Abstract

Classical mechanics assumes that its laws (and specifically the second law of Newton) are independent of spatio-temporal resolutions. To see whether there is an alternative to this assumption we write the energy of a relativistic particle in a finite-difference form, e.g., ɛ=ɛ0[1-(Δx/c Δt)2]1/2. We assume that in the limit Δt→0 the energy ε has a simple pole a/Δt. We show that quantum mechanics in its different formulations (Schrödinger, Feynman, Schwinger, Klein-Gordon, and Dirac) follows in elementary fashion from this assumption. We also rigorously prove that the classical Poisson brackets represent a coefficient in the first power of h in the expansion of the commutator in power series in h. The uncertainty relations are seen as the consequences of well-defined resolution dependence of measurements. In particular, if the simple pole in the energy dependence on Δt is located at α′ m/h, where α′ is the stringy constant, then this dependence yields the uncertainty principle of the string theory. Here α′ m/h can be interpreted as the highest physically allowed temporal resolution