With radian as a unit, should action and angular momentum have the different units?

Question

If one accepts radian as a fundamental unit, does it make sense that action and angular momentum have units differing in radian to the power of one?

The same question applies for energy and torque.

The origin of this is that, we usually write

$h\nu = \hbar\omega$

and also $\nu$ has units of $1/Time$, and $\omega$ has units of Angle/Time.

Interestingly, if so, that will make $\hbar$ and $h$ have different units. The first being the fundamental unit of angular momentum and the second being the fundamental unit of mechanical action.

• At the end it could be that question boils down to whether such distinction is useful (and not self-contradictory)?

Although I don't write it explicitly it usually helps me think that these pairs of quantities have different units.

Answer 1

Radians are a unit, but they are a dimensionless unit. When dealing with explicit angles, it can be useful to maintain the unit of radians. However, when dealing with things that are more complicated, it is extra work to carry around dimensionless units like radians; it adds no particular value; and it is not, in general, even possible to carry such units around consistently. (Is the $x$ in $e^{ix}=\cos x+i\sin x$ supposed to have units of radians? What if $x$ is complex? There is no unique answer.)

Sometimes different names are given to different units when they measure different quantities. Dimensionally, 1 statvolt/cm is the same as 1 Gauss, but they are used to measure electric and magnetic fields, respectively, so we call them by different names. Dimensionally, 1 radian is just that same as the pure number 1, but sometimes it is useful to distinguish between them. However, that distinction cannot be usefully maintained in more complicated calculations, so it is not really meaningful or useful to try to distinguish, say, the units of action from the units of angular momentum.