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.

 

[sir-pinski]

Hello M,

That is a valid simplification. The Lorentz invariant measure is a concept in special relativity that ensures that physical laws and equations remain the same in all inertial reference frames. It is a measure of momentum space that takes into account the Lorentz transformation, which is a mathematical representation of the effects of relative motion on space and time. In your example, by changing the coordinates to q, you are essentially applying the Lorentz transformation to the measure, making it invariant under such transformations. Therefore, your simplification is valid and not cheating. Hope this helps clarify your question.