It is shown that Schroedinger's equation may be derived from three postulates. The first is a kind of statistical metamorphosis of classical mechanics, a set of two relations which are obtained from the canonical equations of particle mechanics by replacing all observables by statistical averages. The second is a local conservation law of probability with a probability current which takes the form of a gradient. The third is a principle of maximal disorder as realized by the requirement of minimal Fisher information. The rule for calculating expectation values is obtained from a fourth postulate, the requirement of energy conservation in the mean. The fact that all these basic relations of quantum theory may be derived from premises which are statistical in character is interpreted as a strong argument in favor of the statistical interpretation of quantum mechanics. The structures of quantum theory and classical statistical theories are compared and some fundamental differences are identified.

/pdf/the-statistical-origins-of-quantum-mechanics-1so1r5opur.pdf

The statistical origins of quantum mechanics

We analyze the Bianchi I cosmology in the presence of a massless scalar field and describe its dynamics via a semiclassical and quantum polymer approach. We study the morphology of the Big Bounce by adopting three different sets of configurational variables: the Ashtekar connections, a set of anisotropic volume-like coordinates and the Universe volume plus two anisotropy coordinates (the latter two sets of variables would coincide in the case of an isotropic Universe). In the semiclassical analysis we demonstrate that the value of the critical matter energy density depends on the Cauchy problem for the dynamics when adopting the Ashtekar connections or the anisotropic volume-like coordinates. On the contrary, when the Universe volume is considered as a configurational coordinate, we are able to derive a polymer-modified Friedmann equation for the Bianchi I model, from which the expression of the critical energy density can be derived. This analysis shows that the Big Bounce has universal features only when the Universe volume is defined on the polymer lattice. Then, a cosmological constant is included in the Ashtekar connections' formulation and some interesting results are mentioned making a comparison between the synchronous dynamics and that one when the scalar field is taken as a relational time. From a pure quantum point of view, we investigate the Bianchi I dynamics in terms of the Ashtekar connections. We apply the ADM reduction of the variational principle and then we quantize the system. We study the resulting Schr\"{o}dinger dynamics, stressing that the behavior of the wave packet peak over time singles out common features with the semiclassical trajectories, confirming the non-universal character of the emerging Big Bounce also on a quantum level. 

/pdf/semiclassical-and-quantum-features-of-the-bianchi-i-215f8tf1.pdf

Semiclassical and quantum features of the Bianchi I cosmology in the polymer representation

We analyze the Bianchi I cosmology in the presence of a massless scalar field and describe its dynamics via a semiclassical and quantum polymer approach. We study the morphology of the Big Bounce by adopting three different sets of configurational variables: the Ashtekar connections, a set of anisotropic volume-like coordinates and the Universe volume plus two anisotropy coordinates (the latter two sets of variables would coincide in the case of an isotropic Universe). In the semiclassical analysis we demonstrate that the value of the critical matter energy density depends on the Cauchy problem for the dynamics when adopting the Ashtekar connections or the anisotropic volume-like coordinates. On the contrary, when the Universe volume is considered as a configurational coordinate, we are able to derive a polymer-modified Friedmann equation for the Bianchi I model, from which the expression of the critical energy density can be derived. This analysis shows that the Big Bounce has universal features only when the Universe volume is defined on the polymer lattice. Then, a cosmological constant is included in the Ashtekar connections' formulation and some interesting results are mentioned making a comparison between the synchronous dynamics and that one when the scalar field is taken as a relational time. From a pure quantum point of view, we investigate the Bianchi I dynamics in terms of the Ashtekar connections. We apply the ADM reduction of the variational principle and then we quantize the system. We study the resulting Schrodinger dynamics, stressing that the behavior of the wave packet peak over time singles out common features with the semiclassical trajectories, confirming the non-universal character of the emerging Big Bounce also on a quantum level.

The statistical origin of quantum mechanics

The Heisenberg equations of motion for a quantum particle of mass $m$ are deduced from the infinitesimal qr-number equations of motion for the particle. The infinitesimal qr-number equations, and hence the standard quantum mechanical equations, are related to the qr-number equations in much the same way as the equations of geometric optics are related to those of wave optics. The qr-number equations of motion for a quantum particle of mass $m$ describe the motion of a lump, given by on open set in the qr-number space of the particle, while the infinitesimal qr-number equations describe the motion of a point-like particle. The qr-number equations of motion are the Hamiltonian equations of motion for a classical particle of mass $m$ expressed in qr-numbers. The proof requires that the particle's position operators have only continuous spectrum and the force functions are smooth.

Quantum dynamics is infinitesimal qr-number dynamics

ψ-function, the Generalized Ehrenfest Theorem re- veals some interesting exact relationships between quantum and classical expectation values. These general results seem to imply a deep-rooted unity (in spite of apparent radical differences in conceptual structures) between classical and quantum mechan- ics. Some significant consequences and important physical insights which follow from the general formulation are discussed with examples. Most im- portant is that it offers, under reasonable approx i- mations, a pure quantum mechanical description of the Stern-Gerlach experiment with realistic inhomo- geneous magnetic field (∇ ∇.B = 0). EVEN almost seventy-five years after the first formul a- tion of quantum mechanics (QM), its relation with clas - sical mechanics (CM) remains obscure. The usual belief that QM has superseded CM is also not well founded. In all cases where a new theory supersedes an old one, there is a universal prescription of recovering the mathematical structure of the old theory in the doma in of its validity, i.e. where it is empirically established. This is true for both special and general theory of rel a- tivity 1 . But for QM, no such universal approximation procedure can be offered. About the success and failure of the two mechanics we can make two positive asser- tions: (i) There are experimental results in the micr o- scopic domain (such as hydr ogen spectrum) or in the macroscopic domain (such as specific heat of so lids at low temperature) which cannot be interpreted within the framework of CM. (ii) In all such cases, the mathemat i- cal formalism of QM provides the correct empirical results. Not a single failure has yet been reported. These empirical evidences show that CM cannot be the universal mechanics and that QM has such a pote n- tiality. However, since it is not yet possible to recover CM in all the domains of its validity, we cannot assert that QM has superseded CM. A second alternative that

Significance of Ehrenfest theorem in quantum-classical relationship

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Significance of Ehrenfest theorem in quantum-classical relationship