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Under a Galilean transformation, the coordinates and momenta of any system transform as: $$ t \rightarrow t',\\ \vec r\:' = \vec r + \vec vt,\\ \vec p\:' = \vec p + m\vec v $$ where $\vec v$ is velocity of frame moving w.r.t it. Now, what will be unitary transformation that can that will carry out this transformation. Let us say the total momentum of the system is $\vec P$ and its mass $M$ and its position $X$. I know how to write a single coordinate translation, but am not able to put all these together.

An Attempt : The transformation should contain :

$ \bf e^{i \vec p.\vec x} $ for translations and $ \bf e^{iHt}$ for time translations, but what about generators of Velocity transformations ?

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  • $\begingroup$ I think this question answers mine, almost : physics.stackexchange.com/q/56024. $\endgroup$
    – user38249
    Commented Feb 13, 2014 at 15:09
  • $\begingroup$ Although, I would be interested understanding how generators of Galilean boosts can be found. $\endgroup$
    – user38249
    Commented Feb 13, 2014 at 15:10

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Time translations are generated by $H$, as expected. The generator for Gallilean boosts $\vec{K}$ is related to the CM operator $\vec{R}=1/M\sum_i m_i\vec{x}_i$, $\vec{K}=M\vec{R}$. See http://en.wikipedia.org/wiki/Schr%C3%B6dinger_group for some references.

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