## 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
:$\mathcal {L} = - m_o c^2 \sqrt {1 - \frac{v^2}{c^2}}$.