In this Physics.SE post, there is a transformation: $$Q = q,$$ $$P = \sqrt{p} - \sqrt{q}.$$
for Hamiltonian $H = \frac{p^2}{2}$. The post discusses the validity of this transformation as a canonical transformation.
But here I'm want to ask if the transformation is valid, because the dimension of $\sqrt{p}$ is not always same with that of $\sqrt{q}$. If these two dimensions are different, how can they be connected through a minus sign, and what is the dimension of the result?
Suppose that instead, one wrote \begin{align} P = a\sqrt{p} - b\sqrt{q} \end{align} such that $a$ and $b$ had dimensions engineered to make the dimensions match in both terms, then there clearly would be no issue.
Now imagine that the values of $a$ and $b$ in a given system of units (say SI units) are $1$; then we obtain the canonical transformation written in the question. As long as we work in this system of units where $a$ and $b$ both have value $1$, then everything will be fine.
Assuming that this has been done is useful because then we don't have to carry around the parameters $a$ and $b$ that would make computations more cumbersome. This is similar to how particle physicists tend to work in units where the value of the speed of light is unity; $c=1$.
