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It may include a rotation of space; a rotation-free Lorentz transformation is called a Lorentz boost. In Minkowski space—the mathematical model of spacetime in special relativity—the Lorentz transformations preserve the spacetime interval between any two events. This property is the defining property of a Lorentz transformation.
is the generator of the boost in the x direction, so the infinitesimal boost is B ( e x , ϕ ) = I − ϕ K x {\displaystyle B(\mathbf {e} _{x},\phi )=I-\phi K_{x}} Now, ϕ is small, so dividing by a positive integer N gives an even smaller increment of rapidity ϕ / N , and N of these infinitesimal boosts will give the original infinitesimal ...
(The other similar generator, K y + J x as well as it and J z comprise altogether the little group of the light-like vector, isomorphic to E(2).) The action of a Lorentz boost in the x-direction on the light-cone and 'celestial circle' in 1+2 spacetime.
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The (homogeneous) Lorentz group is 6-dimensional. The boost generators K and rotation generators J can be combined into one generator for Lorentz transformations; M the antisymmetric angular momentum tensor, with components = =, =. and correspondingly, the boost and rotation parameters are collected into another antisymmetric four-dimensional ...
The boost and rotation generators have representations denoted D(K) and D(J) respectively, the capital D in this context indicates a group representation. For the Lorentz group, the representations D(K) and D(J) of the generators K and J fulfill the following commutation rules.
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