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

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Discussion Overview

The discussion revolves around the concept of isomorphisms in the context of group theory, specifically examining the relationship between the proper orthochronous Lorentz group and the product of two SU(2) groups. Participants explore the physical implications of isomorphisms and the conditions under which certain statements about them hold true.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant seeks clarification on the physical meaning of "isomorphism" and its application to the proper orthochronous Lorentz group being isomorphic to SU(2)xSU(2).
  • Another participant asserts that the generators of the group follow identical algebras, implying a connection between the groups.
  • Contradictory claims arise regarding the isomorphism, with one participant stating that the proper orthochronous Lorentz group is not isomorphic to SU(2)xSU(2).
  • A subsequent reply challenges the initial claim, suggesting that while the statement is not true at the group level, it may hold at the level of Lie algebras after complexification.
  • One participant expresses uncertainty about their ability to explain the complexities involved in the discussion.

Areas of Agreement / Disagreement

Participants express disagreement regarding the isomorphism between the proper orthochronous Lorentz group and SU(2)xSU(2), with some asserting it is not true while others suggest a nuanced view depending on the context (group vs. Lie algebra).

Contextual Notes

The discussion highlights limitations in understanding isomorphisms, particularly regarding the conditions under which certain statements are valid, such as the distinction between group-level and Lie algebra-level isomorphisms.

ChrisVer
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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...
 
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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.
 

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