Fluctuations, Information, Gravity and the Quantum Potential

Abstract

We show how the quantum potential arises in various ways and trace its connection to quantum fluctuations and Fisher information along with its realization in terms of Weyl curvature. It represents a genuine quantization factor for certain classical systems as well as an expression for quantum matter in gravity theories of Weyl-Dirac type. Many of the facts and examples are extracted from the literature (with references cited) and we mainly provide connections and interpretation, with a few new observations. We deliberately avoid ontological and epistemological discussion and resort to a collection of contexts where the quantum potential plays a visibly significant role. In particular we sketch some recent results of F. and A. Shojai on Dirac-Weyl action and Bohmian mechanics which connects quantum mass to the Weyl geometry. Connections a la Santamato of the quantum potential with Weyl curvature arising from a stochastic geometry, are also indicated for the Schrödinger equation (SE) and Klein-Gordon (KG) equation. Quantum fluctuations and quantum geometry are linked with the quantum potential via Fisher information. Derivations of SE and KG from Nottale's scale relativity are sketched along with a variety of approaches to the KG equation. Finally connections of geometry and mass generation via Weyl-Dirac geometry with many cosmological implications are indicated, following M. Israelit and N. Rosen.

Quantum Potential and Symmetries in Extended Phase Space

“…Equation (19a) is the Wigner equation governing the Wigner distribution functions given by equation (19b) [13]. Therefore, we call the new form of the EPS obtained by transformation (13) on the old one for β = α = − 4 Quantum potential and generalization to the EPS a) Quantum potential in q space.…”

“…The interaction of the particle and the field is attributed to a scalar potential that turns out to be quantum potential, whenever, the constraint of the invariance of Schrödinger equation is applied. The connection of quantum potential with quantum fluctuations and quantum geometry in terms of Weyl curvature has been studied extensively by different authors; see, for example F. and A. Shojai [12], Carroll [13] and the references therein. However, unlike the external potential, the quantum potential is not a pre-assigned function of the system coordinates and can only be derived from the wave function of the system [6] or from the corresponding quantum distribution functions used to calculate the average values of the observables.…”

Geometric description of the Schrödinger equation in (3n+1)-dimensional configuration space

“…It is thus natural that there are numerous attempts to reformulate quantum mechanics in a more geometrical way [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20].…”

“…Using the language of [21,22], it was shown that such trajectories naturally arise in the configuration space for the complex Klein Gordon equation. It was further found that the evolution equation for those trajectories can be cast in the form of a geodesics equation in a conformally rescaled configuration space [2,3,7,9,18,19]. Thus, the relativistic Klein Gordon equation can be rewritten in a geometric language with non-trivial trajectories in configuration space.…”

Symmetry Transformation in Extended Phase Space: the Harmonic Oscillator in the Husimi Representation

“…The interaction of the particle and field is attributed to a scalar potential that turns out to be the quantum potential. The connection of the quantum potential with quantum fluctuations and quantum geometry in terms of Weyl's curvature has been studied by F. Shojai and A. Shojai [7] and Carroll [8]. Nevertheless, unlike the external potential, the quantum potential is not a pre-assigned function of the system coordinates and can only be derived from the wave function of the system [5] or from the corresponding quantum distribution functions used to calculate the average values of the observables [9].…”