# A Lorentz invariant phase space - symplectic geometry

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1. Nov 5, 2018

### gasar8

I have an assignment to show that specific intensity over frequency cubed $$\frac{I}{\nu^3},$$ is Lorentz invariant and one of the main topics there is to show that the phase space is Lorentz invariant. I did it by following J. Goodman paper, but my professor wants me to show this in another way, using symplectic geometry (wedge products), which I am not really familiar with and Liouville's theorem. I found this on Wiki stating that "that the Lie derivative of volume form is zero along every Hamiltonian vector field." Does this prove the also Lorentz invariance?

Also I found these notes on symplectic geometry (chapter 13 on p76). Does the statement that $\omega$ is closed (or $d\omega = 0$), mean that it is invariant? Is this proof enough for Lorentz invariance?

Last edited: Nov 5, 2018
2. Nov 10, 2018

### PF_Help_Bot

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