On a newtonian-like formulation of Einstein's relativity and relativistic quantum mechanics

In relativity space-time is described by charts associating to each event four numbers ( x¹, x², x³, x⁴) ≡ ( x, t) referring to a position and a time. These charts are related to each other by the transformations of the inhomogeneous orthochronous Lorentz group. In this article we discuss the possible description of space-time in relativity by charts associating to each event four numbers ( q¹, q², q³, q⁴) ≡ ( q, τ) in such a way that these charts are related to each other by transformations of the inhomogeneous Galilei group. Then τ refers to a universal time and q refers to a position in a three-dimensional space supplied with the geometry of the Euclidean group. It is shown that such a description is obtained by measuring time according to the clocks of a privileged frame and by defining the unit length in an appropriate way. The above discussion is mainly motivated by considerations concerning quantum mechanics. Actually, universal time τ permits a consistent quantization in relativity. Afterwards we formulate a model for the quantum relativistic spinless particle of mass m₀ based on the assumption that the evolution is governed by the universal time τ. Except for the choice of the Hamiltonian, this model is formulated analogously to the corresponding one in the nonrelativistic case, in particular there exists a position observable. Further we compare the above model with the usual relativistic formalisms. For the free particle case our model may be formulated such as to contain Wigner's group theoretical approach to relativistic quantum mechanics. Further the case of the particle interacting with an external electromagnetic field is discussed in detail and the model is finally compared with the usual Klein-Gordon formalism.