Homomorphism SL(2,C) with restricted Lorentz

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In summary, the conversation discusses the statement that \Lambda^{\mu}_{\nu} = \frac{1}{2}Tr(\overline{\sigma}^{\mu}A\sigma^{\nu}A^{\dagger}) is a homomorphism and how this can be seen. The speaker mentions an identity about products of traces and the existence of a map, \phi, which is a homomorphism if \phi(x) \cdot \phi(y) = \phi(x*y). The proof of this is given by writing the traces with summed indices and using the completeness relation for the Pauli matrices. The conversation concludes with the suggestion to prove the completeness relation as an exercise.
  • #1
gentsagree
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We have that

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

I would like to make sense of the statement that this is a homomorphism because the correspondence above is preserved under multiplication.

Can someone clarify how I could see this?
 
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  • #2
Nevermind. While the tool needed was clear from the beginning, i.e. an identity about products of traces, I was completely oblivious to the existence of one. It turns out that

[tex]
\sum_{\mu}Tr(G\sigma_{\mu})Tr(\overline{\sigma}^{\mu}H) = 2 Tr(GH)
[/tex]

While I still don't know the proof of this, this is the correct answer.
 
  • #3
A map of the form:
[itex] \phi : A \rightarrow B[/itex]
[itex] a \rightarrow b=\phi(a)[/itex]
is roughly speaking an homomorphism if for elements [itex]x, y \in A[/itex] the [itex] \phi(x) \cdot \phi(y) = \phi(x*y)[/itex]
and that's why the comment about multiplication.

As for the proof of your last equation in post2, if I recall well you can prove it better by writting the traces with summed indices:
[itex] G_{ai} \sigma_{ia}^\mu \bar{\sigma}^\mu_{jb} H_{bj}[/itex]
and then using seperately the [itex]\sigma^0, \sigma^k[/itex]'s and use their completeness relation (generalization of the : https://en.wikipedia.org/wiki/Pauli_matrices#Completeness_relation)... but I don't really remember the complete proof...
 
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  • #4
yup it's straightforward (5-6 single-expression lines depending on how rigorous you want to be) with the completeness relation I gave you (obviously I just tried it o0))...

So I guess it would be a better "exercise" for you to prove the completeness relation for the pauli matrices that is given in the wiki article I sent you.
 

1. What is Homomorphism SL(2,C) with restricted Lorentz?

Homomorphism SL(2,C) with restricted Lorentz is a mathematical concept that describes the relationship between two mathematical structures called SL(2,C) and the restricted Lorentz group. SL(2,C) is a special linear group of 2x2 complex matrices, while the restricted Lorentz group is a group of transformations that preserve the spacetime interval in special relativity.

2. What is the significance of this concept?

This concept has significance in both mathematics and physics. In mathematics, it helps to understand the structure and properties of SL(2,C) and the restricted Lorentz group. In physics, it has applications in special relativity and quantum mechanics, as these fields heavily rely on the concepts of group theory.

3. How is Homomorphism SL(2,C) with restricted Lorentz related to special relativity?

The restricted Lorentz group is a fundamental group in special relativity, as it describes the transformations that preserve the spacetime interval between events. Homomorphism SL(2,C) is a way to map the elements of the restricted Lorentz group onto the elements of SL(2,C), providing a mathematical framework for studying the symmetries and transformations in special relativity.

4. Can you give an example of Homomorphism SL(2,C) with restricted Lorentz?

One example is the relationship between the restricted Lorentz group and the special unitary group SU(2). SU(2) is a subgroup of SL(2,C), and there exists a homomorphism from the restricted Lorentz group to SU(2). This homomorphism can be used to study the spinor representations of the restricted Lorentz group, which have important applications in quantum mechanics.

5. What are some current research topics related to Homomorphism SL(2,C) with restricted Lorentz?

Some current research topics include the study of representations of the restricted Lorentz group on higher-dimensional spaces, and the application of this concept in quantum gravity and string theory. Other areas of research include the use of Homomorphism SL(2,C) with restricted Lorentz in computer graphics and computer vision, as well as its applications in understanding the symmetries of physical systems.

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