Historical research guide · 0279 · 2009
Logarithmic nonlinearity in theories of quantum gravity: Origin of time and observational consequences
Konstantin G. Zloshchastiev
In one minute
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.
Connection
Chapter 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.
From the source
One 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
In 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
Through 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
That 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
The 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
The 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
The way in
https://arxiv.org/abs/0906.4282How to cite it
Konstantin G. Zloshchastiev (2009) Logarithmic nonlinearity in theories of quantum gravity: Origin of time and observational consequences doi:10.1134/S0202289310040067