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Foundations of Physics

Torsion Fields, Cartan–Weyl Space–Time and State-Space Quantum Geometries, their Brownian Motions, and the Time Variables

Diego Lucio Rapoport

Published Mar 23, 2007• 7 citations
Publisher
DOI
10.1007/s10701-007-9126-8

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Torsion Fields, The Quantum Potential, Cartan-Weyl Space-Time and State-space Geometries and their Brownian Motions

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We review the relation between space-time geometries with torsion fields (the so-called Riemann-Cartan-Weyl (RCW) )geometries) and their associated Brownian motions. In this setting, the metric conjugate of the tracetorsion one-form is the drift vectorfield of the Brownian motions. Thus, in the present approach, Brownian motions, in distinction with Nelson�s Stochastic Mechanics, are spacetime structures. We extend this to the state-space of non-relativistic quantum mechanics and discuss the relation between a noncanonical quantum RCW geometry in state-space associated with the gradient of the quantum-mechanical expectation value of a self-adjoint operator given by the generalized laplacian operator defined by a RCW geometry. We discuss the reduction of the wave function in terms of a RCW quantum geometry in state-space. We characterize the Schroedinger equation for both an observed and unobserved quantum systems in terms of the RCW geometries and Brownian motions. Thus, in this work, the Schroedinger field is a torsion generating field, and the U and R processes, in the sense of Penrose, are associated, the former to spacetime geometries and their associated Brownian motions, and the latter to their extension to the state-space of nonrelativistic quantum mechanics given by the projective Hilbert space. In this setting, the Schroedinger equation can be either linear or nonlinear. We discuss the problem of the many times variables and the relation with dissipative processes. We present as an additional example of RCW geometries and their Brownian motions counterpart, the dynamics of viscous fluids obeying the invariant Navier-Stokes equations. We introduce in the present setting an extension of R. Kiehn�s approach to dynamical systems starting from the notion of the topological dimension of one-forms, to apply it to the trace-torsion one-form whose metric conjugate is the Brownian motion�s drift vectorfield and discuss the topological notion of turbulence. We discuss the relation between our setting and the Nottale theory of Scale Relativity, and the work of Castro and Mahecha in this volume in nonlinear quantum mechanics, Weyl geometries and the quantum potential.

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Summary: We review the relation between space-time geometries with tor-sion fields (the so-called Riemann-Cartan-Weyl (RCW) geometries) and their associated Brownian motions. In this setting, the metric conjugate of the trace-torsion one-form is the drift vector field of the Brownian motions. Thus, in the present approach space-time fluctuations as Brownian motions are-in dis-tinction with Nelson’s Stochastic Mechanics- space-time structures. Thus, space and time have a fractal structure. We discuss the relations with Nottale’s theory of Scale Relativity, which stems from Nelson’s approach. We characterize the Schroedinger equation in terms of the RCW geometries and Brownian motions. In this work, the Schroedinger field is a torsion generating field. The potential functions on Schroedinger equations can be alternatively linear or nonlinear on the wave function, leading to nonlinear and linear creation-annihilation of par-ticles by diffusion systems. We give a brief presentation of the isotopic lift of Quantum Mechanics known as Hadronic Mechanics due to Santilli. We start by giving the isotopic lift (i.e. a non-unitary transformation not identically re-

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Torsion Fields, Propagating Singularities, Nilpotence, Quantum Jumps and the Eikonal Equations

Diego Lucio Rapoport, Daniel M. Dubois

We establish a geometrical theory in terms of torsion fields and their singularities of quantum jumps and of the propagation of wave‐front singularities described by the eikonal equation of geometrical optics basic to Fock’s theory of gravitation and General Relativity. The latter equations correspond to the wavefront propagation for the Maxwell and Einstein equations. We discuss the genesis of spacetime in terms of these singularities and torsion fields. We introduce the class of solutions of the wave propagation (defined in terms of the metric geometry) and the eikonal equations. The lagrangian functional for quantum jumps defined in terms of the quantum potential is introduced. We give a formula that characterizes the quantum jumps in terms of an extension of the argument principle in complex analysis. We show that the wave propagation in terms of the metric geometry under a change of gauge has a natural expression as a wave propagation in terms of the laplacians associated to a torsion geometry of the Cartan‐Weyl type which has an additional interaction first‐order torsion term. In this geometry there is a differential one‐form trace‐torsion term given by the logarithmic differential of (monochromatic) waves. It is shown that quantum jumps are associated with the Cartan‐Weyl geometry, through a torsion potential given by the logarithmic differential of the composition of an analytic function ‐or alternatively a twice differentiable function‐ with a monochromatic wave function. In particular, if follows that monochromatic wave functions generate torsion. The node sets of monochromatic functions are shown to be the locus for quantum jumps. In the case of the metric being Minkowski or positive‐definite, the generalized laplacians corresponding to this torsion geometry, are generators of Brownian motions in which the torsion describes the drift of the Brownian processes. We show that this torsion potential and its singularities due to the nodes of the monochromatic wave functions, gives rise to an Aharonov‐Bohm effect. We briefly indicate the role of quantum jumps in establishing a global time and space coordinates in semiclassical General Relativity. We indicate some relations between the present approach to the geometry of quantum jumps, and the problem of topographical representation in visual perception, the Klein bottle, quantum physics and holography.

AIP conference proceedings•Jan 1, 2010•5 citations
We review the relation between space-time geometries with torsion fields (the so-called Riemann-Cartan-Weyl (RCW) )geometries) and their associated Brownian motions. In this setting, the metric conjugate of the tracetorsion one-form is the drift vectorfield of the Brownian motions. Thus, in the present approach, Brownian motions, in distinction with Nelson�s Stochastic Mechanics, are spacetime structures. We extend this to the state-space of non-relativistic quantum mechanics and discuss the relation between a noncanonical quantum RCW geometry in state-space associated with the gradient of the quantum-mechanical expectation value of a self-adjoint operator given by the generalized laplacian operator defined by a RCW geometry. We discuss the reduction of the wave function in terms of a RCW quantum geometry in state-space. We characterize the Schroedinger equation for both an observed and unobserved quantum systems in terms of the RCW geometries and Brownian motions. Thus, in this work, the Schroedinger field is a torsion generating field, and the U and R processes, in the sense of Penrose, are associated, the former to spacetime geometries and their associated Brownian motions, and the latter to their extension to the state-space of nonrelativistic quantum mechanics given by the projective Hilbert space. In this setting, the Schroedinger equation can be either linear or nonlinear. We discuss the problem of the many times variables and the relation with dissipative processes. We present as an additional example of RCW geometries and their Brownian motions counterpart, the dynamics of viscous fluids obeying the invariant Navier-Stokes equations. We introduce in the present setting an extension of R. Kiehn�s approach to dynamical systems starting from the notion of the topological dimension of one-forms, to apply it to the trace-torsion one-form whose metric conjugate is the Brownian motion�s drift vectorfield and discuss the topological notion of turbulence. We discuss the relation between our setting and the Nottale theory of Scale Relativity, and the work of Castro and Mahecha in this volume in nonlinear quantum mechanics, Weyl geometries and the quantum potential.
Summary: We review the relation between space-time geometries with tor-sion fields (the so-called Riemann-Cartan-Weyl (RCW) geometries) and their associated Brownian motions. In this setting, the metric conjugate of the trace-torsion one-form is the drift vector field of the Brownian motions. Thus, in the present approach space-time fluctuations as Brownian motions are-in dis-tinction with Nelson’s Stochastic Mechanics- space-time structures. Thus, space and time have a fractal structure. We discuss the relations with Nottale’s theory of Scale Relativity, which stems from Nelson’s approach. We characterize the Schroedinger equation in terms of the RCW geometries and Brownian motions. In this work, the Schroedinger field is a torsion generating field. The potential functions on Schroedinger equations can be alternatively linear or nonlinear on the wave function, leading to nonlinear and linear creation-annihilation of par-ticles by diffusion systems. We give a brief presentation of the isotopic lift of Quantum Mechanics known as Hadronic Mechanics due to Santilli. We start by giving the isotopic lift (i.e. a non-unitary transformation not identically re-
We establish a geometrical theory in terms of torsion fields and their singularities of quantum jumps and of the propagation of wave‐front singularities described by the eikonal equation of geometrical optics basic to Fock’s theory of gravitation and General Relativity. The latter equations correspond to the wavefront propagation for the Maxwell and Einstein equations. We discuss the genesis of spacetime in terms of these singularities and torsion fields. We introduce the class of solutions of the wave propagation (defined in terms of the metric geometry) and the eikonal equations. The lagrangian functional for quantum jumps defined in terms of the quantum potential is introduced. We give a formula that characterizes the quantum jumps in terms of an extension of the argument principle in complex analysis. We show that the wave propagation in terms of the metric geometry under a change of gauge has a natural expression as a wave propagation in terms of the laplacians associated to a torsion geometry of the Cartan‐Weyl type which has an additional interaction first‐order torsion term. In this geometry there is a differential one‐form trace‐torsion term given by the logarithmic differential of (monochromatic) waves. It is shown that quantum jumps are associated with the Cartan‐Weyl geometry, through a torsion potential given by the logarithmic differential of the composition of an analytic function ‐or alternatively a twice differentiable function‐ with a monochromatic wave function. In particular, if follows that monochromatic wave functions generate torsion. The node sets of monochromatic functions are shown to be the locus for quantum jumps. In the case of the metric being Minkowski or positive‐definite, the generalized laplacians corresponding to this torsion geometry, are generators of Brownian motions in which the torsion describes the drift of the Brownian processes. We show that this torsion potential and its singularities due to the nodes of the monochromatic wave functions, gives rise to an Aharonov‐Bohm effect. We briefly indicate the role of quantum jumps in establishing a global time and space coordinates in semiclassical General Relativity. We indicate some relations between the present approach to the geometry of quantum jumps, and the problem of topographical representation in visual perception, the Klein bottle, quantum physics and holography.