42 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 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
Hypermutable Non-Synonymous Sites Are under Stronger Negative Selection
Mutation rate varies greatly between nucleotide sites of the human genome and depends both on the global genomic location and the local sequence context of a site. In particular, CpG context elevates the mutation rate by an order of magnitude. Mutations also vary widely in their effect on the molecular function, phenotype, and fitness. Independence of the probability of occurrence of a new mutation's effect has been a fundamental premise in genetics. However, highly mutable contexts may be preserved by negative selection at important sites but destroyed by mutation at sites under no selection. Thus, there may be a positive correlation between the rate of mutations at a nucleotide site and the magnitude of their effect on fitness. We studied the impact of CpG context on the rate of human–chimpanzee divergence and on intrahuman nucleotide diversity at non-synonymous coding sites. We compared nucleotides that occupy identical positions within codons of identical amino acids and only differ by being within versus outside CpG context. Nucleotides within CpG context are under a stronger negative selection, as revealed by their lower, proportionally to the mutation rate, rate of evolution and nucleotide diversity. In particular, the probability of fixation of a non-synonymous transition at a CpG site is two times lower than at a CpG site. Thus, sites with different mutation rates are not necessarily selectively equivalent. This suggests that the mutation rate may complement sequence conservation as a characteristic predictive of functional importance of nucleotide sites