Q:Is it possible to do a coordinate transfomation in momentum space?

[FinalCatch]

Q: How does one do a coordinate transformation in momentum space while insuring conservation of momentum?

I have a several particles with momentum components $P_x , P_y , P_z$.
I would like to rotate the x, y, and z axis. By angle θ in the x/y and angle Θ in the y/z .
So giving new momentum $P_x' , P_y' , P_z'$.

Is it possible to do this an conserve momentum while remaining in the lab frame? (The particles are relativistic but I don't believe this matters). What are the coordinate transformations?

 

[HomogenousCow]

The conserved quantity is a vector, you can rotate the basis vectors however you want and it wouldn't change the magnitude of the vector.

 

[FinalCatch]

just to double check $P^2=P_x^2 +P_y^2 +P_z^2=P_x'^2 +P_y'^2 +P_z'^2$ correct?

 

[Nugatory]

just to double check $P^2=P_x^2 +P_y^2 +P_z^2=P_x'^2 +P_y'^2 +P_z'^2$ correct?

If you've done the transformation to the primed coordinates correctly, yes.

In fact, HomogeneousCow has understated how much is conserved; the direction of the vector is also conserved. Of course it's a bit tricky talking about the "direction" of a vector when you don't have coordinate axes to make angles with - (1,0) in coordinates in which the x-axis points to the northeast is the same vector in the same direction as $\sqrt{2}/2(1,1)$ in coordinates in which the x-axis points east, but it's not obvious at all from the coordinates that that is so.

However, the dot-product of two vectors is invariant under these coordinate transformations, and as the dot-product depends on the angle between the vectors, that gives us a coordinate-independent way of claiiming that direction is also invariant.

 

[dipole]

This article might be worth reading.

http://en.wikipedia.org/wiki/Canonical_transformation