Quantum mechanics on Hilbert manifolds: The principle of functional relativity

Quantum mechanics is formulated as a geometric theory on a Hilbert manifold. Images of charts on the manifold are allowed to belong to arbitrary Hilbert spaces of functions including spaces of generalized functions. Tensor equations in this setting, also called functional tensor equations, describe families of functional equations on various Hilbert spaces of functions. The principle of functional relativity is introduced which states that quantum theory is indeed a functional tensor theory, i.e., it can be described by functional tensor equations. The main equations of quantum theory are shown to be compatible with the principle of functional relativity. By accepting the principle as a hypothesis, we then analyze the origin of physical dimensions, provide a geometric interpretation of Planck's constant, and find a simple interpretation of the two-slit experiment and the process of measurement.Comment: 45 pages, 9 figures, see arXiv:0704.3225v1 for mathematical considerations and http://www.uwc.edu/dept/math/faculty/kryukov/ for related paper

On the motion of macroscopic bodies in quantum theory

Quantum observables can be identified with vector fields on the sphere of normalized states. The resulting vector representation is used in the paper to undertake a simultaneous treatment of macroscopic and microscopic bodies in quantum mechanics. Components of the velocity and acceleration of state under Schr\"odinger evolution are given for a clear physical interpretation. Solutions to Schr\"odinger and Newton equations are shown to be related beyond the Ehrenfest results on the motion of averages. A formula relating the normal probability distribution and the Born rule is found

On the measurement problem for a two-level quantum system

A geometric approach to quantum mechanics with unitary evolution and non-unitary collapse processes is developed. In this approach the Schrodinger evolution of a quantum system is a geodesic motion on the space of states of the system furnished with an appropriate Riemannian metric. The measuring device is modeled by a perturbation of the metric. The process of measurement is identified with a geodesic motion of state of the system in the perturbed metric. Under the assumption of random fluctuations of the perturbed metric, the Born rule for probabilities of collapse is derived. The approach is applied to a two-level quantum system to obtain a simple geometric interpretation of quantum commutators, the uncertainty principle and Planck's constant. In light of this, a lucid analysis of the double-slit experiment with collapse and an experiment on a pair of entangled particles is presented.Comment: for related papers, see http://www.uwc.edu/dept/math/faculty/kryukov

On observation of position in quantum theory

Newtonian and Scr{\"o}dinger dynamics can be formulated in a physically meaningful way within the same Hilbert space framework. This fact was recently used to discover an unexpected relation between classical and quantum motions that goes beyond the results provided by the Ehrenfest theorem. A formula relating the normal probability distribution and the Born rule was also found. Here the dynamical mechanism responsible for the latter formula is proposed and applied to measurements of macroscopic and microscopic systems. A relationship between the classical Brownian motion and the diffusion of state on the space of states is discovered. The role of measuring devices in quantum theory is investigated in the new framework. It is shown that the so-called collapse of the wave function is not measurement specific and does not require a ``concentration" near the eigenstates of the measured observable. Instead, it is explained by the common diffusion of state over the space of states under interaction with the apparatus and the environment. This in turn provides us with a basic reason for the definite position of macroscopic bodies in space

Measurement in classical and quantum physics

It was recently shown that quantum and classical mechanics are related in a deeper and more intimate way than previously thought possible. A geometric framework for both theories that allowed going back and forth between quantum and classical processes was discovered. The quantum-mechanical experiments were presented in a new and illuminating fashion, paving a way to resolve the paradoxes of quantum theory. The downside of the theory is a relatively involved machinery of functional analysis and differential geometry used to derive the results. At the same time, the main conclusions are fundamentally simple and can be presented without complicated math. The goal of this paper is to provide such a ``no math" presentation of the theory

Measurement in classical and quantum physics

It was recently shown that quantum and classical mechanics are related in a deeper and more intimate way than previously thought possible. A geometric framework for both theories that allowed going back and forth between quantum and classical processes was discovered. The quantum-mechanical experiments were presented in a new and illuminating fashion, paving a way to resolve the paradoxes of quantum theory. The downside of the theory is a relatively involved machinery of functional analysis and differential geometry used to derive the results. At the same time, the main conclusions are fundamentally simple and can be presented without complicated math. The goal of this paper is to provide such a ``no math" presentation of the theory