Bundle Description Research Articles

Multicontact formulation for non-conservative field theories

A new geometric structure inspired by multisymplectic and contact geometries, which we call multicontact structure, is developed to describe non-conservative classical field theories. Using the differential forms that define this multicontact structure as well as other geometric elements that are derived from them while assuming certain conditions, we can introduce, on the multicontact manifolds, the variational field equations which are stated using sections, multivector fields, and Ehresmann connections on the adequate fiber bundles. Furthermore, it is shown how this multicontact framework can be adapted to the jet bundle description of classical field theories; the field equations are stated in the Lagrangian and the Hamiltonian formalisms both in the regular and the singular cases.

D-brane description from nontrivial M2-branes

We obtain the bosonic D-brane description of toroidally compactified non-trivial M2-branes with the unique property of having a purely discrete supersymmetric regularized spectrum with finite multiplicity. As a byproduct, we generalize the previous Hamiltonian formulation to describe a M2-brane on a completely general constant quantized background C3 denoted by us as CM2-brane. We show that under this condition, the theory is equivalent to a more restricted one, denoted as an M2-brane with C± fluxes, which has been shown to have good quantum behavior. As a result, the spectral properties of both sectors must be the same. We then obtain its bosonic D-brane description and discover new symmetries. We find that it contains a new symplectic gauge field in addition to the expected U(1) Dirac-Born-Infeld gauge symmetry and a nontrivial U(1) associated with the presence of 2-form fluxes. Its bundle description takes on a new structure in the form of a twisted torus bundle. By turning off some of the fields, the D-brane description of other toroidally nontrivial M2-brane sectors can be obtained from this one. The possibility of reinterpreting these sectors in terms of Dp-brane bound states is discussed. These D-brane descriptions constitute String theory sectors with a quantum consistent uplift to M-theory.

Consequences of a fiber bundle description of a lattice system

Some observable consequences that follow from a fiber bundle description of a tight binding condensed matter system on a lattice are described. The geometrical picture can be extended to describe non-periodic lattice structures, where a single Brillouin zone is predicted.

A Hilbert bundle description of differential K-theory

We give an infinite dimensional description of the differential K-theory of a manifold M. The generators are triples [H,A,ω] where H is a Z2-graded Hilbert bundle on M, A is a superconnection on H and ω is a differential form on M. The relations involve eta forms. We show that the ensuing group is the differential K-group Kˇ0(M). In addition, we construct the pushforward of a finite dimensional cocycle under a proper submersion with a Riemannian structure. We give the analogous description of the odd differential K-group Kˇ1(M). Finally, we give a model for twisted differential K-theory.

Geometric description of multiplier modules for Hilbert $C^{*}$ -modules in simple cases

In this article we suggest a vector bundle description for multiplier modules of vector bundles over noncompact spaces. We prove that the isomorphism classes of multiplier modules are dependent on the isomorphism classes of their underlying modules. This gives a way to evaluate the set of extensions of Hilbert modules in topological terms in simple cases.

Effects of a scalar scaling field on quantum mechanics

This paper describes the effects of a complex scalar scaling field on quantum mechanics. The field origin is an extension of the gauge freedom for basis choice in gauge theories to the underlying scalar field. The extension is based on the idea that the value of a number at one space time point does not determine the value at another point. This, combined with the description of mathematical systems as structures of different types, results in the presence of separate number fields and vector spaces, both as structures, at different space time locations. Complex number structures and vector spaces at each location, are scaled by a complex space time dependent scaling factor. The effect of this scaling factor on several physical and geometric quantities has been described in other work. Here the emphasis is on quantum mechanics of one and two particles, their states and properties. Multiparticle states are also briefly described. The effect shows as a complex, nonunitary, scalar field connection on a fiber bundle description of nonrelativistic quantum mechanics. The lack of physical evidence for the presence of this field so far means that the coupling constant of this field to fermions is very small. It also means that the gradient of the field must be very small in a local region of cosmological space and time. Outside this region there are no restrictions on the field gradient.

Fiber bundle description of number scaling in gauge theory and geometry

This work uses fiber bundles as a framework to describe some effects of number scaling on gauge theory and some geometric quantities. A description of number scaling and fiber bundles over a flat space time manifold, M, is followed by a description of gauge theory. A fiber at point x of M contains a pair of scaled complex number and vector space structures, $C^{c}_{x}\times V^{c}_{x} $ for each c in GL(1,C). A space time dependent scalar field, g, determines, for each x, the scaling value of the vector space structures that contain the values of a vector valued matter field at x. The vertical components of connections between neighboring fibers are taken to be the gradient field A(x)+iB(x), of g. Abelian gauge theory for these fields gives the result that B is massless and no mass restrictions for A. Addition of an electromagnetic field dies not change these results. In the Mexican hat Higgs mechanism B combines with a Goldstone boson to create massive vector bosons, the photon field, and the Higgs field. For geometric quantities the fiber bundle is a tangent bundle with a fiber at point x containing scaled pairs, $R^{r}_{x}\times T^{r}_{x}$ of real number and tangent space structures for each x and and nonnegative real r. B is zero everywhere. The A field affects path lengths and the proper times of clocks along paths. It also appears in the geodesic equation. The lack of physical evidence for the gradient field, A(x)+iB(x) means that it either couples very weakly to matter fields or that it is close to zero for all x in a local region of cosmological space and time. It says nothing about the values outside the local region.

On a geometric realization of C*-algebras

Further to the functional representations of C$^*$-algebras proposed by R. Cirelli, A. Mania and L. Pizzocchero, we consider in this article the uniform K\"ahler bundle (in short, UKB) description of some C$^*$-algebraic subjects. In particular, we obtain an one-to-one correspondence between closed ideals of a C$^*$-algebra $\mathcal{A}$ and full uniform K\"ahler sub bundles over open subsets of the base space of the UKB associated with $\mathcal{A}$. In addition, we will present a geometric description of the pure state space of hereditary C$^*$-subalgebras and show that that if $\mathcal{B}$ is a hereditary C$^*$-subalgebra of $\mathcal{A}$, the UKB of $\mathcal{B}$ is a kind of K\"ahler subbundle of the UKB of $\mathcal{A}$. As a simple example, we consider hereditary C$^*$-subalgebras of the C$^*$-algebra of compact operators on a Hilbert space. Finally, we remark that hereditary C$^*$-subalgebras also naturally can be characterized as uniform holomorphic Hilbert subbundles.

Arbitrary Starting Point of Separation Affects Morphology of the 2 Bundles of Anterior Cruciate Ligament at Insertion Sites

To explore whether changes in the starting point of separation affect the morphology of the 2 bundles of the anterior cruciate ligament (ACL) at insertion. Ten cadaveric knees were used to separate the ACL fibers into 2 bundles from 3 different starting points that were located at the proximal one fifth, two fifths, and three fifths of the inter-bundle mark line. The dividing lines between the 2 bundles at insertion and the area of the anteromedial (AM) bundle resulting from these different separations were compared, with data collected in groups I, II, and III, respectively. The angle of the dividing line and the long axis of the femoral footprint was 67.24 degrees +/- 11.94 degrees, 91.01 degrees +/- 11.16 degrees, and 116.03 degrees +/- 9.01 degrees in groups I, II, and III, respectively, and the percentage of the AM bundle area in the whole femoral footprint was 38.64% +/- 5.55%, 52.22% +/- 6.76%, and 65.09% +/- 4.53%, respectively. At the tibial insertion sites, the angle between the dividing line and the sagittal plane was 110.17 degrees +/- 13.26 degrees, 127.72 degrees +/- 8.94 degrees, and 149.28 degrees +/- 18.80 degrees in groups I, II, and III, respectively, and the percentage of the AM bundle area in the whole footprint was 25.72% +/- 3.82%, 40.41% +/- 3.73%, and 60.56% +/- 6.59%, respectively. There were statistical differences between the angle and the area data of each of the 2 groups at either the tibial or femoral insertion site. In an anatomic study of the ACL, changes in the point from which separation started resulted in noticeably different bundle morphology at the insertion. Proper separation will lead to a more accurate bundle description and, thus, a more accurate location of the tunnels in ACL reconstruction.

Fiber Bundle Description of Flow and Nonlinear Hydrodynamics on Circles

We introduce a differential geometry description of the path lines, stream lines and particles contours in hydrodynamics. We present a generalized form of a Korteweg-de Vries type of equation for the exterior of a circle. Nonlinearities from the boundary conditions, surface tension and the Euler equations are taken into account, but the flow is considered inviscid and irrotational. For the circular case we describe the traveling waves shapes, solitons and the particles trajectories.

Remark on Charge Conjugation in the Non Relativistic Limit

We study the non relativistic limit of the charge conjugation operation $\cal C$ in the context of the Dirac equation coupled to an electromagnetic field. The limit is well defined and, as in the relativistic case, $\cal C$, $\cal P$ (parity) and $\cal T$ (time reversal) are the generators of a matrix group isomorphic to a semidirect sum of the dihedral group of eight elements and $\Z_2$. The existence of the limit is supported by an argument based in quantum field theory. Also, and most important, the limit exists in the context of galilean relativity. Finally, if one complexifies the Lorentz group and therefore the galilean spacetime $x_\mu$, then the explicit form of the matrix for $\cal C$ allows to interpret it, in this context, as the complex conjugation of the spatial coordinates: $\vec{x} \to \vec{x}^*$. This result is natural in a fiber bundle description.

GEOMETRIC ASPECTS OF TIME-DEPENDENT SINGULAR DIFFERENTIAL EQUATIONS

A geometric framework for describing and solving time-dependent implicit differential equations F(t,x,x′) = 0 is studied, paying special attention to the linearly singular case, where F is affine in the velocities: A(t,x)x′ = b(t,x). This framework is based on the jet bundle of a time-dependent configuration space, and is an extension of the geometric framework of the autonomous case. When A is a singular matrix, the solutions can be obtained by means of constraint algorithms, either directly or through an equivalent autonomous system that can be constructed using the vector hull functor of affine spaces. As applications, we consider the jet bundle description of time-dependent lagrangian systems and the Skinner–Rusk formulation of time-dependent mechanics.

Fibre Bundle Formulation of Relativistic Quantum Mechanics: I. Time-dependent Approach

A new fibre bundle formulation of the mathematical base of relativistic quantum mechanics is presented. At the present stage, the bundle form of the theory is equivalent to its conventional one, but it admits new types of generalizations in different directions.In the present, first, part of our investigation we consider the time-dependent or Hamiltonian approach to bundle description of relativistic quantum mechanics. In it the wavefunctions are replaced with (state) liftings of paths or sections along paths of a suitably chosen vector bundle over space-time whose (standard) fibre is the space of the wavefunctions. Now the quantum evolution is described as a linear transportation (by means of the evolution transport along paths in the space-time) of the state liftings/sections in the (total) bundle space. The equations of these transportations turn to be the bundle versions of the corresponding relativistic wave equations.

Singular Lagrangian Systems on Jet Bundles

The jet bundle description of time-dependent mechanics is revisited. The constraint algorithm for singular Lagrangians is discussed and an exhaustive description of the constraint functions is given. By means of auxiliary connections we give a basis of constraint functions in the Lagrangian and Hamiltonian sides. An additional description of constraints is also given considering at the same time compatibility, stability and second-order condition problems. Finally, a classification of the constraints in first and second class is obtained using a cosymplectic geometry setting. Using the second class constraints, a Dirac bracket is introduced, extending the well-known construction by Dirac.

FIBRE BUNDLE FORMULATION OF NONRELATIVISTIC QUANTUM MECHANICS IV: MIXED STATES AND EVOLUTION TRANSPORT'S CURVATURE

We propose a new systematic fibre bundle formulation of nonrelativistic quantum mechanics. The new form of the theory is equivalent to the usual one and is in harmony with the modern trends in theoretical physics and potentially admits new generalizations in different directions. In it the Hilbert space of a quantum system (from conventional quantum mechanics) is replaced with an appropriate Hilbert bundle of states and a pure state of the system is described by a lifting of paths or section along paths in this bundle. The evolution of a pure state is determined through the bundle (analog of the) Schrödinger equation. Now the dynamical variables and density operators are described via liftings of paths or morphisms along paths in suitable bundles. The mentioned quantities are connected by a number of relations derived in this work. The present fourth part of this series is devoted mainly to the fibre bundle description of mixed quantum states. We show that to the conventional density operator there corresponds a unique density lifting of paths for which the corresponding equations of motion are derived. It is also investigated the bundle description of mixed quantum states in the different pictures of motion. We calculate the curvature of the evolution transport and prove that it is curvature free iff the values of the Hamiltonian operator at different moments commute.

FIBRE BUNDLE FORMULATION OF NONRELATIVISTIC QUANTUM MECHANICS V: INTERPRETATION, SUMMARY AND DISCUSSION

We propose a new systematic fibre bundle formulation of nonrelativistic quantum mechanics. The new form of the theory is equivalent to the usual one and is in harmony with the modern trends in theoretical physics and potentially admits new generalizations in different directions. In it the Hilbert space of a quantum system (from conventional quantum mechanics) is replaced with an appropriate Hilbert bundle of states and a pure state of the system is described by a lifting of paths or section along paths in this bundle. The evolution of a pure state is determined through the bundle (analog of the) Schrödinger equation. Now the dynamical variables and density operator are described via liftings of paths or morphisms along paths in suitable bundles. The mentioned quantities are connected by a number of relations derived in this work. This is the fifth, closing, part of our investigation. We briefly discuss the observer's role in the theory and different realizations of the space–time model used as a base space in the bundle approach to quantum mechanics. The exact conditions for the equivalence of Hilbert bundle and Hilbert space formulations of the theory are pointed out. A table of comparison between both descriptions of nonrelativistic quantum mechanics is presented. We discuss some principal moments of the Hilbert bundle description and show that as a scheme it is more general than the Hilbert space one. Different directions for further research are pointed out.

The topological completion of a bilinear form

Let M= M n, m be the Euclidean space R p equipped with a symmetric bilinear form B M of rank p= n+ m and signature n− m. We compactify M so that M c is homogeneous and has as its group of isometries a Lie group whose dimension is the dimension of M plus 2 p+1. We observe that M c is in two ways the total space of a non-trivial sphere bundle with base space real projective space. The compactification is well understood in the classical case when M is Minkowski space. The contribution here is to observe that the construction works generally and that it admits a natural bundle description.

Geometry of multisymplectic Hamiltonian first-order field theories

In the jet bundle description of field theories (multisymplectic models, in particular), there are several choices for the multimomentum bundle where the covariant Hamiltonian formalism takes place. As a consequence, several proposals for this formalism can be stated, and, on each one of them, the differentiable structures needed for setting the formalism are obtained in different ways. In this work we make an accurate study of some of these Hamiltonian formalisms, showing their equivalence. In particular, the geometrical structures (canonical or not) needed for the Hamiltonian formalism, are introduced and compared, and the derivation of Hamiltonian field equations from the corresponding variational principle is shown in detail. Furthermore, the Hamiltonian formalism of systems described by Lagrangians is performed, both for the hyper-regular and almost-regular cases. Finally, the role of connections in the construction of Hamiltonian field theories is clarified.

A conic bundle description of Moishezon twistor spaces without effective divisors of degree one

In [K2] Moishezon twistor spaces over the connected sum \(n{\mathbb{CP}}^2\) (\(n\geq 4\)), which do not contain effective divisors of degree one, were constructed as deformations of the twistor spaces introduced in [LeB]. We study their structure for \(n\geq 4\) by constructing a modification which is a conic bundle over \({\mathbb P}^2\). We show that they are rational. In case n = 4 we give explicit equations for such conic bundles and use them to construct explicit birational maps between these conic bundles and \({\mathbb P}^3\).

Bundle Description of Packing and Dynamics in Polycarbonate Homopolymers, Copolymers, and Blends

Dipolar-rotational spinecho 13C NMR spectra at 15.1 MHz have been obtained for 12 homopolymers, copolymers, and blends of polycarbonate, poly(ether carbonate), poly(ether sulfone), dimethyl polycarbonate, and tetramethyl polycarbonate. The spectra were used to determine the fraction of nonmethylated phenylene rings that are blocked from undergoing flips faster than 10 kHz at 300 K. This fraction is interpreted in terms of a simple two-chain bundle packing model in which empirically determined steric parameters specify the interchain interactions between flexible chain linkers and rigid chain extenders. The success of this bundle description suggests that the glassy state for all of these polycarbonate-like systems has local mainchain orientational order.