Equivalence Principle, Higher Dimensional Moebius Group and the Hidden Antisymmetric Tensor of Quantum Mechanics

Abstract

We show that the recently formulated Equivalence Principle (EP) implies a basic cocycle condition both in Euclidean and Minkowski spaces, which holds in any dimension. This condition, that in one-dimension is sufficient to fix the Schwarzian equation [6], implies a fundamental higher dimensional Moebius invariance which in turn univocally fixes the quantum version of the Hamilton-Jacobi equation. This holds also in the relativistic case, so that we obtain both the time-dependent Schroedinger equation and the Klein-Gordon equation in any dimension. We then show that the EP implies that masses are related by maps induced by the coordinate transformations connecting different physical systems. Furthermore, we show that the minimal coupling prescription, and therefore gauge invariance, arises quite naturally in implementing the EP. Finally, we show that there is an antisymmetric two-tensor which underlies Quantum Mechanics and sheds new light on the nature of the Quantum Hamilton-Jacobi equation.

1+48 pages, LaTeX. Expanded version, two appendices, several comments, including comparison with Einstein Equivalence Principle, added. Typos corrected, one reference added. To appear in CQG

References in corpus (4)

• Generic Bohmian Trajectories of an Isolated Particle
• Duality between coordinates and Dirac field
• Duality of Coordinates and Matter Fields in Curved Spacetime
• Some remarks on time, uncertainty, and spin

Cited by in corpus (43)

• Reconciling Semiclassical and Bohmian Mechanics: V. Wavepacket Dynamics
• Equivalence Postulate and Quantum Origin of Gravitation
• Existential Contextuality and the Models of Meyer, Kent and Clifton
• The Quantum Newton's Law
• Trajectories in the Context of the Quantum Newton's Law
• Superluminality and a Curious Phenomenon in the Relativistic Quantum Hamilton-Jacobi Equation
• The Geometry of Quantum Mechanics
• Comments on Bouda and Djama's "Quantum Newton's law"
• The Geometrical Origin of Dark Energy
• Welcher Weg? A trajectory representation of a quantum Young's diffraction experiment
• OPERA data and The Equivalence Postulate of Quantum Mechanics
• Interference, reduced action, and trajectories
• Energy Quantisation and Time Parameterisation
• A note on the quantum-mechanical Ricci flow
• A mechanics for the Ricci flow
• The high energy limit of the trajectory representation of quantum mechanics
• Quantum-Mechanical Dualities from Classical Phase Space
• On the Ricci flow and emergent quantum mechanics
• Geometric description of the Schrödinger equation in 3n+1 dimensional configuration space
• The Equivalence Postulate of Quantum Mechanics, Dark Energy and The Intrinsic Curvature of Elementary Particles
• Two particle entanglement and its geometric duals
• Quantum Law of Motion: Analysis and Extension to Higher Dimensions
• On the noncommutative eikonal
• Coherent States and Duality
• Quantum States from Tangent Vectors
• A gauge theory of quantum mechanics
• The Quantum Reduced Action In Higher Dimensions
• Duality between the coordinates and wave functions on noncommutative space
• Remarks on osmosis, quantum mechanics, and gravity
• Positive curvature can mimic a quantum
• Hamilton--Jacobi meet Möbius
• Quantum Mechanics from General Relativity and the Quantum Friedmann Equation
• Spinor-Vector Duality and the Swampland
• Quantum Mechanics in symmetry language
• Extended Version of "The Philosophy of the Trajectory Representation of Quantum Mechanics"
• The Möbius Symmetry of Quantum Mechanics
• The Relativistic Quantum Motions
• Complex Moduli of Physical Quanta
• Neutrino Oscillations with Nil Mass
• Universe as Klein-Gordon Eigenstates
• A Quantum is a Complex Structure on Classical Phase Space
• Contribution a la recherche des equations de Mouvement en Mecanique quantique et a une interpretation deterministe
• String Phenomenology: Past, Present and Future Perspectives