Equivalence principle, higher-dimensional Möbius 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 Möbius 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 Schrödinger 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.

Energy quantisation and time parameterisation

“…In this paper we offer a solution to this puzzle. We demonstrate the absence of trajectories in the derivation of the QHJE from point transformations leading to the trivial hamiltonian [11][12][13][14][15][16][17][18]. The basic point is that trajectories can only be defined by time parameterisation of them, and include the Bohm-de Broglie pilot wave representation and Floyd's time parameterisation [19][20][21][22][23][24][25][26][27] by using Jacobi theorem.…”

“…Such a transformation leads to a basic cocycle condition which, in turn, implies that the analog of the Hamilton characteristic function must satisfy the quantum analog of the stationary HJ equation. The derivation extends to the higher dimensional non-stationary case and to the relativistic case as well [18].…”

“…In particular, we will see that the implementation of the cocycle condition fixes gluing conditions on the ratio of two linearly independent solutions of the wave function that can be satisfied only on compact spaces, so that continuous energy spectra arise only in the decompactification limit. The fact that such gluing conditions are a basic step in the geometrical formulation of [11][12][13][14][15][16][17][18], related to Legendre duality [28], and the fact that they led to the quantisation of the energy in the case of confining potentials, suggest that they should be always satisfied. It should be observed that in the case of a free particle the level spacing of the energy spectrum will be determined by the geometry of our three-dimensional space, essentially of the order R −2 , where R is some characteristic cosmological length, and therefore extremely tiny.…”

Complex Moduli of Physical Quanta

“…These facts imply that the concept of a quantum is not absolute, but relative to the quantum theory used to measure it [1]. In particular classical and quantum, for long known to be intimately related [10], are not necessarily always the same for all observers on phase space. In this letter we have analysed the dependence of the notion of a quantum on the complex-differentiable structure chosen on classical phase space.…”

Relativistic quantum motion

“…Deriving quantum mechanics from an equivalence postulate, Faraggi and Matone showed that the Schrödinger wave function must have the form [1,2,3,4] φ(x) = ∂S 0 ∂x…”