Is Representation Theory on Rigged Hilbert Spaces Explored in Textbooks?

Click For Summary
SUMMARY

Representation theory on Rigged Hilbert Spaces (RHS) has not been extensively explored in textbooks, particularly in the context of Quantum Mechanics (QM). The discussion highlights the importance of RHS for implementing symmetries through representation theory of various symmetry groups, including Lie groups such as the Galilei group, Poincaré group, and SO(3). It is established that the universal covering group is what is actually represented, emphasizing the need for further exploration of this topic in academic literature.

PREREQUISITES
  • Understanding of Rigged Hilbert Spaces (RHS)
  • Familiarity with Quantum Mechanics (QM)
  • Knowledge of representation theory and symmetry groups
  • Basic concepts of Lie groups, including SO(3) and SU(n)
NEXT STEPS
  • Research the applications of Rigged Hilbert Spaces in Quantum Mechanics
  • Study representation theory of Lie groups, focusing on the Galilei and Poincaré groups
  • Explore the concept of universal covering groups in mathematical physics
  • Investigate existing literature on the intersection of representation theory and RHS
USEFUL FOR

Physicists, mathematicians, and students interested in advanced Quantum Mechanics, particularly those focusing on the mathematical foundations of quantum theory and symmetry representations.

dextercioby
Science Advisor
Insights Author
Messages
13,409
Reaction score
4,201
Has it been done...? If so, any textbook on it...?

The concept of Rigged Hilbert Spaces is essential to Quantum Mechanics. Symmetries are implemented in physics by doing representation theory of some symmetry groups, almost all of them being Lie groups: Galilei group, Poincaré group, Conformal group, SO(3), SU(n),... We actually know that what really is represented is the universal covering group for each group.

So, since the natural setting for QM is a RHS, the continuous symmetries for a quantum theory need to be represented on this RHS...:smile:

Daniel.
 
Physics news on Phys.org
Well, you know what they say: sometimes silence is an answer.

I can only conclude that representation theory on RHS has not been done yet.

Daniel.
 

Similar threads

Graduate Reality conditions on representations of classical groups
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Adjoint Representation Confusion
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Understanding SU(2) Representations and Their Role in Particle Physics
  • · Replies 2 ·
Replies
2
Views
4K
Graduate Questions about representation theory of Lie algebra
  • · Replies 5 ·
Replies
5
Views
3K
Graduate Unitary representations of Lie group from Lie algebra
  • · Replies 5 ·
Replies
5
Views
3K
Graduate Are there two different adjoint representations used in the SM?
  • · Replies 9 ·
Replies
9
Views
2K
Graduate Representations on Hilbert space
  • · Replies 8 ·
Replies
8
Views
3K
Undergrad Representations in Quantum Physics
  • · Replies 15 ·
Replies
15
Views
3K
Graduate Why is matter-free gravity not ultraviolet finite?
  • · Replies 13 ·
Replies
13
Views
4K
Graduate Shauder basis for Hilbert spaces
  • · Replies 0 ·
Replies
0
Views
772