Realistic physical origin of the quantum observable operator algebra in the frame of the causal stochastic interpretation of quantum mechanics: The relativistic spin-zero case

The deduction by Guerra and Marra of the usual quantum operator algebra from a canonical variable Hamiltonian treatment of Nelson’s hydrodynamical stochastic description of real nonrelativistic Schrödinger waves is extended to the causal stochastic interpretation given by Guerra and Ruggiero and by Vigier of relativistic Klein-Gordon waves. A specific representation shows that the Poisson brackets for canonical hydrodynamical observables become ‘‘averages’’ of quantum observables in the given state. Stochastic quantization thus justifies the standard procedure of replacing the classical particle (or field) observables with operators according to the scheme $p_{\mu }$→-iħ${\partial }_{\mu }$ and $L_{\mu \nu }$→-iħ($x_{\mu }$ ${\partial }_{\nu }$-$x_{\nu }$ ${\partial }_{\mu }$).