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International Journal of Geometric Methods in Modern Physics

Geometrization of the Schrödinger equation: Application of the Maupertuis Principle to quantum mechanics

Antonia Karamatskou, Hagen M. Kleinert

Published May 23, 2014• 8 citations
Publisher

Abstract

In its geometric form, the Maupertuis Principle states that the movement of a classical particle in an external potential V(x) can be understood as a free movement in a curved space with the metric gμν(x) = 2M[V(x) - E]δμν. We extend this principle to the quantum regime by showing that the wavefunction of the particle is governed by a Schrödinger equation of a free particle moving through curved space. The kinetic operator is the Weyl-invariant Laplace–Beltrami operator. On the basis of this observation, we calculate the semiclassical expansion of the particle density.

DOI
10.1142/s0219887814500662

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In its geometric form, the Maupertuis Principle states that the movement of a classical particle in an external potential V(x) can be understood as a free movement in a curved space with the metric gμν(x) = 2M[V(x) - E]δμν. We extend this principle to the quantum regime by showing that the wavefunction of the particle is governed by a Schrödinger equation of a free particle moving through curved space. The kinetic operator is the Weyl-invariant Laplace–Beltrami operator. On the basis of this observation, we calculate the semiclassical expansion of the particle density.
According to the Maupertuis principle, the movement of a classical particle in an external potential V(x)V(x)V(x) can be understood as the movement in a curved space with the metric gμν(x)=2M[V(x)−E]δμνg_{μν}(x)=2M[V(x)-E]δ_{μν}gμν​(x)=2M[V(x)−E]δμν​. We show that the principle can be extended to the quantum regime, i.e., we show that the wave function of the particle follows a Schrödinger equation in curved space where the kinetic operator is formed with the {\it Weyl--invariant Laplace-Beltrami} operator. As an application, we use DeWitt's recursive semiclassical expansion of the time-evolution operator in curved space to calculate the semiclassical expansion of the particle density ρ(x;E)=ρ(x;E)=ρ(x;E)=.
The Schrödinger operators on the Newtonian spacetime are defined in a way which make them independent of the class of inertial observers. In this picture the Schrödinger operators act not on functions on the spacetime but on sections of a certain one-dimensional complex vector bundle—the Schrödinger line bundle . This line bundle has trivializations indexed by inertial observers and is associated with an U (1)-principal bundle with an analogous list of trivializations—the Schrödinger principal bundle . If an inertial frame is fixed, the Schrödinger bundle can be identified with the trivial bundle over spacetime, but as there is no canonical trivialization (inertial frame), these sections interpreted as 'wavefunctions' cannot be viewed as actual functions on the spacetime. In this approach, the change of an observer results not only in the change of actual coordinates in the spacetime but also in a change of the phase of wavefunctions. For the Schrödinger principal bundle, a natural differential calculus for 'wave forms' is developed that leads to a natural generalization of the concept of the Laplace–Beltrami operator associated with a pseudo-Riemannian metric. The free Schrödinger operator turns out to be the Laplace–Beltrami operator associated with a naturally distinguished invariant pseudo-Riemannian metric on the Schrödinger principal bundle. The presented framework does not involve any ad hoc or axiomatically introduced geometrical structures. It is based on the traditional understanding of the Schrödinger operator in a given reference frame—which is supported by producing right physics predictions—and it is proven to be strictly related to the frame-independent formulation of analytical Newtonian mechanics and Hamilton–Jacobi equations that makes a bridge between the classical and quantum theory.
A theory physically equivalent to traditional nonrelativistic quantum mechanics is presented, in which both dynamical and probabilistic concepts enter in a classical way. Particle trajectories are deterministically governed by classical mechanics, only the initial position being at random. Quantum effects are supposed to arise from a modification of the geometry of space, due to the presence of matter. However, unlike gravitational forces, which are related to the metric of space-time, quantum-mechanical forces are proved to be related to the transplantation law of vectors. The resulting geometry of space, in the nonrelativistic limit, is found to be Weyl's geometry. Both particle motion and geometry of space are obtained from a unique averaged least-action principle.
Extension of the Schrodinger equation is submitted by removing its limitations appearing due to the limitations of the formalism of Hamilton, based on which this equation was obtained. For this purpose the problems of quantum mechanics arising from the limitations of classical mechanics are discussed. These limitations, in particular, preclude the use of the Schrodinger equation to describe the time symmetry violation. The extension of the Schrodinger equation is realized based on the principle of duality symmetry. According to this principle the dynamics of the systems is determined by the symmetry of the system and by the symmetry of the space. The extension of the Schrodinger equation was obtained from the dual expression of energy, represented in operator form. For this purpose the independent micro - and macro-variables that determine respectively the dynamics of quantum particle system relative to its center of mass and the movement of the center of mass in space are used. The solution of the extended Schrodinger equation for the system near equilibrium is submitted. The main advantage of the extended Schrodinger equation is that it is applicable to describe the interaction and evolution of quantum systems in inhomogeneous field of external forces.
Abstract This paper reformulates the Schrödinger dynamics in a fully geometric and gauge-theoretic language, where the phase of the quantum wavefunction acts as a U(1) connection and its curvature manifests as measurable interference and geometric deformation. By coupling the connection Aμ=ℏ−1∂μSA_\mu = \hbar^{-1}\partial_\mu SAμ=ℏ−1∂μS to a curved phase metric gμν(n)=gμν+χnAμAν+ηnA2Πμνg^{(n)}_{\mu\nu} = g_{\mu\nu} + \chi_n A_\mu A_\nu + \eta_n A^2 \Pi_{\mu\nu}gμν(n)=gμν+χnAμAν+ηnA2Πμν, the Schrödinger equation acquires a covariant form that unites probability flow, curvature, and analytic hierarchy. The corresponding field strength Fμν=∂μAν−∂νAμF_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\muFμν=∂μAν−∂νAμ defines the geometric curvature of phase space, leading to a Maxwell-like equation for the phase field and an emergent Lorentz-type force on the probability current. The framework provides a direct correspondence between phase holonomy, Cn-carrier parity, and the analytic ζ-layer structure, suggesting a measurable bridge between gauge geometry and transcendental constants in multi-path interferometry.
We analyze the Schrödinger dynamics and the Schrödinger function (or the so-called wavefunction) in the following four aspects. (1) The Schrödinger equation is reconstructed from scratch in the real field only, without referring to Newtonian mechanics nor optics. Only the very simple conditions such as the space-time translational symmetry and the conservation of flux and energy are imposed on the factorization of the density distribution in configuration space, giving rise to a two-dimensional real vector. On returning to the original Schrödinger equation, the imaginary number arises naturally. (2) Like the Langevin equation in a Brownian dynamics, we formulate a single-event path dynamics in quantum mechanics, contrasting with the Schrödinger distribution function. The path thus attained is referred to as one-world path, which represents, for instance, a path of a singly launched electron in the double-slit experiment that leaves a spot at the measurement board, while many of accumulated spots give rise to the fringe pattern. We start from the Feynman-Kac formula to draw a relation between a stochastic dynamics and the parabolic differential equations, to one of which the Schrödinger equation is transformed. (3) To highlight the roles of the flux and energy conservation in the Schrödinger dynamics, we build that the quantum Maupertuis-Hamilton principle, which reveals the symplectic structure in the parameter space. (4) We track how the inherent quantum nature like the Huygens-principle-like properties is built. We show that classical trajectory components in semiclassics are demanded to branch into many coherent pieces beyond the semiclassical regime and dissolve into the deep dynamics of genuine full quantum dynamics.