We propose a new variational principle for out-of-equilibrium dynamic systems that are fundamentally based on the method of Lagrange multipliers applied to the total entropy of an ensemble of particles. However, we use the fundamental equation of thermodynamics on differential forms, considering U and S as 0-forms. We obtain a set of two first order differential equations that reveal the same f...

We show that the probabilistic formalism of QM can be obtained as a special projection of classical statistical mechanics for systems with an infinite number of degrees of freedom. Such systems can be interpreted as classical fields. Thus in our approach QM is a projection of (prequantum) classical statistical field theory (PCSFT). This projection is based on the Taylor expansion of classical p...

This is the written version of a short talk on 10 Conference on Problems and Methods in Mathematical Physics (September 13 17, 1993 in Chemnitz, Germany). A new scheme of the quantization is presented. A realization of the scheme for a particle in n-dimensional space by two-sided convolutions on the Heisenberg group is constructed.

According to Dirac, fundamental laws of Classical Mechanics should be recovered by means of an ”appropriate limit” of Quantum Mechanics. In the same spirit it is reasonable to enquire about the fundamental geometric structures of Classical Mechanics which will survive the appropriate limit of Quantum Mechanics. This is the case for the symplectic structure. On the contrary, such geometric struc...

This is the written version of a short talk on 10th Conference on Problems and Methods in Mathematical Physics (September 13 17, 1993 in Chemnitz, Germany). A new scheme of the quantization is presented. A realization of the scheme for a particle in ndimensional space by two-sided convolutions on the Heisenberg group is constructed.

Compared with introductory physics, relatively little is known about the development of expertise in advanced physics courses, especially in the case of quantum mechanics. Here, we describe a framework for understanding the patterns of student reasoning difficulties and how students develop expertise in quantum mechanics. The framework posits that the challenges many students face in developing...

We develop an approach to the classical limit of quantum theory using the mathematical framework of nonstandard analysis. In this framework innnitesimal quantities have a rigorous meaning, and the quantum mechanical parameter h can be chosen to be such an innnitesimal. We consider those bounded observables which are transformed continuously on the standard (non-innnitesimal) scale by the phase ...

We present in this article all Hamiltonian systems in E(2) that are separable in cartesian coordinates and that admit a third-order integral, both in quantum and in classical mechanics. Many of these superintegrable systems are new, and it is seen that there exists a relation between quantum superintegrable potentials, invariant solutions of the Korteweg-De Vries equation and the Painlevé trans...

Classical contact Lie algebras are the fundamental algebraic structures on the manifolds of contact elements of configuration spaces in classical mechanics. In this paper, we determine the structure of the currently largest known category of contact simple Lie algebras introduced earlier by the second author. These algebras are in general not finitely-graded.

In this paper, I examine whether or not the Hamiltonian and Lagrangian formulations of classical mechanics are equivalent theories. I do so by applying a standard for equivalence that was recently introduced into philosophy of science by Halvorson (2012, 2016) and Weatherall (2016a). This case study yields three general philosophical payoffs. The first is a specific plea for clarity in our disc...