The fussy "roots" of quantum mechanics are traced directly to the Hamilton-Jacobi equation of classical mechanics. It is shown that the Shroedinger equation can be derived from the Hamilton-Jacobi equation. A deep underlying unity of both equations lies in the fact that a unique trajectory of a classical particle is "selected" out of many-continuum paths according to the Principle of least action. We can say that a classical particle has a membership in every path of the above set, which collapses to the winning single trajectory of a real motion. At the same time it can also be said that a quantum mechanical "particle" has different degrees of membership in a set of many-continuum paths where all of them contribute to the dynamics of the quantum mechanical particle. This allows one to provide an interpretation of the wave function as a parameter describing deterministic entity endowed by a fuzzy character. As a logical consequence of such an interpretation the complimentarity principle and the wave-particle duality concept can be abandoned in favor of a fuzzy deterministic microobject. This idea leads to a possibility of a quantum mechanical computer based on the fuzzy logic.
The fussy "roots" of quantum mechanics are traced directly to the Hamilton-Jacobi equation of classical mechanics. It is shown that the Shroedinger equation can be derived from the Hamilton-Jacobi equation. A deep underlying unity of both equations lies in the fact that a unique trajectory of a classical particle is "selected" out of many-continuum paths according to the Principle of least action. We can say that a classical particle has a membership in every path of the above set, which collapses to the winning single trajectory of a real motion. At the same time it can also be said that a quantum mechanical "particle" has different degrees of membership in a set of many-continuum paths where all of them contribute to the dynamics of the quantum mechanical particle. This allows one to provide an interpretation of the wave function as a parameter describing deterministic entity endowed by a fuzzy character. As a logical consequence of such an interpretation the complimentarity principle and the wave-particle duality concept can be abandoned in favor of a fuzzy deterministic microobject. This idea leads to a possibility of a quantum mechanical computer based on the fuzzy logic.
It is shown that quantum mechanics can be regarded as what one might call a "fuzzy" mechanics whose underlying logic is the fuzzy one, in contradistinction to the classical "crisp" logic.Therefore classical mechanics can be viewed as a crisp limit of a "fuzzy" quantum mechanics.Based on these considerations it is possible to arrive at the Schroedinger equation directly from the Hamilton-Jacobi equation.The link between these equations is based on the fact that a unique ( "crisp") trajectory of a classical particle emerges out of a continuum of possible paths collapsing to a single trajectory according to the principle of least action.This can be interpreted as a consequence of an assumption that a quantum "particle" "resides" in every path of the continuum of paths which collapse to a single( unique) trajectory of an observed classical motion.A wave function then is treated as a function describing a deterministic entity having a fuzzy character.As a consequence of such an interpretation, the complimentarity principle and wave-particle duality can be abandoned in favor of a fuzzy deterministic microoobject.
It is shown that quantum mechanics can be regarded as what one might call a "fuzzy" mechanics whose underlying logic is the fuzzy one, in contradistinction to the classical "crisp" logic. Therefore classical mechanics can be viewed as a crisp limit of a "fuzzy" quantum mechanics. Based on these considerations it is possible to arrive at the Schroedinger equation directly from the Hamilton-Jacobi equation. The link between these equations is based on the fact that a unique ("crisp") trajectory of a classical particle emerges out of a continuum of possible paths collapsing to a single trajectory according to the principle of least action. This can be interpreted as a consequence of an assumption that a quantum "particle" "resides" in every path of the continuum of paths which collapse to a single(unique) trajectory of an observed classical motion. A wave function then is treated as a function describing a deterministic entity having a fuzzy character. As a consequence of such an interpretation, the complimentarity principle and wave-particle duality can be abandoned in favor of a fuzzy deterministic microoobject.
The approach proposed in the study is based on the revision of the concept of time as a point on the real axis. It uses the concept of fuzzy time as the set of real numbers with a finite, but not equal to one, function of membership to the time set, i.e. the fuzzy time concept. It is postulated that in fuzzy time t the system dynamics follows from the standard variational principle of the least action and is ordinary Hamilton-Jacobi mechanics. This validates the passage to the limit from fuzzy mechanics to ordinary variational conservative mechanics. The Liouville equation is solved by the method of successive approximations in the time domain of a much larger characteristic scale of fuzziness, using interaction as a small parameter. A standard diagram technique is used. It can be shown that the defuzzification of the Liouville equation inevitably reduces the reversible part in the description to the irreversible evolutionary equation. The latter leads to the second law of thermodynamics. Generalization to the quantum case is possible, i.e. the so-called fuzzy Pauli equation can be drawn.
The goal of this study is to integrate fuzzy mathematics and quantum mechanics to seek deeper cosmic principles. The concept of fuzzy wave function is proposed to represent the fuzzy quantum probability by utilizing fuzzy mathematics to blur the probability (wave) of quantum mechanics. By applying the non-fuzzy formula of fuzzy quantity and the Schrödinger wave equation of quantum mechanics, the membership function equation is constructed to describe the evolution of the fuzzy wave function. The concept of membership degree amplitude is introduced to calculate fuzzy probability amplitude, and the membership degree amplitude equation is derived. Some important concepts in fuzzy mathematics are also illustrated.
First, we show that there exists in classical mechanics three actions corresponding to different boundary conditions: two well-known actions, the Euler-Lagrange classical action S_cl(x,t;x_0), which links the initial position x_0 and its position x at time t, the Hamilton-Jacobi action S(x,t), which links a family of particles of initial action S_0(x) to their various positions x at time t, and a new action, the deterministic action S(x,t;x_0,v_0), which links a particle in initial position x_0 and initial velocity v_0 to its position x at time t. We study, in the semi-classical approximation, the convergence of the quantum density and the quantum action, solutions to the Madelung equations, when the Planck constant h tends to 0. We find two different solutions which depend on the initial density. In the first case, where the initial quantum density is a classical density, the quantum density and the quantum action converge to a classical action and a classical density which satisfy the statistical Hamilton-Jacobi equations. These are the equations of a set of classical particles whose initial positions are known only by the initial density. In the second case where initial density converges to a Dirac density, the density converges to the Dirac function and the quantum action converges to a deterministic action. Therefore we introduce into classical mechanics non-discerned particles, which satisfy the statistical Hamilton-Jacobi-equations and explain the Gibbs paradox, and discerned particles, which satisfy the deterministic Hamilton-Jacobi equations. Finally, we propose an interpretation of the Schrodinger wave function that depends on the initial conditions (preparation). This double interpretation seems to be the interpretation of Louis de Broglie's "double solution" idea.
Some recent experiments claim to show that any model in which a quantum state represents mere information about an underlying physical reality of the system must make predictions which contradict those of quantum theory. The present work undertakes to investigate the issue of reality, treading a more fundamental route from the Hamilton-Jacobi equation of classical mechanics to the Schrodinger equation of quantum mechanics. Motivation for this is a similar approach from the eikonal equation in geometrical optics to the wave equation in electromagnetic theory. We rewrite the classical Hamilton-Jacobi equation as a wave equation and seek to generalise de Broglie's wave particle duality by demanding that both particle and light waves have the freedom of being described by any square-integrable function. This generalisation, which allows superposition also for matter wave functions, helps us to obtain the Schrodinger equation, whose solution can be seen to be as much objective as the classical mechanics wave function. Several other equations which one writes in quantum mechanics, including the eigenvalue equations for observables, series expansion of energy states in terms of eigenstates of observables other than energy, etc., can be written in the classical case too. Absence of any collapse of the wave function, entanglement, etc. in the classical realm have their origin in the nonlinearity of the classical wave equation. These considerations indicate that many of the puzzles in quantum mechanics are present also in classical mechanics in a dormant form, which fact shall help to demystify quantum mechanics to a great extent.