On Weyl geometry, random processes, and geometric quantum mechanics

1992 ◽  
Vol 22 (4) ◽  
pp. 569-615 ◽  
Author(s):  
Carlos Castro
Keyword(s):  
Quantum Mechanics ◽  
Random Processes ◽  
Weyl Geometry ◽  
Geometric Quantum Mechanics

scholarly journals Supreme Theory of Everything

2019 ◽  
Vol 2 (2) ◽  
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Electromagnetic Wave ◽  
Theory Of Relativity ◽  
General Theory Of Relativity ◽  
Wave Particle Duality ◽  
Theory Of Everything ◽  
Parallel Universes ◽  
Celestial Bodies ◽  
Geometric Quantum Mechanics

Not only universe, but everything has general characters as eternal, infinite, cyclic and wave-particle duality. Everything from elementary particles to celestial bodies, from electromagnetic wave to gravity is in eternal motions, which dissects only to circle. Since everything is described only by trigonometry. Without trigonometry and mathematical circle, the science cannot indicate all the beauty of harmonic universe. Other method may be very good, but it is not perfect. Some part is very nice, another part is problematic. General Theory of Relativity holds that gravity is geometric. Quantum Mechanics describes all particles by wave function of trigonometry. In this paper using trigonometry, particularly mathematics circle, a possible version of the unification of partial theories, evolution history and structure of expanding universe, and the parallel universes are shown.

Remarks on geometric quantum mechanics

2004 ◽  
Vol 51 (2) ◽  
pp. 229-243 ◽  
Author(s):  
Alberto Benvegnù ◽  
Nicola Sansonetto ◽  
Mauro Spera
Keyword(s):  
Quantum Mechanics ◽  
Geometric Quantum Mechanics

Connections between Weyl geometry, quantum potential and quantum entanglement

2021 ◽  
Author(s):  
Shi-Dong Liang ◽  
Wenjing Huang
Keyword(s):  
Quantum Mechanics ◽  
Scalar Curvature ◽  
Quantum Entanglement ◽  
Quantum Potential ◽  
Entangled State ◽  
Dimensional System ◽  
Weyl Geometry ◽  
3D Space ◽  
Potential Applications ◽  
Weyl Scalar

The Weyl geometry promises potential applications in gravity and quantum mechanics. We study the relationships between the Weyl geometry, quantum entropy and quantum entanglement based on the Weyl geometry endowing the Euclidean metric. We give the formulation of the Weyl Ricci curvature and Weyl scalar curvature in the n-dimensional system. The Weyl scalar field plays a bridge role to connect the Weyl scalar curvature, quantum potential and quantum entanglement. We also give the Einstein–Weyl tensor and the generalized field equation in 3D vacuum case, which reveals the relationship between Weyl geometry and quantum potential. Particularly, we find that the correspondence between the Weyl scalar curvature and quantum potential is dimension-dependent and works only for the 3D space, which reveals a clue to quantize gravity and an understanding why our space must be 3D if quantum gravity is compatible with quantum mechanics. We analyze numerically a typical example of two orthogonal oscillators to reveal the relationships between the Weyl scalar curvature, quantum potential and quantum entanglement based on this formulation. We find that the Weyl scalar curvature shows a negative dip peak for separate state but becomes a positive peak for the entangled state near original point region, which can be regarded as a geometric signal to detect quantum entanglement.

THE DYNAMICAL EVOLUTION OF GEOMETRIC UNCERTAINTY PRINCIPLE FOR SPIN 1/2 SYSTEM

2021 ◽  
Vol 10 (9) ◽  
pp. 3241-3251
Author(s):  
H. Umair ◽  
H. Zainuddin ◽  
K.T. Chan ◽  
Sh.K. Said Husein
Keyword(s):  
Quantum Mechanics ◽  
Uncertainty Principle ◽  
Uncertainty Relation ◽  
Dynamical Behavior ◽  
Dynamical Evolution ◽  
Riemannian Metric ◽  
Hamiltonian Flow ◽  
Space Dynamics ◽  
Geometric Quantum Mechanics ◽  
Complex Projective

Geometric Quantum Mechanics is a formulation that demonstrates how quantum theory may be casted in the language of Hamiltonian phase-space dynamics. In this framework, the states are referring to points in complex projective Hilbert space, the observables are real valued functions on the space and the Hamiltonian flow is defined by Schr{\"o}dinger equation. Recently, the effort to cast uncertainty principle in terms of geometrical language appeared to become the subject of intense study in geometric quantum mechanics. One has shown that the stronger version of uncertainty relation i.e. the Robertson-Schr{\"o}dinger uncertainty relation can be expressed in terms of the symplectic form and Riemannian metric. In this paper, we investigate the dynamical behavior of the uncertainty relation for spin $\frac{1}{2}$ system based on this formulation. We show that the Robertson-Schr{\"o}dinger uncertainty principle is not invariant under Hamiltonian flow. This is due to the fact that during evolution process, unlike symplectic area, the Riemannian metric is not invariant under the flow.

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