arXiv subjects
Publications and source records attributed to Elemer E Rosinger.
Quantum computation has suggested, among others, the consideration of "non-quantum" systems which in certain respects may behave "quantum-like". Here, what algebraically appears to be the most general possible known setup, namely, of {\it magmas} is used in order to construct "quantum-like" systems. The resulting magmatic composition of systems has as a well known particular case the tensor products.
Entanglement is a well known fundamental resource in quantum information. Here the following question is addressed : which are the deeper roots of entanglement that may help in its better understanding and use ? The answer is that one can reproduce the phenomenon of entanglement in a far more general and simple way, a way that goes much beyond the usual one which is limited to the framework of tensor products of vector spaces. In this general approach to entanglement presented here - and much unlike in the particular setup of tensor products of vector spaces - the spaces involved can be rather arbitrary sets, just as in the case of Cartesian products. In particular, they need not even have any algebraic structure. Thus they do not have to be vector spaces, groups or even semigroups.
Two successive generalizations of the usual tensor products are given. One can be constructed for arbitrary binary operations, and not only for semigroups, groups or vector spaces. The second one, still more general, is constructed for arbitrary generators on sets.
A characterization of congruences in free semigroups is presented.
Part 1 : For more than two millennia, ever since Euclid's geometry, the so called Archimedean Axiom has been accepted without sufficiently explicit awareness of that fact. The effect has been a severe restriction of our views of space-time, a restriction which above all affects Physics. Here it is argued that, ever since the invention of Calculus by Newton, we may actually have {\it empirical evidence} that time, and thus space as well, are not Archimedean. Part 2 : There is an insufficient awareness about the {\it rich and complex} structure of various totally ordered scalar fields obtained through the {\it ultrapower} construction. This rich and complex structure comes from the presence of {\it infinitesimals} in such fields, presence which leads to the fact that such fields are {\it non-Archimedean}. Here, with the concept of {\it walkable world}, which has highly intuitive and pragmatic geometric meaning, the mentioned rich and complex structure is illustrated. The issues presented have relevance for what are usually called the "infinities in physics".
The group invariance of entanglement is obtained within a very general and simple setup of the latter, given by a recently introduced considerably extended concept of tensor products. This general approach to entanglement - unlike the usual one given in the particular setup of tensor products of vector spaces - turns out not to need any specific algebraic structure. The resulting advantage is that, entanglement being in fact defined by a negation, its presence in a general setup increases the chances of its manifestations, thus also its availability as a resource.
A recent general model of entanglement, [5], that goes much beyond the usual one based on tensor products of vector spaces is further developed here. It is shown that the usual Cartesian product can be seen as two extreme particular instances of non-entanglement. Also the recent approach to entanglement in [8] is incorporated in the general model in [5]. The idea pursued is that entanglement is by now far too important a phenomenon in Quantum Mechanics, in order to be left confined to its present exclusive modelling by tensor products. Once this is realized, it turns out that one can quite easily de-entangle entanglement from tensor products, and in fact, one can do so in a large variety of ways. Within such general settings the issue of entanglement becomes connected with the issue of "System versus Subsystem" in General Systems Theory, where synthesizing subsystems into a system may often be less difficult than identifying subsystems in a system.
For a large class of nonlinear evolution PDEs, and more generally, of nonlinear semigroups, as well as their approximating numerical methods, two rather natural stability type convergence conditions are given, one being necessary, while the other is sufficient. The gap between these two stability conditions is analyzed, thus leading to a general nonlinear equivalence between stability and convergence.
Semigroups generated by topological operations such as closure, interior or boundary are considered. It is noted that some of these semigroups are in general finite and noncommutative. The problem is formulated whether they are always finite.
The Principle of Relativity has so far been understood as the {\it 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 {\it one single} type of mathematical entities, namely, the scalars given by the usual continuum of the field $\mathbb{R}$ of real numbers, or by the usual mathematical structures built upon $\mathbb{R}$, such as the scalars given by the complex numbers $\mathbb{C}$, or the vectors in finite dimensional Euclidean spaces $\mathbb{R}^n$, infinite dimensional Hilbert spaces, etc. This paper argues for {\it 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 {\it large variety} of algebras of scalars which extend significantly $\mathbb{R}$ or $\mathbb{C}$, 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 $\mathbb{R}$ and $\mathbb{C}$ as particular cases, have the important feature of typically {\it 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".
Inverse limits, unlike direct limits, can in general be void, [1]. The existence of fixed points for arbitrary mappings $T : X \longrightarrow X$ is conjectured to be equivalent with the fact that related direct limits of all finite partitions of X are not void.
A rather general ergodic type scheme is presented on arbitrary sets X, as they are generated by arbitrary mappings T : X \longrightarrow X. The structures considered on X are given by suitable subsets of the set of all of its finite partitions. Ergodicity is studied not with respect to subsets of X, but with the {\it inverse limits} of families of finite partitions.
It has for ages been a rather constant feature of thinking in science to take it for granted that the respective thinking happens in realms which are totally outside and independent of all the other phenomena that constitute the objects of such thinking. The imposition of this divide on two levels may conflict with basic assumptions of Newtonian and Einsteinian mechanics, as well as with those in Quantum Mechanics. It also raises the question whether the realms in which thinking happens have no any other connection with the realms science deals with, except to host and allow scientific thinking.
The parts contributed by the author in recent discussions with several physicists and mathematicians are reviewed, as they have been occasioned by the 2006 book "The Trouble with Physics", of Lee Smolin. Some of the issues addressed are the possible and not yet sufficiently explored relationship between modern Mathematics and theoretical Physics, as well as the way physicists may benefit from becoming more aware of what at present appear to be certain less than fortunate yet essential differences between modern Mathematics and theoretical Physics, as far as the significant freedom of introducing new fundamental concepts, structures and theories in the former is concerned. A number of modern mathematical concepts and structures are suggested for consideration by physicists, when dealing with foundational issues in present day theoretical Physics. Since here discussions with several persons are reviewed, certain issues may be brought up more than one time. For such repetitions the author ask for the kind understanding of the reader.
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.
The comments relate to the often overlooked, if not in fact intentionally disregarded depths of what the so called internal aspects of mathematical knowledge may involve, depths concerning among others issues such as its unreasonable effectiveness in natural sciences, to use the terms of Eugene Wigner, suggested more than four decades ago.
One of the two basic theorems in [5] on the existence of solutions of PDEs is improved with the use of a group analysis type argument.
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.