Quantization of Relativistic action in multiples of Planck's (constant) Quantum of action

Quantization of Relativistic action in multiples of Planck’s (constant) Quantum of action

Quantization of Relativistic action in multiples of Planck’s (constant) Quantum of action.

a new axiom for special relativity theory.

The third axiom:
Relativistic action is limited to Planck’s Quantum of action.
$\mathcal {S}=\int^{t_f}_{t_i}\mathcal {L}dt=n.h\qquad n \in\mathbb{Z}.$

where the $\mathcal {L}=-m_oc^2\gamma^{-1},$ is the Lagrangian.

action for a point particle in a curved spacetime.
$\mathcal S =-Mc \int ds = -Mc \int_{\xi_0}^{\xi_1}\sqrt{g_{\mu\nu}(x)\frac{dx^\mu(\xi)}{d\xi} \frac{dx^\nu(\xi)}{d\xi}} \ \ d\xi=nh$

Quantization of Nambu–Goto action:

$\mathcal{S} ~=~ -\frac{1}{2\pi\alpha'} \int \mathrm{d}^2 \Sigma \sqrt{{\dot{X}} ^2 - {X'}^2}~=~nh\qquad n \in\mathbb{Z}.$

point:
The action

$S= - E_0 ~ \Delta \tau$

of a relativistic particle is minus the rest energy $E_0=m_0c^2$ times the change $\Delta \tau=\tau_f-\tau_i$ in proper time.

Single relativistic particle

When relativistic effects are significant, the action of a point particle of mass ”m” travelling a world line ”C” parametrized by the proper time $\tau$ is
: $S = - m_o c^2 \int_{C} \, d \tau$ .

If instead, the particle is parametrized by the coordinate time ”t” of the particle and the coordinate time ranges from ”t”₁ to ”t”₂, then the action becomes
: $\int_{t1}^{t2}\mathcal {L} \, dt$

where the Lagrangian is
: $\mathcal {L} = - m_o c^2 \sqrt {1 - \frac{v^2}{c^2}}$ .