Logarithmic nonlinearity in theories of quantum gravity: Origin of time and observational consequences
Konstantin G. Zloshchastiev
Abstract and summary · read the original at the source
In one page
Konstantin Zloshchastiev asks what changes if the quantum wave equation carries one extra term — a logarithm of the wave’s own intensity, the same nonlinearity that describes a Bose-Einstein condensate. Two things follow that are hard to get any other way. First, time appears. The new term behaves like an entropy, and through a theorem about operator algebras it generates the flow we experience as evolution: a fully covariant theory with no clock inside it becomes an ordinary theory with a clock. Second, that clock ticks at a rate that depends on energy, so Lorentz invariance comes out as a low-energy limit rather than an exact law, and the vacuum acquires an effective refractive index. Zloshchastiev then works out what this would do to high-energy cosmic rays and gamma-ray bursts, and notes plainly that a Fermi satellite measurement ruled out his linear approximation while leaving the exact relations standing. In this picture the vacuum is a medium with physics of its own.
Why it matters hereChapter 5 reads the vacuum as a quantum fluid rather than a backdrop, and this paper supplies that reading’s mathematical spine: the logarithmic term is exactly the nonlinearity a condensate obeys. It also hands the site something to point at — if the vacuum is a medium it has a refractive index, and high-energy astrophysics is where that index would first show itself.
What it claims
01One logarithmic term added to the quantum wave equation preserves everything the correspondence principle asks for: general covariance of the classical theory, locality of product states, separability of uncorrelated subsystems, additivity of energy and boundedness from below, and the Planck relation.Section II, Equations 3 to 5 and the five listed properties
Published and peer-reviewed02In the non-relativistic limit that term averages to a temperature multiplied by a Shannon-type entropy assigned to a single quantum particle, measuring how far the particle is smeared over space.Section III, Equation 7
Published and peer-reviewed03Through the Tomita-Takesaki modular group of the von Neumann algebra of observables, the logarithmic term generates the evolution time itself: a covariant theory carrying the nonlinearity but no observer-independent time is equivalent to the linear theory evolving in modular time.Section III, Equations 8 to 11
Published and peer-reviewed04That time parameter is energy-dependent — two systems with different values of the coupling have different time scales — and becomes global only when energies are small compared with the effective quantum gravity scale of order ten to the nineteenth power in GeV, which makes Lorentz invariance an asymptotic low-energy phenomenon rather than an exact symmetry.Section IV, Equations 13 and 14
Published and peer-reviewed05The deformed dispersion relations give the physical vacuum an effective refractive index in Cauchy form, and predict that other things being equal a higher-energy particle travels more slowly, so its mean free path, its lifetime in a high-energy state and its travel distance from the source can all exceed what conventional theory expects.Section IV, Equations 18 to 22; Conclusion
What to watch06The Fermi LAT and GBM measurement of the speed of light bounded the linear velocity dispersion; the author’s reply, added in proof, is that the linear approximation does not apply to extremely ultrarelativistic particles, that the exact non-perturbative relations remain standing, and that the Fermi data instead put bounds on the vacuum’s constant of refraction — the number to watch.Section IV B, Equations 21 and 25; Note added in proof
What to watch
Read it · abstract
Abstract
Starting from a generic generally covariant classical theory we introduce the logarithmic correction to the quantum wave equation. We demonstrate the emergence of the evolution time from the group of automorphisms of the von Neumann algebra governed by this non-linear correction. It turns out that such time parametrization is essentially energy-dependent and becomes global only asymptotically - when the energies get very small comparing to the effective quantum gravity scale. Similar thing happens to the Lorentz invariance - in the resulting theory it becomes an asymptotic low-energy phenomenon. We show how the logarithmic non-linearity deforms the vacuum wave dispersion relations and explains certain features of the astrophysical data coming from recent observations of high-energy cosmic rays. In general, the estimates imply that ceteris paribus the particles with higher energy propagate slower than those with lower one, therefore, for a high-energy particle the mean free path, lifetime in a high-energy state and, therefore, travel distance from the source can be significantly larger than one would expect from the conventional theory. Apart from this, we discuss also the possibility and conditions of the transluminal phenomena in the physical vacuum such as the Cherenkov-type shock waves.
The way in
https://arxiv.org/abs/0906.4282Posted to arXiv under the arXiv.org perpetual non-exclusive licence, so this page carries the summary, the claims and the author’s own abstract, and sends the reader to the source. Published as Gravitation and Cosmology 16, 288–297 (2010); based on a talk at the Gamow’105 memorial conference, Odessa, 2009.
How to cite it
Konstantin G. Zloshchastiev (2009) Logarithmic nonlinearity in theories of quantum gravity: Origin of time and observational consequences. doi:10.1134/S0202289310040067
Where it sits in the curriculum
The vacuum as a quantum fluidWhat the vacuum isThe unified picture