Generalized stochastic mechanics on Riemannian manifolds

Dankel's stochastic mechanics on Riemannian manifolds is generalized such that the diffusion constant becomes a free parameter in the theory.     

Statistical mechanics of holonomic systems as a Brownian motion on smooth manifolds

The statistical mechanics of arbitrary holonomic scleronomous systems subjected to arbitrary external forces is described by specializing the Lagrange and Hamilton equations of motion to those of the Brownian motion on a manifold. In this context, the Klein‐Kramers and Smoluchowski equations are derived in covariant form, and it is demonstrated that these equations have equilibrium solutions corresponding to the Gibbs distribution, in agreement with standard thermodynamics. At last, the Langevin dynamics corresponding to the Smoluchowski limit is found to exactly correspond to the Brownian motion on a smooth manifold. These results find significant applications in the study of several statistical properties of constrained molecular assemblies (e.g. polymers) of interest in chemistry, physics and biology.     

Quantum mechanics and geometric analysis on manifolds

A central idea of modern geometric analysis is the assignment of a geometric structure, usually called thesymbol, to a differential operator. It is known that this operation is closely related to quantum mechanics. For a class of linear operators, including the Dirac operator, a geometric structure, called aco-Riemannian metric, is assigned to such symbols. Certain other topics related to the geometric structure of quantum mechanics, e.g., the symplectic structure of the projective space of Hilbert space, are briefly treated.     

Quantum mechanics on manifolds embedded in Euclidean space

Quantum particles confined to surfaces in higher-dimensional spaces are acted upon by forces that exist only as a result of the surface geometry and the quantum mechanical nature of the system. The dynamics are particularly rich when confinement is implemented by forces that act normal to the surface. We review this confining potential formalism applied to the confinement of a particle to an arbitrary manifold embedded in a higher-dimensional Euclidean space. We devote special attention to the geometrically induced gauge potential that appears in the effective Hamiltonian for motion on the surface. We emphasize that the gauge potential is only present when the space of states describing the degrees of freedom normal to the surface is degenerate. We also distinguish between the effects of the intrinsic and extrinsic geometry on the effective Hamiltonian and provide simple expressions for the induced-scalar potential. We discuss examples including the case of a three-dimensional manifold embedded in a five-dimensional Euclidean space.     

Stochastic quantum mechanics viewed from the language of manuals

The language of manuals may be used to discuss inference in measurement in a general experimental context. Specializing to the context of the frame manual for Hilbert space, this inference leads to state dominance of the inferred state from partial measurements; this in turn, by Sakai's theorem, determines observables which are described by positive operator-valued measures. Symmetries are then introduced, showing that systems of covariance, rather than systems of imprimitivity, are natural objects to study in quantum mechanics. Experiments measuring different polarization components simultaneously are reexamined in this language. Finally, implications of the Naimark extension theorem for the manual approach are investigated.     

Some harmonic maps on pseudo-Riemannian manifolds

Rosen's bimetric theory of gravitation, when equipped with a flat background metric, is known to be realized as a harmonic mapping of Minkowski spacetime into a certain homogeneous space. This paper develops and exploits these facts to provide four classes of explicit solutions to Rosen's field equations. These four classes form the elements from which more general solutions may be formed by a type of superposition. It is also shown how spherical gravitational waves may be explicitly built into these solutions.This paper was presented at the Third Loyola Conference on Quantum Theory and Gravitation, May 29–June 1, 1985, but was received too late for inclusion in the Proceedings issue.     

Lift order problems for ordinary differential equations on manifolds

Fork≥0, let ττ_k:T ^k+1(M)=T(T ^k(M))→T ^k(M) denote the (k+1)th iterated tangent bundle in relation to a base manifoldT ⁰(M)=M. LetV represent a possibly nonstationary vector field overT ^k(M), and letQ be a subset/submanifold inT ^k(M). Sufficient conditions (and, whenV is completely integrable inQ, necessary and sufficient conditions) are established to ensure that all solutionsg toy′=V(t, y) lying entirely inQ have the formG=f ^[k], wheref ^[k] is thekth-order differential lift of a curvef lying inM. The relevance of the issue for higher order dynamical systems (especially in mechanics) is discussed. Higher order involutions and complete vector field lifts are examined from the viewpoint of the differential equations they present. Collateral results on the general solvability of initial value problems are obtained and numerous examples are discussed in detail. To the memory of my teacher and friend M. Kuga (1928–1990).     

Some observations on cochlear mechanics.

A set of experiments was conducted using the M?ssbauer effect to determine the vibratory characteristics of the basilar membrane, Reissner's membrane, the malleus, incus, and oval window in squirrel monkey. A few measurements were also made in guinea pig in the basal cochlear region. The nonlinear vibration properties of the basilar membrane are described in detail for the midfrequency region in the squirrel monkey. Only in this region have nonlinear effects been observed. A comparison of mechanical and neural data indicates good qualitative agreement.     

Some remarks on conformal invariant theories on four-lorentz manifolds

Contrary to the eleven-parameter group consisting of Poincaré-transformations and dilatations, the group of so-called special conformal transformations can act on the Minkowski space only as a local conformal Lie transformation group. We show that the universal covering space of the compactified Minkowski space , together with an appropriate metric \(\tilde g\) on it, form a suitable Lorentz manifold that admits universal covering group of the “conformal group” of as a transitive Lie transformation group. This group respects the causality notion on usually defined on a Lorentz manifold. However, possesses only seven isometries in contrast to the well-known ten isometries on the Minkowski space , which correspond to conservation of energy-momentum and angular-momentum.     

On the statistical mechanics of elastic manifolds with random interaction

We study the statistical mechanics of D-dimensional elastic manifolds, interacting via randomly distributed forces. It is shown, that this model can be mapped onto the statistical mechanics of disorder-induced roughening of a D-dimensional interface with D transverse degrees of freedom in a disordered medium. The roughness exponent ζ for the lateral deformations is calculated for different kinds of elastic response of the manifolds. Pis’ma Zh. éksp. Teor. Fiz. 64, No. 11, 801–806 (10 December 1996) Published in English in the original Russian journal. Edited by Steve Torstveit.     

Some potentials for the curvature tensor on three-dimensional manifolds

We study equations of Riemann–Lanczos type on three dimensional manifolds. Obstructions to global existence for global Lanczos potentials are pointed out. We check that the imposition of the original Lanczos symmetries on the potential leads to equations which do not have a determined type, leading to problems when trying to prove global existence. We show that elliptic equations can be obtained by relaxing those symmetry requirements in at least two different ways, leading to global existence of potentials under natural conditions. A second order potential for the Ricci tensor is introduced.     

Some remarks on classical representations of quantum mechanics

It is shown that, to a certain extent, the statistical framework of Hilbert-space quantum mechanics can be reformulated in classical terms.     

Bicovariant differential calculus on quantum groups and wave mechanics

The bicovariant differential calculus on quantum groups being defined by Woronowicz and later worked out explicitly by Carow-Watamura et at. and Juro for the real quantum groupsSU _q (N) andSO _q (N) through a systematic construction of the bicovariant bimodules of these quantum groups is reviewed forSU _q (2) andSO _q (N). The resulting vector fields build representations of the quantized universal enveloping algebras acting as covariant differential operators on the quantum groups and their associated quantum spaces. As an application a free particle stationary wave equation on quantum space is formulated and solved in terms of a complete set of energy eigenfunctions.Presented at the Colloquium on the Quantum Groups, Prague, 18–20 June 1992.     

Capacities as conformal quasi-invariants on pseudo-riemannian manifolds

A class of capacities is introduced on pseudo-riemannian manifolds. They arise as a natural counterpart of the well-known plane quasiconformal capacities and their higher dimensional analogues which have been studied extensively in the recent years by F.W. Gehring, R. Kühnau and others. The capacities in question are shown to be either conformal invariants or conformal quasi-invariants, and, in the latter case, exact bounds are established. We thus arrive at the notion of quasiconformal mappings of pseudo-riemannian manifolds, which correspond to the inhomogeneous media. These mappings are studied briefly and the physical interpretation of some of the capacities in question is also given.     

Some aspects of gravitational quantum mechanics

String theory, quantum geometry, loop quantum gravity and black hole physics all indicate the existence of a minimal observable length on the order of Planck length. This feature leads to a modification of Heisenberg uncertainty principle. Such a modified Heisenberg uncertainty principle is referred as gravitational uncertainty principle(GUP) in literatures. This proposal has some novel implications on various domains of theoretical physics. Here, we study some consequences of GUP in the spirit of Quantum mechanics. We consider two problem: a particle in an one-dimensional box and free particle wave function. In each case we will solve corresponding perturbational equations and compare the results with ordinary solutions.     

Some remarks on a model of supersymmetric quantum mechanics

We develop a recently proposed model within supersymmetric quantum mechanics that puts a group structure on the creation and annihilation operators. We apply the scheme to a variety of quantum mechanical problems and work out a two-term energy recursion equation when the overall group structure isU(1, 1).