Recently I stumbled upon an apparently trivial question: what is the origin of quantum of action (Plank's constant h). Originally, Plank, Einstein, & others were talking about energy quantization Delta E = h omega, with h already figuring as a universal constant. Later (1916) there appeared the Wilson-Zommerfeld quantum condition (often referred to as the Bohr-Zommerfeld condition) Int (p dq) = n h, with the integral taken over the period of a conservative system. This has a graphical interpretation as the quantization of phase space, which has dimensionality of action, and the Int itself is = S, the part of classical action. Hence the universally used expression "quantum of action", which I myself have used in my QM class without giving it more thought. But then I was reminded that general definition of action includes the temporal part as well, according to which the complete action is dS = pdx - Hdt, where H is the classical Hamiltonian. In that
case the Bohr-Sommerfeld condition only proves the quantization of the quantity S + Int H dt. How come then did Plank, Sommerfeld and others conclude that h was a quantum of action?
Also, is there a way to derive quantization of action within the framework of the new quantum theory, by bringing in the indeterminacy relation between time and energy, Delta E Delta t > h/2 pi (apart from Delta p Delta x > h/2 pi). If we additionally require Int Hdt = m h with integer m, so we would have S= j h with integer j = n - m. This also seems reasonable. But I have never seen in literature any explicit mentioning of these complications. Any thoughts about this? Or any references to a sources discussing this questions?