Applications of Invariant Theory to Quantum Physics

In summary, the discussion centered around using invariant theory of Lie Groups to understand quantum physics. It was mentioned that one can treat quantum theory as applied group theory, specifically by defining observables and their representations using group theory. Examples were given, such as deriving the Schrödinger or Pauli Equation from the demand for a unitary ray representation of the Galileo group. The spinning top and selection rules for electromagnetic transitions were also mentioned as applications of invariant theory to quantum physics. However, the main challenge lies in choosing interesting examples rather than finding them.
  • #1
timb00
15
0
Hey everybody,

I have to give a talk in our seminar on invariant theory of Lie Groups. And I'm now looking for
easy applications of invariant theory to quantum physics. I want to present them to
motivate the discussion.

I would be lucky if someone of you has an idea where I can found examples.

Best wishes, Timb00
 
Physics news on Phys.org
  • #2
In a sense the problem is not to find examples but to restrict yourself to interesting ones. One can treat quantum theory in its very foundations as applied (Lie-)group theory! The very definition of observables and their representation in the quantum theoretical formalism can be motivated by the application of group theory: In a sense this refines the correspondence principle of "canonical quantization" to define quantum mechanical systems by unitary (ray) representations of the corresponding symmetry group or a subgroup of it of the classical system. E.g., you can derive the Schrödinger or Pauli Equation from the demand that quantum systems live in Newtonian spacetime and thus the quantum theory should admit a unitary ray representation of the Galileo group underlying the structure of spacetime. For special relativity, the same is true for the Poincare group.

A nice example is the quantization of the spinning top, which doesn't work out right in the naive canonical quantization, but one has to use the representation theory of the rotation group to get the correct description (see, e.g., Hagen Kleinert's book on path integrals, where this is given as an example for the group-theoretical approach to quantization of classical systems).

Another application are selection rules for electromagnetic transitions, the Wigner-Eckart theorem, etc. etc. As I said, the problem is to restrict oneself to interesting examples rather than to find some :-)).
 

1. What is Invariant Theory?

Invariant Theory is a branch of mathematics that studies symmetries and transformations of mathematical objects, such as matrices and polynomials. It focuses on identifying properties that remain unchanged under these transformations, known as invariants.

2. How is Invariant Theory related to Quantum Physics?

Invariant Theory has been applied to various areas of physics, including quantum mechanics. In quantum physics, symmetries and invariants play a crucial role in understanding the behavior of particles and systems, and Invariant Theory provides a mathematical framework for studying these concepts.

3. What are some specific applications of Invariant Theory to Quantum Physics?

Some specific applications of Invariant Theory to Quantum Physics include the study of symmetries in quantum systems, such as rotational and translational invariance. It has also been used to analyze the behavior of particles under the influence of electromagnetic fields.

4. Can Invariant Theory help in solving problems in Quantum Physics?

Yes, Invariant Theory has been used to solve various problems in Quantum Physics, such as determining the energy levels of a quantum system or predicting the behavior of particles in certain environments. It provides a powerful tool for analyzing and understanding the underlying symmetries and invariants in quantum systems.

5. Are there any limitations to using Invariant Theory in Quantum Physics?

While Invariant Theory has proven to be a valuable tool in studying quantum systems, it also has its limitations. Some quantum systems may not exhibit obvious symmetries or invariants, making it challenging to apply Invariant Theory. Additionally, the complexity of certain quantum systems may make it difficult to use Invariant Theory to derive analytical solutions.

Similar threads

  • Quantum Physics
Replies
1
Views
643
Replies
0
Views
411
  • Quantum Physics
Replies
5
Views
1K
  • Quantum Physics
2
Replies
41
Views
3K
Replies
15
Views
1K
Replies
36
Views
3K
  • Quantum Physics
Replies
7
Views
940
  • Sticky
  • Quantum Physics
Replies
1
Views
5K
  • Quantum Physics
Replies
3
Views
2K
Replies
1
Views
228
Back
Top