Hermann Minkowski Research Articles

Possible Minkowskian language in two-level systems

One hundred years ago, in 1908, Hermann Minkowski completed his proof that Maxwell’s equations are covariant under Lorentz transformations. During this process, he introduced a four-dimensional space called the Minkowskian space. In 1949, P.A.M. Dirac showed the Minkowskian space can be handled with the light-cone coordinate system with squeeze transformations. While the squeeze is one of the fundamental mathematical operations in optical sciences, it could serve useful purposes in two-level systems. Some possibilities are considered in this report. It is shown possible to cross the light-cone boundary in optical and two-level systems while it is not possible in Einstein’s theory of relativity.

Weimar Physics: Sommerfeld’s Seminar and the Causality Principle

The development of quantum mechanics in the period 1920–1927 posed challenges to the understanding of causality and its incorporation into the new dynamics. Though aware of these challenges to the classical concepts of causality and to the conservation laws of energy and momentum, Arnold Sommerfeld and the members of his seminar never wavered in their commitment to the conservation laws because of their belief in the “preestablished harmony” between mathematics and physics that Felix Klein, David Hilbert, and Hermann Minkowski had championed. I survey these developments and trace how the concept of causality was reformulated in scattering theory by Gregor Wentzel, Enrico Fermi, and Giulio Racah.

The Place of “Place” in Jewish Tradition

The Place of “Place” in Jewish Tradition Nina Beth Cardin Since the beginning of time, humans have had a problem with land. The first dilemma, of course, was the very mystery of land. That is, how did land itself, the ground we walk on, come into being? All relationship with and attitude toward the land would flow from this answer. The Torah addresses that problem with its opening line: God created the heavens and the earth. That being the case, we can then conclude that God owns the land. All human claims to the land, therefore, are grounded in this basic knowledge. God first owned the land. It did not originally belong to us. Therefore, we must justify somehow any subsequent claim to use or ownership we may profess. But, how do we do that? What are our rights to the land? How are they earned? How conferred? How secured? In addition we need to ask, even after justifying a claim to the land, what is the status of the land? Does this land/all land have a spiritual dimension? Or is it just a prop of life, a platform on which human history is played out? Can we do anything we want with it? Is it possible to lose our claim to land? If so, how? In which case what happens to the land? Does all land have the same meaning, or are some places different, holier, than others? These questions about the status of land underlie the world’s most pressing issues: political, economical and spiritual. They affect questions of property ownership and economic distribution of resources, environmental injustice, geopolitical warfare, resource and waste management, environmental ethics, and so on. And while Judaism is richly steeped in conversations on these subjects, with laws stipulating the proper use and distribution of land rights, the welfare of animals, appreciation of nature as witness to the majesty of God, etc., I am going to largely avoid these for the purposes of this paper and focus on the place of “place” in Jewish tradition. As noted above, humans cannot ignore the experience and discourse of land. After all, we cannot exist without a place to be. All our experience is bound up with place. One of the first memories we recall about a remarkable or tragic event is where we were when it happened. Einstein’s teacher, Hermann Minkowski, captured it succinctly when he said: “Nobody ever noticed a place except at a time, or a time except at a place.” Jewish tradition, especially in its biblical stories, intimately combines event and place. Indivisible from the telling of our sacred history is the story of land itself. For the Jewish people, land is not just a backdrop, not a prop, but a partner, a covenantal character in our long unfolding sacred saga. Cultures with autochthonous myths had it much easier. If the land spit you out, gave birth to you, you could claim that you belonged to that land, and that land belonged to you. (Remnants of this tradition remain in force even today. Citizenship is often determined not by parental status but by location at birth.) But a creation story that speaks of human beings created by God and then placed on the earth, or even created from the earth but not in a geographically identifiable or reclaimable spot, makes it problematic to claim possession and rights to a particular place. The Bible clearly wants to state that land is God’s and any use or claim we humans make on it must be somehow divinely justifiable. Even more, the story of the Jewish people begins with Abraham outside the land that his descendants would eventually call their own. Yet an enduring claim to a particular place is essential for a tradition like Judaism that is built on peoplehood. Even more telling, then, is the fact that the people were not born of, or in, the land. The Torah clearly wants to teach that no matter how central Israel:the land is to Israel:the people, occupancy in that land is a gift, not a given. The land belongs to God, and was gifted as part of the covenant, a dynamic...

Egg-Forms and Measure-Bodies: Different Mathematical Practices in the Early History of the Modern Theory of Convexity

ArgumentTwo simultaneous episodes in late nineteenth-century mathematical research, one by Karl Hermann Brunn (1862–1939) and another by Hermann Minkowski (1864–1909), have been described as the origin of the theory of convex bodies. This article aims to understand and explain (1) how and why the concept of such bodies emerged in these two trajectories of mathematical research; and (2) why Minkowski's – and not Brunn's – strand of thought led to the development of a theory of convexity. Concrete pieces of Brunn's and Minkowski's mathematical work in the two episodes will, from the perspective of the above questions, be presented and analyzed with the use of the methodological framework of epistemic objects, techniques, and configurations as adapted from Hans-Jörg Rheinberger's work on empirical sciences to the historiography of mathematics by Moritz Epple. Based on detailed descriptions and a comparison of the objects and techniques that Brunn and Minkowski studied and used in these pieces it will be concluded that Brunn and Minkowski worked in different epistemic configurations, and it will be argued that this had a significant influence on the mathematics they developed for those bodies, which can provide answers to the two research questions listed above.

It's likely that times are changing: A century ago, mathematician Hermann Minkowski famously merged space with time, establishing a new foundation for physics; today physicists are rethinking how the two should fit to together

It's likely that times are changing: A century ago, mathematician Hermann Minkowski famously merged space with time, establishing a new foundation for physics; today physicists are rethinking how the two should fit to together

Hermann Minkowski’s approach to physics

Hermann Minkowski’s contributions to relativity theory are extraordinary from a disciplinary standpoint, in that so few mathematicians in Wilhelmine Germany took an interest in theoretical physics. Based on correspondence, lectures, research notes and the accounts of his students and colleagues, Minkowski’s interest in mechanics and physics is retraced, from his early career to his discovery of spacetime.

Ann. Phys. 9‐10/2008: Special Issue: The Minkowski spacetime of special relativity theory ‐ 100 years after its discovery

Annalen der PhysikVolume 520, Issue 8 p. 523-523 Recent and forthcoming publication in Annalen der Physik Ann. Phys. 9-10/2008: Special Issue: The Minkowski spacetime of special relativity theory - 100 years after its discovery First published: 24 October 2008 https://doi.org/10.1002/andp.20085200802AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Graphical Abstract On 21 September 2008 is the centennial for Hermann Minkowski's famous popular lecture in Cologne on “Space and Time”. Therein he demonstrated that the spacetime of Einstein's special relativity theory can be described by means of a four-dimensional (pseudo-)Euclidean geometry. The September/October 2008 issue of the Annalen is dedicated to the memory of Minkowski. Historical papers (Damour, Kastrup), papers extending special relativity (Mashhoon, Cacciatori-Gorini-Kamenshchik), papers on electrodynamics in Minkowski space (Itin-Friedman, Obukhov, Sihvola-Lindell), and an essay on the Weinberg-Witten theorem (Loebbert) convey ideas of the modern developments initiated by Minkowski's speech. Volume520, Issue8August 2008Pages 523-523 RelatedInformation

From measuring tool to geometrical object: Minkowski’s development of the concept of convex bodies

The present paper is concerned with the emergence of the modern theory of convex sets. Whereas the special instances of what we today understand by convex sets, such as the circle or regular polygons, have been studied throughout the history of mathematics, the modern theory understood as the systematic study of sets characterised exclusively by the property of convexity began only towards the end of the nineteenth century from where it developed into one of the many new disciplines of twentieth century mathematics. Today the theory of convexity is considered a central theory not least due to its expansion into almost all important areas of mathematics such as geometry, analysis, and applied mathematics.1 According to the history2 presented in textbooks on the theory of convexity the German mathematician Karl Hermann Brunn (1862–1939) was the first to engage in such systematic studies. His studies were then followed by the work of another German mathematician, Hermann Minkowski (1864–1909), who developed the theory further and explored some of its many applications. Examining this history one realises that Minkowski did not know about Brunn’s work until after he himself had begun his own investigations of what led to the theory of convexity. And even though Bonnesen

Planting in his Neighbor's Garden: David Hilbert and Early G�ttingen Quantum Physics

David Hilbert (1862-1943) played an important role in establishing quantum physics in Gottingen. I analyze the ways in which his influence was decisive by comparison with Woldemar Voigt (1850-1919). Voigt was the leading Gottingen theoretical physicist before the arrival of Peter Debye (1884-1966), who was appointed to a new professorship in 1914 at Hilbert's instigation. I portray the Gottingen mathematicians, above all Hermann Minkowski (1864-1909) and David Hilbert, as planting the seeds for the blossoming of quantum physics under their student Max Born (1882-1970) in the 1920s.

Piercing d -Intervals

A (homogeneous) d-interval is a union of d closed intervals in the line. Using topological methods, Tardos and Kaiser proved that for any finite collection of d-intervals that contains no k + 1 pairwise disjoint members, there is a set of O(dk) points that intersects each member of the collection. Here we give a short, elementary proof of this result. A (homogeneous) d-interval is a union of d closed intervals in the line. Let H be a finite collection of d-intervals. The transversal number τ(H) of H is the minimum number of points that intersect every member of H. The matching number ν(H) of H is the maximum number of pairwise disjoint members of H. Gyarfas and Lehel [3] proved that τ ≤ O(νd!) and Kaiser [4] proved that τ ≤ O(d2ν). His proof is topological, applies the Borsuk-Ulam theorem and extends and simplifies a result of Tardos [5]. Here we give a very short, elementary proof of a similar estimate, using the method of [2]. Theorem 1 Let H be a finite family of d-intervals containing no k + 1 pairwise disjoint members. Then τ(H) ≤ 2d2k. Proof. Let H′ be any family of d-intervals obtained from H by possibly duplicating some of its members, and let n denote the cardinality of H′. Note that H′ contains no k + 1 pairwise disjoint members. Therefore, by Turan’s Theorem, there are at least n(n − k)/(2k) unordered intersecting pairs of members of H′. Each such intersecting pair supplies at least 2 ordered pairs (p, I), where p is an end point of one of the intervals in a member of H′, I is a different member of H′, and p lies in I. Since there are altogether at most 2dn possible choices for p, there is such a point that lies in at least n(n−k) k2dn members of H ′ besides the one in which it is an endpoint of an interval, showing that there is a point that lies in at least n 2dk of the members of H ′. This implies that for any rational weights on the members of H there is a point that lies in at least a fraction 1 2dk of the total weight. By the min-max theorem it follows that there is a collection of m points so that each member of H contains at least m/(2dk) of them, and thus contains an interval that contains at least m/(2d2k) of the points. Order the points from left to right, and take the set of all points whose rank in this ordering is divisible by dm/(2d2k)e. This is a set of at most 2d2k points that intersects each member of H, completing the proof. 2 Remarks. It may be possible to improve the constant factor in the above proof. Kaiser’s estimate is indeed better by roughly a factor of 2; τ(H) ≤ (d2 − d+ 1)ν(H). It will be interesting to decide if the quadratic dependence on d is indeed best possible. Higher dimensional extensions are possible, using the techniques in [2], [1]. ∗Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel. Research supported in part by a USA-Israeli BSF grant and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University. Email: noga@math.tau.ac.il.

The influence of David Hilbert and Hermann Minkowski on Einstein's views over the interrelation between physics and mathematics

In the early years of his scientific career, Albert Einstein considered mathematics to be a mere tool in the service of physical intuition. In later years, he came to consider mathematics as the very source of scientific creativity. A main motive behind this change was the influence of two prominent German mathematicians: David Hilbert and Hermann Minkowski.

Hermann Minkowski and the postulate of relativity

Le mathematicien Minkowski (H.M.) a tenu un role important dans l'histoire des theories de la relativite d'Einstein en adaptant le langage et les equations mathematiques a la theorie de la pesanteur defendue par Einstein

My Encounters—as a Physicist—with Mathematics

Mathematical physics is not a discipline with its own identity. Rather there is mathematics and there is physics, and their cyclical relationship enjoys periods of cooperation interspersed with periods of mutual indifference. New initiatives leading to rapid development in physics frequently are accompanied by mathematical innovation. Examples from the past are the construction of particle mechanics and differential calculus by Isaac Newton and Gottfried Leibniz, developments in general relativity and differential geometry at the time of Albert Einstein and Hermann Minkowski, and advances in group theory and analysis following the invention of quantum mechanics and quantum field theory.

Artin’s conjecture for primitive roots

In his preface to Diophantische Approximationen, Hermann Minkowski expressed the conviction that the “deepest interrelationships in analysis are of an arithmetical nature.” Gauss described one such remarkable interrelationship in articles 315–317 of his Disquisitiones Arithmeticae. There, he asked why the decimal fraction of 1/7 has period length 6: $$ x^2 + y^2 = 5 $$ (1) whereas 1/11 has period length of only 2: $$ x - y = 1 $$ (2) Why does 1/99007599, when written as a binary fraction (that is, expanded in base 2), have a period of nearly 50 million 0s and 1s? To answer these questions, Gauss introduced the concept of a primitive root.

Number theory and the real world

Number theory has been considered since time immemorial to be the very paradigm of pure (some would say useless) mathematics. According to Carl Friedrich Gauss, the "Princeps Mathemat icorum", "mathematics is the queen of sc iences--and number theory is the queen of mathemat ics" . In fact, in Chinese the very name of mathematics is Number Science. What could be more beautiful than a deep, satisfying relation between whole numbers? (One is almost tempted to call them wholesome numbers.) Indeed, it is hard to come up with a more appropriate designation than their learned name: the integers, meaning the "untouched ones". How high they rank, in the realms of pure thought and aesthetics, above their lesser brethren: the real and complex numbers, whose first names virtually exude unsavory involvement with the complex realities of everyday life! Yet the theory of integers can provide totally unexpected answers to real-world problems. In fact, discrete mathematics is taking on an ever more important role. If nothing else, the advent of the digital computer and digital communication has seen to that. But even earlier, in physics, the emergence of quantum mechanics and discrete elementary particles put a premium on the methods and, indeed, the spirit of discrete mathematics. In mathematics proper, Hermann Minkowski, in the preface to his introductory book on number theory, Diophantische Approximationen, published in 1907 (the year he gave special relativity its proper four-dimensional clothing in preparation for its journey into general covariance and cosmology) expressed his conviction that the "deepest interrelationships in analysis are of an arithmetical nature". Or, on another occasion: "The primary source (Urquell) of all of mathematics are the integers". Yet much of our schooling concentrates on analysis and other branches of continuum mathematics to the virtual exclusion of number theory, group theory, combinatorics and graph theory. As an illustration, at a recent symposium on information theory, the author met several young women, formally trained in mathematics and working in the field of primality testing, who, in all their studies up to the Ph.D., had not heard a single lecture on number theory! Or, to give an earlier example, when Werner Heisenberg discovered "matrix" mechanics in 1925, he didn't know what a matrix was (Max Born had to tell him), and neither Heisenberg nor Born knew what to make of the appearance of matrices in the context of the atom. (David Hilbert is reported to have told them to go look for a differential equation with the same eigenvalues, if that would make them happier. They did not follow Hilbert's well-meant advice and thereby may have missed discovering the Schr6dinger wave equation.) Integers have repeatedly played a crucial role in the

Aus den wissenschaftlichen anfängen hermann minkowskis

The inspiration and core of this paper is a previously unknown letter from H. Weber to R. Dedekind, which sheds information on Hermann Minkowski's mathematical studies and investigations while he was still a student. The article begins with descriptions of the Gymnasium that Minkowski attended at Königsberg, and his teachers there. Minkowski's first introduction to higher mathematics is studied, along with his arithmetical investigations before he composed the memoir submitted for the Prize of the Academy of Paris in 1883. After observations on Minkowski's studies at the University of Königsberg, the polemics surrounding his prize from the Paris Academy are discussed, taking into consideration new documents which are quoted in the text.

Minkowski reduction of integral matrices

In 1905 Hermann Minkowski introduced his theory of reduction of positive definite quadratic forms. Recently, Hans J. Zassenhaus has suggested that this theory can be applied to the problem of row reduction of matrices of integers. Computational investigations have shown that for matrices with more columns than rows, the number of steps required for reduction decreases drastically. In this paper it is proved that as the number of columns increases, the probability that a matrix is Minkowski reduced approaches one. This fact is the motivation behind the introduction of a modified version of Minkowski reduction, resulting in a reduction procedure more suitable for computation.

Hermann Minkowski and Einstein's special theory of relativity

With his work on the general theory of relativity in the period 1912-1916 Einstein thought that he was introduced for the first time to the heuristic power of mathematics for formulating new physical theories. Before 1912 he avoided complicated or innovative mathematics.1 In particular, unlike many other mathematicians and physicists Einstein remained unimpressed with the physical ideas in Hermann Minkowski's theory of matter and four-dimensional spacetime, developed during the period 1907-08. Einstein believed instead that Minkowski had introduced a useful mathematical formalism that was not of central

Perfect Codes in the Lee Metric and the Packing of Polyominoes

Perfect Codes in the Lee Metric and the Packing of Polyominoes

On differential geometry in the large. I. Minkowski’s problem

Introduction. Hermann Minkowski,t in a fundamental paper on convex bodies, proposed the following Problem (M): to determine a convex, threedimensional body B whose surface admits of a given Gaussian curvature K(n) >0, assigned as a continuous function of the direction of the interior normal n to the surface. After having stated three obviously necessary conditions for the function K(n), Minkowski proceeds to solve an analogous Problem (M') for polyhedra. He then considers a passage to the limit among the solutions of problems (M') approximating (M), and establishes their convergence to a convex body Bo. This construction leaves open the question as to whether Bo is a solution of Minkowski remarks that B, if it exists, is, to within a translation, the uniquely determined solution of a certain third Problem of the calculus of variations and that Bo, too, is a solution of (M). Now if we assume (H): the surface of Bo is differentiable to a sufficiently high order, then Bo solves However, Minkowski does not discuss (H), but proves instead that the mixed volume V(Bo, Bo, C) of Bo with an arbitrary convex body C may be computed as though Bo were a solution of (M) and that, furthermore, a convex body is, except for a translation, uniquely determined by its mixed volume with the totality of convex bodies. While later authors have modified Minkowski's methods, there has been no improvement of his results as far as the hypothesis (H) is concerned. Thus Minkowski's results are open to the same criticism that could be raised against the early solutions of Plateau's problem: namely, that instead of solving the proposed problem, a more general problem is treated whose solution coincides with that of the former only if the latter solution satisfies certain highly restrictive conditions, and no indication is presented that these conditions are actually satisfied. The present paper contains a solution of (M) for the case of analytic K(n). It does not involve the Brunn-Minkowski inequalities, nor, indeed, the idea of mixed volume. It uses instead the author's results on elliptic and