Expectation values of minimum-length Ricci scalar

In this paper, we consider a specific model, implementing the existence of a fundamental limit distance [Formula: see text] between (space or time separated) points in spacetime, which in the recent past has exhibited the intriguing feature of having a minimum-length Ricci scalar [Formula: see text] that does not approach the ordinary Ricci scalar [Formula: see text] in the limit of vanishing [Formula: see text]. [Formula: see text] at a point has been found to depend on the direction along which the existence of minimum distance is implemented. Here, we point out that the convergence [Formula: see text] in the [Formula: see text] limit is anyway recovered in a relaxed or generalized sense, which is when we average over directions, this suggesting we might be taking the expectation value of [Formula: see text] promoted to be a quantum variable. It remains as intriguing as before the fact that we cannot identify (meaning this is much more than simply equating in the generalized sense above) [Formula: see text] with [Formula: see text] in the [Formula: see text] limit, namely, when we get ordinary spacetime. Thing is like if, even when [Formula: see text] (read here the Planck length) is far too small to have any direct detection of it feasible, the intrinsic quantum nature of spacetime might anyway be experimentally at reach, witnessed by the mentioned special feature of Ricci, not fading away with [Formula: see text] (i.e. persisting when taking the [Formula: see text] limit).

Image processing via simulated quantum dynamics

We apply simulated quantum evolution to image processing, and examine its practicality in the context of image denoising. More specifically, in our approach image processing consists of three stages: First, a digitized gray-scale image is represented as a quantum variable – typically, a density matrix. Second, the quantum variable is evolved via the Markovian master equation in Lindblad form. Third, the quantum variable is back-converted into an image. Numerical experiments indicate remarkable denoising results are obtained in this way for a suitable choice of flow parameters. To our knowledge the proposed image processing technique is conceptually new.

Evidence for the s count macrogene

Background: A macrogene is defined here as a gene on which successive mutations incrementing a repeat count produces successive punctuated evolutionary events in species that are homogeneous for it. The set of repeat count on the asp (abnormal spindle) family of gene is thought to affect brain size in mammals. Corticogenesis requires two integer valued (quantum) variables, the f and s counts, to determine the number of division cycles during the first and second phases, respectively, of neuron production in the cerebral cortex. Quantum ‘extra’ neuron theory hypothesizes that increments in a quantum variable, the n count, cause punctuated encephalization events in species that are homogenous for it. There is evidence in six pairs of inbred mice strains for one or more major genes affecting brain size. Results: The s count is probably equal to the n count plus a positive integer. The calculated n counts are different in three of the four pairs of strains studied where encephalization data has been previously published. Five different n counts have been found in eleven mouse strains. The difference between the n counts of humans and mice is about 25. Conclusions: Encephalization in mammals may be caused by a macrogene that determines the s count. This theory can be tested by determining the s counts of the various mice strains. However, the asp family of gene is probably not the s count macrogene because the difference in the asp counts of humans and mice of 13 (= 74 – 61) is much smaller than the difference in their s counts of around 25.

Evidence for the s count macrogene

Background: A macrogene is defined here as a gene on which successive mutations incrementing a repeat count produces successive punctuated evolutionary events in species that are homogeneous for it. The set of repeat count on the asp (abnormal spindle) family of gene is thought to affect brain size in mammals. Corticogenesis requires two integer valued (quantum) variables, the f and s counts, to determine the number of division cycles during the first and second phases, respectively, of neuron production in the cerebral cortex. Quantum ‘extra’ neuron theory hypothesizes that increments in a quantum variable, the n count, cause punctuated encephalization events in species that are homogenous for it. There is evidence in six pairs of inbred mice strains for one or more major genes affecting brain size. Results: The s count is probably equal to the n count plus a positive integer. The calculated n counts are different in three of the four pairs of strains studied where encephalization data has been previously published. Five different n counts have been found in eleven mouse strains. The difference between the n counts of humans and mice is about 25. Conclusions: Encephalization in mammals may be caused by a macrogene that determines the s count. This theory can be tested by determining the s counts of the various mice strains. However, the asp family of gene is probably not the s count macrogene because the difference in the asp counts of humans and mice of 13 (= 74 – 61) is much smaller than the difference in their s counts of around 25.

A SCHRÖDINGER BLACK HOLE AND ITS ENTROPY

We discuss the conditions under which one can expect to have the usual identification of black hole entropy with the area of the horizon. We then construct an example in which the actual presence of the event horizon on a given hypersurface depends on a quantum event in which a certain quantum variable acquires a value and which occurs in the future of the given hypersurface. This situation indicates that there is something fundamental that is missing in the most popular of the current approaches towards the construction of a theory of quantum gravity, or, alternatively, that there is something fundamental that we do not understand about entropy in general, or at least in its association with black holes.

GENERAL RELATIVITY AND QUANTUM MECHANICS

Usual quantum mechanics requires a fixed background spacetime geometry and its associated causal structure. A generalization of the usual theory may therefore be needed at the Planck scale for quantum theories of gravity in which spacetime geometry is a quantum variable. The elements of generalized quantum theory are briefly reviewed and illustrated by generalizations of usual quantum theory that incorporate spacetime alternatives, gauge degrees of freedom, and histories that move forward and backward in time. A generalized quantum framework for cosmological spacetime geometry is sketched. This theory is in fully four-dimensional form and free from the need for a fixed causal structure. Usual quantum mechanics is recovered as an approximation to this more general framework that is appropriate in those situations where spacetime geometry behaves classically.

The elimination of the nodes in quantum mechanics

The laws of classical mechanics must be generalised when applied to atomic systems, the generalisation being that the commutative law of multiplication, as applied to dynamical variables, is to be replaced by certain quantum con­ditions, which are just sufficient to enable one to evaluate xy - yx when x and y are given. It follows that the dynamical variables cannot be ordinary numbers expressible in the decimal notation (which numbers will be called c-numbers), but may be considered to be numbers of a special kind (which will be called q-numbers), whose nature cannot be exactly specified, but which can be used in the algebraic solution of a dynamical problem in a manner closely analogous to the way the corresponding classical variables are used. The only justification for the names given to dynamical variables lies in the analogy to the classical theory, e. g. , if one says that x, y, z are the Car­tesian co-ordinates of an electron, one means only that x, y, z are q-numbers which appear in the quantum solution of the problem in an analogous way to the Cartesian co-ordinates of the electron in the classical solution. It may happen that two or more q-numbers are analogous to the same classical quantity (the analogy being, of course, imperfect and in different respects for the different q-numbers), and thus have claims to the same name. This occurs, for instance, when one considers what q-numbers shall be called the frequencies of a multiply periodic system, there being orbital frequencies and transition frequencies, either of which correspond in certain respects to the classical frequencies. In such a case one must decide which of the properties of the classical variable are dynamically the most important, and must choose the q-number which has these properties to be the corresponding quantum variable.