Starting from the quantization of the action variable as a basic principle, and regarding the momentum and the energy of a particle as random functions of coordinates I arrive at the quantum mechanical formalism and the rule for probabilities in quantum mechanics. The discussion involves the correspondence principle and the assumption that the space-time symmetries known in classical physics are valid in the quantum domain.
Starting from the quantization of the action variable as a basic principle, and regarding the momentum and the energy of a particle as random functions of coordinates I arrive at the quantum mechanical formalism and the rule for probabilities in quantum mechanics. The discussion involves the correspondence principle and the assumption that the space-time symmetries known in classical physics are valid in the quantum domain.
Following previous works by E. Prugovečki [Physica A 91A, 202 (1978) and Stochastic Quantum Mechanics and Quantum Space–time (Reidel, Dordrecht, 1986)] on common features of classical and quantum mechanics, we develop a unified mathematical framework for classical and quantum mechanics (based on L2-spaces over classical phase space), in order to investigate to what extent quantum mechanics can be obtained as a simple modification of classical mechanics (on both logical and analytical levels). To obtain this unified framework, we split quantum theory in two parts: (i) general quantum axiomatics (a system is described by a state in a Hilbert space, observables are self-adjoints operators, and so on) and (ii) quantum mechanics proper that specifies the Hilbert space as L2(Rn); the Heisenberg rule [pi,qj]=−iℏδij with p=−iℏ∇, the free Hamiltonian H=−ℏ2Δ/2m and so on. We show that general quantum axiomatics (up to a supplementary “axiom of classicity”) can be used as a nonstandard mathematical ground to formulate physical ideas and equations of ordinary classical statistical mechanics. So, the question of a “true quantization” with “ℏ” must be seen as an independent physical problem not directly related with quantum formalism. At this stage, we show that this nonstandard formulation of classical mechanics exhibits a new kind of operation that has no classical counterpart: this operation is related to the “quantization process,” and we show why quantization physically depends on group theory (the Galilei group). This analytical procedure of quantization replaces the “correspondence principle” (or canonical quantization) and allows us to map classical mechanics into quantum mechanics, giving all operators of quantum dynamics and the Schrödinger equation. The great advantage of this point of view is that quantization is based on concrete physical arguments and not derived from some “pure algebraic rule” (we exhibit also some limit of the correspondence principle). Moreover spins for particles are naturally generated, including an approximation of their interaction with magnetic fields. We also recover by this approach the semi-classical formalism developed by E. Prugovečki [Stochastic Quantum Mechanics and Quantum Space–time (Reidel, Dordrecht, 1986)].
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.
Classical mechanics involves position and momentum variables that must be special coordinates chosen to promote to suitable quantum operators. Since classical variables may be broadly chosen, only unique variables should be chosen. We will outline how the favored variables and their suitable quantum operators is guaranteed to assure a truly valid quantization. Invalid quantizations may be mistaken for valid ones, which then leads to incorrect physics. Besides particle examples, there is also a brief run-through for fields and gravity.
A type of mechanics will be presented that possesses some distinctive properties. On the one hand, its physical description & rules of operation are readily comprehensible & intuitively clear. On the other, it fully satisfies all observable predictions of non-relativistic quantum mechanics. Within it, particles exist at points in space, follow continuous, piecewise differentiable paths, and their linear momentum is equal to their mass times their velocity along their path. Yet the probabilities for position and momentum, conditioned on the state of the particle's environment, follow the rules of quantum theory. Indeed, all observable consequences of quantum theory are satisfied; particles can be entangled, have intrinsic spin, this spin is not local to the particle, particle identity can effect probabilities, and so forth. All the rules of quantum mechanics are obeyed, and all arise in a straightforward fashion. After this is established, connections will be drawn out between this type of mechanics and other types of quantum worlds; those that obey Bohmian mechanics, stochastic mechanics, the many worlds interpretation, and physical collapse. In the final section, a relativistic version of the mechanics will be presented.
Quantum Mechanics is a successful model to describe physical phenomena. We show that a simple set of postulates about the description of objects in Nature allows the derivation of the principles of QM, transforming this theory into a deductive approach to Physics. The main postulate is: Objects are described as energy distributions in Space–Time. 1 Quantum Mechanics from START It is universally accepted that Quantum Mechanics is a successful model to describe physical phenomena, in particular of the electronic structure of atoms, molecules and solids. As such it has been reformulated in several different forms (see for example: Nine formulations of quantum mechanics, Daniel F. Styer et al. [15]). The aim of the present paper is to show that a simple set of postulates about the description of objects in Nature allows the derivation of the principles of QM, transforming this theory into a deductive approach to Physics. In our approach the main postulate is: Objects are described as energy distributions in Space–Time. The theory is in fact a systematic mathematical analysis of this postulate and of other general considerations about action, energy–momentum and the role of the theory as a useful description of Nature. The paper also shows the fundamental and basic status of Density Functional Theory, also contained in our approach. The initial mathematical structure is a geometric (quadratic form based) union of space, time and action. In physics bundles of trajectories are used to describe the objects of nature: distribution of heights over a surface, a real object is described as a bundle of space-time trajectories, here we use a description as a bundle of action valves in space-time (energy densities in space following light-like lines). The author is fully aware that the presentation is strongly guided by the knowledge of the usual computational approach to study atoms, molecules and condensed matter electronic structure.
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.