In 1960–1962, E. Kähler enriched É. Cartan’s exterior calculus, making it suitable for quantum mechanics (QM) and not only classical physics. His “Kähler-Dirac” (KD) equation reproduces the fine structure of the hydrogen atom. Its positron solutions correspond to the same sign of the energy as electrons.The Cartan-Kähler view of some basic concepts of differential geometry is presented, as it explains why the components of Kähler’s tensor-valued differential forms have three series of indices. We (...) demonstrate the power of his calculus by developing for the electron’s and positron’s large components their standard Hamiltonian beyond the Pauli approximation, but without resort to Foldy-Wouthuysen transformations or ad hoc alternatives (positrons are not identified with small components in K ähler’s work). The emergence of negative energies for positrons in the Dirac theory is interpreted from the perspective of the KD equation. Hamiltonians in closed form (i.e. exact through a finite number of terms) are obtained for both large and small components when the potential is time-independent.A new but as yet modest new interpretation of QM starts to emerge from that calculus’ peculiarities, which are present even when the input differential form in the Kähler equation is scalar-valued. Examples are the presence of an extra spin term, the greater number of components of “wave functions” and the non-association of small components with antiparticles. Contact with geometry is made through a Kähler type equation pertaining to Clifford-valued differential forms. (shrink)

The theory of exterior differential systems plays a crucial role in Cartan’s whole mathematical production. As he once recognized, all the germs of his subsequent work were contained there. Indeed, it provided him with powerful technical tools that turned out to be very useful in many different fields such as the theory of partial differential equations, the theory of infinite dimensional Lie groups (Lie pseudogroups) and differential geometry. Nevertheless, scarce attention has been paid to this area of historical research (...) thus far. Although authoritative scholars have investigated the foundation of exterior differential calculus in Cartan’s early papers, no specific analysis of Cartan’s subsequent works laying the foundations of what nowadays is known as the Cartan–Kähler theory has been yet provided. This article represents a first attempt to remedy this unsatisfactory state of affairs by focusing on Cartan’s work on Pfaffian systems at the very beginning of the past century. The analysis of Cartan’s relevant papers is preceded by a description of the historical context in which such contributions were conceived. In this respect, special emphasis will be put on some works by Engel and von Weber on Pfaffian systems, which laid the basis for the subsequent geometrical developments of Cartan’s theory of exterior differential systems. (shrink)

Husserl's transcendental phenomenology is not a mere egology, but gets its concrete accomplishment only as a phenomenology of 'transcendental intersubjectivity'. However, the subjective centers of any transcendentality and thus of every constitution — even of intersubjectivity itself — have to be such unities as Leibniz' 'monads', that is, individually concrete subjects producing all their representations of one another completely out of themselves, respectively. Thus the problem arises, how the genuine transcendental status of each monadic subject in all its constitutive achievements (...) could be maintained so that they develop their own intentional lives in a universal mutual accordance, without presupposing again a supra-monadic ground for their 'harmony', i.e. a higher-levelled ordering unity. I shall argue that th peculiar Husserlian 'transcendental monadology' is inevitably bound to this intrinsic paradox which can at best be reduced to a mere postulate but never be resolved. (shrink)

In as far as every philosophy appears at a certain point in time it later inevitably becomes a historical fact. If its content is then to be recognizable, if it is to be reclaimed as philosophy in the present, then we ourselves must have a philosophical conception of history and, furthermore, a conception of philosophy's own historicity. Only since Kant has modern metaphysics expressly and methodically developed such a conception within its systematic claim to truth. Moreover, for the presentday philosophy (...) and its relationship to a completed metaphysics such a conception is indispensable - under the presupposition that the history of philosophy is to be taken seriously as philosophy. (shrink)

The intention of my comments is mainly to draw attention to a necessary distinction between that prereflective cogito of post‐metaphysical subjectivity that is analysed in Westphal's paper and the subject of the cogito that can be identified and verified as the very principle of modern philosophy from Descartes to Hegel, namely, as the subject of reason. This means first of all to step back from the conviction, taken as self‐evident, that the subject of reason — and thereby the truth claims (...) of that entire philosophical epoch — are illusionary, that is, without any right of their own. Instead we should be ready to ask how it is brought about philosophically that the subject is “shattered,” “humiliated,” “declared forfeit,” etc. My thesis is that post‐metaphysical subjectivity with its contaminated opacity can be made understandable in principle out of the endogenous crisis of the fully developed “absolute” subject of reason, if this crisis is carried out and decided as the transformation of the subject from its absolute to its decentered status. (shrink)

In this paper I attempt to analyze the basic tension in Kant's philosophy of right, i.e. within the conceptual framework of his Transcendental Idealism. Right as the embodiment of laws of freedom applicable to mortal-rational agents bridges the gap between their noumenal character and empirical-natural existence. How can this gap be bridged? A priori juridical laws are not sufficient to provide a complete classification for the empirical realm, exhausting all possible positive rights. Furthermore, according to Kant, Right must be basically (...) justifiable by reason. Yet this justification is merely a necessary condition for obligation. In order to develop any real obligation, it must be complemented by a non-deducible, empirical-positive aspect . Hence only as a result of actually exercising individual freedom does the mutual necessitation to recognize individual freedom for all arise. The historical-factual formation of civil social orders, though subject to a priori foundation, is thus unavoidably marked by contingency. Nonetheless, because of its a priori character, the idea of a society of free individuals becomes the standard of critique for every empirical instantiation of a civil social order. Recht als Inbegriff der Gesetze der Freiheit endlich-vernünftiger Subjekte leistet eine Vermittlung zwischen dem noumenalen Wesen und der emprisch-naturalen Existenz dieser Subjekte. Wie ist diese Vermittlung möglich? Eine vollständige Einteilung des Empirischen unter Rechtsgesetzen a priori ist unmöglich. Was Recht ist, ist nach Kant zwar nur in der Vernunft begründbar, aber dieser Grund bleibt nur notwendige Bedingung, zu der stets eine nicht deduzierbare, empirisch-positive Seite hinzukommen muß, um tatsächliche Verbindlichkeit zu erhalten : Erst aufgrund einer faktischen Ausübung der Freiheit entsteht die wechselseitige Nötigung, die individuelle Freiheit als Freiheit aller anzuerkennen. Die historisch-faktische Gestaltung von Rechtsgemeinschaften bleibt somit auch unter der Bedingung apriorischer Begründung mit unaufhebbarer Kontingenz behaftet; doch zugleich gewinnt die Idee einer Gemeinschaft freier Individuen kraft ihrer Apriorität eine kritische Funktion gegenüber jeglicher empirischer Instanz. (shrink)

On November 21, 1994, Werner Marx passed away peacefully in the place he loved so well, his apartment in the Schloß in Bollschweil. Professor Marx was born in 1910 in Mulheim, Germany. He studied law and philosophy in Berlin, Freiburg, and Bonn before completing his state examination and doctorate in law in 1933. In the same year, he was removed from civil service and from an apprentice judgeship by the Nazis. After this, he emigrated first to Palestine and then in (...) 1938 to New York, where he took up academic studies again. After receiving an M.A. in economics, he completed the Ph.D. in philosophy, working with Kurt Reizler and Karl Löwith at the German University in Exile, later to become the New School for Social Research, and writing a dissertation on Aristotle’s ontology. During this period, he renewed his interest in Hegel and German Idealism, and also in Heidegger’s critique of the tradition and attempts to make “another beginning” in philosophy. Marx began teaching at the New School in 1949. After the appearance of his classic study, Heidegger and the Tradition, he was named successor to the Husserl-Heidegger chair in philosophy and director of Philosophical Seminar I in Freiburg, which he held from 1964 until his retirement in 1979. He also served as Director of the Husserl-Archives in Freiburg until his death. (shrink)

In an accompanying paper (I), it is shown that the basic equations of the theory of Lorentzian connections with teleparallelism (TP) acquire standard forms of physical field equations upon removal of the constraints represented by the Bianchi identities. A classical physical theory results that supersedes general relativity and Maxwell-Lorentz electrodynamics if the connection is viewed as Finslerian. The theory also encompasses a short-range, strong, classical interaction. It has, however, an open end, since the source side of the torsion field equation (...) is not geometric.In this paper, Kaehler's partial geometrization of the Dirac equation is taken as a starting point for the development of fully geometric Dirac equations via the correspondence principle given in I. For this purpose, Kaehler's calculus (where the spinors are differential forms) is generalized so that it also applies when the torsion is not zero. The point is then made that the forms can take values in tangent Clifford algebras rather than in tensor algebras. The basic “Eigenschaft” of the Kaehler calculus also is examined from the physical perspective of dimensional analysis.Geometric Dirac equations of great structural simplicity are finally inferred from the standard Dirac equation by using the aforementioned correspondence principle. The realm of application of the Dirac theory is thus enriched in principle, though only at an abstract level at this point: the standard spinors, which are scalar-valued forms in the Kaehler version of that theory, become Clifford-valued. In addition, the geometrization of the Dirac equation implies a geometrization of the Dirac current. When this current is replaced in the field equations for the torsion, the theory of Paper I becomes fully geometric. (shrink)

A new axiomatic treatment of equilibrium thermodynamics—thermostatics—is presented. The equilibrium states of a thermal system are assumed to be represented by a differentiable manifold of dimensionn + 1 (n finite). The empirical temperature is defined by the notion of thermal equilibrium. Empirical entropy is shown to exist for all systems with the property that the total work delivered along closed adiabats is zero. Absolute entropy and temperature follow from the additivity of heat and energy for two separate systems in thermal (...) equilibrium considered as a whole. The absolute temperature is defined up to a multiplicative constant. The exterior differentiable calculus of Cartan is introduced and in a subsequent paper its use for the derivation of standard results in thermostatics will be explained. (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)

Kottler, Cartan, and van Dantzig independently uncovered a key property of the Maxwell equations, which, in retrospect, is instrumental for treating noninertial situations. The essence of this KCD procedure is outlined. Present traditions incompatible with the KCD procedure are identified. KCD predicts a rotation-induced magnetoelectric effect in vacuum, as verified by the experiments of Kennard and Pegram. The description of nonvacuum situations still has some unresolved differences awaiting further experimental delineation. Explicit calculations and technical specifications of experiments receive references (...) to the literature. The emphasis of presentation stresses the conceptual reorganization necessary for lifting the Kennard-Pegram experiments out of their present state of obscurity. The question of whether or not KCD is a physically viable counterpart of the Lagrange-Hamilton-Jacobi procedure in mechanics is contingent on the outcome of more detailed experimentation of the Kennard-Pegram type. (shrink)

This handbook with exercises reveals the mathematical beauty of formalisms hitherto mostly used for software and hardware design and verification.

The epsilon calculus is a logical formalism developed by David Hilbert in the service of his program in the foundations of mathematics. The epsilon operator is a term-forming operator which replaces quantifiers in ordinary predicate logic. Specifically, in the calculus, a term εx A denotes some x satisfying A(x), if there is one. In Hilbert's Program, the epsilon terms play the role of ideal elements; the aim of Hilbert's finitistic consistency proofs is to give a procedure which removes (...) such terms from a formal proof. The procedures by which this is to be carried out are based on Hilbert's epsilon substitution method. The epsilon calculus, however, has applications in other contexts as well. The first general application of the epsilon calculus was in Hilbert's epsilon theorems, which in turn provide the basis for the first correct proof of Herbrand's theorem. More recently, variants of the epsilon operator have been applied in linguistics and linguistic philosophy to deal with anaphoric pronouns. (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 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)

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.