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

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In summary, the conversation discusses the concept of isomorphism in relation to the proper orthochronous Lorentz group and SU(2)xSU(2). It is mentioned that the two groups are not isomorphic at the group level, but they are almost isomorphic at the level of Lie algebras. The conversation ends with the suggestion of a longer explanation to clarify the concept further.
  • #1
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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  • #2
It means that the generators of the group follow identical algebras
 
  • #3
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).
 
  • #4
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.
 
  • #5
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.
 

1. What is an isomorphism?

An isomorphism is a bijective function that preserves the structure of a mathematical object between two different sets. In other words, it is a mapping that maintains the relationships and operations within the sets.

2. What is the significance of SU(2)xSU(2) and Lorentz group in isomorphisms?

SU(2)xSU(2) is a mathematical structure that represents the symmetries of a four-dimensional space-time, while the Lorentz group is a fundamental group in the theory of relativity. These groups are important in isomorphisms because they allow us to understand the transformations between different reference frames.

3. How does an isomorphism relate to physical examples?

In physics, isomorphisms can be used to understand the underlying symmetries and transformations within physical systems. This can help us make predictions and solve complex problems in fields such as particle physics and quantum mechanics.

4. Can you provide a physical example of an isomorphism using SU(2)xSU(2) and the Lorentz group?

One physical example of an isomorphism using these groups is the relationship between spin-1/2 particles and the Lorentz group. Spin-1/2 particles have a symmetry that is isomorphic to SU(2)xSU(2), and the Lorentz group can be used to describe the transformations of these particles in different reference frames.

5. How do isomorphisms contribute to our understanding of the physical world?

Isomorphisms are powerful tools in physics because they allow us to connect different mathematical structures and understand the underlying symmetries and transformations in the physical world. They also help us make predictions and solve complex problems in various fields of physics.

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