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)

Hamilton shows how awareness of the uses of space -- in particular uses of space in which to stage an event of any kind -- enable spectators to pick out characters, props, and the like across performances within production runs, across production runs, and even across productions employing different scripts. The key ideas of object identification are taken both from the philosophical and the empirical literature and are treated as epistemic ideas rather than metaphysical conceptions.

The visual arts include painting, sculpture, photography, video, and film. But many people would argue that music is the universal or only art of sound. In the modernist era, Western art music has incorporated unpitched sounds or ‘noise’, and I pursue the question of whether this process allows space for a non-musical soundart. Are there non-musical arts of sound—is there an art phonography, for instance, to parallel art photography? At the same time, I attempt a characterization of music, contrasting acoustic, (...) aesthetic, and acousmatic accounts. My view is that there is some truth in all of these. I defend the claim that music is an art with a small ‘a’—a practice involving skill or craft whose ends are essentially aesthetic, that especially rewards aesthetic attention—whose material is sounds exhibiting tonal organization. But acoustic and acousmatic accounts help to distinguish between music and non-musical soundart, since music must have a preponderance of tones for its material. (shrink)

A fundamental discussion in lower-level undergraduate neuroscience and psychology courses is Descartes’s “radical” or “mind-body” dualism. According to Descartes, our thinking mind, the res cogitans, is separate from the body as physical matter or substance, the res extensa. Since the transmission of sensory stimuli from the body to the mind is a physical capacity shared with animals, it can be confused, misled, or uncertain (e.g., bodily senses imply that ice and water are different substances). True certainty thus arises from within (...) the mind and its capacity to doubt physical stimuli. Since this doubting mind is a thinking thing that is distinct from bodily stimuli, truth and certainty are reached through the doubting mind as cogito ergo sum, or the certainty of itself as it thinks: hence Descartes’s famous maxim, I think, therefore I am. However, in the last century of Western philosophy, with nervous system investigation, and with recent advances in neuroscience, the potential avenues to explore student’s understanding of the epistemology and effects of Cartesian mind-body dualism has expanded. This article further explores this expansion, highlighting pedagogical practices and tools instructors can use to enhance a psychology student’s understanding of Cartesian dualistic epistemology, in order to think more critically about its implicit assumptions and effects on learning. It does so in two ways: first, by offering instructors an alternative philosophical perspective to dualistic thinking: a mind-body holism that is antithetical to the assumed binaries of dualistic epistemology. Second, it supplements this philosophical argument with a practical component: simple mind-body illusions that instructors may use to demonstrate contrary epistemologies to students. Combining these short philosophical and neuroscience arguments thereby acts as a pedagogical tool to open new conceptual spaces within which learning may occur. (shrink)

Exemplars have the power to help people navigate various levels of moral struggle, from the relatively straightforward problem of lacking motivation to the much deeper problem of failing to see the moral realities that surround us. But there are also serious moral risks in the appeal to exemplars: we romanticize them, we make use of them in authoritarian ways, and we tend to forget how our choice of exemplars is conditioned by oppressive cultural formations. I argue that we need to (...) develop a social model of exemplarity, attuned to social contexts of our exemplars themselves as well as the social processes of constructing and appealing to exemplars. More particularly, I argue that we need to develop space for thinking about exemplary groups, not just exemplary individuals, in order to develop the strengths and avoid the weaknesses in exemplarist moral theories. (shrink)

The authors of this essay reflect on the experience of co-teaching a course on the history of genetics and race. The collaboration has pushed them both—a historian of science and a biologist—to consider how to make space for moral and scientific judgment in a history classroom. Drawing on examples from the course, they argue that it is possible to encourage social action and thoughtful critiques of past and current science without succumbing to a whiggish narrative of progress.

Current struggles over laws prohibiting and criminalizing the public disclosure of violence in the spaces of animal use in the US have underscored the centrality of exposés to animal activism. This article complicates the activist belief in the power of exposure—“If slaughterhouses had glass walls...”—by drawing on the insights of Jean-Paul Sartre and Simone de Beauvoir concerning the prevalence of bad faith in systems of oppression and exploitation. I describe four forms of bad faith common to these systems, and (...) offer suggestions for exposés of the animal enterprise modeled on Sartre’s and de Beauvoir’s “engaged exposés.”. (shrink)

This chapter contains section titled: Identifying Characters, Events, and Other Objects in Narrative Performances Re‐Identification of Characters and Other Objects in Narrative Performances The Special Nature of Theatrical (Uses of) Space: Performances and Performance Space Cross‐Performance Re‐Identification Identifying and Re‐Identifying Objects in Non‐Narrative Performances Added Benefits of the Demonstrative and Recognition based Approach to Identification and Re‐Identification Theatrical Performance as a Fully Independent Practice.

In this study, the author uses ethnographic and interview data from a women's floor in a university residence hall to examine how some heterosexual women's gender strategies contribute to their homophobia. The author describes a prevailing heterosexual erotic market on campus—the Greek party scene—and the status hierarchy linked to it. Within this hierarchy, heterosexual women assign lesbians low rank because of their assumed disinterest in the erotic market and perceived inability to acquire men's erotic attention. Active partiers invest more in (...) this social world and prefer higher levels of social distance from lesbians than do others. These women also engage in same-sex eroticism primarily designated for a male audience. They define their behaviors as heterosexual, reducing the spaces in which lesbians can be comfortable. Finally, the author concludes by discussing the unique nature of women's homophobia and the links between sexism and heterosexism. (shrink)

The consequences of restricting or not restricting the right to freedom of assembly are potentially magnified in transitional societies. Yet determining whether such consequences are indeed ‘harmful’, and whether their cost should be borne despite the harms caused, requires the elaboration of criteria which define what are valid and relevant harms. While a human rights framework can perform this task, open-textured rights standards prescribe neither the threshold of legal intervention nor the goals of transition. By extension, the rule of law—underpinned (...) by this rights discourse—is silent about whether liberal or communitarian ideals should inform the reconstruction of public space in conflicted or nascent democracies. Illustrated by analysis of legal interventions in parade disputes in Northern Ireland, this article argues that the rule of law is necessarily orientated by ethical consensus about its scope. Furthermore, this consensus operates as a restraint upon the degree of normative discontinuity permitted during transitional compromises. The article frames the ethical options in terms of three liberty-limiting principles—the argument from democracy, the argument for toleration, and the argument for recognition. Each suggests different parameters for the transitional project and for the role of law within it. (shrink)

The body is the canvas on which the female experience is painted and through which female identity is often understood. The female body is a slate on which a patriarchal story has been written, scarred onto the flesh. For Simone Weil metaxu was simultaneously that which separated and connected, so for instance the wall between two prison cells cuts off the prisoners but was also the means by which they communicated by knocking on that wall. Could the body be that (...) metaxu all at once separating us and connecting us to the Divine? The nature of metaxu is that it offers a route not just for the individual soul but for the souls of others to travel. If all peoples renounce their outer shell – and by this I do not mean their bodies but rather the bodies or clothes or ideals written upon those people by patriarchy or capitalism or colonialism – if those real bodies come to the broken body of Christ [which is stripped and scarred] standing in solidarity with and mimicking this broken body, without the things which have been put upon their bodies but with scarred flesh showing, then this is where metaxu is possible for here we see a renouncing of ‘self’ as created by patriarchy and all that is left is a space that is the place where we are authentically self and God dwells. (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.

The Philosophy Outreach Project gets high school students across Indiana thinking. POP creates alternative spaces for learning in classrooms, clubs, online, and conference settings. By curating philosophical content and fostering philosophical discussion, POP provides high school students with tools and a platform to engage with each other and the world. POP is run by three teams of Ball State students with a variety of different interests and backgrounds. POP's team includes students studying philosophy, psychology, English, communications, criminal justice, and (...) more. This diversity of interests and knowledge allows for an impressive level of collaboration, which is ultimately what makes POP successful in promoting and facilitating pre-college philosophy. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:The Moral Dynamics of Economic Life: An Extension and Critique of Caritas in Veritate ed. by Daniel K. Finn, and: Rethinking Poverty: Income, Assets, and the Catholic Social Justice Tradition by James P. BaileyBrian HamiltonReview of The Moral Dynamics of Economic Life: An Extension and Critique of Caritas in Veritate EDITED BY DANIEL K. FINN New York: Oxford University Press, 2012. 166 pp. $85.35Review of Rethinking Poverty: Income, (...) Assets, and the Catholic Social Justice Tradition JAMES P. BAILEY Notre Dame, IN: University of Notre Dame Press, 2010. 176 pp. $30.00The academic study of Catholic social teaching has flowered in recent years. The moral vision elaborated over a century of official Catholic documents on social and political issues has proven attractive to a wide variety of scholars, providing a point of reference for interdisciplinary conversation. By bringing that vision into conversation with their respective disciplines, such scholars have gone a long way toward addressing one of the most common complaints about Catholic social teaching: that its proposals are too general and abstract. Two recent books, both working at the intersection of theology and economics, are a case in point.The Moral Dynamics of Economic Life, edited by Daniel Finn, synthesizes a conversation held in Rome in 2010 under the aegis of the Institute of Advanced Catholic Studies and the Pontifical Council for Justice and Peace. Theologians, philosophers, public policy experts, lawyers, businesspeople, and Church leaders gathered to discuss the significance of Benedict XVI’s social encyclical, Caritas in veritate, for the US context. Rather than simply reprinting the papers they discussed, the participants’ contributions were split up and reorganized along thematic lines. There are eleven chapters—on topics like reciprocity, development, and polarization in US political discourse—each composed of a series of two- to four-page essays from different scholars. It is a creative and surprisingly effective structure. No single argument is developed at any length, of course. But the essays, punchy and provocative, accomplish far more than one would think possible in so short a space. Taken together, they define the contours of a much richer conversation than a series of independent papers could have done. [End Page 205]James Bailey’s Rethinking Poverty provides a more sustained argument. Working within the moral framework provided by Catholic social teaching, Bailey argues that US public policy regarding poverty needs to focus less on income relief and more on asset ownership and development. He lays out his basic case for the “asset paradigm” in chapter 1. In the current system, asset-building assistance is provided almost exclusively to the nonpoor rather than to the poor since it is mediated primarily through the federal income tax—which the poor do not pay. Income assistance to the poor, though crucial in the short term, does nothing to address the root causes of endemic poverty in the United States. In fact, the regressive structure of asset-building assistance has itself played an important role in perpetuating and exacerbating the wealth gap in this country. In the next two chapters, Bailey lays out two complementary moral frameworks that support a policy shift toward an asset paradigm: Catholic social teaching and the “capabilities approach” developed by Martha Nussbaum and Amartya Sen. In the fourth chapter, Bailey surveys the history of racialized asset discrimination in US policy, demonstrating that the denial of asset-based assistance has been instrumental in creating enduring wealth disparities between white and black communities. In the fifth chapter, Bailey concludes by suggesting that an older, neglected American policy tradition did privilege an assets-based approach to poverty relief, and that tradition continues to be a viable option.Both books succeed, in my judgment, in rendering the principles and priorities of Catholic social teaching more concrete. Bailey’s main goal, obviously, is to recommend a specific policy orientation, and Moral Dynamics is full of practical suggestions: we should encourage conversations between theology and business schools at Catholic universities (89, 135); we should give more attention to empirical measures of well-being and development beyond narrowly economic ones like the gross domestic product (92–98); we should work to redefine the standard of “reasonableness” in the common... (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)

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)

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)

Figure emblématique de l’art de l’installation, Ann Hamilton crée depuis près de quarante ans des environnements immersifs complexes et poétiques, constitués d’objets et de matériaux divers convoquant tous les sens du spectateur et dont les surprenantes associations, ou leur importante accumulation, provoquent chez ce dernier de multiples réponses affectives et cognitives, bien souvent difficiles à décrire ou à restituer par le simple langage. Absorbé par la présence physique et sensible de ce qui l’entoure, le spectateur est invité à s’abandonner (...) aussi longtemps que possible dans ce qu’il éprouve, à se mettre à l’écoute de la situation et du savoir intuitif que lui livre son corps en interaction avec le lieu. L’expérience de l’œuvre, pour Ann Hamilton, est en effet avant tout un « acte d’attention ». Cette notion d’attention est particulièrement importante, à une époque où la surenchère de stimuli, l’immédiateté et l’accélération qui caractérisent les rythmes de nos vies de plus en plus désincarnées ne nous permettent plus de percevoir et de donner sens à ce que nous ressentons, ni à ce qui nous entoure. Phénomène singulier d’ouverture à soi, aux autres et au monde, l’expérience des installations d’Ann Hamilton engage un certain mode d’être, une « qualité de présence » dont dépendent nos capacités de mémoire, de pensée, d’imagination et d’action. À partir de l’analyse de quelques grandes installations, de tropos à the event of the thread, il s’agira d’établir plus précisément les processus phénoménologiques et cognitifs mis en œuvre chez le spectateur par le dispositif immersif, et d’en éclairer les enjeux. For nearly forty years, Ann Hamilton, an iconic figure in the art of installation, has been creating complex and poetic immersive environments, made up of objects and various materials which summon all the viewer’s senses. The astonishing associations and significant accumulation trigger multiple emotional and cognitive responses, often difficult to put simply into words. Absorbed by the physical and sensitive presence of what surrounds them, the viewers are invited to abandon themselves as long as possible in what they are experiencing as well as pay attention to the situation and the intuitive knowledge they receive from their own physical interaction with the space. For Ann Hamilton, experiencing the art is above all an “act of attention”. The notion of attention is particularly important at a time when continuous stimuli, immediacy and acceleration, which characterize the rhythms of our evermore fragmented lives, no longer allow us to perceive and give meaning to what we feel, or what surrounds us. Ann Hamilton’s installations call for an openness to oneself, to others and to the world. They involve a certain mode of being, a “quality of presence” upon which our ability to remember, think, imagine and act depends. Based on the observation of several large installations, we will explore and analyze the phenomenological and cognitive processes which occur as the viewer is immersed in her works. (shrink)

In the many worlds community there seems to exist a belief that the physics of quantum theory is completely defined by it’s Hamilton operator given in an abstract Hilbert space, especially that the position basis may be derived from it as preferred using decoherence techniques.We show, by an explicit example of non-uniqueness, taken from the theory of the KdV equation, that the Hamilton operator alone is not sufficient to fix the physics. We need the canonical operators $\hat{p}$ , (...) $\hat{q}$ as well. As a consequence, it is not possible to derive a “preferred basis” from the Hamilton operator alone, without postulating some additional structure like a “decomposition into systems”. We argue that this makes such a derivation useless for fundamental physics. (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)

Hamilton-Jacobi theory is applied to find appropriate canonical transformations for the calculation of the phase-space path integrals of the relativistic particle equations. Hence, canonical transformations and Hamilton-Jacobi theory are also introduced into relativistic quantum mechanics. Moreover, from the classical physics viewpoint, it is very interesting to find and to solve the Hamilton-Jacobi equations for the relativistic particle equations.

Abstract:This study analyses how the migrant experience is reflected in the geography of Hugo Hamilton's work of fiction entitled Hand in the Fire (2010). The novel is told from the point of view of Vid, a Serbian immigrant who is trying to settle in Ireland. The young carpenter endeavours to fit in Irish society through his friendship with a young Dublin lawyer, Kevin Concannon, who tells him that a true friend would put his “hand in the fire” for you. (...) Through the portrayal of these displaced characters, Hamilton explores his favourite themes of memory and belonging. This article focuses on the places depicted in the novel, from the Serbia that Vid has left in his past to the modern-day Dublin and the west coast of Ireland, in order to analyse how the settings of the narrative reflect the characters’ state of mind. Using Deleuze and Guattari's concepts of de/re-territorialization and of striated/smooth spaces, the examination of the geographies of the novel allows for a better understanding of the issues at stake for the characters. The liminal, deterritorialized figure of the immigrant represented by Vid is echoed by the smooth islands of the novel. In an archipelagic country where the coastline stands for the visual border, it is no wonder that the transnational protagonist is drawn to the Irish shores. On the edge of two cultures, Vid, who lives in Dún Laoghaire, south of Dublin, is gradually moving west to the striated landscape of Connemara, where he investigates on Kevin's aunt who drowned and was washed ashore the Aran Islands. (shrink)

Let $V_\propto$ be a fixed, fully effective, infinite dimensional vector space. Let $\mathscr{L}(V_\propto)$ be the lattice consisting of the recursively enumerable (r.e.) subspaces of $V_\propto$ , under the operations of intersection and weak sum (see § 1 for precise definitions). In this article we examine the algebraic properties of $\mathscr{L}(V_\propto)$ . Early research on recursively enumerable algebraic structures was done by Rabin [14], Frolich and Shepherdson [5], Dekker [3], Hamilton [7], and Guhl [6]. Our results are based upon the (...) more recent work concerning vector spaces of Metakides and Nerode [12], Crossley and Nerode [2], Remmel [15], [16], and Kalantari [8]. In the main theorem below, we extend a result of Lachlan from the lattice E of r.e. sets to $\mathscr{L}(V_\propto)$ . We define hyperhypersimple vector spaces, discuss some of their properties and show if $A, B \in \mathscr{L}(V_\propto)$ , and A is a hyperhypersimple subspace of B then there is a recursive space C such that A + C = B. It will be proven that if $V \in \mathscr{L}(V_\propto)$ and the lattice of superspaces of V is a complemented modular lattice then V is hyperhypersimple. The final section contains a summary of related results concerning maximality and simplicity. (shrink)

In modern physics nonlinear systems with a lot of heterogeneity and anisotropy which are in strong fields and processes of crossings of electromechanical, spinor-mechanical, termo-magnetic and other ones are actual. Correct building with the help of mathematics of the given theories is possible only with the attraction geometry of the stratified spaces. The geometry of the stratified spaces chow its power by examining these systems and processes. Noncontradictional, covarianty theory of the single whole field of matter can be (...) built only with the attraction the geometry of the stratified spaces. Introduction the stratified spaces into physics of geometry is a continuation of works of Hamiltone, Langrange, Aler and other scientists. The problem is multivariate spaces of internal degrees offreedom. The ideas, positions of geometry of stratified spaces and their supplements demand a new philosophical sense of understanding. (shrink)

We present a general derivation of the Duffin-Kemmer-Petiau (D.K.P) equation on the relativistic phase space proposed by Bohm and Hiley. We consider geometric algebras and the idea of algebraic spinors due to Riesz and Cartan. The generators βμ (p) of the D.K.P algebras are constructed in the standard fashion used to construct Clifford algebras out of bilinear forms. Free D.K.P particles and D.K.P particles in a prescribed external electromagnetic field are analized and general Liouville type equations for these cases (...) are obtained. Choosing particular values for the label p we classify the different types of the D.K.P Liouville operators. (shrink)

Some arguments, based on the possible spontaneous violation of the cosmological principle (represented by the observed large-scale structures of galaxies), on the Cartan geometry of simple spinors, and on the Fock formulation of hydrogen atom wave equation in momentum space, are presented in favor of the hypothesis that space-time and momentum space should be both conformally compactified and should both originate from the two four-dimensional homogeneous spaces of the conformai group, both isomorphic (S 3 ×S 1)/Z 2 and (...) correlated by conformal inversion, but should not necessarily be identified with them. Within this framework, the possible common origin for the SO(4) symmetry underlying the geometrical structure of the Universe, of Kepler orbits, and of the H atom is discussed. One of the consequences of the proposed hypothesis could be that any quantum field theory should be naturally free from both infrared and ultraviolet divergences. But then physical spaces defined as those where physical phenomena may be best described through some set of fields, could be different from those homogeneous spaces. A simple, exactly soluble, toy model, valid for a two-dimensional spacetime, is presented where the conjectured conformally compactified homogeneous spaces are both isomorphic to (S 1 ×S 1)/Z 2,while the possible physical spaces could be two finite lattices which are dual since Fourier transforms, represented by finite, discrete, sums, may be well defined on them. (shrink)

The symplectic quantization scheme proposed for matter scalar fields in the companion paper (Gradenigo and Livi, arXiv:2101.02125, 2021) is generalized here to the case of space–time quantum fluctuations. That is, we present a new formalism to frame the quantum gravity problem. Inspired by the stochastic quantization approach to gravity, symplectic quantization considers an explicit dependence of the metric tensor gμν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{\mu \nu }$$\end{document} on an additional time variable, named intrinsic time at variance (...) with the coordinate time of relativity, from which it is different. The physical meaning of intrinsic time, which is truly a parameter and not a coordinate, is to label the sequence of gμν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{\mu \nu }$$\end{document} quantum fluctuations at a given point of the four-dimensional space–time continuum. For this reason symplectic quantization necessarily incorporates a new degree of freedom, the derivative g˙μν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\dot{g}}_{\mu \nu }$$\end{document} of the metric field with respect to intrinsic time, corresponding to the conjugated momentum πμν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{\mu \nu }$$\end{document}. Our proposal is to describe the quantum fluctuations of gravity by means of a symplectic dynamics generated by a generalized action functional A[gμν,πμν]=K[gμν,πμν]-S[gμν]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}[g_{\mu \nu },\pi _{\mu \nu }] = {\mathcal {K}}[g_{\mu \nu },\pi _{\mu \nu }] - S[g_{\mu \nu }]$$\end{document}, playing formally the role of a Hamilton function, where S[gμν]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S[g_{\mu \nu }]$$\end{document} is the standard Einstein–Hilbert action while K[gμν,πμν]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {K}}[g_{\mu \nu },\pi _{\mu \nu }]$$\end{document} is a new term including the kinetic degrees of freedom of the field. Such an action allows us to define an ensemble for the quantum fluctuations of gμν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{\mu \nu }$$\end{document} analogous to the microcanonical one in statistical mechanics, with the only difference that in the present case one has conservation of the generalized action A[gμν,πμν]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}[g_{\mu \nu },\pi _{\mu \nu }]$$\end{document} and not of energy. Since the Einstein–Hilbert action S[gμν]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S[g_{\mu \nu }]$$\end{document} plays the role of a potential term in the new pseudo-Hamiltonian formalism, it can fluctuate along the symplectic action-preserving dynamics. These fluctuations are the quantum fluctuations of gμν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{\mu \nu }$$\end{document}. Finally, we show how the standard path-integral approach to gravity can be obtained as an approximation of the symplectic quantization approach. By doing so we explain how the integration over the conjugated momentum field πμν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{\mu \nu }$$\end{document} gives rise to a cosmological constant term in the path-integral approach. (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.

Gravitomagnetic equations result from applying quaternionic differential operators to the energy–momentum tensor. These equations are similar to the Maxwell’s EM equations. Both sets of the equations are isomorphic after changing orientation of either the gravitomagnetic orbital force or the magnetic induction. The gravitomagnetic equations turn out to be parent equations generating the following set of equations: the vorticity equation giving solutions of vortices with nonzero vortex cores and with infinite lifetime; the Hamilton–Jacobi equation loaded by the quantum potential. This (...) equation in pair with the continuity equation leads to getting the Schrödinger equation describing a state of the superfluid quantum medium ; gravitomagnetic wave equations loaded by forces acting on the outer space. These waves obey to the Planck’s law of radiation. (shrink)

The modern use of algebra to describe geometric ideas is discussed with particular reference to the constructions of Grassmann and Hamilton and the subsequent algebras due to Clifford. An Einstein addition law for nonparallel boosts is shown to follow naturally from the use of the representation-independent form of the geometric algebra of space-time.

The idea of obtaining a pilot-wave quantum theory on a lattice with discrete time is presented. The motion of quantum particles is described by a \-distributed Markov chain. Stochastic matrices of the process are found by the discrete version of the least-action principle. Probability currents are the consequence of Hamilton’s principle and the stochasticity of the Markov process is minimized. As an example, stochastic motion of single particles in a double-slit experiment is examined.

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)

Contrary to the predominant way of doing physics, we claim that the geometrical structure of a general differentiable space-time manifold can be determined from purely dynamical considerations. Anyn-dimensional manifoldV a has associated with it a symplectic structure given by the2n numbersp andx of the2n-dimensional cotangent fiber bundle TVn. Hence, one is led, in a natural way, to the Hamiltonian description of dynamics, constructed in terms of the covariant momentump (a dynamical quantity) and of the contravariant position vectorx (a geometrical quantity). (...) That is, the Hamiltonian description furnishes a natural way of relating dynamics and geometry. Thus, starting from the Hamiltonian state function (for a particle)—taken as the fundamental dynamical entity—we show that general relativistic physics implies a general pseudo-Riemannian geometry, whereas the physics of the special theory of relativity is tied up with Minkowski space-time, and nonrelativistic dynamics is bound up to Newton-Cartan space-time. (shrink)

The revised Robertson's test theory of special relativity (SR) has been constructed upon a family of sets of passive coordinate transformations in flat space-time [J. G. Vargas and D. G. Torr,Found. Phys., 16, 1089 (1986)]. In the same paper, it has also been shown that the boosts depend in general on the velocities of the two frames involved and not only on their relative velocity. The only exception to this is SR, if one has previously used an appropriate constraint to (...) remove the other relativities—like Galilean relativity—from the family.In this paper we look at these coordinate transformations in the only way there is to do so, namely as transformations in a seven-dimensional “Cartan Space” (Cartan first considered this in his dealings with Newtonian kinematics). In this space, the boosts only depend on the relative velocity of the frames. The passive coordinate transformations in each set are shown to have a nonlinear group structure isomorphic to that of the Poincaré group.The existence of a preferred frame, except in SR, makes the active transformations inequivalent to the passive ones. It is shown that the composite active-passive transformations act on a ten-dimensional space and that each member set of the family also has a group structure. As a result, one ends up with a family of mutually isomorphic 9-parameter (homogeneous) supergroups and a family of mutually isomorphic (9 + 4)-parameter (inhomogeneous) supergroups. The presence of extra parameters could be looked upon as “internal” degrees of freedom, which are, however, an offshoot of the Robertson space-time. (shrink)

The concept of quantum information is introduced as both normed superposition of two orthogonal sub-spaces of the separable complex Hilbert space and in-variance of Hamilton and Lagrange representation of any mechanical system. The base is the isomorphism of the standard introduction and the representation of a qubit to a 3D unit ball, in which two points are chosen. The separable complex Hilbert space is considered as the free variable of quantum information and any point in it (a wave (...) function describing a state of a quantum system) as its value as the bound variable. A qubit is equivalent to the generalization of ‘bit’ from the set of two equally probable alternatives to an infinite set of alternatives. Then, that Hilbert space is considered as a generalization of Peano arithmetic where any unit is substituted by a qubit and thus the set of natural number is mappable within any qubit as the complex internal structure of the unit or a different state of it. Thus, any mathematical structure being reducible to set theory is re-presentable as a set of wave functions and a subspace of the separable complex Hilbert space, and it can be identified as the category of all categories for any functor represents an operator transforming a set (or subspace) of the separable complex Hilbert space into another. Thus, category theory is isomorphic to the Hilbert-space representation of set theory & Peano arithmetic as above. Given any value of quantum information, i.e. a point in the separable complex Hilbert space, it always admits two equally acceptable interpretations: the one is physical, the other is mathematical. The former is a wave function as the exhausted description of a certain state of a certain quantum system. The latter chooses a certain mathematical structure among a certain category. Thus there is no way to be distinguished a mathematical structure from a physical state for both are described exhaustedly as a value of quantum information. This statement in turn can be utilized to be defined quantum information by the identity of any mathematical structure to a physical state, and also vice versa. Further, that definition is equivalent to both standard definition as the normed superposition and in-variance of Hamilton and Lagrange interpretation of mechanical motion introduced in the beginning of the paper. Then, the concept of information symmetry can be involved as the symmetry between three elements or two pairs of elements: Lagrange representation and each counterpart of the pair of Hamilton representation. The sense and meaning of information symmetry may be visualized by a single (quantum) bit and its interpretation as both (privileged) reference frame and the symmetries of the Standard model. (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)

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)

In a previous paper, an elementary and thoroughly arithmetical proof of Fermat’s last theorem by induction has been demonstrated if the case for “n = 3” is granted as proved only arithmetically (which is a fact a long time ago), furthermore in a way accessible to Fermat himself though without being absolutely and precisely correct. The present paper elucidates the contemporary mathematical background, from which an inductive proof of FLT can be inferred since its proof for the case for “n (...) = 3” has been known for a long time. It needs “Hilbert mathematics”, which is inherently complete unlike the usual “Gödel mathematics”, and based on “Hilbert arithmetic” to generalize Peano arithmetic in a way to unify it with the qubit Hilbert space of quantum information. An “epoché to infinity” (similar to Husserl’s “epoché to reality”) is necessary to map Hilbert arithmetic into Peano arithmetic in order to be relevant to Fermat’s age. Furthermore, the two linked semigroups originating from addition and multiplication and from the Peano axioms in the final analysis can be postulated algebraically as independent of each other in a “Hamilton” modification of arithmetic supposedly equivalent to Peano arithmetic. The inductive proof of FLT can be deduced absolutely precisely in that Hamilton arithmetic and the pransfered as a corollary in the standard Peano arithmetic furthermore in a way accessible in Fermat’s epoch and thus, to himself in principle. A future, second part of the paper is outlined, getting directed to an eventual proof of the case “n=3” based on the qubit Hilbert space and the Kochen-Specker theorem inferable from it. (shrink)

Philosophers of physics and physicists have long been intrigued by the analogies and disanalogies between gravitational theories and gauge theories. Indeed, repeated attempts to collapse these disanalogies have made us acutely aware that there are fairly general obstacles to doing so. Nonetheless, there is a special case space-time dimensions) in which gravity is often claimed to be identical to a gauge theory. I subject this claim to philosophical scrutiny in this article. In particular, I analyse how the standard disanalogies can (...) be overcome in dimensions, and consider whether really licenses the interpretation of gravity as a gauge theory. Our conceptual analysis reveals more subtle disanalogies between gravity and gauge, and connects these to interpretive issues in classical and quantum gravity. 1 Introduction1.1 Motivation1.2 Prospectus2 Disanalogies3 Three-dimensional gravity and gauge3.1 gravity3.2 Chern–Simons3.2.1 Cartan geometry3.2.2 Overcoming obst-gauge via Cartan connections3.3 Disanalogies collapsed4 Two More Disanalogies4.1 What about the symmetries?4.2 The phase spaces of the two theories5 Summary and Conclusion. (shrink)

A fiber bundle constructed over spacetime is used as the basic underlying framework for a differential geometric description of extended hadrons. The bundle has a Cartan connection and possesses the de Sitter groupSO(4, 1) as structural group, operating as a group of motion in a locally defined space of constant curvature (the fiber) characterized by a radius of curvatureR≈10−13 cm related to the strong interactions. A hadronic matter field ω(x, ζ) is defined on the bundle space, withx the spacetime (...) coordinate and ζ varying in the local fiber. The components of a generalized tensor current ℑ μab (M) (x) are introduced, involving a bilinear expression in the fields ω(x, ζ) and ωΔ(x, ζ) integrated over the local fiber at the pointx. This hadronic matter current is considered as a source current for the underlying fiber geometry by coupling it in a gauge-invariant manner to the curvature expressions derived from the bundle connection coefficients, which are associated here with strong interaction effects, i.e., play the role of meson fields induced in the geometry. Studying discrete symmetry transformations between the 16 differently soldered Cartan bundles, a generalized matter-antimatter conjugation Ĉ is established which leaves the basic current-curvature equations Ĉ-invariant. The discrete symmetry transformation Ĉ turns out to be the direct product of an ordinary charge conjugation for the Dirac point-spinor part of ω(x, ζ) and an internal $\hat P\hat T$ transformation applied globally on the bundle to the fiber (i.e., de Sitter) part of ω(x, ζ). (shrink)