Bundle Formalism Research Articles

Relating gauge gravity to string theory through obstruction

Abstract In this article we provide a more detailed account of the geometry and topology of the composite bundle formalism introduced by Tresguerres in Phys. Rev.
D 66 (2002) 064025~\cite{Tresguerres2002} to accommodate gravitation as a gauge theory. In the first half of the article we identify a global structure required by the composite construction which not only exposes how the ordinary frame and tangent bundle expected in general relativity arise but provides the link or connection between these ordinary bundles and the gauge bundles of the composite bundle construction. In the second half of the article we discuss implications of this method of constructing gravity as a fiber bundle for the global structure of spacetime. We find that the underlying manifold of the composite bundle construction is expected to admit, not only a spin structure but also a string structure. As a consequence of our work we are able to extend past work on global structures of physically reasonable spacetime manifolds. It has been shown that in four spacetime dimensions, that if an oriented, Lorentzian, four dimensional manifold is stably casual that it is parallizable, and hence admits a spin structure which allows for chiral spinors. We may now add to this that such a manifold also admits a string structure.

The metaphysics of fibre bundles

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.

Riemannian Calculus of Variations Using Strongly Typed Tensor Calculus

In this paper, the notion of strongly typed language will be borrowed from the field of computer programming to introduce a calculational framework for linear algebra and tensor calculus for the purpose of detecting errors resulting from inherent misuse of objects and for finding natural formulations of various objects. A tensor bundle formalism, crucially relying on the notion of pullback bundle, will be used to create a rich type system with which to distinguish objects. The type system and relevant notation is designed to “telescope” to accommodate a level of detail appropriate to a set of calculations. Various techniques using this formalism will be developed and demonstrated with the goal of providing a relatively complete and uniform method of coordinate-free computation. The calculus of variations pertaining to maps between Riemannian manifolds will be formulated using the strongly typed tensor formalism and associated techniques. Energy functionals defined in terms of first-order Lagrangians are the focus of the second half of this paper, in which the first variation, the Euler–Lagrange equations, and the second variation of such functionals will be derived.

An introduction to the relativistic kinetic theory on curved spacetimes

This article provides a self-contained pedagogical introduction to the relativistic kinetic theory of a dilute gas propagating on a curved spacetime manifold (M,g) of arbitrary dimension. Special emphasis is made on geometric aspects of the theory in order to achieve a formulation which is manifestly covariant on the relativistic phase space. Whereas most previous work has focussed on the tangent bundle formulation, here we work on the cotangent bundle associated with (M,g) which is more naturally adapted to the Hamiltonian framework of the theory. In the first part of this work we discuss the relevant geometric structures of the cotangent bundle T*M, starting with the natural symplectic form on T*M, the one-particle Hamiltonian and the Liouville vector field, defined as the corresponding Hamiltonian vector field. Next, we discuss the Sasaki metric on T*M and its most important properties, including the role it plays for the physical interpretation of the one-particle distribution function. In the second part of this work we describe the general relativistic theory of a collisionless gas, starting with the derivation of the collisionless Boltzmann equation for a neutral simple gas. Subsequently, the description is generalized to a charged gas consisting of several species of particles and the general relativistic Vlasov-Maxwell equations are derived for this system. The last part of this work is devoted to a transparent derivation of the collision term, leading to the general relativistic Boltzmann equation on (M,g). The meaning of global and local equilibrium and the stringent restrictions for the existence of the former on a curved spacetime are discussed. We close this article with an application of our formalism to the expansion of a homogeneous and isotropic universe filled with a collisional simple gas and its behavior in the early and late epochs. [abbreviated version]

Covariant-differential formulation of Lagrangian field theory

Building on the Utiyama principle we formulate an approach to Lagrangian field theory in which exterior covariant differentials of vector-valued forms replace partial derivatives, in the sense that they take up the role played by the latter in the usual jet bundle formulation. Actually a natural Lagrangian can be written as a density on a suitable “covariant prolongation bundle”; the related momenta turn out to be natural vector-valued forms, and the field equations can be expressed in terms of covariant exterior differentials of the momenta. Currents and energy-tensors naturally also fit into this formalism. The examples of bosonic fields and spin one-half fields, interacting with non-Abelian gauge fields, are worked out. The “metric-affine” description of the gravitational field is naturally included, too.

Quantum particles in general spacetimes: A tangent bundle formalism

Using tangent bundle geometry we construct an equivalent reformulation of classical field theory on flat spacetimes which simultaneously encodes the perspectives of multiple observers. Its generalization to curved spacetimes realizes a new type of nonminimal coupling of the fields and is shown to admit a canonical quantization procedure. For the resulting quantum theory we demonstrate the emergence of a particle interpretation, fully consistent with general relativistic geometry. The path dependency of parallel transport forces each observer to carry their own quantum state; we find that the communication of the corresponding quantum information may generate extra particles on curved spacetimes. A speculative link between quantum information and spacetime curvature is discussed which might lead to novel explanations for quantum decoherence and vanishing interference in double-slit or interaction-free measurement scenarios, in the mere presence of additional observers.

In-Rem Property in Adam Smith's Lectures on Jurisprudence

For decades, both economists and legal scholars have regularly used the metaphor of a ‘bundle of rights’ to describe property. In recent and growing literature, the bundle formulation has been the target of a number of critiques, some of which point to Adam Smith, among others, as a source for an alternative perspective on property. This paper examines Smith's work and argues that Smith is an appropriate authority and focal figure for a traditional, in-rem view of property. It illustrates how Smith's usage of concepts in the civil-law tradition, his appreciation for Hume, and especially his unique theory of the impartial spectator, together make clear that Smith's understanding of property is irreconcilable with the modern bundle formulation.

Bosonization of Fermionic Fields and Fermionization of Bosonic Fields

In this paper using the Clifford and spin-Clifford bundles formalism we show how Weyl and Dirac equations formulated in the spin-Clifford bundle may be written in an equivalent form as generalized Maxwell like form formulated in the Clifford bundle. Moreover, we show how Maxwell equation formulated in the Clifford bundle formalism can be written as an equivalent equation for a spinor field in the spin-Cillford bundle. Investigating the details of such equivalences this exercise shows explicitly that a fermionic field is equivalent (in a precise sense) to an equivalence class of well defined boson fields and that a bosonic field is equivalent to a well defined equivalence class of fermionic fields. These equivalences may be called the bosonization of fermionic fields and the fermionization of bosonic fields.

A categorical equivalence between generalized holonomy maps on a connected manifold and principal connections on bundles over that manifold

A classic result in the foundations of Yang-Mills theory, due to Barrett [Int. J. Theor. Phys. 30, 1171–1215 (1991)], establishes that given a “generalized” holonomy map from the space of piece-wise smooth, closed curves based at some point of a manifold to a Lie group, there exists a principal bundle with that group as structure group and a principal connection on that bundle such that the holonomy map corresponds to the holonomies of that connection. Barrett also provided one sense in which this “recovery theorem” yields a unique bundle, up to isomorphism. Here we show that something stronger is true: with an appropriate definition of isomorphism between generalized holonomy maps, there is an equivalence of categories between the category whose objects are generalized holonomy maps on a smooth, connected manifold and whose arrows are holonomy isomorphisms, and the category whose objects are principal connections on principal bundles over a smooth, connected manifold. This result clarifies, and somewhat improves upon, the sense of “unique recovery” in Barrett’s theorems; it also makes precise a sense in which there is no loss of structure involved in moving from a principal bundle formulation of Yang-Mills theory to a holonomy, or “loop,” formulation.

Superluminal Motion of Free Spin-half Particles in a Fiber Bundle Formalism

Superluminal Motion of Free Spin-half Particles in a Fiber Bundle Formalism

Notes on Conservation Laws, Equations of Motion of Matter, and Particle Fields in Lorentzian and Teleparallel de Sitter Space-Time Structures

We discuss the physics of interacting fields and particles living in a de Sitter Lorentzian manifold (dSLM), a submanifold of a 5-dimensional pseudo-Euclidean (5dPE) equipped with a metric tensor inherited from the metric of the 5dPE space. The dSLM is naturally oriented and time oriented and is the arena used to study the energy-momentum conservation law and equations of motion for physical systems living there. Two distinct de Sitter space-time structuresMdSLandMdSTPare introduced given dSLM, the first equipped with the Levi-Civita connection of its metric field and the second with a metric compatible parallel connection. Both connections are used only as mathematical devices. Thus, for example,MdSLis not supposed to be the model of any gravitational field in the General Relativity Theory (GRT). Misconceptions appearing in the literature concerning the motion of free particles in dSLM are clarified. Komar currents are introduced within Clifford bundle formalism permitting the presentation of Einstein equation as a Maxwell like equation and proving that in GRT there are infinitely many conserved currents. We prove that in GRT even when the appropriate Killing vector fields exist it is not possible to define a conserved energy-momentum covector as in special relativistic theories.

Jacobi multipliers, non-local symmetries, and nonlinear oscillators

Constants of motion, Lagrangians, and Hamiltonians admitted by a family of relevant nonlinear oscillators are derived using a geometric formalism. The theory of the Jacobi last multiplier allows us to find Lagrangian descriptions and constants of the motion. An application of the jet bundle formulation of symmetries of differential equations is presented in the second part of the paper. After a short review of the general formalism, the particular case of non-local symmetries is studied in detail by making use of an extended formalism. The theory is related to some results previously obtained by Krasil’shchik, Vinogradov, and coworkers. Finally, the existence of non-local symmetries for such two nonlinear oscillators is proved.

Elko Spinor Fields and Massive Magnetic Like Monopoles

In this paper we recall that by construction Elko spinor fields of λ and ρ types satisfy a coupled system of first order partial differential equations (csfopde) that once interacted leads to Klein-Gordon equations for the λ and ρ type fields. Since the csfopde is the basic one and since the Klein-Gordon equations for λ and ρ possess solutions that are not solutions of the csfopde for λ and ρ we infer that it is legitimate to attribute to those fields mass dimension 3/2 (as is the case of Dirac spinor fields) and not mass dimension 1 as previously suggested in recent literature (see list of references). A proof of this fact is offered by deriving the csfopde for the λ and ρ from a Lagrangian where these fields have indeed mass dimension 3/2. Taking seriously the view that Elko spinor fields due to its special properties given by their bilinear invariants may be the description of some kind of particles in the real world a question then arises: what is the physical meaning of these fields? Here we proposed that the fields λ and ρ serve the purpose of building the fields $\mathcal {K}\in \mathcal {C}\ell ^{0} (M,\eta )\otimes {\mathbb {R}}_{1,3}^{0}$ and $\mathcal {M}\in \sec \mathcal {C}\ell ^{0}(M,\eta ) \otimes {\mathbb {R}}_{1,3}^{0}$ (see (37)). These fields are electrically neutral but carry magnetic like charges which permit them to couple to a $su(2)\simeq {\textit {spin}}_{3,0}\subset {\mathbb {R}}_{3,0}^{0}$ valued potential $\mathcal {A}\in \sec {\textstyle \bigwedge \nolimits ^{1}} T^{\ast }M\otimes {\mathbb {R}}_{3,0}^{0}$ . If the field $\mathcal {A}$ is of short range the particles described by the $\mathcal {K}$ and $\mathcal {M}$ fields may be interacting and forming condensates of zero spin particles analogous to dark matter, in the sense that they do not couple with the electromagnetic field (generated by charged particles) and are thus invisible. Also, since according to our view the Elko spinor fields as well as the $\mathcal {K}$ and $\mathcal {M}$ fields are of mass dimension 3/2 we show how to calculate the correct propagators for the $\mathcal {K}$ and $\mathcal {M}$ fields. We discuss also the main difference between Elko and Majorana spinor fields, which are kindred since both belong to class five in Lounesto classification of spinor fields. Most of our presentation uses the representation of spinor fields in the Clifford bundle formalism, which makes very clear the meaning of all calculations.

The Hole Argument and Some Physical and Philosophical Implications.

This is a historical-critical study of the hole argument, concentrating on the interface between historical, philosophical and physical issues. Although it includes a review of its history, its primary aim is a discussion of the contemporary implications of the hole argument for physical theories based on dynamical, background-independent space-time structures.The historical review includes Einstein’s formulations of the hole argument, Kretschmann’s critique, as well as Hilbert’s reformulation and Darmois’ formulation of the general-relativistic Cauchy problem. The 1970s saw a revival of interest in the hole argument, growing out of attempts to answer the question: Why did three years elapse between Einstein’s adoption of the metric tensor to represent the gravitational field and his adoption of the Einstein field equations?The main part presents some modern mathematical versions of the hole argument, including both coordinate-dependent and coordinate-independent definitions of covariance and general covariance; and the fiber bundle formulation of both natural and gauge natural theories. By abstraction from continuity and differentiability, these formulations can be extended from differentiable manifolds to any set; and the concepts of permutability and general permutability applied to theories based on relations between the elements of a set, such as elementary particle theories.We are closing with an overview of current discussions of philosophical and physical implications of the hole argument.

Jet bundle formulation of infinite-dimensional port-Hamiltonian systems using differential operators

We consider infinite-dimensional port-Hamiltonian systems described on jet bundles. Based on a power balance relation we introduce the port-Hamiltonian system representation using differential operators regarding the structural mapping, the dissipation mapping and the input mapping. In contrast to the well-known representation on the basis of the underlying Stokes–Dirac structure our approach is not necessarily based on using energy-variables which leads to a different port-Hamiltonian representation of the analyzed partial differential equations. The presented constructions will be specialized to mechanical systems to which class also the presented examples belong.

Generation of a New Coupled Ultra-Short Pulse System from a Group Theoretical Viewpoint: the Cartan Ehresman Connection

Based upon the group theoretical jet bundle formalism introduced by Wahlquist and Estabrook for discussing the complete integrability of soliton systems, we investigate the prolongation structure of Wadati—Konno—Ichikawa isospectral evolution equations. As a result, we unearth a new physical coupled system entailing a hidden structural symmetry SL(3, R) arising in the description of ultra-short pulse propagation in optical nonlinear media. As a matter of fact, we depict a graphical representation of one-breather and two-breather ultra-short pulses in motion with a non-zero angular momentum. By extending the previous study to multidimensional symmetry SL(n, R), we unearth a more general class of multicomponent coupled nonlinear ultra-short pulse system with its associated inverse scattering formulation particularly useful in soliton theory.

Jet bundle formalism towards the extension of the stretched rope equation

In this paper, based upon a jet bundle formalism, we investigate the connection and the curvature associated to the stretched rope equation. Following the prolongation structure analysis due to Wahlquist and Estabrook combined to Cartan–Ehresmann connection with structure group SL(3,R), we derive an interesting coupled system dubbed as the two-component stretched rope equation which physically and geometrically arises from the curve motion flow in Euclidean space.

Pair and impair, even and odd form fields, and electromagnetism

AbstractIn this paper after reviewing the Schouten and de Rham definition of impair and pair differential form fields (not to be confused with differential form fields of even and odd grades) we prove that in a relativistic spacetime it is possible (despite claims in contrary) to coherently formulate electromagnetism (and we believe any other physical theory) using only pair form fields or, if one wishes, using pair and impair form fields together, in an appropriate way. Those two distinct descriptions involve only a mathematical choice and do not seem to lead to any observable physical consequence if due care is taken. Moreover, we show in details that a formulation of electromagnetic theory in the Clifford bundle formalism of differential forms where the two Maxwell equations of the so called metric‐free approach becomes a single equation is compatible with both formulations of electromagnetism just mentioned above. In addition we derive directly from Maxwell equation the density of force (coupling of the electromagnetic field with the charge current) that is a postulate in the free metric approach to electromagnetism. We recall also a formulation of the engineering version of Maxwell equations using electric and magnetic fields as objects of the same nature, i.e., without using polar and axial vectors.

Pair and impair, even and odd form fields, and electromagnetism

In this paper after reviewing the Schouten and de Rham definition of impair and pair differential form fields (not to be confused with differential form fields of even and odd grades) we prove that in a relativistic spacetime it is possible (despite claims in contrary) to coherently formulate electromagnetism (and we believe any other physical theory) using only pair form fields or, if one wishes, using pair and impair form fields together, in an appropriate way. Those two distinct descriptions involve only a mathematical choice and do not seem to lead to any observable physical consequence if due care is taken. Moreover, we show in details that a formulation of electromagnetic theory in the Clifford bundle formalism of differential forms where the two Maxwell equations of the so called free metric approach becomes a single equation is compatible with both formulations of electromagnetism just mentioned above. Moreover we derive directly from Maxwell equation the density of force (coupling of the electromagnetic field with the charge current) that is a postulate in the free metric approach to electromagnetism. We recall also a formulation of the engineering version of Maxwell equations using electric and magnetic fields as objects of the same nature, i.e., without using polar and axial vectors.

The effective lorentzian and teleparallel spacetimes generated by a free electromagnetic field

In this paper we show that a free electromagnetic field living in Minkowski spacetime generates an effective Weitzenbock or an effective Lorentzian spacetime whose properties aredetermined in details. These results are possible because we found using the Clifford bundle formalism the noticeable result that the energy-momentum densities of a free electromagnetic field are sources of the Hodge duals of exact 2-form fields which satisfy Maxwell like equations.