1,393 research outputs found

### 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 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

### Geometric derivation of quantum uncertainty

Quantum observables can be identified with vector fields on the sphere of
normalized states. Consequently, the uncertainty relations for quantum
observables become geometric statements. In the Letter the familiar uncertainty
relation follows from the following stronger statement: Of all parallelograms
with given sides the rectangle has the largest area.Comment: see http://depts.uwc.edu/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

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