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 allows one to use semiclassical models for the quantum system simulation. The quantization principle modifies as well. Namely, quantum system stationary conditions are such conditions, at which test particles motion is Lyapunov stable.
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 allows one to use semiclassical models for the quantum system simulation. The quantization principle modifies as well. Namely, quantum system stationary conditions are such conditions, at which test particles motion is Lyapunov stable.
The basic principles of the quantum mechanics in the K-field formalism (see author's papers) are stated in the paper. K-field formalism arises from geometric generalization of de Broglie postulate. So, the quantum theory equations (including well-known Schrodinger, Klein-Gordon and quadratic Dirac equations) are obtained as the free wave equations on a manifold metrizing force interactions of particles. In this paper, describing wave properties of particles we will restricted ourself with construction special geometric formulation of force interactions.
The formulation of quantum mechanics in terms of the picture of trajectory ensemble or of the hydrodynamical picture, developed in a previous article, is further supplemented for some points. Individual trajectories for some typical non-stationary state are explicitly obtained, illustrating that they generally show complicated fluctuations due to the action of quantum potential. Next, emphasis is placed on the fact that the Schrödinger field admits a mechanical model. Also the Klein-Gordon field is represented with a quasi-irrotational flow under some additional scalar field. This representation of Klein-Gordon field suggests its possible generalization to include the vortical motion. Such treatments of interacting Klein-Gordon and electromagnetic fields involve the formalism of Dirac's recent classical electrodynamics as a limiting case. Some remarks are given on this Dirac's theory. Finally, the picture of trajectory ensemble is generalized so as to represent wave equations with sources. In an appendix criticism is given on the attempt to interpret quantum mechanics in terms of Markoff process.
Derivation of Klein–Gordon equation Goals and prior knowledge The main wave equations of relativistic quantum mechanics are the Klein–Gordon, Dirac, Weyl , and Rarita–Schwinger equations. The principal aim of this chapter is to derive these, by means of the EPI principles (3.16) and (3.18). The derivations generally follow Frieden (1995), or Frieden and Soffer (1995), with numerous additions. In addition, the Schrödinger wave equation (SWE), which is non -relativistic, is also to be derived. Since rigorous use of EPI gives only relativistic wave equations (Chapter 3), EPI can only derive the SWE in some approximate sense. This is directly carried through in Appendix D, or by taking the well-known non-relativistic limit of the Klein–Gordon equation in Appendix G. Also, in Appendix D a simple EPI derivation of classical mechanics is obtained. In particular, in the limit h → 0 the EPI extremum condition (3.16) is found to give Newton's second law , and the zero-condition (3.18) gives the virial theorem , of classical mechanics. Other derived results are Eq. (4.17) – the equivalence of mass, momentum, and energy; the constancy of the Planck constant h (Sec. 4.1.14); the Compton wavelength as an ultimate resolution length (Sec. 4.1.17); and the Heisenberg uncertainty principle (Sec. 4.3). It is important to state what is assumed a priori . The relativistic properties of space and time coordinates were derived in Sec. 3.5. Hence, these are assumed here. Also, appropriate units for the concepts of mass and energy are assumed to be known, although the basic relation Eq. (4.17) between them is not presumed.
The quantum theory of relativistic particles, based on the first quantization technique similar to that used by Schroedinger and Dirac in formulating quantum mechanics, is reconsidered on the basis of a photon-like dispersion relation corresponding to the energy conservation equation of Einstein's special relativity. First, scalar quantum mechanics of particles operating with their wave functions is discussed. Using the first quantization of the photon-like dispersion relation, very simple new derivation of the Dirac equation is given. Then, vector field quantum mechanics is introduced, which defines vector fields associated with the relativistic particle. Basic equations for the vector-field quantum mechanics, similar to the source-free Maxwell equations, are derived. Following these equations, the particle's de Broglie wave can be considered as the transversal electromagnetic wave. Therefore, the wave-particle duality can be redefined as the electromagnetic wave-particle duality. Relationships between the scalar and vector field quantum mechanics are analyzed.
We propose six principles as the fundamental principles of quantum mechanics: principle of space and time, Galilean principle of relativity, Hamilton's principle, wave principle, probability principle, and principle of indestructibility and increatiblity of particles. We deductively develop the formalism of quantum mechanics on the basis of them: we determine the form of the Lagrangian that satisfies requirements of these principles, and obtain the Schroedinger equation from the Lagrangian. We also derive the canonical commutation relations. Then we adopt the following four guide lines. First, we do not premise the relations between dynamical variables in classical mechanics. Second, we define energy, momentum, and angular momentum as the constants of motion that are derived from homogeneity and isotropy in space and time on the basis of principle of space and time. Since energy and momentum are quantitatively defined in classical mechanics, we define them in quantum mechanics so that the corresponding conservation laws are satisfied in a coupling system of a quantum particle and a classical particle. Third, we define Planck's constant and the mass of a particle as proportionality constants between energy and frequency due to one of Einstein-de Broglie formulas and between momentum and velocity, respectively. We shall obtain the canonical commutation relations and the Schroedinger equation for a particle in an external field in the definitive form. We shall also prove that relations between dynamical variables in quantum mechanics have the same forms for those in classical mechanics.
Solitary-particle quantum mechanics' inherent compatibility with special relativity is implicit in Schroedinger's postulated wave-function rule for the operator quantization of the particle's canonical three-momentum, taken together with his famed time-dependent wave-function equation that analogously treats the operator quantization of its Hamiltonian. The resulting formally four-vector equation system assures proper relativistic covariance for any solitary-particle Hamiltonian operator which, together with its canonical three-momentum operator, is a Lorentz-covariant four-vector operator. This, of course, is always the case for the quantization of the Hamiltonian of a properly relativistic classical theory, so the strong correspondence principle definitely remains valid in the relativistic domain. Klein-Gordon theory impairs this four-vector equation by iterating and contracting it, thereby injecting extraneous negative-energy solutions that are not orthogonal to their positive-energy counterparts of the same momentum, thus destroying the basis of the quantum probability interpretation. Klein-Gordon theory, which thus depends on the square of the Hamiltonian operator, is as well thereby cut adrift from Heisenberg's equations of motion. Dirac theory confuses the space-time symmetry of the four-vector equation system with such symmetry for its time component alone, which it fatuously imposes, thereby breaching the strong correspondence principle for the free particle and imposing the starkly unphysical momentum-independence of velocity. Physically sensible alternatives, with external electromagnetic fields, to the Klein-Gordon and Dirac equations are derived, and the simple, elegant symmetry-based approach to antiparticles is pointed out.