- 63

- 0

**1. Homework Statement**

I have an assignment to prove that specific intensity over frequency cubed is Lorentz invariant. One of the main tasks there is to prove the invariance of phase space [itex]d^3q \ d^3p[/itex] and I am trying to prove it with symplectic geometry. I am following Jorge V. Jose and Eugene J. Salentan: Classical dynamics.

**2. Homework Equations**

[tex]\omega = dq^{\alpha} \wedge dp_{\alpha}\\

v = d^3q \ d^3p = \frac{1}{n!} \omega^{\wedge n} = \frac{1}{n!} \omega \wedge \omega \wedge \cdots \wedge \omega = dq^1 \wedge dp_1 \wedge \cdots \wedge dq^n \wedge dp_n,[/tex]

of course in my case [itex]n=3[/itex].

**3. The Attempt at a Solution**

The book says that this [itex]v[/itex] is invariant under canonical transformations, because [itex]\omega[/itex] is. I am now wondering if this is enough also for the Lorentz invariance?