Why the Six Generators of the Restrict Lorentz Group

[martyf]

Why the six generator of the restrict lorentz group are the three rotation's generator(angular momentum) and the three boost's generator?

 

[Dale]

The Lorentz group is a group of linear transforms, so it preserves the origin. If you want a more general transform that includes translations as well as rotations and boosts, then you want the Poincare group. That is a group of affine transforms, so it doesn't preserve the origin.

 

[Fredrik]

Why the six generator of the restrict lorentz group are the three rotation's generator(angular momentum) and the three boost's generator?

What exactly is it that you would like to know? There are at least four different things that could be your main concern: The number of independent parameters, the interpretation of the parameters as boosts and rotations, the topology of the group (i.e. what well-known set it can be continuously and bijectively mapped onto), and the commutation relations satisfied by the generators.

The number of independent parameters follows immediately from the condition $\Lambda^T\eta\Lambda=\eta$. The fact that 3 parameters correspond to rotation parameters follow from the fact that restricted Lorentz transformations that leave $x^0$ unchanged are rotations (the components of such a $\Lambda$ that aren't on the 0th row or 0th column form a 3x3 orthogonal matrix). The fact that 3 parameters correspond to a velocity change follow from the fact that $\Lambda$ takes the time axis to some other straight line to the origin. (The slope of that line can be interpreted as a speed, and its projection onto the x-y-z hyperplane defines a direction). The topology stuff and the commutation relations involve too much typing for me to include those details here. You can find them in lots of books, e.g. Weinberg's QFT book (vol.1, the appendix to chapter 2).

 

[martyf]

Thank's!
I wanted to know the imterpretation of the parameters as boosts and rotations!