Isomorphisms Explained: A Physical Example with SU(2)xSU(2) and Lorentz Group

  • Thread starter Thread starter ChrisVer
  • Start date Start date
  • Tags Tags
    Groups
ChrisVer
Science Advisor
Messages
3,372
Reaction score
465
I am not really sure whether this topic belongs here or not, but since my example will be a certain one I will proceed here...

Someone please explain me what "isomorphism" means physically? For example what is the deal in saying that the proper orthochronous Lorentz group is isomorphic to SU(2)xSU(2)?
I can understand as far that this example breaks everything in left-right handed movers, but I just can't generalize it in everything when we are talking about isomorphisms...
 
Physics news on Phys.org
It means that the generators of the group follow identical algebras
 
ChrisVer said:
For example what is the deal in saying that the proper orthochronous Lorentz group is isomorphic to SU(2)xSU(2)?

The proper orthochronous Lorentz group is not isomorphic to SU(2)xSU(2).
 
George Jones said:
The proper orthochronous Lorentz group is not isomorphic to SU(2)xSU(2).
Don't you think it would be even more useful, George, if you told us what it IS isomrphic to.
 
Maybe I should have stayed quiet. There is a fair bit going on, and I am not sure that I can explain all of it well.

The statement in the OP is not true at the group level, but it is almost true at the level of Lie algebras (generators), i.e., it is true after the relevant real Lie algebras have been complexified.

I might try a longer explanation later.
 

Thread 'Toponium Discovered'

Toponium is a hadron which is the bound state of a valance top quark and a valance antitop quark. Oversimplified presentations often state that top quarks don't form hadrons, because they decay to bottom quarks extremely rapidly after they are created, leaving no time to form a hadron. And, the vast majority of the time, this is true. But, the lifetime of a top quark is only an average lifetime. Sometimes it decays faster and sometimes it decays slower. In the highly improbable case that...
View full post »

Thread 'Dirac-Delta from Normalization of Continuous Eigenfunctions'

I'm following this paper by Kitaev on SL(2,R) representations and I'm having a problem in the normalization of the continuous eigenfunctions (eqs. (67)-(70)), which satisfy \langle f_s | f_{s'} \rangle = \int_{0}^{1} \frac{2}{(1-u)^2} f_s(u)^* f_{s'}(u) \, du. \tag{67} The singular contribution of the integral arises at the endpoint u=1 of the integral, and in the limit u \to 1, the function f_s(u) takes on the form f_s(u) \approx a_s (1-u)^{1/2 + i s} + a_s^* (1-u)^{1/2 - i s}. \tag{70}...
View full post »

Similar threads

A How to Multiply SU(4)XSU(2) Matrices to Form a 8x8 Matrix?
Replies
6
Views
2K
5 rep of SU(5) under SU(3)XSU(2)XU(1)
Replies
4
Views
5K
Is Chirality Necessary in Theories of Physics?
Replies
1
Views
3K
A Learn SU Groups in QCD: Chiral Symmetry, 8 Goldstone Bosons & More
Replies
8
Views
3K
Standard Model decompositions of larger group representations?
Replies
4
Views
2K
B What Are the Key Questions Surrounding SU(3)xSU(2)xU(1) Symmetry Groups?
2
Replies
52
Views
15K
A SU(5), 'Standard Model decomposition', direct sum etc.
Replies
23
Views
5K
I Understanding Lorentz Groups and some key subgroups
Replies
3
Views
2K
Meaning of SO(4) - SU(2)xSU(2)
Replies
12
Views
15K
A Breaking of ##SU(2) \times SU(2)## to ##SU(2)##
Replies
1
Views
2K

Hot Threads

Recent Insights

Back
Top