We present the Dirac and Laplacian operators on Clifford bundles over space–time, associated to metric compatible linear connections of Cartan–Weyl, with trace-torsion, Q. In the case of nondegenerate metrics, we obtain a theory of generalized Brownian motions whose drift is the metric conjugate of Q. We give the constitutive equations for Q. We find that it contains Maxwell’s equations, characterized by two potentials, an harmonic one which has a zero field (Bohm-Aharonov potential) and a coexact term that generalizes the (...) Hertz potential of Maxwell’s equations in Minkowski space.We develop the theory of the Hertz potential for a general Riemannian manifold. We study the invariant state for the theory, and determine the decomposition of Q in this state which has an invariant Born measure. In addition to the logarithmic potential derivative term, we have the previous Maxwellian potentials normalized by the invariant density. We characterize the time-evolution irreversibility of the Brownian motions generated by the Cartan–Weyl laplacians, in terms of these normalized Maxwell’s potentials. We prove the equivalence of the sourceless Maxwell equation on Minkowski space, and the Dirac-Hestenes equation for a Dirac-Hestenes spinor field written on Minkowski space provided with a Cartan–Weyl connection. If Q is characterized by the invariant state of the diffusion process generated on Euclidean space, then the Maxwell’s potentials appearing in Q can be seen alternatively as derived from the internal rotational degrees of freedom of the Dirac-Hestenes spinor field, yet the equivalence between Maxwell’s equation and Dirac-Hestenes equations is valid if we have that these potentials have only two components corresponding to the spin-plane. We present Lorentz-invariant diffusion representations for the Cartan–Weyl connections that sustain the equivalence of these equations, and furthermore, the diffusion of differential forms along these Brownian motions. We prove that the construction of the relativistic Brownian motion theory for the flat Minkowski metric, follows from the choices of the degenerate Clifford structure and the Oron and Horwitz relativistic Gaussian, instead of the Euclidean structure and the orthogonal invariant Gaussian. We further indicate the random Poincaré–Cartan invariants of phase-space provided with the canonical symplectic structure. We introduce the energy-form of the exact terms of Q and derive the relativistic quantum potential from the groundstate representation. We derive the field equations corresponding to these exact terms from an average on the invariant state Cartan scalar curvature, and find that the quantum potential can be identified with 1 / 12R(g), where R(g) is the metric scalar curvature. We establish a link between an anisotropic noise tensor and the genesis of a gravitational field in terms of the generalized Brownian motions. Thus, when we have a nontrivial curvature, we can identify the quantum nonlocal correlations with the gravitational field. We discuss the relations of this work with the heat kernel approach in quantum gravity. We finally present for the case of Q restricted to this exact term a supersymmetric system, in the classical sense due to E.Witten, and discuss the possible extensions to include the electromagnetic potential terms of Q. (shrink)

We review the relation between spacetime geometries with trace-torsion fields, the so-called Riemann–Cartan–Weyl (RCW) geometries, and their associated Brownian motions. In this setting, the drift vector field is the metric conjugate of the trace-torsion one-form, and the laplacian defined by the RCW connection is the differential generator of the Brownian motions. We extend this to the state-space of non-relativistic quantum mechanics and discuss the relation between a non-canonical quantum RCW geometry in state-space associated with the gradient of the quantum-mechanical (...) expectation value of a self-adjoint operator given by the generalized laplacian operator defined by a RCW geometry. We discuss the reduction of the wave function in terms of a RCW quantum geometry in state-space. We characterize the Schroedinger equation in terms of the RCW geometries and Brownian motions. Thus, in this work, the Schroedinger field is a torsion generating field, both for the linear and non-linear cases. We discuss the problem of the many times variables and the relation with dissipative processes, and the role of time as an active field, following Kozyrev and a recent experiment in non-relativistic quantum systems. We associate the Hodge dual of the drift vector field with a possible angular-momentum source for the phenomenae observed by Kozyrev. (shrink)

We present the unification of Riemann–Cartan–Weyl (RCW) space-time geometries and random generalized Brownian motions. These are metric compatible connections (albeit the metric can be trivially euclidean) which have a propagating trace-torsion 1-form, whose metric conjugate describes the average motion interaction term. Thus, the universality of torsion fields is proved through the universality of Brownian motions. We extend this approach to give a random symplectic theory on phase-space. We present as a case study of this approach, the invariant Navier–Stokes (...) equations for viscous fluids, and the kinematic dynamo equation of magnetohydrodynamics. We give analytical random representations for these equations. We discuss briefly the relation between them and the Reynolds approach to turbulence. We discuss the role of the Cartan classical development method and the random extension of it as the method to generate these generalized Brownian motions, as well as the key to construct finite-dimensional almost everywhere smooth approximations of the random representations of these equations, the random symplectic theory, and the random Poincaré–Cartan invariants associated to it. We discuss the role of autoparallels of the RCW connections as providing polygonal smooth almost everywhere realizations of the random representations. (shrink)

From Einstein's point of view, his General Relativity Theory had strengths as well as failings. For him, its shortcoming mainly was that it did not unify gravitation and electromagnetism and did not provide solutions to field equations which can be interpreted as particle models with discrete mass and charge spectra, As a consequence, General Relativity did not solve the quantum problem, either. Einstein tried to get rid of the shortcomings without losing the achievements of General Relativity Theory. Stimulated by papers (...) of Weyl 465) and Eddington 194), from 1923 onward, he believed that, to reach this goal, one has to transit to space–times which possess more comprehensive geometrical structures than the Riemann space–time. This was the beginning of a decade's lasting search for a unitary field theory. We describe this exciting part of the history of physics, discuss achievements and failures of this development, and ask how these early attempts of a unified theory strike us today. Taking into account the fact that the Equivalence Principle only speaks for a geometrization of gravitation, we consider an alternative way to give those non-Riemannian structures which were introduced by the unitary field approach a physical meaning, namely the meaning of a generalized gravitational field. This is interesting since there are arguments in favor of such a generalization of General Relativity Theory, e.g., the problems the latter theory meets with if one tries to quantize it and to unify gravitation with other interactions. (shrink)

According to Kant, the axioms of intuition, i.e. space and time, must provide an organization of the sensory experience. However, this first orderliness of empirical sensations seems to depend on a kind of faculty pertaining to subjectivity, rather than to the encounter of these same intuitions with the real properties of phenomena. Starting from an analysis of some very significant developments in mathematical and theoretical physics in the last decades, in which intuition played an important role, we argue that nevertheless (...) intuition comes into play in a fundamentally different way to that which Kant had foreseen: in the form of a formal or “categorical” yet not sensible intuition. We show further that the statement that our space is mathematically three-dimensional and locally Euclidean by no means follows from a supposed a priori nature of the sensible or subjective space as Kant claimed. In fact, the three-dimensional space can bear many different geometrical and topological structures, as particularly the mathematical results of Milnor, Smale, Thurston and Donaldson demonstrated. On the other hand, it has been stressed that even the phenomenological or perceptual space, and especially the visual system, carries a very rich geometrical organization whose structure is essentially non-Euclidean. Finally, we argue that in order to grasp the meaning of abstract geometric objects, as n-dimensional spaces, connections on a manifold, fiber spaces, module spaces, knotted spaces and so forth, where sensible intuition is essentially lacking and where therefore another type of mathematical idealization intervenes, we need to develop a new form of intuition. (shrink)

In the range of his intellectual interests and the profundity of his mathematical thought Hermann Weyl towered above his contemporaries, many of whom viewed him with awe. This volume, the most ambitious study to date of Weyl's singular contributions to mathematics, physics, and philosophy, looks at the man and his work from a variety of perspectives, though its gaze remains fairly steadily fixed on Weyl the geometer and space‐time theorist. Structurally, the book falls into two parts, described in the general (...) introduction by the editor: Part 1 contains four essays on particular aspects of Weyl's work, highlighting ideas he developed in various editions of his classic Raum‐Zeit‐Materie. Part 2 presents a lengthy study by Robert Coleman and Herbert Korté covering nearly the whole gamut of Weyl's mathematical research, an impressive feat. Both in the introduction as well as in footnotes to the articles Erhard Scholz's editorial voice chimes in discreetly, helping tie all five studies together.Coleman and Korté begin chronologically with Weyl's early work in analysis and the modern theory of Riemann surfaces before turning to differential geometry, unified field theory, and the space problem, a topic they use as a springboard for a discussion of their own recent work on the foundations of space‐time. They then take up Weyl's shift to group representation theory and its applications to quantum mechanics, ending with his much earlier research on the structure of the continuum. All of these topics are well handled, but the authors' own agendas coupled with their penchant for overlooking chronology in order to package Weyl's work into neat little bundles leave one feeling rather stranded and far removed from the sources of Weyl's inspiration. Moreover, the narrative style makes this part of the volume read like a technical appendix, albeit a most informative one. Readers who tackle Scholz's far more contextualized essay will be amply rewarded by comparing his views with the opinions set forth by Coleman and Korté in Part 2.Scholz gives a masterful account of Weyl's intellectual journeys from 1917 to 1925 in a study that serves as a fulcrum for the entire volume. Drawing on a number of recently published studies, including his own, on the interplay between mathematics and physics inspired by Einstein's theory of general relativity, Scholz describes how Weyl responded to this challenge by developing a truly infinitesimal space‐time geometry that generalized classical Riemannian geometry. Although unconvinced by Einstein's critique of his unified field theory, Weyl shifted his focus from this realm to the classical space problem, analyzed earlier with more primitive techniques by Hermann Helmholtz and Sophus Lie. In this connection, it should be mentioned that Thomas Hawkins has given a probing analysis of Weyl's related work on the representation of Lie groups in his tour‐de‐force work, Emergence of the Theory of Lie Groups . Scholz argues that Weyl's struggle to tame his modernized version of the space problem stemmed from a deep‐seated belief in his geometrical ideas, which in turn were nourished by philosophical musings. By demonstrating the closely related conceptual links that motivated Weyl's research in infinitesimal geometry, space‐time physics, and the foundations of mathematics, Scholz nicely illuminates the underlying fabric of epistemological concerns that occupied Weyl's attention during this fertile period.The three remaining essays in Part 1 focus on other aspects of Weyl's work in mathematical physics and cosmology. Skuli Sigurdsson's “Journeys in Spacetime” offers a broad interpretation of Weyl's career, one that emphasizes Weyl's sensitivity to cultural tensions as reflected in his philosophical roots, which combined phenomenology with facets of German idealism. Shaken by the annihilation of cultural values in Nazi Germany, Weyl became deeply aware of the gulf that separated his earlier life in Göttingen and Zurich from the one he took up at Princeton's Institute for Advanced Study in 1933. He tried to adapt, but felt out of place in an Anglo‐American scientific culture openly hostile toward metaphysics and speculative philosophy. Sigurdsson stresses these tensions, contrasting the introspective, creative individual against the backdrop of the collective in the age of the machine, but without spelling out which collective were most important for him. Wolfgang Pauli thought he knew and, like Einstein before him, he had no compunction about bluntly telling Weyl he was a mathematician, not a physicist.Pauli's opinions notwithstanding, Weyl did far more than just dabble around the mathematical edges of the new physics. If Coleman and Korté perhaps press their case for his visionary accomplishments too far, Norbert Straumann's essay “Ursprünge der Eichtheorien” suggests why Weyl's reputation among physicists has risen steadily ever since the advent of Yang‐Mills theory in the 1950s. In the course of describing Weyl's adaptation of his gauge transformation formalism to Dirac's electron theory, Straumann sheds considerable light on Pauli's role as self‐appointed watchman guarding the disciplinary boundary that separated theoretical physics from physical mathematics . He further suggests that disciplinary jealousy was a major reason why Pauli dismissed Weyl's two‐component formalism for spinors out of hand.In the realm of cosmology, on the other hand, Weyl's work has long since passed into the dustbins of history, as Hubert Goenner remarks in recounting a fascinating chapter in the infancy of space‐time physics. While doing so, Goenner shows how initially Weyl almost slavishly adopted what Einstein called Mach's principle, which asserts that the metric structure of space‐time is solely determined by the distribution of matter in the universe. This notion was quickly challenged by Willem De Sitter, who showed that Einstein's matter‐free field equations admitted a global solution with non‐zero constant curvature. Both Einstein and Weyl tried to argue that invisible masses must be present just over the “spatial horizon” of De Sitter's world in order to account for its curvature. Goenner meticulously analyzes the physical and mathematical issues at stake in this debate, stressing how Weyl gradually moved away from a strong physical interpretation to one in which mathematics models rather than physics models simply reveal natural phenomena. He argues further that Weyl's cosmological principle arose as the final expression of his search for a deeper physical meaning.Given the quality of these essays, it is regrettable that this book contains so little about Weyl's professional career, a weakness the editor could have redressed at least partially in his general introduction. This omission is all the more unfortunate given the dearth of readily accessible information about Weyl's life available elsewhere. For however mundane his outward existence may have been, the reader cannot be expected to appreciate the interplay between the world Weyl knew and his creative responses to it without fairly detailed knowledge of his biography. Shorn from these contexts, it becomes difficult to form a flesh‐and‐blood image of Weyl beyond the cliché‐ridden stereotype that sees him as a “heroic thinker in the grand German tradition.” While none of the authors falls into this trap, the collective impression they leave suggests a most enigmatic figure. Either Weyl the man tends to get lost in the shadows of his collected scientific output or he appears as a mystic loner, an outcast who abhorred the machine age in which he lived. Closer attention to the people in his life would no doubt produce a very different picture of the man and his interests. This major lacuna notwithstanding, the present volume will surely remain an indispensable resource for any future investigations of Weyl's staggering intellectual achievements. (shrink)

Evans developed a classical unified field theory of gravitation and electromagnetism on the background of a spacetime obeying a Riemann-Cartan geometry. This geometry can be characterized by an orthonormal coframe ϑ α and a (metric compatible) Lorentz connection Γ α β . These two potentials yield the field strengths torsion T α and curvature R α β . Evans tried to infuse electromagnetic properties into this geometrical framework by putting the coframe ϑ α to be proportional to four extended electromagnetic (...) potentials $\mathcal{A}^{\alpha }$ ; these are assumed to encompass the conventional Maxwellian potential A in a suitable limit. The viable Einstein-Cartan (-Sciama-Kibble) theory of gravity was adopted by Evans to describe the gravitational sector of his theory. Including also the results of an accompanying paper by Obukhov and the author, we show that Evans’ ansatz for electromagnetism is untenable beyond repair both from a geometrical as well as from a physical point of view. As a consequence, his unified theory is obsolete. (shrink)

Is Einstein's metric theory of gravitation to be quantized to yield a complete and logically consistent picture of the geometry of the real world in the presence of quantized material sources? To answer this question, we give arguments that there is a consistent way to extend general relativity to small distances by incorporating further geometric quantities at the level of the connection into the theory and introducing corresponding field equations for their determination, allowing thereby the metric and the Levi-Civita connection (...) to remain classical quantities. The dualism between matter and geometry is extended to quantized fields with the help of a Hibert bundle ℋ raised over a Riemann-Cartan spacetime. Quantized subnuclear matter fields (generalized quantum mechanical wave functions) are sections on ℋ which determine generalized bilinear currents acting as sourc currents for the bundle geometry at small distances. The established dualism between matter and the underlying bundle geometry contains general relativity as a classical part. (shrink)

The history and philosophy of science are destined to play a fundamental role in an epoch marked by a major scientific revolution. This ongoing revolution, principally affecting mathematics and physics, entails a profound upheaval of our conception of space, space–time, and, consequently, of natural laws themselves. Briefly, this revolution can be summarized by the following two trends: by the search for a unified theory of the four fundamental forces of nature, which are known, as of now, as gravity, electromagnetism, and (...) strong and weak nuclear forces; by the search for new mathematical concepts capable of elucidating and therefore explaining such a relationship. In fact, the first search is essentially dependent on the second; that is to say, that in order for a new theory of physics to come to light, the development of a deeper geometric theory capable of explaining the structure of space–time on a quantum scale appears to be necessary. On careful consideration, we notice that both of these developments converge in the direction of a unitary and fundamental tendency of modern science—which is the geometrization of theoretical physics and of natural sciences. This new emergent situation carries within it a profound conceptual change, affecting the way in which relations are conceived of, first and foremost, between mathematics and physics. This new paradigm can be summed up by the intimately interdependent points: the immense variety of physical phenomena and of natural forms follows from the equally infinite variety of geometric and topological objects that can be made out in space and from which space is made up; the second point, which ensues from the former one and which is of great historical and epistemological significance, is that mathematics is involved in rather than applied to phenomena. In other words, phenomena are effects that emerge from the geometrical structure of space–time. There is no doubt that this new conception of the relationship between the universe of mathematical ideas and objects and the world of natural phenomena is the true scientific revolution of our century, of great conceptual importance, and consequently, capable of changing our view of science and of nature at one and the same time. It is all at once of a scientific, philosophical and aesthetic order. (shrink)

In recent years, the ideas of the mathematician Bernhard Riemann have come to the fore as one of Deleuze's principal sources of inspiration in regard to his engagements with mathematics, and the history of mathematics. Nevertheless, some relevant aspects and implications of Deleuze's philosophical reception and appropriation of Riemann's thought remain unexplored. In the first part of the paper I will begin by reconsidering the first explicit mention of Riemann in Deleuze's work, namely, in the second chapter of Bergsonism. In (...) this context, as I intend to show first, Deleuze's synthesis of some key features of the Riemannian theory of multiplicities is entirely dependent, both textually and conceptually, on his reading of another prominent figure in the history of mathematics: Hermann Weyl. This aspect has been largely underestimated, if not entirely neglected. However, as I attempt to bring out in the second part of the paper, reframing the understanding of Deleuze's philosophical engagement with Riemann's mathematics through the Riemann–Weyl conjunction can allow us to disclose some unexplored aspects of Deleuze's further elaboration of his theory of multiplicities and profound confrontation with contemporary science. This finally permits delineation of a correlation between Deleuze's plane of immanence and the contemporary physico-mathematical space of fundamental interactions. (shrink)

The main features of how to build a Born’s Reciprocal Gravitational theory in curved phase-spaces are developed. By recurring to the nonlinear connection formalism of Finsler geometry a generalized gravitational action in the 8D cotangent space (curved phase space) can be constructed involving sums of 5 distinct types of torsion squared terms and 2 distinct curvature scalars ${\mathcal{R}}, {\mathcal{S}}$ which are associated with the curvature in the horizontal and vertical spaces, respectively. A Kaluza-Klein-like approach to the construction of the curvature (...) of the 8D cotangent space and based on the (torsionless) Levi-Civita connection is provided that yields the observed value of the cosmological constant and the Brans-Dicke-Jordan Gravity action in 4D as two special cases. It is found that the geometry of the momentum space can be linked to the observed value of the cosmological constant when the curvature in $\mathit{momentum}$ space is very large, namely the small size of P is of the order of $( 1/R_{\mathit{Hubble}})$ . Finally we develop a Born’s reciprocal complex gravitational theory as a local gauge theory in 8D of the $\mathit{deformed}$ Quaplectic group that is given by the semi-direct product of U(1,3) with the $\mathit{deformed}$ (noncommutative) Weyl-Heisenberg group involving four $\mathit{noncommutative}$ coordinates and momenta. The metric is complex with symmetric real components and antisymmetric imaginary ones. An action in 8D involving 2 curvature scalars and torsion squared terms is presented. (shrink)

Brouwer’s intuitionism was a far-reaching attempt to reform the foundations of mathematics. While the mathematical community was reluctant to accept Brouwer’s work, its response to later-developed brands of intuitionism, such as those presented by Hermann Weyl and Arend Heyting, was different. The paper accounts for this difference by analyzing the intuitionistic versions of Brouwer, Weyl, and Heyting in light of a two-tiered model of the body and image of mathematical knowledge. Such a perspective provides a richer account of each story (...) and points to a possible connection between the community’s reaction and the changes each mathematician had proposed. (shrink)

To represent extension of objects in particle physics, a modified Weyl theory is used by gauging the curvature radius of the local fibers in a soldered bundle over space-time possessing a homogeneous space G/H of the (4, 1)-de Sitter group G as fiber. Objects with extension determined by a fundamental length parameter R0 appear as islands D(i) in space-time characterized by a geometry of the Cartan-Weyl type (i.e., involving torsion and modified Weyl degrees of freedom). Farther away from the domains (...) D(i), space-time is identified with the pseudo-Riemannian space of general relativity. Extension and symmetry breaking are described by a set of additional fields ( $(\tilde \xi ^a (x),R(x)) \in {G \mathord{\left/ {\vphantom {G H}} \right. \kern-0em} H}x R^ +$ , given as a section on an associated bundle $\tilde E(B,{G \mathord{\left/ {\vphantom {G H}} \right. \kern-0em} H}x R^ + ,\tilde G)$ over space-time B with structural group $\tilde G$ = G ⊗ D(1), where D(1) is the dilation group. Field equations for the quantities defining the underlying bundle geometry and for the fields $\tilde \xi ^a (x)$ are established involving matter source currents derived from a generalized spinor wave function. Einstein's equations for the metric are regarded as the part of the $\tilde G$ -gauge theory related to the Lorentz subgroup H of G exhibiting thereby the broken nature of the $\tilde G$ -symmetry for regions outside the domains D(i). (shrink)

Hermann Weyl as a founding father of field theory in relativistic physics and quantum theory always stressed the internal logic of mathematical and physical theories. In line with his stance in the foundations of mathematics, Weyl advocated a constructivist approach in physics and geometry. An attempt is made here to present a unified picture of Weyl's conception of space-time theories from Riemann to Minkowski. The emphasis is on the mathematical foundations of physics and the foundational significance of a constructivist philosophical (...) point of view. I conclude with some remarks on Weyl's broader philosophical views. (shrink)

In 1922, Cartan introduced in differential geometry, besides the Riemannian curvature, the new concept of torsion. He visualized a homogeneous and isotropic distribution of torsion in three dimensions (3d) by the “helical staircase”, which he constructed by starting from a 3d Euclidean space and by defining a new connection via helical motions. We describe this geometric procedure in detail and define the corresponding connection and the torsion. The interdisciplinary nature of this subject is already evident from Cartan’s discussion, since he (...) argued—but never proved—that the helical staircase should correspond to a continuum with constant pressure and constant internal torque. We discuss where in physics the helical staircase is realized: (i) In the continuum mechanics of Cosserat media, (ii) in (fairly speculative) 3d theories of gravity, namely (a) in 3d Einstein-Cartan gravity—this is Cartan’s case of constant pressure and constant intrinsic torque—and (b) in 3d Poincaré gauge theory with the Mielke-Baekler Lagrangian, and, eventually, (iii) in the gauge field theory of dislocations of Lazar et al., as we prove for the first time by arranging a suitable distribution of screw dislocations. Our main emphasis is on the discussion of dislocation field theory. (shrink)

A book-length study of Riemann's multi-dimensional work (in Spanish), which considers his contributions to physics, philosophy and mathematics. Plus a bi-lingual edition (German-Spanish) of some of his landmark papers: the lecture on geometry, with Weyl's comments; the paper introducing the Riemann Conjecture, part of his 1857 paper on function theory; all of the philosophical fragments, etc. These different contributions, and their interconnections, are carefully studied in the introductory essay of 150 pages.

I address a question recently raised by Simon Saunders concerning the relationship between the space-time structure of Newton-Cartan theory and that of what I will call “Maxwell-Huygens space-time.” This discussion will also clarify a connection between Saunders’s work and a recent paper by Eleanor Knox.

We discuss the concepts of Weyl and Riemann frames in the context of metric theories of gravity and state the fact that they are completely equivalent as far as geodesic motion is concerned. We apply this result to conformally flat spacetimes and show that a new picture arises when a Riemannian spacetime is taken by means of geometrical gauge transformations into a Minkowskian flat spacetime. We find out that in the Weyl frame gravity is described by a scalar field. We (...) give some examples of how conformally flat spacetime configurations look when viewed from the standpoint of a Weyl frame. We show that in the non-relativistic and weak field regime the Weyl scalar field may be identified with the Newtonian gravitational potential. We suggest an equation for the scalar field by varying the Einstein-Hilbert action restricted to the class of conformally-flat spacetimes. We revisit Einstein and Fokker’s interpretation of Nordström scalar gravity theory and draw an analogy between this approach and the Weyl gauge formalism. We briefly take a look at two-dimensional gravity as viewed in the Weyl frame and address the question of quantizing a conformally flat spacetime by going to the Weyl frame. (shrink)

We show that in 1929 Cartan and Einstein almost produced a theory in which the electromagnetic (EM) field constitutes the time-like 2-form part of the torsion of Finslerian teleparallel connections on pseudo-Riemannian metrics. The primitive state of the theory of these connections would not, and did not, permit Cartan and Einstein to realize how their torsion field equations contained the Maxwell system and how the Finslerian torsion contains the EM field. Cartan and Einstein discussed curvature field equations, though (...) failing to focus on the fact that teleparallelism automatically implies gravitational field equations with torsion terms as source, both in first and second order. We further show that the first-order contribution of the EM field to the source of the gravitational field may play havoc with the remeasurement of Newton's gravitational constant, even if the experiment is electrically grounded. These results are also used as support for the thesis that there is an alternative to the present way of dealing with the great theoretical questions of physics. On the practical side, the inconveniences faced in measuring G may be greatly compensated by the possibility of manipulating spacetime with electric fields at the first-order level. (shrink)

We construct a notion of teleparallelization for Newton-Cartan theory, and show that the teleparallel equivalent of this theory is Newtonian gravity; furthermore, we show that this result is consistent with teleparallelization in general relativity, and can be obtained by null-reducing the teleparallel equivalent of a five-dimensional gravitational wave solution. This work thus strengthens substantially the connections between four theories: Newton-Cartan theory, Newtonian gravitation theory, general relativity, and teleparallel gravity.

It is now well-known that Newton–Cartan theory is the correct geometrical setting for modelling the quantum Hall effect. In addition, in recent years edge modes for the Newton–Cartan quantum Hall effect have been derived. However, the existence of these edge modes has, as of yet, been derived using only orthodox methodologies involving the breaking of gauge-invariance; it would be preferable to derive the existence of such edge modes in a gauge-invariant manner. In this article, we employ recent work by Donnelly (...) and Freidel in order to accomplish exactly this task. Our results agree with known physics, but afford greater conceptual insight into the existence of these edge modes: in particular, they connect them to subtle aspects of Newton–Cartan geometry and pave the way for further applications of Newton–Cartan theory in condensed matter physics. (shrink)

Les variations de Gilles Châtelet Sur une petite phrase de Riemann, montrent que loin de constituer une variante de la « philosophie baroque » telle que la définit Deleuze, son maniérisme épistémologique le rapproche étonnamment de l’attention aux « modes de données » et aux « modes de visée » caractéristique de la phénoménologie transcendantale. Comme Hermann Weyl, il voit à l’œuvre dans la phénoménologie husser-lienne un approfondissement de l’esthétique transcendantale kantienne, en la tenant pour une approche philosophique pertinente et (...) ajustée à la science de son temps.Châtelet’s variation on a note by Riemann, shows that, far from constituting a variant of the “Baroque philosophy” such as Deleuze defines it, his epistemological mannerism pays attention to “modes of givenness” and “modes of constitution”, which is characteristic of transcendental phe-nomenology. This explains why he considers, like Hermann Weyl, phe-nomenology as a deepening of Kant’s transcendental aesthetics and as a philosophical approach, relevant and adapted to the science of his time. It is in this perspective that we should hear their joint statements that transcendental phenomenology is epistemologically contemporary to special and general relativity. Châtelets Variation auf einer kleinen Phrase von Rie-mann zeigt, daß sein erkenntnistheoretischer Manierismus, eine Variante der nach Definition „barocken Philosophie“ von Deleuze weit entfernt zu sein, im Gegenteil in der Nähe von transzendentale Phänomenologie und seine Aufmerksamkeit auf die „Gegebenheitsmodi“ und „Konstitution-sweise“ steht. Überdies betrachtet er, wie Hermann Weyl, die Phänomenol-ogie und die Vertiefung der transzendentalen Ästhetik als philosophischen Ansatz für die Wissenschaft seiner Zeit. Es ist in dieser Perspektive, dass wir ihre gemeinsamen Aussagen hören, dass die Kantische Ästhetik in Husserls vertieft wird, und dass die transzendentale Phänomenologie epistemologisch zeitgenössisch ist für speziell und allgemeine Relativität.This article is in French. (shrink)

In 1918, Emmy Noether published a (now famous) theorem establishing a general connection between continuous 'global' symmetries and conserved quantities. In fact, Noether's paper contains two theorems, and the second of these deals with 'local' symmetries; prima facie, this second theorem has nothing to do with conserved quantities. In the same year, Hermann Weyl independently made the first attempt to derive conservation of electric charge from a postulated gauge symmetry. In the light of Noether's work, it is puzzling that Weyl's (...) argument uses local gauge symmetry. This paper explores the relationships between Weyl's work, Noether's two theorems, and the modern connection between gauge symmetry and conservation of electric charge. This includes showing that Weyl's connection is essentially an application of Noether's second theorem, with a novel twist. (shrink)

The Riemann–Roch theorem for algebraic curves is derived from a theorem for Girard quantales. Serre duality is shown to be a quantalic phenomenon. An example provides a Girard quantale satisfying the Riemann–Roch theorem, where the associated curve is non-connected and irreducible.

The analysis of the measurement of gravitational fields leads to the Rosenfeld inequalities. They say that, as an implication of the equivalence of the inertial and passive gravitational masses of the test body, the metric cannot be attributed to an operator that is defined in the frame of a local canonical quantum field theory. This is true for any theory containing a metric, independently of the geometric framework under consideration and the way one introduces the metric in it. Thus, to (...) establish a local quantum field theory of gravity one has to transit to non-Riemann geometry that contains (beside or instead of the metric) other geometric quantities. From this view, we discuss a Riemann–Cartan and an affine model of gravity and show them to be promising candidates of a theory of canonical quantum gravity. (shrink)

Recently discovered correspondence from Oskar Becker to Hermann Weyl sheds new light on Weyl's engagement with Husserlian transcendental phenomenology in 1918-1927. Here the last two of these letters, dated July and August, 1926, dealing with issues in the philosophy of mathematics are presented, together with background and a detailed commentary. The letters provide an instructive context for re-assessing the connection between intuitionism and phenomenology in Weyl's foundational thought, and for understanding Weyl's term ‘symbolic construction’ as marking his own considered position (...) in the foundational controversy of the 1920s. In addition, they reveal Weyl's hitherto unknown objections to Becker's detailed attempt (Mathematische Existenz, 1927) to ground the transfinite phenomenologically. (shrink)

We study a Weyl geometric scalar tensor theory of gravity with scalar field \ and scale invariant “aquadratic” kinematical Lagrange density. The Weylian scale connection in Einstein gauge induces an additional acceleration. In the weak field, static, low velocity limit it acquires the deep MOND form of Milgrom/Bekenstein’s gravity. The energy momentum of \ leads to another add on to Newton acceleration. Both additional accelerations together imply a MOND-ian phenomenology of the model. It has unusual transition functions \, \nu _w\). (...) They imply higher phantom energy density than in the case of the more common MOND models with transition functions \, \, \mu _2\). A considerable part of it is due to the scalar field’s energy density which, in our model, gives a scale and generally covariant expression for the self-energy of the gravitational field. (shrink)

The traditional standard theory of quantum mechanics is unable to solve the spin–statistics problem, i.e. to justify the utterly important “Pauli Exclusion Principle” but by the adoption of the complex standard relativistic quantum field theory. In a recent paper :858–873, 2015) we presented a proof of the spin–statistics problem in the nonrelativistic approximation on the basis of the “Conformal Quantum Geometrodynamics”. In the present paper, by the same theory the proof of the spin–statistics theorem is extended to the relativistic domain (...) in the general scenario of curved spacetime. The relativistic approach allows to formulate a manifestly step-by-step Weyl gauge invariant theory and to emphasize some fundamental aspects of group theory in the demonstration. No relativistic quantum field operators are used and the particle exchange properties are drawn from the conservation of the intrinsic helicity of elementary particles. It is therefore this property, not considered in the standard quantum mechanics, which determines the correct spin–statistics connection observed in Nature :858–873, 2015). The present proof of the spin–statistics theorem is simpler than the one presented in Santamato and De Martini :858–873, 2015), because it is based on symmetry group considerations only, without having recourse to frames attached to the particles. Second quantization and anticommuting operators are not necessary. (shrink)

Evans attempted to develop a classical unified field theory of gravitation and electromagnetism on the background of a spacetime obeying a Riemann-Cartan geometry. In an accompanying paper I, we analyzed this theory and summarized it in nine equations. We now propose a variational principle for a theory that implements some of the ideas that have been (imprecisely) indicated by Evans and show that it yields two field equations. The second field equation is algebraic in the torsion and we can resolve (...) it with respect to the torsion. It turns out that for all physical cases the torsion vanishes and the first field equation, together with Evans’ unified field theory, collapses to an ordinary Einstein equation. (shrink)

An analysis of the extension of the Hawking-Penrose singularity theorem to Riemann-Cartan U4 space-times with torsion and spin density is undertaken. The minimal coupling principle in U4 is used to formulate a new expression for the convergence condition autoparallels in Einstein-Cartan theory. The Gödel model with torsion is given as an example.

Thetranslation Chern-Simons type three-formcoframe∧torsion on a Riemann-Cartan spacetime is related (by differentiation) to the Nieh-Yan fourform. Following Chandia and Zanelli, two spaces with nontrivial translational Chern-Simons forms are discussed. We then demonstrate, first within the classical Einstein-Cartan-Dirac theory and second in the quantum heat kernel approach to the Dirac operator, how the Nieh-Yan form surfaces in both contexts, in contrast to what has been assumed previously.

The foundations of a theory of nonminimal coupling of matter and the gravitational field in the framework of Riemannian (or Riemann-Cartan) geometry are presented. In the absence of matter, the Einstein vacuum field equations hold. In order to allow for a Newtonian limit, the theory contains a new parameter l0 of dimension length. For systems with finite total mass, l0 is set equal to the Schwarzschild radius.

To make sense of what Gilles Deleuze understands by a mathematical concept requires unpacking what he considers to be the conceptualizable character of a mathematical theory. For Deleuze, the mathematical problems to which theories are solutions retain their relevance to the theories not only as the conditions that govern their development, but also insofar as they can contribute to determining the conceptualizable character of those theories. Deleuze presents two examples of mathematical problems that operate in this way, which he considers (...) to be characteristic of a more general theory of mathematical problems. By providing an account of the historical development of this more general theory, which he traces drawing upon the work of Weierstrass, Poincaré, Riemann, and Weyl, and of its significance to the work of Deleuze, an account of what a mathematical concept is for Deleuze will be developed. (shrink)

The analysis of the measurement of gravitational fields leads to the Rosenfeld inequalities. They say that, as an implication of the equivalence of the inertial and passive gravitational masses of the test body, the metric cannot be attributed to an operator that is defined in the frame of a local canonical quantum field theory. This is true for any theory containing a metric, independently of the geometric framework under consideration and the way one introduces the metric in it. Thus, to (...) establish a local quantum field theory of gravity one has to transit to non-Riemann geometry that contains (beside or instead of the metric) other geometric quantities. From this view, we discuss a Riemann–Cartan and an affine model of gravity and show them to be promising candidates of a theory of canonical quantum gravity. (shrink)

Automatic conservation of energy-momentum and angular momentum is guaranteed in a gravitational theory if, via the field equations, the conservation laws for the material currents are reduced to the contracted Bianchi identities. We first execute an irreducible decomposition of the Bianchi identities in a Riemann-Cartan space-time. Then, starting from a Riemannian space-time with or without torsion, we determine those gravitational theories which have automatic conservation: general relativity and the Einstein-Cartan-Sciama-Kibble theory, both with cosmological constant, and the nonviable pseudoscalar model. The (...) Poincaré gauge theory of gravity, like gauge theories of internal groups, has no automatic conservation in the sense defined above. This does not lead to any difficulties in principle. Analogies to 3-dimensional continuum mechanics are stressed throughout the article. (shrink)

I offer a critical review of several different conceptions of the activity of foundational research, from the time of Gauss to the present. These are (1) the traditional image, guiding Gauss, Dedekind, Frege and others, that sees in the search for more adequate basic systems a logical excavation of a priori structures, (2) the program to find sound formal systems for so-called classical mathematics that can be proved consistent, usually associated with the name of Hilbert, and (3) the historicist alternative, (...) guiding Riemann, Poincaré, Weyl and others, that seeks to perfect available conceptual systems with the aim to avoid conceptual limitations and expand the range of theoretical options. I shall contend that, at times, assumptions about the foundational enterprise emerge from certain dogmas that are frequently inherited from previous, outdated images. To round the discussion, I mention some traits of an alternative program that investigates the epistemology of mathematical knowledge. (shrink)

Motivated by the search for a Hamiltonian formulation of Einstein equations of gravity which depends in a minimal way on choices of coordinates, nor on a choice of gauge, we develop a multisymplectic formulation on the total space of the principal bundle of orthonormal frames on the 4-dimensional space-time. This leads quite naturally to a new theory which takes place on 10-dimensional manifolds. The fields are pairs of \,\varpi )\), where \\) is a 1-form with coefficients in the Lie algebra (...) of the Poincaré group and \ is an 8-form with coefficients in the dual of this Lie algebra. The dynamical equations derive from a simple variational principle and imply that the 10-dimensional manifold looks locally like the total space of a fiber bundle over a 4-dimensional base manifold. Moreover this base manifold inherits a metric and a connection which are solutions of a system of Einstein–Cartan equations. (shrink)

The Cartan equations defining simple spinors (renamed “pure” by C. Chevalley) are interpreted as equations of motion in compact momentum spaces, in a constructive approach in which at each step the dimensions of spinor space are doubled while those of momentum space increased by two. The construction is possible only in the frame of the geometry of simple or pure spinors, which imposes contraint equations on spinors with more than four components, and then momentum spaces result compact, isomorphic to invariant-mass-spheres (...) imbedded in each other, since the signatures result steadily Lorentzian; starting from dimension four (Minkowski) up to dimension ten with Clifford algebra ℂℓ(1, 9), where the construction naturally ends. The equations of motion met in the construction are most of those traditionally postulated ad hoc: from Weyl equations for neutrinos (and Maxwell's) to Majorana ones, to those for the electroweak model and for the nucleons interacting with the pseudoscalar pion, up to those for the 3 baryon-lepton families, steadily progressing from the description of lower energy phenomena to that of higher ones. The 3 division algebras: complex numbers, quaternions and octonions appear to be strictly correlated with Clifford algebras and then with this spinor-geometrical approach, from which they appear to gradually emerge in the construction, where they play a basic role for the physical interpretation: at the third step complex numbers generate U(1), possible origin of the electric charge and of the existence of charged—neutral fermion pairs, explaining also easily the opposite charges of proton-electron. Another U(1) appears to generate the strong charge at the fourth step. Quaternions generate the signature of space-time at the first step, the SU(2) internal symmetry of isospin and, in the gauge term, the SU(2) L one, of the electroweak model at the third step; they are also at the origin of 3 families; in number equal to that of quaternion imaginary units. At the fifth and last step octonions generate the SU(3) internal symmetry of flavour, with SU(2) isospin subgroup and, in the gauge term, the one of color, correlated with SU(2) L of the electroweak model. These 3 division algebras seem then to be at the origin of charges, families and of the groups of the Standard model. In this approach there seems to be no need of higher dimensional (>4) space-time, here generated by the four Poincaré translations, and dimensional reduction from ℂℓ(1,9) to ℂℓ(1,3) is equivalent to decoupling of the equations of motion. This spinor-geometrical approach is compatible with that based on strings, since these may be expressed bilinearly (as integrals) in terms of Majorana–Weyl simple or pure spinors which are admitted by ℂℓ(1, 9) = R(32). (shrink)

It is shown in the present paper that the transformation relating a parallel transported vector in a Weyl space to the original one is the product of a multiplicative gauge transformation and a proper orthochronous Lorentz transformation. Such a Lorentz transformation admits a spinor representation, which is obtained and used to deduce the transportation properties of a Weyl spinor, which are then expressed in terms of a composite gauge group defined as the product of a multiplicative gauge group and the (...) spinor group. These properties render a spinor amenable to its treatment as a particle coupled to a multidimensional gauge field in the framework of the Kaluza–Klein formulation extended to multidimensional gauge fields. In this framework, a fiber bundle is constructed with a horizontal, base space and a vertical, gauge space, which is a Lie group manifold, termed its structure group. For the present, the base is the Minkowski spacetime and the vertical space is the composite gauge group mentioned above. The fiber bundle is equipped with a Riemannian structure, which is used to obtain the classical description of motion of a spinor. In its classical picture, a Weyl spinor is found to behave as a spinning charged particle in translational motion. The corresponding quantum description is deduced from the Klein–Gordon equation in the Riemann spaces obtained by the methods of path-integration. This equation in the present fiber bundle reduces to the equation for a Weyl spinor, which is close to but differs somewhat from the squared Dirac equation. (shrink)

The foundations of Wesson’s induced matter theory are analyzed. It is shown that the empty—without matter—5-dimensional bulk must be regarded as a Weylian space rather than as a Riemannian one. Revising the geometry of the bulk, we have assumed that a Weylian connection vector and a gauge function exist in addition to the metric tensor. The framework of a Weyl–Dirac version of Wesson’s theory is elaborated and discussed. In the 4-dimensional hypersurface (brane), one obtains equations describing both fields, the gravitational (...) and the electromagnetic. The result is a geometrically based unified theory of gravitation and electromagnetism with mass and current induced by the bulk. In special cases on obtains on the brane the equations of Einstein–Maxwell, or these of the original induced matter theory. (shrink)

Extension is probably the most general natural property. Is it a fundamental property? Leibniz claimed the answer was no, and that the structureless intuition of extension concealed more fundamental properties and relations. This paper follows Leibniz's program through Herbart and Riemann to Grassmann and uses Grassmann's algebra of points to build up levels of extensions algebraically. Finally, the connection between extension and measurement is considered.

This is a basically expository article, with some new observations, tracing connections of the quantum potential to Fisher information, to Kähler geometry of the projective Hilbert space of a quantum system, and to the Weyl-Ricci scalar curvature of a Riemannian flat spacetime with quantum matter.

The traditional standard quantum mechanics theory is unable to solve the spin–statistics problem, i.e. to justify the utterly important “Pauli Exclusion Principle”. A complete and straightforward solution of the spin–statistics problem is presented on the basis of the “conformal quantum geometrodynamics” theory. This theory provides a Weyl-gauge invariant formulation of the standard quantum mechanics and reproduces successfully all relevant quantum processes including the formulation of Dirac’s or Schrödinger’s equation, of Heisenberg’s uncertainty relations and of the nonlocal EPR correlations. When the (...) conformal quantum geometrodynamics is applied to a system made of many identical particles with spin, an additional constant property of all elementary particles enters naturally into play: the “intrinsic helicity”. This property, not considered in the Standard Quantum Mechanics, determines the correct spin–statistics connection observed in Nature. (shrink)

A rigorous ab initio derivation of the (square of) Dirac’s equation for a particle with spin is presented. The Lagrangian of the classical relativistic spherical top is modified so to render it invariant with respect conformal changes of the metric of the top configuration space. The conformal invariance is achieved by replacing the particle mass in the Lagrangian with the conformal Weyl scalar curvature. The Hamilton-Jacobi equation for the particle is found to be linearized, exactly and in closed form, by (...) an ansatz solution that can be straightforwardly interpreted as the “quantum wave function” of the 4-spinor solution of Dirac’s equation. All quantum features arise from the subtle interplay between the conformal curvature acting on the particle as a potential and the particle motion which affects the geometric “pre-potential” associated to the conformal curvature itself. The theory, carried out here by assuming a Minkowski metric, can be easily extended to arbitrary space-time Riemann metric, e.g. the one adopted in the context of General Relativity. This novel theoretical scenario appears to be of general application and is expected to open a promising perspective in the modern endeavor aimed at the unification of the natural forces with gravitation. (shrink)

We present some interesting connections between PT symmetry and conformal symmetry. We use them to develop a metricated theory of electromagnetism in which the electromagnetic field is present in the geometric connection. However, unlike Weyl who first advanced this possibility, we do not take the connection to be real but to instead be PT symmetric, with it being \ rather than \ itself that then appears in the connection. With this modification the standard minimal coupling of electromagnetism to fermions (...) is obtained. Through the use of torsion we obtain a metricated theory of electromagnetism that treats its electric and magnetic sectors symmetrically, with a conformal invariant theory of gravity being found to emerge. An extension to the non-Abelian case is provided. (shrink)

Lautman synthétise dans ce rapport quelques idées centrales qui seront par la suite développées dans ses Thèses . Il s’agit d’un manuscrit inédit, qui semble être le premier texte scientifique du jeune philosophe. Lautman étudie le local et le global suivant Galois, Riemann, Hilbert et Cartan, et propose une hypothèse sur les rapports structurels généraux du local et du global, qui préfigure l’essor de la théorie des faisceaux, laquelle apparaîtra une dizaine d’années après.Lautman synthetizes some of the main ideas that (...) will be developped in his Thesis . It is an unpublished manuscript, which may be the first scientific text of the young philosopher. Lautman studies the local/global dialectics, following Galois, Riemann, Hilbert and Cartan, and proposes an hypothesis on general structural relations between local and global, very close to the sheaf theory paradigm which will emerge ten years later. (shrink)

Nous analysons le développement mathématique et la signification épistémologique du mouvement de géométrisation de la physique théorique, à partir des travaux fondamentaux d’E. Cartan et de H. Weyl jusqu’aux théories de jauge non-abéliennes récentes. Le principal propos de cet article est d'étudier ces développements qui ont été inspirés par les tentatives de résoudre l'un des problèmes centraux de la physique théorique au siècle dernier, c’est-à-dire comment arriver à concilier la relativité générale et la théorie quantique des champs dans un cadre (...) théorique unitaire du monde physique. Ces développements ont produit un changement conceptuel profond concernant tout particulièrement la façon de concevoir les rapports entre structures mathématiques et phénomènes physiques. Nous mettons l’accent sur les points suivants : plutôt que de simplement s’appliquer aux phénomènes, les mathématiques sont impliquées dans leur constitution, autrement dit, les phénomènes sont autant d’effets qui émergent de la structure géométrique de l’espace-temps ; la structure géométrique et topologique de l’espace-temps est à l’origine de la dynamique de ce dernier; les symétries « internes » dictent les différentes interactions entre forces et entre particules ; l’invariance de jauge est un principe « universel » régissant les forces fondamentales et les interactions entre les champs de matière. (shrink)

The purpose of this paper is to evaluate the `Lorentzian Pedagogy' defended by J.S. Bell in his essay ``How to teach special relativity'', and to explore its consistency with Einstein's thinking from 1905 to 1952. Some remarks are also made in this context on Weyl's philosophy of relativity and his 1918 gauge theory. Finally, it is argued that the Lorentzian pedagogy---which stresses the important connection between kinematics and dynamics---clarifies the role of rods and clocks in general relativity.