Applications of Invariant Theory to Quantum Physics

[timb00]

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

 

[vanhees71]

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 :-)).