Unification of classic and quantum mechanics

This paper summarizes and generalizes a recently proposed mathematical framework that unifies the standard formalisms of special relativity and quantum mechanics. The framework is based on Hilbert spaces H of functions of four space-time variables x,t, furnished with an additional indefinite inner product invariant under Poincar\'e transformations, and isomorphisms of these spaces that preserve the indefinite metric. The indefinite metric is responsible for breaking the symme...

General Theory of Relativity and Quantum theory gives two different description of the same mother nature in the big and small scale respectively. Mathematical languages of these two theories are entirely different, one is geometric while the other one is probabilistic. This curious feature makes the merging of these theories (quantum-gravity) considerably difficult. In this manuscript, we explore quantum-gravity in a dynamical system perspective. For example, in the standard...

An approach to a Unified Field Theory (UFT) is developed as an attempt to establish unification of the Theory of Quantum Fields (QFT) and General Theory of Relativity (GTR) on the background of a covariant differential calculus. A dual State Vector field (DSV)consisting of covariant and contravariant N-component functions of variables of a N-dimensional unified manifod (UM)is introduced to represents matter. DSV is supposed to transform in a way distinct from that of the diff...

The author discusses particular solutions of a second order equation designated by source equation. This equation is special because the metric of the space where it is written is influenced by the solution, rendering the equation recursive. The recursion mechanism is established via a first order equation which bears some resemblance to Dirac equation. In this paper the author limits the discussion to solutions with constant norm but makes use of 4-dimensional hypercomplex n...

In the last 100 years, the most important equations in physics are Maxwell's equations for electrodynamics, Einstein's equation for gravity, Dirac's equation for the electron and Yang-Mills equation for elementary particles. Do these equations follow a common principle and come from a single theory? Despite intensive efforts to unify gravity and the particle interactions in the last 30 years, the goal is still to be achieved. Recent theories have not answered any question in ...

The basic principles of the quantum mechanics in the K-field formalism are stated in the paper. The basic distinction of this theory arises from that the quantum theory equations (including well-known Schrodinger, Klein-Gordon and quadratic Dirac equations) are obtained from de Broglie postulate geometric generalization. Rather, they are obtained as the free wave equations on a manifold metrizing force interactions of particles. Such view on the quantum theory basic equations...

Alternative versions of the Klein-Gordon and Dirac equations in a curved spacetime are got by applying directly the classical-quantum correspondence.

General relativity and quantum mechanics are perhaps the two most successful theories of the XXth century. Despite their impressive accurate predictions, they are both valid at their own scales and do not seem to be expressible using the same framework. It is commonly accepted that in order to create a consistent theory of both quantum mechanics and gravity, it is required to quantize the gravitational field. In the present paper, another path is taken on which the Einstein f...

The idea of wave mechanics leads naturally to assume the well-known relation $E=\hbar \omega $ in the specific form $H=\hbar W$, where $H$ is the classical Hamiltonian of a particle and $W$ is the dispersion relation of the sought-for wave equation. We derive the expression of $H$ in a curved spacetime with an electromagnetic field. Then we derive the Dirac equation from factorizing the polynomial dispersion equation corresponding with $H$. Conversely, summarizing a recent wo...

This is an introduction to quantum gravity, aimed at a fairly general audience and concentrating on what have historically two main approaches to quantum gravity: the covariant and canonical programs (string theory is not covered). The quantization of gravity is discussed by analogy with the quantization of the electromagnetic field. The conceptual and technical problems of both approaches are discussed, and the paper concludes with a discussion of evidence for quantum gravit...