Decompose SL(2C) Matrix: Real Parameters from Complex

[DrDu]

TL;DR Summary

How can an SL(2,C) matrix be decomposed into a product of a boost along the z-axis and a pure rotation?

Hi,

suppose I am given an SL(2C) matrix of the form $\exp(i\alpha/2 \vec{t}\cdot\vec{\sigma})$ where $\alpha$ is the complex rotation angle, $\vec{t}$ the complex rotation axis and $\vec{\sigma}$ the vector of the three Pauli matrices.
I would like to decompose this vector into $\exp(i\beta/2 \vec{q}\cdot\vec{\sigma})\exp(\gamma\sigma_z)$, where now the rotation angle $\beta$, axis $\vec{q}$ and the boost parameter $\gamma$ are all real.
Is there a non-brain damaged way to do this? This isn't homework related.

Thank you!

 

[fresh_42]

Just a thought, and maybe a silly one as I think in terms of algebra and less in terms of physics.
$\vec{t}\cdot\vec{\sigma}$ is a complex number, hence $\exp(i\alpha/2 \vec{t}\cdot\vec{\sigma})$ is of the form $e^{x+iy}=e^x\cdot e^{iy}$. In case $\vec{t}\cdot\vec{\sigma}$ is real, which I assume, then $\beta =2\alpha /(\vec{t}\cdot\vec{\sigma}), \gamma =0$ and $\vec{q}$ any vector such that $\vec{q}\cdot\vec{\sigma}=1.$

 

[DrDu]

Hi fresh, thank you. No, t sigma is not real. I now saw that my initial assumption is incorrect. You can't decompose a general Lorentz transformation into a boost along the z axis and a rotation. However, you can decompose it into a boost (along a direction which has to be determined) and a rotation. This is in principle a polar decomposition. I found some information in be book by Sexl and Urbantke.

 

[vanhees71]

I'm not sure, but couldn't one write down the series of the matrix exponential and somehow sort out the real and imaginary parts?

 

[fresh_42]

Hi fresh, thank you. No, t sigma is not real. I now saw that my initial assumption is incorrect. You can't decompose a general Lorentz transformation into a boost along the z axis and a rotation. However, you can decompose it into a boost (along a direction which has to be determined) and a rotation. This is in principle a polar decomposition. I found some information in be book by Sexl and Urbantke.

This all sounds like the decomposition of a matrix into its toral (diagonalizable) and nilpotent (upper triangular) part. Within Lie algebras it is called Malcev decomposition or Jordan-Chevalley. I asssume the latter holds for groups, too.

 

[DrDu]

Of course, but this wouldn't provide much insight.

This all sounds like the decomposition of a matrix into its toral (diagonalizable) and nilpotent (upper triangular) part. Within Lie algebras it is called Malcev decomposition or Jordan-Chevalley. I asssume the latter holds for groups, too.

Yes, it is similar. In this case it is the polar decomposition, i.e. the decomposition of a normal matrix into the product of a hermitian and a unitary one.