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latentcorpse
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Define [itex]B( \theta, \vec{n} ) \in SL( 2 , \mathbb{C} )[/itex] by
[itex]B( \theta , \vec{n}) = \cosh { \frac{1}{2} \theta} + \vec{\sigma} \cdot \vec{n} \sinh{ \frac{1}{2} \theta}[/itex] where [itex]\vec{n}^2 =1 [/itex]
Show that this corresponds to a Lorentz boost with velocity [itex]\vec{v}=\tanh{ \theta} \vec{n}[/itex]. Show that
[itex] ( 1 + \frac{1}{2} \vec{sigma} \cdot \delta \vec{v}) B(\theta, \vec{n}) = B( \theta' , \vec{n}' ) R[/itex]
where, to 1st order in [itex]\delta \vec{v}[/itex],
[itex]\theta' = \theta + \delta \vec{v} \cdot \vec{n}[/itex], [itex]\vec{n}'=\vec{n} \coth{\theta} (\delta \vec{v} - \vec{n} \vec{n} \cdot \delta \vec{v})[/itex]
and [itex]R[/itex] is an infinitesimal rotation given by
[itex]R= 1 + \tanh{\frac{1}{2} \theta} \frac{1}{2}i ( \delta \vec{v} \times \vec{n} ) \cdot \vec{\sigma} = 1 + \frac{\gamma}{\gamma + 1} \frac{1}{2} i ( \delta \vec{v} \times \vec{v} ) \cdot \vec{\sigma}[/itex]
and [itex]\gamma = ( 1 - \vec{v}^2) ^{-\frac{1}{2}}[/itex]
Show that [itex]\vec{v}' = \vec{v} + \delta \vec{v} - \vec{v} \vec{v} \cdot \delta \vec{v}[/itex]
Note that [itex]\vec{\sigma} \cdot \vec{a} \vec{\sigma} \cdot \vec{b} = \vec{a} \cdot \vec{b} 1 + i \vec{\sigma} \cdot ( \vec{a} \times \vec{b} )[/itex]I don't understand how to go about the first bit here - what do I need to do in order to show this is a Lorentz boost?
Thanks!
[itex]B( \theta , \vec{n}) = \cosh { \frac{1}{2} \theta} + \vec{\sigma} \cdot \vec{n} \sinh{ \frac{1}{2} \theta}[/itex] where [itex]\vec{n}^2 =1 [/itex]
Show that this corresponds to a Lorentz boost with velocity [itex]\vec{v}=\tanh{ \theta} \vec{n}[/itex]. Show that
[itex] ( 1 + \frac{1}{2} \vec{sigma} \cdot \delta \vec{v}) B(\theta, \vec{n}) = B( \theta' , \vec{n}' ) R[/itex]
where, to 1st order in [itex]\delta \vec{v}[/itex],
[itex]\theta' = \theta + \delta \vec{v} \cdot \vec{n}[/itex], [itex]\vec{n}'=\vec{n} \coth{\theta} (\delta \vec{v} - \vec{n} \vec{n} \cdot \delta \vec{v})[/itex]
and [itex]R[/itex] is an infinitesimal rotation given by
[itex]R= 1 + \tanh{\frac{1}{2} \theta} \frac{1}{2}i ( \delta \vec{v} \times \vec{n} ) \cdot \vec{\sigma} = 1 + \frac{\gamma}{\gamma + 1} \frac{1}{2} i ( \delta \vec{v} \times \vec{v} ) \cdot \vec{\sigma}[/itex]
and [itex]\gamma = ( 1 - \vec{v}^2) ^{-\frac{1}{2}}[/itex]
Show that [itex]\vec{v}' = \vec{v} + \delta \vec{v} - \vec{v} \vec{v} \cdot \delta \vec{v}[/itex]
Note that [itex]\vec{\sigma} \cdot \vec{a} \vec{\sigma} \cdot \vec{b} = \vec{a} \cdot \vec{b} 1 + i \vec{\sigma} \cdot ( \vec{a} \times \vec{b} )[/itex]I don't understand how to go about the first bit here - what do I need to do in order to show this is a Lorentz boost?
Thanks!