Recently, Dewar (2019) has suggested that one can apply the strategy of 'sophistication' - as exemplified by sophisticated substantivalism as a response to the diffeomorphism invariance of General Relativity - to gauge theories such as electrodynamics. This requires a shift to the formalism of fibre bundles. In this paper, I develop and defend this suggestion. Where my approach differs from previous discussions is that I focus on the metaphysical picture underlying the fibre bundle formalism. In particular, I (...) aim to affirm the physical reality of gauge properties. I argue that this allows for a local and separable explanation of the Aharonov-Bohm effect. Its puzzling features are explained by a form of holism inherent to fibre bundles. (shrink)

This paper examines the so-called "gauge argument" sometimes used by physicists to motivate the introduction of gauge fields, here facilitated by an informal exposition of the fiber bundle formalism. The discussion suggests some preliminary ways of understanding the connection between gauge fields and interactions.

Those enlightened philosophers of physics acknowledging some manner of descent from Kant’s ‘Copernican Revolution’ have long found encouragement and inspiration in the writings of Roberto Torretti. In this tribute, I focus on his “perspective on Kant’s perspective on objectivity” (2008), a short but highly stimulating attempt to extract the essential core of the Kantian doctrine that ‘objects of knowledge’ are constituted, not given, or with Roberto’s inimitable pungency, that “objectivity is an achievement, not a gift.” That essential core Roberto locates (...) in the Kantian notion of apperception, or self-activity, manifested in cognition in the idea of combination (Verbindung) or composition, which, Kant tells us, “among all ideas … is the one that is not given through objects, but can only be performed by the subject itself, because it is an act of self-activity” (B 130). I first rehearse Roberto’s proposal for how an imaginative interplay between sensibility and understanding can be fashioned via the productive imagination or power of reflective judgment (of the third Critique). In this way, the notion of composition in general, unfettered from needless period constraints issuing in “pure forms of sensibility” and “pure concepts of the understanding”, can be seen as the intellectual motor for the “free creation” of concepts celebrated by Einstein and others, furnishing structural scaffolding required to articulate and display physical objects and processes, a conceptual panoply that “cannot be fished out of the stream of impressions”. Roberto emphasizes that historical case studies are needed to evaluate his proposal, suggesting one himself, the continuous conceptual development inaugurated by Riemann’s Habilitätionsschrift (1854) resulting, some hundred years later, in the fiber bundle formalism of modern differential geometry and topology. I sketch a related suggestion, that the gauge groups of modern particle physics are the outcome of a similar line of conceptual advance, a structural scaffolding saving the phenomena of high-energy experiment within the framework of ‘effective field theory.’. (shrink)

A draft of a possible comparison between the use made of mathematics in classical field theories and in quantum mechanics is presented. Hilbert’s space formalism, although not only elegant and powerful but intuitive as well, does not give us a spatio-temporal representation of physical events. The picture of the electromagnetic field as an entity which is real in itself– i.e., as a wave without support – fostered by the emergence of special relativity can be seen as the first step, (...) favored by many physicists and philosophers, of a gradual “escape” from intuition into a purely mathematical representation of the external world. After the introduction, in recent theoretical physics, of fiber bundle formalism the classical notion of field acquires a new spatio-temporal intuitiveness. This intuitiveness is clearly foreshadowed in the Kantian and Meinongian analysis of the notion of magnitude. At the end of the paper we show that, contrary to what happens in quantum mechanics, mathematics plays a truly explicative role in general relativity, without any loss of spatio-temporal intuitiveness. (shrink)

This paper presents two interpretations of the fiber bundle formalism that is applicable to all gauge field theories. The constructionist interpretation yields a substantival spacetime. The analytic interpretation yields a structural spacetime, a third option besides the familiar substantivalism and relationalism. That the same mathematical formalism can be derived in two different ways leading to two different ontological interpretations reveals the inadequacy of pure formal arguments.

Philosophy and Phenomenological Research, EarlyView.

In this review paper, we discuss how gravity and spin can be obtained as the realization of the local Conformal-Affine group of symmetry transformations. In particular, we show how gravitation is a gauge theory which can be obtained starting from some local invariance as the Poincaré local symmetry. We review previous results where the inhomogeneous connection coefficients, transforming under the Lorentz group, give rise to gravitational gauge potentials which can be used to define covariant derivatives accommodating minimal couplings of matter, (...) gauge fields (and then spin connections). After we show, in a self-contained approach, how the tetrads and the Lorentz group can be used to induce the spacetime metric and then the Invariance Induced Gravity can be directly obtained both in holonomic and anholonomic pictures. Besides, we show how tensor valued connection forms act as auxiliary dynamical fields associated with the dilation, special conformal and deformation (shear) degrees of freedom, inherent to the bundle manifold. As a result, this allows to determine the bundle curvature of the theory and then to construct boundary topological invariants which give rise to a prototype (source free) gravitational Lagrangian. Finally, the Bianchi identities, the covariant field equations and the gauge currents are obtained determining completely the dynamics. (shrink)

I articulate and discuss a geometrical interpretation of Yang–Mills theory. Analogies and disanalogies between Yang–Mills theory and general relativity are also considered.

In this paper I discuss Tim Maudlin’s attempt to reject the theory of universals based on the interpretation of gauge theories in the fiber bundle framework. The project is novel and assuring, but, I argue, it is vulnerable to several objections stemming from both metaphysics and physics. I complement his project by emphasizing two missing elements: first, a commitment to realism; second, the fundamentality or non-fundamentality of gauge theories.

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)

We use the notion of polar duality from convex geometry and the theory of Lagrangian planes from symplectic geometry to construct a fiber bundle over ellipsoids that can be viewed as a quantum-mechanical substitute for the classical symplectic phase space. The total space of this fiber bundle consists of geometric quantum states, products of convex bodies carried by Lagrangian planes by their polar duals with respect to a second transversal Lagrangian plane. Using the theory of the (...) John ellipsoid we relate these geometric quantum states to the notion of “quantum blobs” introduced in previous work; quantum blobs are the smallest symplectic invariant regions of the phase space compatible with the uncertainty principle. We show that the set of equivalence classes of unitarily related geometric quantum states is in a one-to-one correspondence with the set of all Gaussian wavepackets. We emphasize that the uncertainty principle appears in this paper as geometric property of the states we define, and is not expressed in terms of variances and covariances, the use of which was criticized by Hilgevoord and Uffink. (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 geometro-stochastic quantization of a gauge theory based on the (4,1)-de Sitter group is presented. The theory contains an intrinsic elementary length parameter R of geometric origin taken to be of a size typical for hadron physics. Use is made of a soldered Hilbert bundle ℋ over curved spacetime carrying a phase space representation of SO(4, 1) with the Lorentz subgroup related to a vierbein formulation of gravitation. The typical fiber of ℋ is a resolution kernel Hilbert space (...) ℋ $_{\bar \eta }^{(\rho )} $ constructed in terms of generalized coherent states $\bar \eta $ ρ related to the principal series of unitary irreducible representations of SO(4, 1), namely de Sitter horospherical waves for spinless particles characterized by the parameter ρ. The framework is, finally, extended to a quantum field-theoretical formalism by using bundles with Fock space fibers constructed from ℋ $_{\bar \eta }^{(\rho )} $. (shrink)

Bundles of C*-algebras can be used to represent limits of physical theories whose algebraic structure depends on the value of a parameter. The primary example is the ℏ→0 limit of the C*-algebras of physical quantities in quantum theories, represented in the framework of strict deformation quantization. In this paper, we understand such limiting procedures in terms of the extension of a bundle of C*-algebras to some limiting value of a parameter. We prove existence and uniqueness results for such extensions. (...) Moreover, we show that such extensions are functorial for the C*-product, dynamical automorphisms, and the Lie bracket (in the ℏ→0 case) on the fiber C*-algebras. (shrink)

In this paper we formulate Einstein's gravitational theory with the Clifford bundle formalism. The formalism suggests interpreting the gravitational field in the sense of Faraday, i.e., with the field residing in Minkowski spacetime. We succeeded in discovering the condition for this interpretation to hold. For the variables that play the role of the gravitational field in our theory, the Lagrangian density turns out to be of the Yang-Mills type (with an auto-interaction plus gauge-fixing terms). We give a (...) brief comparison of our theory with other field theories of the gravitational field in the flat Minkowski spacetime. (shrink)

The geometro-stochastic quantization of a gauge theory for extended objects based on the (4, 1)-de Sitter group is used for the description of quantized matter in interaction with gravitation. In this context a Hilbert bundle ℋ over curved space-time B is introduced, possessing the standard fiber ℋ $_{\bar \eta }^{(\rho )} $ , being a resolution kernel Hilbert space (with resolution generator $\tilde \eta $ and generalized coherent state basis) carrying a spin-zero phase space representation of G=SO(4, 1) (...) belonging to the principal series of unitary irreducible representations determined by the parameter ρ. The bundle ℋ, associated to the de Sitter frame bundle P(B, G), provides a geometric arena with built-in fundamental length parameter R (taken to be of the order of 10−13 cm characterizing hadron physics) yielding, in the presence of gravitation, a quantum kinematical framework for the geometro-stochastic description of spinless matter described in terms of generalized quantum mechanical wave functions, Ψ x ρ (ξ, ζ), defined on #x210B;. By going over to a nonlinear realization of the de Sitter group with the help of a section ξ(x) on the soldered bundle E, associated to P, with homogeneous fiber V′4⋍G/H, one is able to recover gravitation in a de Sitter gauge invariant manner as a gauge theory related to the Lorentz subgroup H of G. ξ(x) plays the dual role of a symmetry-reducing and an extension field. After introducing covariant bilinear source currents in the fields Ψ x ρ (ξ, ζ) and their adjoints determined by G-invariant integration over the local fibers in ℋ, a quantum fiber dynamical (QFD) framework is set up for the dynamics at small distances in B determining the geometric quantities beyond the classical metric of Einstein's theory through a set of current-curvature field equations representing the source equations for axial vector torsion and the de Sitter boost contributions to the bundle connection (the latter defining the soldering forms of the Cartan connection in P(B, G) in the nonlinear gauge). The presented bundle framework yields a theory for quantized material objects in interaction with gravitation, the long-range metrical part of which remains classical. (shrink)

Like moths attracted to a bright light, philosophers are drawn to glitz. So in discussing the notions of ‘gauge’, ‘gauge freedom’, and ‘gauge theories’, they have tended to focus on examples such as Yang–Mills theories and on the mathematical apparatus of fibre bundles. But while Yang–Mills theories are crucial to modern elementary particle physics, they are only a special case of a much broader class of gauge theories. And while the fibre bundle apparatus turned out, in retrospect, to be (...) the right formalism to illuminate the structure of Yang–Mills theories, the strength of this apparatus is also its weakness: the fibre bundle formalism is very flexible and general, and, as such, fibre bundles can be seen lurking under, over, and around every bush. What is needed is an explanation of what the relevant bundle structure is and how it arises, especially for theories that are not initially formulated in fibre bundle language. Here I will describe an approach that grows out of the conviction that, at least for theories that can be written in Lagrangian/Hamiltonian form, gauge freedom arises precisely when there are Lagrangian/Hamiltonian constraints of an appropriate character. This conviction is shared, if only tacitly, by that segment of the physics community that works on constrained Hamiltonian systems. (shrink)

Historical work on the emergence of sheaf theory has mainly concentrated on the topological origins of sheaf cohomology in the period from 1945 to 1950 and on subsequent developments. However, a shift of emphasis both in time-scale and disciplinary context can help gain new insight into the emergence of the sheaf concept. This paper concentrates on Henri Cartan’s work in the theory of analytic functions of several complex variables and the strikingly different roles it played at two stages of the (...) emergence of sheaf theory: the definition of a new structure and formulation of a new research programme in 1940–1944; the unexpected integration into sheaf cohomology in 1951–1952. In order to bring this two-stage structural transition into perspective, we will concentrate more specifically on a family of problems, the so-called Cousin problems, from Poincaré (1883) to Cartan. This medium-term narrative provides insight into two more general issues in the history of contemporary mathematics. First, we will focus on the use of problems in theory-making. Second, the history of the design of structures in geometrically flavoured contexts—such as for the sheaf and fibre-bundle structures—which will help provide a more comprehensive view of the structuralist moment, a moment whose algebraic component has so far been the main focus for historical work. (shrink)

Those looking for holism in contemporary physics have focused their attention primarily on quantum entanglement. But some gauge theories arguably also manifest the related phenomenon of nonseparability. While the argument is strong for the classical gauge theory describing electromagnetic interactions with quantum “particles”, it fails in the case of general relativity even though that theory may also be formulated in terms of a connection on a principal fiber bundle. Anandan has highlighted the key difference in his analysis of (...) a supposed gravitational analog to the Aharonov-Bohm effect. By contrast with electromagnetism in the original Aharonov-Bohm effect, gravitation is separable and exhibits no novel holism in this case. Whether the nonseparability of classical gauge theories of non-gravitational interactions is associated with holism depends on what counts as the relevant part-whole relation. Loop representations of quantized gauge theories of non- gravitational interactions suggest that these conclusions about holism and nonseparability may extend also to quantum theories of the associated fields. (shrink)

After decades of neglect philosophers of physics have discovered gauge theories--arguably the paradigm of modern field physics--as a genuine topic for foundational and philosophical research. Incidentally, in the last couple of years interest from the philosophy of physics in structural realism--in the eyes of its proponents the best suited realist position towards modern physics--has also raised. This paper tries to connect both topics and aims to show that structural realism gains further credence from an ontological analysis of gauge theories--in particular (...) U(1) gauge theory. In the first part of the paper the framework of fiber bundle gauge theories is briefly presented and the interpretation of local gauge symmetry will be examined. In the second part, an ontological underdetermination of gauge theories is carved out by considering the various kinds of non-locality involved in such typical effects as the Aharonov-Bohm effect. The analysis shows that the peculiar form of non-separability figuring in gauge theories is a variant of spatiotemporal holism and can be distinguished from quantum theoretic holism. In the last part of the paper the arguments for a gauge theoretic support of structural realism are laid out and discussed. (shrink)

I defend the interpretation of the Aharonov-Bohm effect originally advanced by Aharonov and Bohm, i.e., that it is caused by an interaction between the electron and the vector potential. The defense depends on taking the fiber bundle formulation of electrodynamics literally, or almost literally.

It has been argued within some philosophy of quantum gravity circles that endorsing Lewisian modal metaphysics is incompatible with endorsing the fundamental physical ontology of any quantum gravity theory. Speaking concisely, the unsolvable tension would be between Lewis' metaphysical commitment to the fundamentality of space and time, and the physical lesson of quantum gravity about the disappearance of space and time from the fundamental structure of the world. In this essay I argue against the idea that the tension is unsolvable. (...) The analysis does not apply to quantum gravity in general, but only to the specific perspective delivered by quantum string theory. In the first two sections I describe what I think is the general formal payoff of Lewisian possible worlds semantics, and more importantly the general metaphysical payoff generated by some Lewisian metaphysical applications of that formal framework. Lewisian possible worlds semantics is not among the topics of this essay, still a concise presentation of its basic features is required for a more accurate description of the type of modal metaphysics Lewis delivers by applying that formal machinery. The moral of the story in the first two sections is that an attempt of preserving crucial features of Lewisian modal metaphysics, although endorsing the quantum string theory lesson on spacetime non fundamentality, would be not only possible, but also philosophically worthy in a general sense. Then, section three is divided in three parts. In the first part I summarize some formal tools of the methodology used elsewhere - (Vistarini, Routledge, 2019, chapter 6)- to argue for string theory background independence. More precisely, I briefly re-describe some crucial features of a topological-vector bundle structure there defined on a "space" associated to the theory. Such structure is there shown to carry physical and dynamical information about quantum strings systems supporting the thesis of background independence of quantum strings laws (string theory background independence is not a topic of this essay, and the construction of the full formal structure as well as the full argumentative line can be found in the work cited above). The rationale behind the choice of including these formal tools is explained in the second and third parts of section three: that very same topological-fiber bundle structure associated to string theory, if read differently, is shown to also carry structural properties of the Lewisian metaphysical scheme. More precisely, in the second part of section three I attempt to show that such topological-fiber bundle structure owns a topology which qualifies as a plausible accessibility structure extending the Lewisian similarity accessibility. To avoid confusion, I will denote the extended similarity order with S*, or S*-similarity. The latter has some interesting properties. It is quantitatively describable, and even more interestingly its spectrum of variation is a countable one. Then, insofar Lewisian similarity order can be regained from the S*-similarity, one might say that the latter is a metaphysical accessibility relation underlying the former because "more fundamental". As we will see, the explanation of what "more fundamental" means in this metaphysical context mainly involves the idea that the formal nature of S*-similarity, in particular its property of having a countable spectrum of variation, calibrates a metaphysical similarity that reflects the discreteness of the fundamental quantum nature of reality. The extended modal metaphysics with accessibility structure S*-similarity is informed by quantum string physics. Within the extended scheme, the Lewisian claim "things could have been different in countless ways" (Lewis 1973) is an approximate one because only permitted by ordinary language. As soon as one refines the accessibility structure by using the formal language of quantum string theory, it appears things could have been different in countable ways. But the Lewisian variety of alternative states of affairs can be regained from the underlying countable variety via a specific interpretation of the Humean supervenience thesis, one that changes the metaphysical status of the thesis' content and restricts the thesis' domain of application. Finally, in the third part of section three I attempt to show that the same topological-fiber bundle structure used to extend Lewisian similarity has a bundle component producing the formal scheme for a tentative extension of Lewisian nomological accessibility through the lens of string dualities. This will prove to be a non trivial task, and the Lewisian accessibility structure will be regained from the extended one only partially. Finally, the essay contains two short appendices summarizing two argumentative lines outside the main topic of the essay. They briefly describe different ways in which the non fundamentality of spatiotemporal structures can be found in quantum string theory. These sections can be skipped. Nevertheless, they might contribute to understand more robustly the general view endorsed by this essay, i.e. that there isn't any unsolvable tension between Lewis' commitment to the fundamentality of spatiotemporal relations and quantum gravity's lesson about their non fundamentality. -/- . (shrink)

Although the language-related fiber pathways in the human brain, such as the superior longitudinal fasciculus and arcuate fasciculus, are already well-known, understanding more sophisticated cortical regions connected by the fiber tracts is essential to scrutinize the structural connectivity of language circuits. With the regions of interest that were selected based on the Brainnetome atlas, the fiber orientation distribution estimation method for tractography was used to produce further elaborate connectivity information. The results indicated that both fiber bundles (...) had two distinct connections with the prefrontal cortex. The SLF-II and dorsal AF are mainly connected to the rostrodorsal part of the inferior parietal cortex and lateral part of the fusiform gyrus with the inferior frontal junction, respectively. In contrast, the SLF-III and ventral AF were primarily linked to the anterior part of the supramarginal gyrus and superior part of the temporal cortex with the inferior frontal cortex, including the Broca's area. Moreover, the IFJ in the PFC, which has rarely been emphasized as a language-related subregion, also had the strongest connectivity with the previously known language-related subregions among the PFC; consequently, we proposed that these specific regions are interconnected via the SLF and AF within the PFC, IPC, and temporal cortex as language-related circuitry. (shrink)

Differential tractography and correlation tractography are new tractography modalities to study neuronal changes in brain diseases, but their performances in detecting neuronal injuries are yet to be investigated in patients with mild traumatic brain injury. Here we investigated the white matter injury in mTBI patients using differential and correlation tractography. The diffusion MRI was acquired at 33 mTBI patients and 31 health controls. 7 of the mTBI patients had one-year follow-up scans, and differential tractography was used to evaluate injured (...) class='Hi'>fiber bundles on these 7 patients. All subjects were evaluated using digital symbol substitution test and trail making test A, and the correlation tractography was performed to explore the exact pathways related to the cognitive performance. Our results showed that differential tractography revealed neuronal changes in the corpus callosum in all 7 follow-up mTBI patients with FDR between 0.007 and 0.17. Further, the correlation tractography showed that the splenium of the corpus callosum, combined with the right superior longitudinal fasciculus and right cingulum, were correlated with DSST in the acute mTBI patients. The cognitive impairment findings in the acute stage and the longitudinal findings in the corpus callosum in the chronic stage of mTBI patients suggest that differential tractography and correlation tractography are valuable tools in the diagnostic and prognostic evaluation of neuronal injuries in mTBI patients. (shrink)

This chapter explains how mindmelding — the direct experience by one person of another's conscious states — is in fact possible. The temporal lobes causally interact with the prefrontal lobes by way of fiber bundles that run underneath the cortical surface. This provides the perfect first experiment in mindmelding: to ‘branch’ those fiber bundles and run the other end into the brain of another person. Evidence is provided that these bundles have close connections to consciousness, in that whatever (...) affects them has immediate effects on consciousness. Then, before responding to several objections, the chapter considers another issue brought up by these experiments — the question of the relation between mindmelding and mindreading. Is mindmelding similar to mindreading? Does the existence of a mindreading system help us achieve mindmelding? (shrink)

Purpose: We aimed to find out the distributed functional connectome of white matter in patients with functional dyspepsia.Methods: 20 patients with FD and 24 age- and gender-matched healthy controls were included into the study. The functional connectome of white matter and graph theory were used to these participants. Two-sample t-test was used for the detection the abnormal graph properties in FD. Pearson correlation was used for the relationship between properties and the clinical and neuropshychological information.Results: Patients with FD and healthy (...) controls showed small-world properties in functional connectome of white matter. Compared with healthy controls, the FD group showed decreased global properties. Four pairs of fiber bundles that are connected to the frontal lobe, insula, and thalamus were affected in the FD group. Duration and Pittsburgh Sleep Quality Index positively correlated with the betweenness centrality of white matter regions of interest.Conclusion: FD patients turned to a non-optimized functional organization of WM brain network. Frontal lobe, insula, and thalamus were key regions in brain information exchange of FD. It provided some novel imaging evidences for the mechanism of FD. (shrink)

The eliminative view of gauge degrees of freedom—the view that they arise solely from descriptive redundancy and are therefore eliminable from the theory—is a lively topic of debate in the philosophy of physics. Recent work attempts to leverage properties of the QCD $$\theta _{\text {YM}}$$ θ YM -term to provide a novel argument against the eliminative view. The argument is based on the claim that the QCD $$\theta _{\text {YM}}$$ θ YM -term changes under “large” gauge transformations. Here we review (...) geometrical propositions about fiber bundles that unequivocally falsify these claims: the $$\theta _{\text {YM}}$$ θ YM -term encodes topological features of the fiber bundle used to represent gauge degrees of freedom, but it is fully gauge-invariant. Nonetheless, within the essentially classical viewpoint pursued here, the physical role of the $$\theta _{\text {YM}}$$ θ YM -term shows the physical importance of bundle topology (or superpositions thereof) and thus counts against (a naive) eliminativism. (shrink)

This paper presents two interpretations of the fiber bundle fonnalism that is applicable to all gauge field theories. The constructionist interpretation yields a substantival spacetime. The analytic interpretation yields a structural spacetime, a third option besides the familiar substantivalism and relationalism. That the same mathematical fonnalism can be derived in two different ways leading to two different ontological interpretations reveals the inadequacy of pure fonnal arguments.

A survey is given of the concepts of interaction (force) and matter, i.e., of process and substance. The development of these concepts, first in antiquity, then in early modern times, and finally in the contemporary system of quantum field theory is described. After a summary of the basic phenomenological attributes (coupling strengths, symmetry quantities, charges), the common ground of concepts of quantum field theory for both interactions and matter entities is discussed. Then attention is focused on the gauge principle which (...) has been developed to describe all interaction fields in the same way, and hopefully to unite them all into one unified field. While a similar unification of all fundamental types of matter fields (quarks and leptons) into one family may be possible (SU 5), there still remains at this level a duality between interaction quanta (bosons with spin 1) and matter particles (fermions with spin 1/2). Whether this duality may be removed in some future supersymmetric theory is not discussed in this paper. Nor is Quantum Gravitation discussed, though the analogy of the gauge principle for the three fundamental non-gravitational interactions (hadronic, electromagnetic and weak) to Einstein's principle of equivalence for gravitation in spacetime is noted. However, the equivalence concept is applied not to spacetime but to the internal spaces for the matter (or charge) fields which are the sources between which the fundamental interactions operate. The gauge principle states that a change in the measures of the internal space charge of gauge or phases of the matter fields is equivalent to, and can be compensated by, suitably introduced interaction fields. From such an interaction field, the gauge potential field in the internal space, one may derive a gauge force field by exterior differentiation.Geometrically, the collection of all internal spaces, one over each point of spacetime, constitutes a fiber bundle. The gauge potential field represents a connection on the fiber bundle, and the gauge force field is the curvature (calculated by taking the exterior derivative of the connection and adding to it the exterior product of the connection with itself). Thus, just as gravitational force is interpreted as spacetime curvature, so the three other fundamental forces are interpretable as internal space curvature. The Standard Model which unites the three non-gravitational fields into an SU c 3 ×SU 2×U 1 structure, and the grand unified model, SU 5, are discussed briefly, and difficulties are noted. Finally it is suggested that a composite model, based on more subtle structure, may be needed to remove the present obscurities and difficulties that stand in the way of a unified theory. (shrink)

In a model of extended particles described by Minkowski space-time variables x, de Sitter internal variables ξ, a physical wave Ψ x (ξ) representing the proper characteristics of the particles, and a functional wave X [ Ψ ] giving previsions, we study functional propagation of X in the space of physical waves (as advocated by a quantum functional theory) as well as the nonlinear realization of the internal de Sitter group on its Lorentz subgroup (introduced by Drechsler). The first study (...) is undertaken in a special instance Ψ x (ξ) = ξ(x), while in the second the general structure of the model is adopted and curved space-time treated, but the functional propagation is not considered. A fiber bundle structure and an induced representation method are used. Propagators are derived, a quantum version of a variant of Drechsler's theory is obtained, and a nonlinear version of our model is constructed. (shrink)

The issue of the intrinsic nonlocality of quantum mechanics raised by J. S. Bell is examined from the point of view of the recently developed method of geometro-stochastic quantization and its applications to general relativistic quantum theory. This analysis reveals that a distinction should be made between the topological concept of locality used in formulating relativistic causality and a type of geometric locality based on the concept of fiber bundle, which can be used in extending the strong equivalence (...) principle to the quantum domain. Both play an essential role in formulating a notion of geometro-stochastic propagation based on quantum diffusions, which throws new light on the EPR paradox, on the origin of the arrow of time, and on other fundamental issues in quantum cosmology and the theory of measurement. (shrink)

This paper offers an alternative view of spacetime different from both substantivalism and relationism. Using basic ideas underlying the fiber bundle formulation of field theories, it illustrates the function of spacetime in individuating local fields. As the system of numerical identities for entities that we can individually refer to, spacetime is an intrinsic, indispensable, and inalienable structure of the physical world with distinct entities.

We review the issues of nonseparability and seemingly acausal propagation of information in EPR, as displayed by experiments and the failure of Bell's inequalities. We show that global effects are in the very nature of the geometric structure of modern physical theories, occurring even at the classical level. The Aharonov-Bohm effect, magnetic monopoles, instantons, etc. result from the topology and homotopy features of the fiber bundle manifolds of gauge theories. The conservation of probabilities, a supposedly highly quantum effect, (...) is also achieved through global geometry equations. The EPR observables all fit in such geometries, and space-time is a truncated representation and is not the correct arena for their understanding. Relativistic quantum field theory represents the global action of the measurement operators as the zero-momentum (and therefore spatially infinitely spread) limit of their wave functions (form factors). We also analyze the collapse of the state vector as a case of spontaneous symmetry breakdown in the apparatus-observed state interaction. (shrink)

A geometro-differential quantum theory of extended particles is presented. The geometrical selling is that of Hilbert fiber bundles whose base manifolds are pseudo-Riemannian space-times of points χ which are interpreted as partial aspects of physical reality (the extended particle). The fibers are carrier spaces of induced (internal configuration and momentum) representations of the structural group (the de Sitter group here). Sections of these bundles are seen as physical representations of the particle, and their values in the fibers are interpreted (...) as states of internally localized modes composing the particle. This geometrical structure is “analogous” to that of a geometro-stochastic quantization developed in recent years. The physical interpretation is a combination of those of an old functional quantum theory and of an induced representation scheme based on an interpretation of intertwining, between configuration and momentum representations, as localization, materialization, and propagation of particles. Our model is applied in two cases: (1) Induced representation is applied in both space-time and internal space: both have de Sitter symmetry whose connection is ignored. Intertwining is considered in both spaces in a composed fashion. (2) For generic spacetimes and when the connection is taken into account, intertwining is applied only in the internal space. Parallel transport is combined with intertwining for a redefinition of localization, materialization, and propagation (the latter in a path integral context inspired from geometro-stochastic quantization). (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)

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)

Summary The philosophical implications associated with the choice of a particular geometry required for the formulation of a dynamics at subnuclear distances are discussed. A dualism between geometry and matter — the former identified with a fiber bundle of Cartan type raised over space-time, the latter represented by a generalized quantum mechanical wave function — is presented as a possible framework for the dynamics of strongly interacting particles at distances of 10-13 cm.

Four empirically equivalent versions of general relativity, namely standard GR, Lorentz-invariant gravitational theory,and the gravitational gauge theories of the Lorentz and translation groups, are investigated in the form of a case study for theory underdetermination. The various ontological indeterminacies (both underdetermination and inscrutability of reference) inherent in gravitational theories are analyzed in a detailed comparative study. The concept of practical underdetermination is proposed, followed by a discussion of its adequacy to describe scientific progress.

Our purpose in this paper is to delineate an ontology for quantum mechanics that results adequate to the formalism of the theory. We will restrict our aim to the search of an ontology that expresses the conceptual content of the recently proposed modal-Hamiltonian interpretation, according to which the domain referred to by non-relativistic quantum mechanics is an ontology of properties. The usual strategy in the literature has been to focus on only one of the interpretive problems of the theory (...) and to design an interpretation to solve it, leaving aside the remaining difficulties. On the contrary, our aim in the present work is to formulate a “global” solution, according to which different problems can be adequately tackled in terms of a single ontology populated of properties, in which systems are bundles of properties. In particular, we will conceive indistinguishability between bundles as a relation derived from indistinguishability between properties, and we will show that states, when operating on combinations of indistinguishable bundles, act as if they were symmetric with no need of a symmetrization postulate. (shrink)

Our purpose in this paper is to delineate an ontology for quantum mechanics that results adequate to the formalism of the theory. We will restrict our aim to the search of an ontology that expresses the conceptual content of the recently proposed modal-Hamiltonian interpretation, according to which the domain referred to by non-relativistic quantum mechanics is an ontology of properties. The usual strategy in the literature has been to focus on only one of the interpretive problems of the theory (...) and to design an interpretation to solve it, leaving aside the remaining difficulties. On the contrary, our aim in the present work is to formulate a “global” solution, according to which different problems can be adequately tackled in terms of a single ontology populated of properties, in which systems are bundles of properties. In particular, we will conceive indistinguishability between bundles as a relation derived from indistinguishability between properties, and we will show that states, when operating on combinations of indistinguishable bundles, act as if they were symmetric with no need of a symmetrization postulate. (shrink)

The constrained Hamiltonian formalism is recommended as a means for getting a grip on the concepts of gauge and gauge transformation. This formalism makes it clear how the gauge concept is relevant to understanding Newtonian and classical relativistic theories as well as the theories of elementary particle physics; it provides an explication of the vague notions of “local” and “global” gauge transformations; it explains how and why a fibre bundle structure emerges for theories which do not wear (...) their bundle structure on their sleeves; it illuminates the connections of the gauge concept to issues of determinism and what counts as a genuine “observable”; and it calls attention to problems which arise in attempting to quantize gauge theories. Some of the limitations and problematic aspects of the formalism are also discussed. (shrink)

The paper deals with the issue of tying (as well as bundling) practices which are applied by dominant undertakings and which, under certain circumstances, can be considered as abuses of a dominant position. The authors describe the concept of tying, indicate its types, and reveal its economic aspects, since all these issues have a certain impact on the legal assessment of tying practices. The authors conclude that the European Commission (the Commission) and the European Community (EC) courts have usually been (...) too formalistic and unreasonably hostile to tying practices and have not paid due attention to the impact of tying on competition and the beneficial features of tying. Thus certain cases have frequently and rather easily been treated as abuses of a dominant position. A more realistic and effect-based approach was taken by the Commission and the courts in the Microsoft case, where certain criteria for the assessment of tying were formulated. Still, the Microsoft case reflects a rather strict attitude towards tying practices, especially in technological tie-ins. (shrink)

We investigated how the structure of the brain network relates to the stability of perceptual alternation in binocular rivalry. Historically, binocular rivalry has provided important new insights to our understandings in neuroscience. Although various relationships between the local regions of the human brain structure and perceptual switching phenomena have been shown in previous researches, the global organization of the human brain structural network relating to this phenomenon has not yet been addressed. To approach this issue, we reconstructed fiber-tract bundles (...) using diffusion tensor imaging and then evaluated the correlations between the speeds of perceptual alternation and fractional anisotropy (FA) values in each fiber-tract bundle integrating among 84 brain regions. The resulting comparison revealed that the distribution of the global organization of the structural brain network showed positive or negative correlations between the speeds of perceptual alternation and the FA values. First, the connections between the subcortical regions stably were negatively correlated. Second, the connections between the cortical regions mainly showed positive correlations. Third, almost all other cortical connections that showed negative correlations were located in one central cluster of the subcortical connections. This contrast between the contribution of the cortical regions to destabilization and the contribution of the subcortical regions to stabilization of perceptual alternation provides important information as to how the global architecture of the brain structural network supports the phenomenon of binocular rivalry. (shrink)

In this AV column we will have a look at the DUMAND project, a new $10 million detector funded by the US Department of Energy for the detection of ultra-high energy neutrinos. DUMAND stands for Deep Underwater Muon And Neutrino Detector. It is now under construction in Hawaii and will come into operation in 1993-94. It is to be placed almost 3 miles deep on a level stretch of Pacific Ocean bottom about 18 miles west of Keahole Point on the (...) Island of Hawaii. Floats anchored to the ocean bottom about 40 meters apart and arranged in an octagon around a central junction point will support nine long vertical strings of sensitive light detectors. DUMAND will be connected to its land-based laboratory by bundles of fiber optics cables. AT&T will lay the cables to shore, and the US Navy's manned deep submersible DSV Sea Cliff will be used to connect, service, and repair the parts of the detector. (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)

The present paper attempts to look at on the genealogy of both shared intentionality and collective intentionality, comparing Michael Tomasello’s concept with Max Scheler’s threedimensional concept of intentionality: ens amans, ens volens, ens cogitans, as affective, conative, and cognitive intentionality. I focus on various forms of affective collective intentionality — Schelerian forms of sympathy — to show collective subjectivity from the whole spectrum of emotional intentionality, presented by Scheler’s example of parents standing over the corpse of a child. Even though (...) Tomasello’s works seem to empirically corroborate Scheler’s intuitions about the emotional genealogy of collective intentionality, they will differ in the horizons within they locate intentionality. In the case of the evolutionary psychology of Tomasello, we can talk about the horizon of cooperation, in the case of Scheler’s Scheler’s phenomenology of acts about the horizon of responsibility or co-responsibility, which gives intentionality its unique character. The similarities of both concepts concern the following pillars: 1) genealogy of intentionality covering the dimension of affective intentionality, conative intentionality, and the level of cognitive intentionality; 2) Imitation or, as Scheler would say, following someone. Because a person is recognized by the author of Formalism as an act, or a bundle of acts, the way to understanding and communication with another person is the maieutic coperformance of their acts — i.e., imitation. The maieutic co-execution of acts of others triggers the constitution process of a person, both on an individual and community level. We can speak, in the case of Tomasello, about the ontho- and sociogenetic function of co-executing acts or imitating; however, in the case of Scheler, we are dealing with the clearly axiological nature of such a constitution of both the individual and collective subjectivity ; 3) collaborative engagement as a driving force behind collective intentionality in one case in form of co-responsibility, a nature of a collective person ; in the other case in form of collaboration developing intentionality to various units of community life. (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)

Logic and linguistics have engaged in a many-faceted dialogue since the very beginnings of both disciplines in Antiquity. While participants may have had diverse views over the ages, arguably, the dialogue has always revolved around the relationship between human thought and natural language. While there are those who see these two domains as one and the same, or as a case of one-directional influence , we beg to differ. To us, the long historical tradition of authors such as Arnauld, Boole, (...) Turing, or Jespersen demonstrates the much richer perspectives on language and reasoning that are needed, including connections with intelligence and computability.Another major historical theme in the above dialogue are similarities or convergences between natural and artificial languages. It is clear that natural languages are not only semantically broader, but also much closer to human communication than artificial languages, and so linguistic grammars have long dealt with a larger set of problems than logical formalism. This specal issue of JoLLI consists six selected papers presented in the "Logic and Linguisics" Workshop in UNI-LOG 2013 in Rio de Janeiro, plus two papers by invited authors: Johan van Benthem, and Marcus Kracht and Uno Klein. It contains a number of samples of what is intriguing and promising work at the interfaces of logic and language. (shrink)