Geometrization of quantum mechanics

Quantum mechanics is cast into a classical Hamiltonian form in terms of a symplectic structure, not on the Hilbert space of state-vectors but on the more physically relevant infinite-dimensional manifold of instantaneous pure states. This geometrical structure can accommodate generalizations of quantum mechanics, including the nonlinear relativistic models recently proposed. It is shown that any such generalization satisfying a few physically reasonable conditions would reduce to ordinary quantum mechanics for states that are near the vacuum. In particular the origin of complex structure is described.     

Geometrization of the electromagnetic field and dark matter

Fundamental concepts, symmetries and dynamic equations of the theory of dark matter are derived from the simple relation: everything in the concept of space and the concept of space in everything. It is shown that the electromagnetic field is the singlet state of the dark matter field and, hence, the last may be considered as a generalized electromagnetic field (shortly gef) and a simple solution is given to the old problem of connecting the electromagnetic field with geometric properties of the physical manifold itself. It is shown that gauge fixing renders the generalized electromagnetic field effectively massive while the Maxwell electromagnetic field remains massless. To learn more about interactions between matter and dark matter on the microscopic level (and to recognize the fundamental role of internal symmetry in this case), the general covariant Dirac equation is derived and its natural generalization is considered. The experiment is suggested to test the formulated theory.     

Geometrization of spin and the Weyssenhoff fluid conjecture

We show that the logical treatment of spinning perfect fluids occurs in a metrictorsion space-time with or without mass conservation. The consequence is the geometrization of spin; that is, we obtain the Weyssenhoff form relating spin and the trace-free torsion without any ad hoc assumption.     

Geometrization of the Electro-weak Model Bosonic Component

In this work, we develop a geometrical unification theory for gravity and the electro-weak model in a Kaluza-Klein approach; in particular, from the curvature dimensional reduction Einstein–Yang–Mills action is obtained. We consider two possible space-time manifolds: 1) V ⁴⊗ S ¹⊗ S ² where isospin doublets are identified with spinors; 2) V ⁴⊗ S ¹⊗ S ³ in which both quarks and leptons doublets can be recast into the same spinor, such that the equal number of quark generations and leptonic families is explained. Finally a self-interacting complex scalar field is introduced to reproduce the spontaneous symmetry breaking mechanism; in this respect, at the end, we get an Higgs fields whose two components have got opposite hypercharges.     

Geometrization in the Yang-Mills extended supergravity and Klein-Kaluza versions

We relate personal encounters of three kinds with geometrical approaches, in the development of a relativistic quantum field theory of the fundamental interactions—including interactions with Nathan Rosen. We characterize the geometrical structures involved and discuss the more recent attempts to develop a unified theory based on a Klein-Kaluza contraction of the eightfold extended supergravity.Invited paper, dedicated to Nathan Rosen on the occasion of his seventy-fifth birthday.On leave from Tel Aviv University, Israel, and the University of Texas at Austin, Texas.     

Geometrization of String Theory Gravity

It is shown that the low energy string theory Lagrangian can be interpreted as pure gravity. In particular, it is shown that the Lagrangian is simply R, the curvature scalar of spacetime with torsion, and unlike in previous work, the covariant derivative of the metric tensor vanishes. As a consequence, it is shown that the physical origin of the scalar field results from the pseudoscalar invariant. This yields, for the first time, a definite physical origin of the dilaton field in four dimensions.     

Geometrization of Mass in General Relativity

In this paper we will extend the notion of tangent bundle to a Z ₂ graded tangent bundle. This graded bundle has a Lie algebroid structure and we can develop notions semi-Riemannian metrics, Levi-Civita connection, and curvature, on it. In case of space-times manifolds, even part of the tangent bundle is related to space and time structures (gravity) and odd part is related to mass distribution in space-time. In this structure, mass becomes part of the geometry, and Einstein field equation can be reconstructed in a new simpler form. The new field equation is purely geometric.     

Geometrization of magnetohydrodynamic equations via lie groups

The equations of magnetohydrodynamics of a perfect fluid are classified with respect to the Coriolis parameter, and all essentially different solutions of rank one are indicated. The geometry of streamlines is discussed.     

Geometrization in the Yang-Mills,extended supergravity,and Klein-Kaluza versions

We relate personal encounters of three kinds with geometrical approaches in the development of a relativistic quantum field theory of the fundamental interactions—including interactions with Nathan Rosen. We characterize the geometrical structures involved and discuss the more recent attempts to develop a unified theory based on a Klein-Kaluza contraction of the eightfold extended supergravity.Publisher's note: The full paper will appear inFoundations of Physics, Volume 15, Number 3, March 1985.     

Geometrization of the classical field theory and its application in Yang-Mills field theory

The paper contains presentation of the finite-dimensional approach to the classical field theory based on the geometry of differential manifolds and forms. Geometrical construction of a symplectic structure and Poisson brackets on the space of initial conditions are realized. This space is not a manifold but it can be furnished with a structure of a differential space.The structural n+1 form for the Yang-Mills field theory is constructed. This gives automatically equations of motion and equations for initial conditions. The parasymplectic structure is computed. The directions of degeneration appear to be exactly the directions of infinitesimal gauge transformations. The Poisson bracket for Yang-Mills field theory is obtained.     

Geometrization of the physics with teleparallelism. I. The classical interactions

A connection viewed from the perspective of integration has the Bianchi identities as constraints. It is shown that the removal of these constraints admits a natural solution on manifolds endowed with a metric and teleparallelism. In the process, the equations of structure and the Bianchi identities take standard forms of field equations and conservation laws.The Levi-Civita (part of the) connection ends up as the potential for the gravity sector, where the source is geometric and tensorial and contains an explicit gravitational contribution.Nonlinear field equations for the torsion result. In a low-energy approximation (linearity andlow energy-momentumtransfer), the postulate that only charge and velocities contribute to the source transforms these equations into the Maxwell system. Moreover, the affine geodesics become the equations of motion of special relativity with Lorentz force in the same approximation [J. G. Vargas,Found. Phys. 21, 379 (1991)]. The field equations for the torsion must then be viewed as applying to an electromagnetic/strong interaction.A classical unified theory thus arises where the underlying geometry confers their contrasting characters to Maxwell-Lorentz electrodynamics and to an Einstein's-like theory of gravity. The highly compact field equations must, however, be developed in phase-spacetime, since the connection is velocity-dependent, i.e., Finsler-like.Further opportunities for similarities with present-day physics are discussed: (a) teleparallelism allows for the formulation of the torsion sector of the theory as a flat space theory with concomitant point-dependent transformations; (b) spinors should replace Lorentz frames in their role as the subjects to which the connection refers; (c) the Dirac equation consistent with the frame bundle for a velocity-dependent metric with Lorentz signature generates a weak-like interaction in the torsion sector.Work done at the Department of Mathematics and Physics of the Interamerican University of Puerto Rico, San German, Puerto Rico 00683.     

Geometrization of metric boundary data for Einstein’s equations

The principle part of Einstein equations in the harmonic gauge consists of a constrained system of 10 curved space wave equations for the components of the space-time metric. A well-posed initial boundary value problem based upon a new formulation of constraint-preserving boundary conditions of the Sommerfeld type has recently been established for such systems. In this paper these boundary conditions are recast in a geometric form. This serves as a first step toward their application to other metric formulations of Einstein’s equations.     

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.     

Generalization and applications of the Sasakawa theory

The Sasakawa theory of scattering is phrased in the form of a Fredhohn reduction technique for integral equations possessing a fixed-point singularity in their kernels. This permits the generalization of this theory to a large variety of scattering integral equations. Some specific applications include the two-particle off-shell and multichannel scattering problems. In the first instance a rank-three approximation to the fully off-shell transition matrix is derived which is exact on and half-off shell, satisfies off-shell unitarity, and which possesses no unphysical singularities. In the second problem it is shown how the method leads to the generation of a unitary approximation to the multichannel amplitudes.     

Noether恒等式的推广及其应用

从系统的作用量在普遍的定域和非定域变换下的性质出发,导出了含非定城变换的广义Noether恒等式.将其用于高阶微商杨—Mills场论,求出了有别于BRS荷的新PBRS守恒荷和非定域变换下的新守恒荷.     

Generalization of the Noether's Identities and Application

Starting from the transformation property of the action integral of a system under the local and non-local transformation, we derive the generalized Noether's identities connecting with non-local transformation. The applications of the theory to the Yang-Mills field with high-order derivatives are presented. A new conservative PBRS charge is found which differs from BRS conservative charge. The other conservative charge connecting with non-local transformation is also obtained.     

Generalization of the Darwin Lagrangian

Starting from a Lagrangian treatment of classical electrodynamics and using simple heuristic arguments, an effective generalization of the Darwin Lagrangian for point charges moving with arbitrary velocities (v<c) is obtained. In addition, another, the simplesta priori Lagrangian (preserving the most important features of standard classical theory) is proposed.