Is Simplifying Lorentz Invariant Measures by Coordinate Change Valid?

[mtak0114]

Hi

I have a question about Lorentz invariant measures,
consider an integral of the form:
$$\int d\mu(p) f(\Lambda^{-1}p)$$

where $$d\mu(p) = d^3{\bf p}/(2\pi)^3(2p_0)^3$$ is the Lorentz invariant measure.

Now to simplify this I can make a change of coordinates

$$\int d\mu(\Lambda q) f(q)$$

can I then simplify this such that:

$$\int d\mu(q) f(q)$$

because this is Lorentz invariant or am I cheating?

thanks

M

 

[hamster143]

Yes, you can. Compare with Euclidean 2D case where $$d\mu = r dr d\phi$$ and $$\Lambda$$ is a rotation about the center.