replies
5
Views
756
I Quantum number and group theory
    • Quantum Physics
Replies
10
Views
1K
A Understanding time translations in Ballentine
    • Quantum Physics
Replies
2
Views
812
I Fundamental representation and adjoint representation
    • Quantum Physics
Replies
27
Views
928
I Gauge and Phase Symmetries in Quantum Systems: Exploring the Confusion
    • Quantum Physics
Replies
6
Views
653
I Significance of the Exchange Operator commuting with the Hamiltonian
    • Quantum Physics
Replies
9
Views
1K
I Calculating group representation matrices from basis vector/function
    • Quantum Physics
Replies
6
Views
867
I The general structure of relativistic QFTs
    • Quantum Physics
3
Replies
87
Views
5K

Hot Threads

Recent Insights

Back
Top