Thick brane worlds arising from pure geometry

Quark-gluon transport theory I. The classical theory

Regge theory and particle physics

On a macroscopic level we take general relativity as the appropriate theory of space-time and gravity. We will argue that, on a more microscopic level, in the Compton wavelength regime of elementary particles, there are good reasons for suspecting the presence of a torsion of space-time. A corresponding gaugetheoretical formalism related to the Poincare group is reviewed, and the kinematical consequences of the presence of a torsion are worked out. In particular we discuss the operational meaning and the measurability of torsion. The dynamics of torsion is left for a forthcoming article.

On the Kinematics of the Torsion of Space-time

We present a comparative analysis of localization of 4D gravity on a non 2 -symmetric scalar thick brane in both a 5-dimensional riemannian space time and a pure geometric Weyl integrable manifold in which variations in the length of vectors during parallel transport are allowed and a geometric scalar field is involved in its formulation. This work was mainly motivated by the hypothesis which claims that Weyl geometries mimic quantum behaviour classically. We start by obtaining a classical 4-dimensional Poincar? invariant thick brane solution which does not respect 2 -symmetry along the (non-)compact extra dimension. This field configuration reproduces the 2 -symmetric solutions previously found in the literature, in both the Riemann and the Weyl frames, when the parameter k 1 = 1. The scalar energy density of our field configuration represents several series of thick branes with positive and negative energy densities centered at y 0 . Thus, our field configurations can be compared with the standard Randall-Sundrum thin brane case. The only qualitative difference we have encountered when comparing both frames is that the scalar curvature of the riemannian manifold turns out to be singular for the found solution, whereas its weylian counterpart presents a regular behaviour. By studying the transverse traceless modes of the fluctuations of the classical backgrounds, we recast their equations into a Sch?dinger's equation form with a volcano potential of finite bottom (in both frames). By solving the Sch?dinger equation for the massless zero mode m 2 = 0 we obtain a single bound state which represents a stable 4-dimensional graviton in both frames. We also get a continuum gapless spectrum of KK states with positive 0$> m 2>0 that are suppressed at y 0 , turning into continuum plane wave modes as y approaches spatial infinity. We show that for the considered solution to our setup, the potential is always bounded and cannot adopt the form of a well with infinite walls; thus, we do not get a discrete spectrum of KK states, and we conclude that the claim that weylian structures mimic, classically, quantum behaviour does not constitute a generic feature of these geometric manifolds.

4D gravity localized in non Bbb Z2 -symmetric thick branes

A massless Weyl-invariant dynamics of a scalar, a Dirac spinor, and electromagnetic fields is formulated in a Weyl space, W4, allowing for conformal rescalings of the metric and of all fields with nontrivial Weyl weight together with the associated transformations of the Weyl vector fields κμ, representing the D(1) gauge fields, with D(1) denoting the dilatation group. To study the appearance of nonzero masses in the theory the Weyl symmetry is broken explicitly and the corresponding reduction of the Weyl space W4 to a pseudo-Riemannian space V4 is investigated assuming the breaking to be determined by an expression involving the curvature scalar R of the W4 and the mass of the scalar, selfinteracting field. Thereby also the spinor field acquires a mass proportional to the modulus Φ of the scalar field in a Higgs-type mechanism formulated here in a Weyl-geometric setting with Φ providing a potential for the Weyl vector fields κμ. After the Weyl-symmetry breaking, one obtains generally covariant and U(1) gauge covariant field equations coupled to the metric of the underlying V4. This metric is determined by Einstein's equations, with a gravitational coupling constant depending on Φ, coupled to the energy momentum tensors of the now massive fields involved together with the (massless) radiation fields.

http://cds.cern.ch/record/346081/files/9802044.pdf

Broken Weyl invariance and the origin of mass

A gauge theory of strong interaction is developed based on fields defined on a fiber bundle. The structural group of the bundle is taken to be the L4,1 de Sitter group. An internal variable e, varying in the fiber over a space-time point x, is introduced as a means to describe – with the help of a semiclassical wave function PS(x, e) defined on the bundle space – the internal structure of extended hadrons in a framework using differential geometric techniques. Three basic nonlinear wave equations for PS(x, e) are established which are of integro-differential type. The nonlinear coupling terms in these de Sitter gauge invariant equations represent physically a generalized spin orbit coupling or a generalized spin spin coupling for the motion taking place in the fiber. The motivation for using a bigger space for the definition of hadronic matter wave functions as well as the implications of this geometric approach to strong interaction physics is discussed in detail, in particular with respect to the problem of hadronic constituents. The proposed fiber bundle formalism allows a dynamical description of extended structures for hadrons without implying the necessity of introducing any constituents.

Wave Equations on a de Sitter Fiber Bundle

In the generalized Kaluza-Klein theories a (4 + N)-dimensional Riemannian space is used as a configuration space to unify gravitation with a nonabelien gauge symmetry characterized by a gauge group G of order N. The original metric space V4+N, however, is broken down in these theories to a local product structure M4 × VN (M4 being Minkowski space-time) leading thereby to a description in terms of a fiber bundle over space-time V4 with structural group G. In this paper we base the discussion from the very beginning on a bundle structure as the underlying geometric stratum and treat the quantum mechanical aspect of matter by representing it in the form of a generalized wave function, ϕ(x, X), defined as a section on a fiber bundle associated to a principal G-bundle over space time having a homogeneous space of the group G as fiber. To make contact with gravitation we consider various soldered bundles possessing the Poincare group, G = ISO(3, 1), as structural or gauge group and define Poincare gauge fields as generalized matter fields on them. After reviewing the formulation of general relativity as a gauge theory of the Lorentz group we turn to the affine case and present a geometric formulation of a Poincare gauge theory based on a Riemann-Cartan space-time U4 with axial vector torsion. Two sets of field equations are discussed relating the material source quantities to the geometric fields. Besides Einstein's equations coupling the metric to the classical distribution of energy and momentum of matter a second set of nonlinear field equations is set up relating a bilinear current in the Poincare gauge fields to the axial vector torsion field of the underlying space-time geometry. Finally, a breaking of the Poincare symmetry is introduced in a manner involving the generalized matter field ϕ(x, X) freezing thereby the translational gauge degrees of freedom, however, leaving the Lorentz gauge symmetry unaffected.

Poincaré gauge theory, gravitation, and transformation of matter fields

The model of the quantum relativistic rotator is defined by three correspondences: (1) the correspondence to a classical relativistic rotator when the quantum description goes over into the classical description (classical limit), (2) the correspondence to an elementary particle when the structure is ignored (elementary limit), and (3) the correspondence to a nonrelativistic quantum rotator (a rigid rotating string) in the nonrelativistic limit. The dynamics is given by a Hamiltonian which is obtained from a constraint relation that leads to a phenomenologically acceptable mass-spin trajectory relation. From the equation of motion it follows that the expectation value of the particle position spirals with approximately the velocity of light about the direction of the momentum, which is also the direction in which the center of mass propagates. The radius of this helical motion (i.e., the "size" of the rotator), as obtained from the phenomenological mass spectrum, is of the order of 10 -13 cm.

Relativistic rotator. I. Quantum observables and constrained Hamiltonian mechanics

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
 
ab
 (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).

Wolfgang Drechsler

Papers

Broken Weyl invariance and the origin of mass

Wave Equations on a de Sitter Fiber Bundle

Poincaré gauge theory, gravitation, and transformation of matter fields

Relativistic rotator. I. Quantum observables and constrained Hamiltonian mechanics

Currents in a theory of strong interaction based on a fiber bundle geometry