Geometric Field Theory Research Articles

Continuum elasticity with topological defects, including dislocations and extra-matter

The elasticity of continuous media with topological defects is described naturally by differential geometry, since it relates metric to strain. We construct a geometrical field theory, identifying disclinations, dislocations and extra-matter defects with the curvature, torsion and nonmetricity tensors, respectively. Connection and metric are given explicitly in the presence of dislocations and extra-matter. The density of extra-matter is a scalar source of isotropic strain, described by a local length scale or gauge. The logarithm of the gauge is related to the density of extra-matter by a Poisson equation. The corresponding integral equation, similar to Gauss' law in electrostatics, measures the amount of extra-matter contained inside a contour.

Improved prediction of differential subsidence caused by underground mining

Abstract Observations in the field demonstrate that there exist discrepancies between the differential movements predicted from theory and those experienced in practice. From the point of view of structural damage, such discrepancies should be eliminated in order to decrease subsidence damage associated with mining. Using non-linear geometrical field theory, the deformation factors are modified and the limitation of linear elastic theory is established. Only when the rotation angles are smaller than 4° is the application of conventional theory reasonable. A new deformation index, the mean rotation angle, is proposed. The study shows that there exists a relation between the mean rotation angle and the surface cracks. Incompatibility of rotation can initiate cracking. The outcome of the work presented will be helpful in improving the calculation accuracy of predicted differential movements, especially for steeper coal seam extraction, extraction under mountains, ‘top coal caving’ of thick seams, and mining in high relief areas.

Two-Spinors, Field Theories and Geometric Optics in Curved Spacetime

A partly new approach to 2-spinor geometry, recently developed, turns out to yield a naturally integrated formulation of Einstein–Cartan and Maxwell–Dirac fields, and to be suitable for describing several topics in field theories which are relevant to covariant quantization on curved spacetime. Here, first a revised version of the above mentioned geometric tools and field theory is presented, in which the geometric data consist only of a complex vector bundle S → M with two-dimensional fibres (all the needed structures are functorially derived from S; any considered object which is not a functorial construction is assumed to be a dynamical field). Then various developments are considered, concerning some features of the electromagnetic gauge and a partly new covariant description of electroweak geometry and fields. Finally, a 2-spinor treatment of optical geometry and geometric optics in curved spacetime is examined, leading to a description of the photon field as a section of the vertical bundle of the bundle of Riemann spheres associated with S.

Functional analysis on the eve of the 21st century: in honor of the eightieth birthday of I. M. Gelfand (2 volumes), edited by S. Gindikin, J. Lepowsky & R. L. Wilson. Pp. 292, 323. DM148 (2 volumes). 1995. ISBN 0-8176-3755-9, 0-8176-3855-5 (Birkhäuser).

Functional analysis on the eve of the 21st century: in honor of the eightieth birthday of I. M. Gelfand (2 volumes), edited by S. Gindikin, J. Lepowsky & R. L. Wilson. Pp. 292, 323. DM148 (2 volumes). 1995. ISBN 0-8176-3755-9, 0-8176-3855-5 (Birkhäuser). - Volume 81 Issue 490

Classical gravity and quantum matter fields in unified field theory

The Einstein-Schrodinger purely affine field theory of the non-symmetric field provides canonical field equations without constraints. These equations imply the Heisenberg-Pauli commutation rules of quantum field theory. In the Schrodinger gauging of the Einstein field coordinatesUkli=Γkli−δliΓkmm, this unified geometric field theory becomes a model of the coupling between a quantized Maxwellian field in a medium and classical gravity. Therefore, independently of the question as to the physical truth of this model, its analysis performed in the present paper demonstrates that, in the framework of a quantized unified field theory, gravity can appear as a genuinely classical field.

General covariance, new variables, and dynamics without dynamics.

We give an example of a generally covariant geometric field theory which leads to the standard Gaussian and supermomentum constraints but which lacks the super-Hamiltonian constraint. This theory is closely linked to Ashtekar’s canonical formulation of general relativity. We discuss the Dirac constraint quantization of our model and comment on the issues which it raises regarding the quantization of generally covariant systems.

Shapiro-Virasoro amplitude and string field theory.

One of the embarrassments of covariant string field theory has been the glaring failure to derive the Shapiro-Virasoro amplitude. Modular invariance appears explicitly violated: either the fundamental region is overcounted an infinite number of times, or it is undercounted because of a missing region. We try to approach this problem from a fresh point of view. Conventional wisdom holds that, in string field theory, the Veneziano amplitude can only be derived either in light-cone string field theory or Witten's string field theory. We show that this firmly held belief is actually wrong, that the Veneziano amplitude can actually be derived using vertices of arbitrary lengths. This is a highly nontrivial calculation. Using third elliptic integrals we show that a series of ``miracles'' occurs which allow us to cancel scores of unwanted terms in the measure, leaving us with the correct Koba-Nielsen variable and measure. We give three independent proofs of our result. When we generalize our results to closed-string scattering with arbitrary lengths, we find a new surprise, that we can successfully derive the Shapiro-Virasoro amplitude as long as a crucial four-string interaction term is added. We check by explicit computer calculation that we reproduce the correct region of integration for the four-closed-string amplitude. Crucial to the theory is the existence of the missing four-string tetrahedron graph, which precisely fills the missing integration region. We comment on the implications of this for geometric string field theory.

DERIVING THE FOUR-STRING AND OPEN-CLOSED STRING INTERACTIONS FROM GEOMETRIC STRING FIELD THEORY

One of the baffling questions concerning the covariant open string field theory is why there are two distinct BRST theories and why the four-string interaction appears in one version but not the other. We solve this mystery by showing that both theories are gauge-fixed versions of a higher gauge theory, called the geometric string field theory, with a new field, a string vierbein [Formula: see text], which allows us to gauge the string length and σ-parametrization. By fixing the gauge, we can derive the “endpoint gauge” (the covariantized light cone gauge), the “midpoint gauge” of Witten, or the “interpolating gauge” with arbitrary string lengths. We show explicitly that the four-string interaction is a gauge artifact of the geometric theory (the counterpart of the four-fermion instantaneous Coulomb term of QED). By choosing the interpolating gauge, we produce a new class of four-string interactions which smoothly interpolate between the endpoint gauge and the midpoint gauge (where it vanishes). Similarly, we can extract the closed string as a bound state of the open string, which appears in the endpoint gauge but vanishes in the midpoint gauge. Thus, the four-string and open-closed string interactions do not have to be added to the action as long as the string vierbein is included.

On large deformation unilateral contact problem with friction (I)—Incremental variational equations

Based on the theory and technique of nonlinear geometric field theory of continuum, a more general incremental variational equation for elastic and plastic large deformation in co-moving coordinate is established in this paper. An expression for two and three-dimensional continua is derived, and the incremental variational equation for large deformation of changing boundary contact and the variational inequality in rate form are obtained, which provides the theoretical basis for the computation of elastic-plastic large deformation contact problem with friction.

On the objective stress rate in co-moving coordinate system

The objective stress rate is a rather important problem in mechanics of finite deformation. In this paper, the objective stress rate in co-moving coordinate is derived by applying nonlinear geometric field theory of deformation. Problems, such as large extension coupled with rotation, and large shear deformation, are exemplified by using the new formula. Comparing with Jaumann's stress rate and other formulae presented in current literature, the new result appears to be the reasonable one in co-moving coordinate system.

Geometric derivation of string field theory from first principles: Closed strings and modular invariance.

We present an entirely new approach to closed-string field theory, called Igeometric string field theory R, which avoids the complications found in Becchi-Rouet-Stora-Tyutin string field theory (e.g., ghost counting, infinite overcounting of diagrams, midpoints, lack of modular invariance). Following the analogy with general relativity and Yang-Mills theory, we define a new infinite-dimensional local gauge group, called the unified string group, which uniquely specifies the connection fields, the curvature tensor, the measure and tensor calculus, and finally the action itself. Geometric field theory, when gauge fixed, yields an entirely new class of gauges called the interpolating gauge which allows us to smoothly interpolate between the midpoint gauge and the end-point gauge (``covariantized light-cone gauge''). We can show that geometric string field theory reproduces one copy of the Shapiro-Virasoro model. Surprisingly, after the gauge is broken, a new Iclosed four-string interactionR emerges as the counterpart of the instantaneous four-fermion Coulomb term in QED. This term restores modular invariance and precisely fills the missing region of the complex plane.

Modular-invariant closed-string field theory.

Attempts so far at constructing a covariant closed-string field theory have been frustrated by the fact that modular invariance always appears to be violated. At both the tree and loop levels, moduli space is either overcounted an infinite number of times, or undercounted because of a missing region. We solve this problem by demonstrating that a new Iclosed four-string interactionR is necessary to reproduce the closed-string amplitude which precisely fills the missing region. This closed four-string interaction, which has the topology of a tetrahedron, is predicted by geometric string field theory. The tetrahedron graph is generated by gauge fixing the geometric theory's local gauge group, the unified string group, and is the exact counterpart of the instantaneous four-fermion Coulomb term found in QED. We prove the existence of this tetrahedron graph both analytically and by direct computer calculation and show that it is the key to reproducing the Shapiro-Virasoro amplitude.

Why are there two BRST string field theories?

One of the mysteries of string field theory is why there are two distinct BRST field theories. We solve this puzzle by (1) explicitly constructing a geometric vertex that smoothly interpolates between the other theories, (2) proving that this geometric vertex links the other two by a one-dimensional reparametrization, (3) proving that both BRST theories are thus gauge-fixed versions of a higher, geometric field theory. The geometric theory also explains the origin of the baffling four-string interaction, which now emerges as a gauge artifact, the counterpart of the four-fermion instantaneous Coulomb term of QED.

Generalization of general relativity to the space of closed strings

A geometrical string field theory is constructed on the space of bosonic closed strings. Einstein gravity is reproduced in a part of the massless sector at the level of the classical lagrangian.

New Weyl theory: Geometrization of electromagnetism and gravitation: Motivations and classical results

A geometrical unified field theory of electromagnetism and gravitation is developed in a Weyl space-time. The integrability conditions of the field equations cast the laws of classical perfect fluids under electromagnetic interactions. The purely gravitational limit of the theory is Einstein's General Relativity and the purely electromagnetic case coincides with the predictions of Maxwell's theory.

The parallel transport of a vector: Its physical meaning in three geometrical unified field theories

The physical interpretation of the parallel transport of a vector is analyzed in three classical geometric unified field theories: the theory of physical space and the theories of Weyl and Infeld. In physical space it is found to be a Fermi-Walker transport in the presence of electromagnetic interactions.

Energy-momentum tensors and stress tensors in geometric field theories

Lagrangian field theories of geometric objects are the most natural framework for investigating the notion of general covariance. We discuss here geometric theories of interacting fields, depending on Lagrangians of arbitrary order, and we give general definitions of energy flow, partial energy flows, energy-momentum tensors, and stress tensors. We also investigate the role which energy-momentum tensors and stress tensors play in formulating the natural conservation laws associated with the second theorem of Noether. Examples of application may be found elsewhere.

Geometrical optics reviewed: A new light on an old subject

The traditional view of geometrical optics as the ray-tracing procedure for the design of mirror, prism, and lens systems of uniform optical components, has become enlarged in recent years. This has been brought about by the applications of electromagnetic systems at ever decreasing wavelengths on the one hand, and by new optical components, such as optical fibers and integrated optical elements, on the other. The range of geometrical optics now includes such concepts as optics-in-the-large, wide-angle and aspheric designs, geodesic optics, and the optics of nonuniform media. The introduction of electromagnetic field theory in this area involves effects such as diffraction, fields near caustics, and evanescent fields. These are also being discussed in a ray optical interpretation, usually presented as the first-order term in an asymptotic expansion. The optical processes have, in turn, introduced some new geometrical theorems and matrix methods which have applications in the overall geometrical optics science. This paper sets out to review these methods and, in view of the commonly expressed compatibility between the two aspects, to investigate the applications of the geometrical theory to the electromagnetic field description. Indications are given as to possible methods available for the definition of a geometrical electromagnetic field theory which would contain the geometrical theories of the optical field in a covariant transformation theory of the electromagnetic field. It is shown that this is compatible with current methods being applied to the unity of all physics in which geometrical optics is the first-order approximation occupying a central position.

Solutions with both magnetic and electric charges in a (4+N)-dimensional geometric field theory

We study classical solutions with both magnetic and electric charges in a ($4+N$)-dimensional geometric field theory. Then we choose the group SO(3,1) for the internal symmetry and try to find the solutions under the ansatz that they are invariant with respect to the diagonal subgroup of $\mathrm{SO}{(3,1)}_{M} \ensuremath{\bigotimes} \mathrm{SO}{(3,1)}_{I}$ in order to get relativistically covariant solutions, where $\mathrm{SO}{(3,1)}_{M}$ and $\mathrm{SO}{(3,1)}_{I}$ represent SO(3,1) for rotations of the Minkowski space and for the internal symmetry, respectively. Thus we obtain solutions having a covariant expression for the electromagnetic field associated with magnetic and electric charges moving with constant velocity. Particles with both magnetic and electric charges are usually referred to as dyons. Hence our solutions give a specific realization of dyons in a uniform motion. As a result of the incorporation of the internal symmetry $\mathrm{SO}{(3,1)}_{I}$, furthermore, it is found that they have a new pole in addition to the magnetic and electric charges. The gauge fields associated with this pole contribute negative energy. Then there exists a relation among the three kinds of charges, which gives a lower bound to the absolute value of the electric charge. Finally, we discuss the stability of our solutions.

Erratum to: Geometric field theories with a given set of constant solutions

Erratum to: Geometric field theories with a given set of constant solutions