Proving the Isomorphism Property of the Spinor Map in SL(2,C) and SO(3,1)

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In summary, the conversation discusses the homomorphism between SL(2,C) and SO(3,1) and the explicit form of the isomorphism, where \textbf{x} is a 2x2 matrix of SL(2,C) and x^{\mu} a 4-vector of SO(3,1). The linear map (spinor map) \textbf{x}\rightarrow\textbf{x}'=A\textbf{x}A^{\dagger} is also mentioned, as well as the relationship between 4-vectors on the SO(3,1) side. The main topic of discussion is proving \phi(AB)=\phi(A)\phi(B), and a reference is given
  • #1
gentsagree
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In the context of the homomorphism between SL(2,C) and SO(3,1), I have that

[tex]\textbf{x}=\overline{\sigma}_{\mu}x^{\mu}[/tex]

[tex]x^{\mu}=\frac{1}{2}tr(\sigma^{\mu}\textbf{x})[/tex]

give the explicit form of the isomorphism, where [itex]\textbf{x}[/itex] is a 2x2 matrix of SL(2,C) and [itex]x^{\mu}[/itex] a 4-vector of SO(3,1).

Considering the linear map (the spinor map)

[tex]\textbf{x}\rightarrow\textbf{x}'=A\textbf{x}A^{\dagger}[/tex]

one can show that the 4-vectors on the SO(3,1) side are also linearly related by

[tex]x'^{\mu}=\phi(A)^{\mu}_{\nu}x^{\nu}[/tex]

where it is easy to show that

[tex]\phi(A)^{\mu}_{\nu}=\frac{1}{2}tr(\sigma^{\mu}A\overline{\sigma}_{\nu}A^{\dagger})[/tex]

I understand all this, but I want to prove that [itex]\phi(AB)=\phi(A)\phi(B)[/itex]. How would I go about doing this? I tried a few things but not very successfully.
 
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  • #2
I can't copy paste the proof here, I can only tell you where to find it: Mueller-Kirsten + Wiedemann's <Supersymmetry> (WS, 1987), pages 66 and 67.
 
  • #3
Thanks a lot dextercioby, the book is really helpful!
 

What is a group homomorphism?

A group homomorphism is a function between two groups that preserves the group structure. This means that the operation of the first group is preserved when the function is applied to it, resulting in the same operation as if it were applied to the second group.

What is the difference between a group homomorphism and an isomorphism?

The main difference between a group homomorphism and an isomorphism is that a group homomorphism does not necessarily preserve the group's identity element, while an isomorphism does. Additionally, an isomorphism is a bijective homomorphism, meaning that it has a one-to-one correspondence between the elements of the two groups.

How do you prove that a function is a group homomorphism?

To prove that a function is a group homomorphism, you need to show that it preserves the group operation. This means that for any two elements in the first group, the function applied to their operation should result in the same operation as the function applied to the individual elements in the second group.

What is the importance of group homomorphisms in mathematics?

Group homomorphisms are important in mathematics because they allow us to understand the relationship between different groups. They also help us to classify and categorize groups, and can be used to solve mathematical problems involving groups.

Can a group homomorphism be both injective and surjective?

Yes, it is possible for a group homomorphism to be both injective and surjective. This type of homomorphism is called an isomorphism, and it is a bijective homomorphism that preserves the identity element and the group operation. In other words, it is a one-to-one correspondence between the elements of the two groups, and the group structures are preserved.

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