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Journal of Yiyang Teachers College

Four-Dimensional Generalized Symmetrical Formalism in Classical Physics and Quantum Mechanics and Mechanical Wave Theory

Yi‐Fang Chang

Published Jan 1, 2002• 0 citations
OpenAlex

Abstract

Combining the known results in mechanics and relativity, the unified four-dimensional generalized formalism of various equations and formulas in classical physics and quantum mechanics are discussed. In usual cases, they possess good symmetry. From this some new results are derived. But under a certain condition the symmetry is violated. Moreover, the mechanical wave theory, new operators and nonlinear equations are connected.

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Combining the known results in mechanics and relativity, the unified four-dimensional generalized formalism of various equations and formulas in classical physics and quantum mechanics are discussed. In usual cases, they possess good symmetry. From this some new results are derived. But under a certain condition the symmetry is violated. Moreover, the mechanical wave theory, new operators and nonlinear equations are connected.
First, various dualities are summarized. Next, based on the universal wave- particle duality, along an opposite direction of the developed quantum mechanics, we use a method where the wave quantities frequency v and wave length  are replaced on various mechanical equations, and may derive some new results. It is called the mechanical wave theory. From this we derive new operators which represent more physical quantities. Further, we propose some nonlinear equations and their solutions, which may be probably applied to quantum theory.
Various dualities are summarized. Based on the universal wave-particle duality, along an opposite direction of the developed quantum mechanics, we use a method where the wave quantities frequency and wave length are replaced on various mechanical equations, and may be derive some new results. It is called the mechanical wave theory. From this we derive new operators which represent more physical quantities. Further, we propose some nonlinear equations and their solutions, which may be probably applied to quantum theory.
The work can be considered as an essay on mathematical and conceptual structure of nonrelativistic quantum mechanics (QM) which is related here to some other (more general, but also to more special and “approximative”) theories. QM is here primarily reformulated in an equivalent form of a Poisson system on the phase space consisting of density matrices, where the “observables”, as well as “symmetry generators” are represented by a specific type of real valued (densely defined) functions, namely the usual quantum expectations of corresponding selfadjoint operators. It is shown in this paper that inclusion of additional (“nonlinear”) symmetry generators (i.e. “Hamiltonians”) into this reformulation of (linear) QM leads to a considerable extension of the theory: two kinds of quantum “mixed states” should be distinguished, and operator ‐ valued functions of density matrices should be used in the r ˆ ole of “nonlinear observables”. A general framework for physical theories is obtained in this way: By different choices of the sets of “nonlinear observables” we obtain, as special cases, e.g. classical mechanics on homogeneous spaces of kinematical symmetry groups, standard (linear) QM, or nonlinear extensions of QM; also various “quasiclassical approximations” to QM are all subtheories of the presented extension of QM - a version of the extended quantum mechanics (EQM). A general interpretation scheme of EQM extending the usual statistical interpretation of QM is also proposed. Eventually, EQM is shown to be (included into) a C -algebraic (hence linear) quantum theory. Mathematical formulation of these theories is presented. The presentation includes an analysis of problems connected with differentiation on infinite ‐ dimensional manifolds, as well as a solution of some problems connected with the work with only densely defined unbounded real‐valued functions on the (infinite dimensional) “phase space” which correspond to unbounded operators (generators) and to their nonlinear generalizations. Also “nonlinear deformations” of unitary representations of kinematical symmetry Lie groups are introduced. Possible applications are briefly discussed, and some specific examples are presented. The text contains also brief reviews of Hamiltonian classical mechanics, as well as of QM. Mathematical appendices make the paper nearly selfcontained.
It is well known that nonrelativistic quantum mechanics presents a clear asymmetry between space and time. Much of this asymmetry is attributed to the lack of Lorentz invariance of the theory. Nonetheless, a recent work [Phys. Rev. A \textbf{95}, 032133 (2017)] showed that even though this is partially true, there is a broader physical scenario in which space and time can be handled in nonrelativistic quantum theory in a more symmetric way. In this space-time-symmetric formalism, an additional Hilbert space is defined so that time is raised to the status of operator and position becomes a parameter. As a consequence, the Hilbert space now requires a space-conditional quantum state governed by a new quantum dynamics. In this manuscript, we reveal some physical and mathematical properties of the space-time-symmetric formalism such as: symmetries between the Hamilton-Jacobi and the space-conditional equation; the general solution for a time-independent potential; and a new Lagrangian for a spinless particle in one dimensional. Finally, we present the space-conditional equation for a particle under the effect of an electromagnetic field, and the gauge invariance of this equation is proved.
Classical mechanics and quantum mechanics have long been framed as binary opposites—but this dichotomy has yielded no real benefits, only persistent confusion. The fundamental equations of quantum mechanics cannot logically demarcate a clear boundary between the quantum and classical realms; in fact, the mass m and potential energy V in the Schrödinger equation seamlessly traverse the divide between the microscopic and macroscopic. The electromagnetic potential energy in the Schrödinger equation can be replaced with the potential energy of interactive forces, which can further be converted into force itself using the relation F = V/r. Derivations have now produced a force operator and quantum mechanical equations that explicitly incorporate force—including a quantum formulation of Newton’s second law. These results provide direct evidence that quantum and classical mechanics are not incompatible, but complementary. By resolving the historical absence of “force” as a formal concept in quantum mechanics, we bridge the conceptual gap between macroscopic classical frameworks and microscopic quantum theory, challenging long-held perceptions of their irreconcilability. This unified approach explains why nanoscale particles exhibit quantum properties, simplifies calculations of quantum forces, and lays the groundwork for a generalized wave mechanics that unifies mechanical descriptions across all scales.