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arXiv:1405.7693

Geometrical Formulation of Quantum Mechanics

Shiv R. Vatsya

Published May 29, 2014• 0 citations
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Abstract

Hamilton's action principle is formulated and extended in conformity with the gauge transformations underlying Weyl's geometry. The extended principle characterizes infinitely many equally likely trajectories with a particle traveling along a randomly selected one. Available similar formulations do not conform as directly to the gauge transformations as the present one. Also, they have not paid much attention to the path-independent, assigned gauges. The freedom available in assigning these gauges is exploited here by defining them in terms of the configuration, and interactions of the observing system with the observed one. Impact of the method of observation on its outcome is described in terms of the assigned gauges so defined and illustrated with examples. A wavefunction is defined in a simply connected region essentially as an aggregate of the gauge transformations over all trajectories; equivalently, an aggregate of the Weyl-lengths acquired by a unit vector transported along all trajectories from everywhere. This representation is similar to Feynman's path integral representation differing only in that it incorporates the assigned gauges yielding an adjusted wavefunction that includes the impact of an observing system on the observed one. Probability density is shown to be a uniquely defined gauge invariant quantity but at the expense of the information about the observable effects contained in the gauge factors, assigned and otherwise. The particle trajectories defined here are thus shown to provide additional significant information about a system than provided by the wavefunction together with the probability density. Present description of the impact of method of observation on its outcome is compared with the descriptions according to the existing major representations of quantum mechanics.

DOI
10.48550/arxiv.1405.7693

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Hamilton's action principle is formulated and extended in conformity with the gauge transformations underlying Weyl's geometry. The extended principle characterizes infinitely many equally likely trajectories with a particle traveling along a randomly selected one. Available similar formulations do not conform as directly to the gauge transformations as the present one. Also, they have not paid much attention to the path-independent, assigned gauges. The freedom available in assigning these gauges is exploited here by defining them in terms of the configuration, and interactions of the observing system with the observed one. Impact of the method of observation on its outcome is described in terms of the assigned gauges so defined and illustrated with examples. A wavefunction is defined in a simply connected region essentially as an aggregate of the gauge transformations over all trajectories; equivalently, an aggregate of the Weyl-lengths acquired by a unit vector transported along all trajectories from everywhere. This representation is similar to Feynman's path integral representation differing only in that it incorporates the assigned gauges yielding an adjusted wavefunction that includes the impact of an observing system on the observed one. Probability density is shown to be a uniquely defined gauge invariant quantity but at the expense of the information about the observable effects contained in the gauge factors, assigned and otherwise. The particle trajectories defined here are thus shown to provide additional significant information about a system than provided by the wavefunction together with the probability density. Present description of the impact of method of observation on its outcome is compared with the descriptions according to the existing major representations of quantum mechanics.
Formulations of quantum mechanics incorporating the Weyl gauge transformations are studied in this article and developed further. In the process, impact of the method of observation on its outcome is interpreted in terms of the assigned gauges by incorporating properties of the corresponding experimental arrangement in defining them. Further, the assigned gauge is explicitly incorporated in the Feynman path integral formulation of quantum mechanics. The resulting wavefunction, which is not uniquely defined, represents a gauge equivalence class. The representative wavefunction is still obtained by the original path integral method. Methods to obtain the pertinent information about the assigned gauges supplementing the representative wavefunction are discussed. The probability density is shown to be a uniquely defined gauge invariant quantity but at the expense of some information describing the observable effects contained in gauge factors. In the standard quantum mechanics, a wavefunction is assumed to be defined within a phase factor while the probability density is phase-independent, paralleling these results. Also, the path integral method is used to deduce the Klein-Gordon equation for the representative wavefunction in the Riemannian spaces in a more streamlined manner than the previous derivations.
An extension of the classical action principle obtained in the framework of the gauge transformations, is used to describe the motion of a particle. This extension assigns many, but not all, paths to a particle. Properties of the particle paths are shown to impart wave like behaviour to a particle in motion and to imply various other assumptions and conjectures attributed to the formalism of Quantum Mechanics. The Klein-Gordon and other similar equations are derived by incorporating these properties in the path-integral formalism.
We present a consistent and comprehensive treatise on the foundations of the extended Hamilton–Lagrange formalism — where the dynamical system is parametrized along a general system evolution parameter s, and the time t is treated as a dependent variable t(s) on equal footing with all other configuration space variables qi(s). In the action principle, the conventional classical action L1d t is then replaced by the generalized action L1d s, with L and L1denoting the conventional and the extended Lagrangian, respectively. It is shown that a unique correlation of L1and L exists if we refrain from performing simultaneously a transformation of the dynamical variables. With the appropriate correlation of L1and L in place, the extension of the formalism preserves its canonical form. In the extended formalism, the dynamical system is described as a constrained motion within an extended space. We show that the value of the constraint and the parameter s constitutes an additional pair of canonically conjugate variables. In the corresponding quantum system, we thus encounter an additional uncertainty relation. As a consequence of the formal similarity of conventional and extended Hamilton–Lagrange formalisms, Feynman's nonrelativistic path integral approach can be converted on a general level into a form appropriate for relativistic quantum physics. In the emerging parametrized quantum description, the additional uncertainty relation serves as the means to incorporate the constraint and hence to finally eliminate the parametrization. We derive the extended Lagrangian L1of a classical relativistic point particle in an external electromagnetic field and show that the generalized path integral approach yields the Klein–Gordon equation as the corresponding quantum description. We furthermore derive the space–time propagator for a free relativistic particle from its extended Lagrangian L1. These results can be regarded as the proof of principle of the relativistic generalization of Feynman's path integral approach to quantum physics.
A gauge-invariant average action principle is presented that permits the formulation of a completely classical theory which is proved to be equivalent to Klein-Gordon quantum mechanics. In this approach, here called geometric quantum mechanics, the particle motion as well as the space-time geometry are determined simultaneously from the same average action principle. Quantum effects are proved to be related to space-time affine connections rather than to space-time metric tensor components. In this way geometric quantum mechanics is made compatible with axiomatic approaches to both space-time structure and probability calculus.
The path integral formulation of quantum mechanics constructs the propagator by evaluating the action S for all classical paths in coordinate space. A corresponding momentum path integral may also be defined through Fourier transforms in the endpoints. Although these momentum path integrals are especially simple for several special cases, no one has, to my knowledge, ever formally constructed them from all classical paths in momentum space. I show that this is possible because there exists another classical mechanics based on an alternate classical action R. Hamilton's Canonical equations result from a variational principle in both S and R. S uses fixed beginning and ending spatial points while R uses fixed beginning and ending momentum points. This alternative action's classical mechanics also includes a Hamilton-Jacobi equation. I also present some important points concerning the beginning and ending conditions on the action necessary to apply a Canonical transformation. These properties explain the failure of the Canonical transformation in the phase space path integral. It follows that a path integral may be constructed from classical position paths using S in the coordinate representation or from classical momentum paths using R in the momentum representation. Several example calculations are presented that illustrate the simplifications and practical advantages made possible by this broader view of the path integral. In particular, the normalized amplitude for a free particle is found without using the Schrodinger equation, the internal spin degree of freedom is simply and naturally derived, and the simple harmonic oscillator is calculated.
Many quantization schemes rely on analogs of classical mechanics where the connections with classical mechanics are indirect. In this work I propose a new and direct connection between classical mechanics and quantum mechanics where the quantum mechanical propagator is derived from a variational principle. I identify this variational principle as a generalized form of Hamilton's principle. This proposed variational principle is unusual because the physical system is allowed to have imperfect information, i.e., there is incomplete knowledge of the physical state. Two distribution functionals over possible generalized momentum paths a[p(t)] and generalized coordinates paths b[q(t)] are defined. A generalized action is defined that corresponds to a contraction of a[p(t)], b[q(t)], and a matrix of the action evaluated at all possible p and q paths. Hamilton's principle is the extremization of the generalized action over all possible distributions. The normalization of the two distributions allows their values to be negative and they are shown to be the real and imaginary parts of the complex amplitude. The amplitude in the Feynman path integral is shown to be an optimal vector that extremizes the generalized action. This formulation is also shown to be directly applicable to statistical mechanics and I show how irreversible behavior and the micro-canonical ensemble follows immediately.