User blogs

Surprising Properties of Non-Archimedean Field Extensions of the Real Numbers (by Elemer E. Rosinger): Abstract: This, under the present form, is a replacement that is a two part paper in which the new second part was brought together with my recently posted arxiv paper, upon the suggestion of the arxiv moderators. http://prespacetime.com/index.php/pst/article/view/160

Quantum Foundations: Is Probability Ontological? (by Elemer E. Rosinger): Abstract: It is argued that the Copenhagen Interpretation of Quantum Mechanics, founded ontologically on the concept of probability, may be questionable in view of the fact that within Probability Theory itself the ontological status of the concept of probability has always been, and is still under discussion. http://prespacetime.com/index.php/pst/article/view/161

Leaving the Aristotelean Realm: Some Comments Inspired by the Articles of Elemer E. Rosinger (by Matti Pitkänen): Abstract: In the following I represent some comments on the articles of Elemer Rosinger as a physicist from the point of view of Topological Geometrodynamics. The construction of ultrapower fields (loosely surreals) is associated with physics using the close analogies with gauge theories, gauge invariance, and with the singularities of classical fields. Non-standard numbers are compared with the numbers generated by infinite primes and it is found that the construction of infinite primes, integers, and rationals has a close similarity with construction of the generalized scalars. The construction replaces at the lowest level the index set Ʌ= N of natural numbers with algebraic numbers A, Frechet filter of N with that of A, and R with unit circle S1 represented as complex numbers of unit magnitude. At higher levels of the hierarchy generalized -possibly infinite and infinitesimal- algebraic numbers emerge. This correspondence maps a given set in the dual of Frechet filter of A to a phase factor characterizing infinite rational algebraically so that correspondence is like representation of algebra. The basic difference between two approaches to infinite numbers is that the counterpart of infinitesimals is infinitude of real units with complex number theoretic anatomy: one might loosely say that these real units are exponentials of infinitesimals. http://prespacetime.com/index.php/pst/article/view/162

Space-Time Foam Differential Algebras of Generalized Functions and a Global Cauchy-Kovalevskaia Theorem (by Elemer E. Rosinger): Abstract: The new global version of the Cauchy-Kovalevskaia theorem presented here is a strengthening and extension of the regularity of similar global solutions obtained earlier by the author. Recently the space-time foam differential algebras of generalized functions with dense singularities were introduced. A main motivation for these algebras comes from the so called space-time foam structures in General Relativity, where the set of singularities can be dense. A variety of applications of these algebras have been presented elsewhere, including in de Rham cohomology, Abstract Differential Geometry, Quantum Gravity, etc. Here a global Cauchy-Kovalevskaia theorem is presented for arbitrary analytic nonlinear systems of PDEs. The respective global generalized solutions are analytic on the whole of the domain of the equations considered, except for singularity sets which are closed and nowhere dense, and which upon convenience can be chosen to have zero Lebesgue measure. In view of the severe limitations due to the polynomial type growth conditions in the definition of Colombeau algebras, the class of singularities such algebras can deal with is considerably limited. Consequently, in such algebras one cannot even formulate, let alone obtain, the global version of the Cauchy-Kovalevskaia theorem presented in this paper. http://prespacetime.com/index.php/pst/article/view/157

Brief Lecture Notes on Self-Referential Mathematics and Beyond (by Elemer E. Rosinger): Abstract: Recently delivered lectures on Self-Referential Mathematics, [2], at the Department of Mathematics and Applied Mathematics, University of Pretoria, are briefly presented. Comments follow on the subject, as well as on Inconsistent Mathematics. http://prespacetime.com/index.php/pst/article/view/158

String Theory: A Mere Prelude to Non-Archimedean Space-Time (by Elemer E. Rosinger): Abstract: It took two millennia after Euclid and until in the early 1880s, when we went beyond the ancient axiom of parallels, and inaugurated geometries of curved spaces. In less than one more century, General Relativity followed. At present, physical thinking is still beheld by the yet deeper and equally ancient Archimedean assumption which entraps us into the limited view of "only one walkable world". In view of that, it is argued with some rather easily accessible mathematical support that Theoretical Physics may at last venture into the Non-Archimedean realms. http://prespacetime.com/index.php/pst/article/view/153

Cosmic Contact to Be, or Not to Be Archimedean (by Elemer E. Rosinger): Abstract: This is a two part paper which discusses various issues of cosmic contact related to what so far appears to be a self-imposed censorship implied by the customary acceptance of the Archimedean assumption on space-time. http://prespacetime.com/index.php/pst/article/view/154

From Reference Frame Relativity to Relativity of MathematicalModels: Relativity Formulas in a Variety of Non-Archimedean Setups (by Elemer E. Rosinger): Abstract: Galilean Relativity and Einstein’s Special and General Relativity showed that the Laws of Physics go deeper than their representations in any given reference frame. Thus covariance or independence of Laws of Physics with respect to changes of reference frames became a fundamental principle. So far, all of that has only been expressed within one single mathematical model, namely, the traditional one built upon the usual continuum of the field R of real numbers, since complex numbers, finite dimensional Euclidean spaces, or infinite dimensional Hilbert spaces, etc., are built upon the real numbers. Here, following [55], we give one example of how one can go beyond that situation and study what stays the same and what changes in the Laws of Physics, when one models them within an infinitely large variety of algebras of scalars constructed rather naturally. Specifically, it is shown that the Special Relativistic addition of velocities can naturally be considered in any of infinitely many reduced power algebras, each of them containing the usual field of real numbers and which, unlike the latter, are non-Archimedean. The nonstandard reals are but one case of such reduced power algebras, and are as well non-Archimedean. Two surprising and strange effects of such a study of the Special Relativistic addition of velocities are that one can easily go beyond the velocity of light, and rather dually, one can as easily end up frozen in immobility, with zero velocity. Both of these situations, together with many other ones, are as naturally available, as the usual one within real numbers. http://prespacetime.com/index.php/pst/article/view/155

How Far Should the Principle of Relativity Go? (by Elemer E. Rosinger): Abstract: The Principle of Relativity has so far been understood as the covariance of laws of Physics with respect to a general class of reference frame transformations. That relativity, however, has only been expressed with the help of one single type of mathematical entities, namely, the scalars given by the usual continuum of the field R of real numbers, or by the usual mathematical structures built upon R, such as the scalars given by the complex numbers C, or the vectors in finite dimensional Euclidean spaces Rn, infinite dimensional Hilbert spaces, etc. This paper argues for progressing deeper and wider in furthering the Principle of Relativity, not by mere covariance with respect to reference frames, but by studying the possible covariance with respect to a large variety of algebras of scalars which extend significantly R or C, variety of scalars in terms of which various theories of Physics can equally well be formulated. First directions in this regard can be found naturally in the simple Mathematics of Special Relativity, the Bell Inequalities in Quantum Mechanics, or in the considerable amount of elementary Mathematics in finite dimensional vector spaces which occurs in Quantum Computation. The large classes of algebras of scalars suggested, which contain R and C as particular cases, have the important feature of typically no longer being Archimedean, see Appendix, a feature which can prove to be valuable when dealing with the so called "infinities" in Physics. The paper has a Comment on the so called "end of time". http://prespacetime.com/index.php/pst/article/view/148

George Boole and the Bell Inequalities (by Elemer E. Rosinger): Abstract: As shown by Pitowsky, the Bell inequalities are related to certain classes of probabilistic inequalities dealt with by George Boole, back in the 1850s. Here a short presentation of this relationship is given. Consequently, the Bell inequalities can be obtained without any assumptions of physical nature, and merely through mathematical argument. http://prespacetime.com/index.php/pst/article/view/149

Which Are the Maximal Ideals? (by Elemer E. Rosinger): Abstract: Ideals of continuous functions which satisfy an off diagonality condition proved to be important connected with the solution of large classes of nonlinear PDEs, and more recently, in General Relativity and Quantum Gravity. Maximal ideals within those which satisfy that off diagonality condition are important since they lead to differential algebras of generalized functions which can handle the largest classes of singularities. The problem of finding such maximal ideals satisfying the off diagonality condition is formulated within some background detail, and commented upon. http://prespacetime.com/index.php/pst/article/view/150

Heisenberg Uncertainty in Reduced Power Algebras (by Elemer E. Rosinger): Abstract: The Heisenberg uncertainty relation is known to be obtainable by a purely mathematical argument. Based on that fact, here it is shown that the Heisenberg uncertainty relation remains valid when Quantum Mechanics is re-formulated within far wider frameworks of scalars, namely,within one or the other of the infinitely many reduced power algebras which can replace the usual real numbers R, or complex numbers C. A major advantage of such a re-formulation is, among others, the disappearance of the well known and hard to deal with problem of the so called "infinities in Physics". The use of reduced power algebras also opens up a foundational question about the role, and in fact, about the very meaning and existence, of fundamental constants in Physics, such as Planck’s constant h. A role, meaning, and existence which may, or on the contrary, may not be so objective as to be independent of the scalars used, be they the usual real numbers R, complex numbers C, or scalars given by any of the infinitely many reduced power algebras, algebras which can so easily be constructed and used. http://prespacetime.com/index.php/pst/article/view/151