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The Classical Newton-Lagrange equations of motion represent the fundamental physical law of mechanics. Their traditional Lagrangian and/or Hamiltonian precursors when available are essential in the context of quantization. However, there are situations that lack Lagrangian and/or Hamiltonian settings. This paper discusses a description of classical dynamics and presents some irresponsible speculations about its quantization by introducing a certain canonical two-form ?. By its construction ? embodies kinetic energy and forces acting within the system (not their potential). A new type of variational principle employing differential two-form ? is introduced. Variation is performed over "umbilical surfaces" instead of system histories. It provides correct Newton-Lagrange equations of motion. The quantization is inspired by the Feynman path integral approach. The quintessence is to rearrange it into an "umbilical world-sheet" functional integral in accordance with the proposed variational principle. In the case of potential-generated forces, the new approach reduces to the standard quantum mechanics. As an example, Quantum Mechanics with friction is analyzed in detail.

The present paper proposes canonical Dirac framework adapted for application to the electronic processes in solid state. The concern is a spatially periodic structure of atoms distinguished by birth and annihilation of particle states excited due to interaction with the electromagnetic field. This implies replacing the conventional energy-momentum relation specific of the canonical Dirac framework and permissible for particle physics by a case specific relation available for the solid state. The advancement is a unified and consistent mathematical framework incorporating the Hilbert space, the quantum field, and the special relativity. Essential details of the birth and annihilation of the particle states are given by an illustrative two-band model obeying basic laws of quantum mechanics, special relativity, and symmetry principles maintained from the canonical Dirac framework as a desirable property and as a prerogative for the study of the particle coupling and correlation. ; The present research has been supported by the Institute of Solid State Physics, the University of Latvia in the frames of the ERA-NET HarvEnPiez project (No. ES RTD/2016/15); Institute of Solid State Physics, University of Latvia as the Center of Excellence has received funding from the European Union's Horizon 2020 Framework Programme H2020-WIDESPREAD-01-2016-2017-TeamingPhase2 under grant agreement No. 739508, project CAMART²

Spatially homogeneous cosmological models of Bianchi types I, V and IX (as well as their isotropic counterparts : the Einstein-de Sitter, open and closed Friedmann-Robertson-Walker (FRW) models, respectively), filled with a perfect fluid with an equation of state of the type p = (γ - 1 )ρ (γ = constant comprised between 1 and 2) are quantized in the framework of an ADM-type canonical quantization scheme introduced by Demaret and Moncrief.
This method which generalizes a method due to Lund and originally restricted to space-times filled with a pressureless fluid (p = 0) is based on Schutz's Hamiltonian theory of a relativistic perfect fluid, extending to general relativity Seliger and Whitham's velocity-potential version of classical hydrodynamics. It leads to a Schrödinger-like equation for the wave function of the universe, which admits a true physical interpretation in terms of probability as in orthodox quantum mechanics.
Contrary to Misner's early results based on a different quantization scheme, our method leads to the conclusion that, apart from a set of measure zero of models comprising FRW models as well as Bianchi models filled with a fluid with a stiff equation of state (p = ρ), all Bianchi models are non-singular, in the sense that the quantum wave function becomes zero at the classical singularity. Such a non-singular model can be interpreted as a contracting universe bouncing into an expanding universe, due to quantum gravitational fluctuations. A similar conclusion of non-singularity of a quantum Bianchi I universe has recently been obtained by Narlikar by using a different method of quantization, i.e. the method of path integration.

Elementary interactions are formulated according to the principle of minimal interaction although paying special attention to symmetries. In fact, we aim at rewriting any field theory on the framework of Lie groups, so that, any basic and fundamental physical theory can be quantized on the grounds of a group approach to quantization. In this way, connection theory, although here presented in detail, can be replaced by "jet-gauge groups" and "jet-diffeomorphism groups." In other words, objects like vector potentials or vierbeins can be given the character of group parameters in extended gauge groups or diffeomorphism groups. As a natural consequence of vector potentials in electroweak interactions being group variables, a typically experimental parameter like the Weinberg angle ϑW is algebraically fixed. But more general remarkable examples of success of the present framework could be the possibility of properly quantizing massive Yang–Mills theories, on the basis of a generalized Non-Abelian Stueckelberg formalism where gauge symmetry is preserved, in contrast to the canonical quantization approach, which only provides either renormalizability or unitarity, but not both. It proves also remarkable the actual fixing of the Einstein Lagrangian in the vacuum by generalized symmetry requirements, in contrast to the standard gauge (diffeomorphism) symmetry, which only fixes the arguments of the possible Lagrangians. © 2021, The Author(s). ; I would thank all people who have contributed somehow to the development of the research reflected in the present document, and very particularly to the coauthors of different papers, that is, in chronological order: José A. de Azcárraga, Kurt W. Wolf, Robin W. Tucker, Juan Bisquert, José Navarro-Salas, Santiago García, Alfonso Ramírez, Renatte Loll, José M. Cerveró, John R. Klauder, Miguel Navarro, Julio Guerrero, Manuel Calixto, Giuseppe Marmo, José L. Jaramillo, Eduardo Sánchez-Sastre, Francisco F. López-Ruiz, Francisco Cossío and Dieter Schuch. My gratitude to the Universidad de Salamanca, Valencia and Granada, The Imperial College of Science, Technology and Medicine, and the Instituto de Astrofísica de Andalucía (IAA-CSIC). ; With funding from the Spanish government through the Severo Ochoa Centre of Excellence accreditation SEV-2017-0709. ; Peer reviewed