Lorentz Transformations boost?

  • #1
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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!
 

Answers and Replies

  • #2
fzero
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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!
[tex]B( \theta, \vec{n} )[/tex] is an element of a representation of [tex]SL( 2 , \mathbb{C} )[/tex]. What vector space does the representation act on? Is there a map from Minkowski space to this vector space that you know of?
 
  • #3
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[tex]B( \theta, \vec{n} )[/tex] is an element of a representation of [tex]SL( 2 , \mathbb{C} )[/tex]. What vector space does the representation act on? Is there a map from Minkowski space to this vector space that you know of?
well [itex]SL(2, \mathbb{C})[/itex] is just matrices so surely they can act on any vector in [itex]\mathbb{R}^2[/itex]?
 

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