Invariance of measure under Lorentz transformations without the four dimensional expression

Ticciati's book Quantum Field Theory for Mathematicians states that the invariance of the measure $$\int\frac{\text{d}^3\vec{k}}{2\sqrt{\left\|\vec{k}\right\|^2+\mu^2}}$$ can be shown by direct computation of the Jacobian, i.e. without using the invariant expression $$d^4k\delta(k^2-\mu^2)\Theta(k_0)$$ (see Homework 1.4.7). He states this is achieved by an "intelligent" factorization of the Lorentz transformation. Can somebody point me in the correct direction? Thanks!

Since every orthochronous Lorentz transformation can be decomposed into a product of proper rotations and a Lorentz boost in the $$x^1$$ direction, and the measure is clearly invariant under proper rotations, it is enough to consider Lorentz transformations $$\tilde{k}^r=\Lambda^r_\mu k^\mu$$ of the form $$\tilde{k}^1=-\omega\left(\vec{k}\right)\sinh(\lambda)+k^1\cosh(\lambda),$$ $$\tilde{k}^2=k^2$$ and $$\tilde{k}^3=k^3$$. The Jacobian of this transformation is clearly $$\det(\Lambda^r_s)=\cosh(\lambda)$$. However I cannot show that $$\omega\left(\vec{l}\right)=\cosh(\lambda)\omega\left(\vec{k}\right)$$. In fact, since $$k$$ is a vector $$\omega\left(\vec{k}\right)$$ goes to $$\cosh(\lambda)\omega\left(\vec{k}\right)-\sinh(\lambda)k^1$$. Does anyone know how to go around this?

Notation: $$\omega\left(\vec{k}\right):=\sqrt{\left\|\vec{k}\right\|^2+\mu^2}=k^0$$

• Perhaps he is referring to the fact that any Lorentz transformation is a product of at least one boost and at least one rotation. Therefore, if you show the measure is invariant under boosts and rotations separately then you have shown it for all Lorentz transformations. – Luke Pritchett Feb 13 at 3:21
• Oh, that may be the way to go. Rotations have unit determinant and are orthogonal. Thus, they preserve both $\text{d}^3k$ and $\left\|\vec{k}\right\|$. We thus only have to prove the theorem for boosts. Moreover, since a Lorentz transformation (of course, in this whole discussion we only care about those of the proper orthochronous type) can always be decomposed as a product of two rotations and a boost in the $x$ direction, we only need to show the theorem holds for such a boost. It then should become a fairly straightforward algebraic exercise. – Iván Mauricio Burbano Feb 13 at 3:31