We present the space‐time and Hilbert‐state space quantum geometries and their associated Brownian motions. We discuss the problem of the reduction of the wave function associated to these geometries and their Brownian motions.
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-
Abstract Our purpose here is to draw attention to an interesting relationship that can be shown to hold between (a) information geometry, and (b) quantum geometry. By ‘information geometry’, we mean the natural geometry associated with families of probability distributions. This is a subject that goes back more than half a century to the pioneering work of Rao (1945), who showed that the Fisher information matrix associated with a parametrised family of probability dis tributions gives rise to a natural Riemannian metric on the parameter space. This then allows one to speak in a precise way of the ‘distance’ between two probability distributions, an idea that turns out to be useful in problems of statistical inference. An extensive literature has thus developed, following this line of enquiry, on applications of differential geometry to statistics (see, e.g., Amari (1985), Barndorff-Nielsen et al. (1986), Murray and Rice (1993)). By ‘quantum geometry’, on the other hand, we mean the geometry of the manifold of states associated with a given quantum mechanical system. In particular, we consider the manifold of ‘pure’ states associated with a given complex Hilbert space. This is the space of ‘rays’ through the origin in the Hilbert space, which has the structure of a complex projective space. It is known that the usual rules of quantum mechanics allow one to construct a natural metric on this space, namely the Fubini-Study metric, the specification of which is equivalent to all the familiar structure associated with ordinary quantum mechanics.
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.