Electromagnetic and Force Field Equations

In classical physics the electromagnetic equations are described by Maxwell's equations. Maxwell's equations proved to be invariant under gauge, or Lorentz transformations. Also, Einstein's equations of the special theory of relativity are invariant under Lorentz transformations. On the other hand classical mechanics and quantum mechanics laws are invariant under Galilean transformations. This means that, there are two different dynamical structures describing our universe. Einstein's unified field theory failled in putting our universe in one dynamical structure. New electromagnetic and force field equations are going to be derived. They have the same shape like Maxwell's equations, but with different dynamical structure. Those equations are invariant under Galilean transformations and in the density matrix formalism of quantum mechanics.     

Necessary and sufficient conditions for the existence of a Lagrangian in field theory. I. Variational approach to self-adjointness for tensorial field equations

In this paper we identify some of the most significant references on the inverse problem of the calculus of variations for single integrals and initiate the study of the generalization of the underlying methodology to classical field theories. We first classify Lorentz-covariant tensorial field equations into nonlinear, quasi-linear, and semilinear forms, and then introduce their systems of equations of variation and adjoint systems. The necessary and sufficient conditions for the self-adjointness of class $\text{C}$², regular, tensorial, nonlinear, quasi-linear and semilinear forms are worked out. We study the Lagrange equations, their system of equations of variations (Jacobi equations) and their adjoint system by proving that, for class $\text{C}$⁴ and regular Lagrangian densities, they are always self-adjoint. We then introduce a concept of analytic representation which occurs when the Lagrange equations coincide with the field equations up to equivalence transformations and refine the definition by particularizing it as direct or indirect and ordered or nonordered. Some of the conventional cases of tensorial fields are considered and we prove, in particular, that the conventional representation of the complex scalar field in interaction with the electromagnetic field is of the ordered indirect type. For the objective of identifying our program we recall the two classes of equivalence transformations of the Lagrangian densities which are primarily used nowadays, namely, the Lorentz (coordinate) transformations and the gauge transformations (transformations of fields within a fixed coordinate system), and postulate the existence of a third class, which we term isotopic transformations of the Lagrangian density and which consist of equivalence transformations within a fixed coordinate system and gauge. We finally outline the objectives of our program, which essentially consist of the identification of the necessary and sufficient conditions for the existence of a Lagrangian in field theories and their first application to the transformation theory within the framework of our variational approach to self-adjointness.     

Covariant Canonical Formalism of Fields

The canonical formalism of fields consistentwith the covariance principle of special relativity isgiven here. The covariant canonical transformations offields are affected by 4-generating functions. All dynamical equations of fields, e.g., theHamilton, Euler–Lagrange, and other fieldequations, are preserved under the covariant canonicaltransformations. The dynamical observables are alsoinvariant under these transformations. The covariantcanonical transformations are therefore fundamentalsymmetry operations on fields, such that the physicaloutcomes of each field theory must be invariant under these transformations. We give here also thecovariant canonical equations of fields. These equationsare the covariant versions of the Hamilton equations.They are defined by a density functional that is scalar under both the Lorentz and thecovariant canonical transformations of fields.     

Certain properties of the externally invariant Maxwell equations and physical meaning of the external-transformation group

A general vector potential derived for the Maxwell equations in their general form is used to find and analyze the tensor for the angular momentum density. With j = 0, the general Maxwell equations are invariant with respect to a four-parameter group of external displacement, whose addition to the group of external transformations leads to field equations which are invariant with respect to the ten-parameter group of external transformations. For integral quantities, the vector of the total field energy-momentum and the tensor of the total angular momentum, transformations from the inhomogeneous group of external transformations are equivalent to transformations from the inhomogeneous Lorentz group.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii Fizika, No. 11, pp. 53–57, November, 1969.The authors thank D. D. Ivanenko, A. E. Levashov, and the participants of the Ivanovo interinstitute seminar on mathematical physics for discussion of these results.     

Unified Field Theory From Enlarged Transformation Group. The Covariant Derivative for Conservative Coordinate Transformations and Local Frame Transformations

Pandres has developed a theory in which the geometrical structure of a real four-dimensional space-time is expressed by a real orthonormal tetrad, and the group of diffeomorphisms is replaced by a larger group called the conservation group. This paper extends the geometrical foundation for Pandres’ theory by developing an appropriate covariant derivative which is covariant under all local Lorentz (frame) transformations, including complex Lorentz transformations, as well as conservative transformations. After defining this extended covariant derivative, an appropriate Lagrangian and its resulting field equations are derived. As in Pandres’ theory, these field equations result in a stress-energy tensor that has terms which may automatically represent the electroweak field. Finally, the theory is extended to include 2-spinors and 4-spinors.     

Quaternion Gravi-Electromagnetism

Defining the generalized charge, potential, current and generalized fields as complex quantities where real and imaginary parts represent gravitation and electromagnetism respectively, corresponding field equation, equation of motion and other quantum equations are derived in manifestly covariant manner. It has been shown that the field equations are invariant under Lorentz as well as duality transformations. It has been shown that the quaternionic formulation presented here remains invariant under quaternion transformations.     

Symmetry group of generalized vector field equations

The equations describing an electromagnetic field, a Yang-Mills massless field, and a free massive vector field are generalized in a quaternion setting. The generalized equations are invariant under a six-parameter group of transformations, which do not affect the space-time coordinates. In application to the generalized Maxwell equations the indicated group is isomorphic to Zaitsev's group of outer transformations of the electromagnetic field variables.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 45–48, August, 1977.The author is indebted to S. I. Kruglov, Yu. A. Kurochkin, and E. A. Tolkachev for a critical and stimulating discussion of the present results.     

Bäcklund transformations and new solutions of nonlinear wave equations in four-dimensional space-time

Bäcklund transformations for several nonlinear field equations in four-dimensional space-time relating two solutions of the same equation (symmetry), or two different equations (dynamical), are given. These transformations can be used to generate new families of solutions and infinitely many conservation laws for nonlinear equations.Bäcklund transformations and solutions of nonlinear equations have been studied extensively in one-space and one-time dimension. We give here a fairly general method for a class of equations in four-dimensional space-time which paves the way for many further generalizations.Supported in part by the U.S. National Academy of Sciences Foundation Grant No. INT 73-20002 A01 (formerly GF-41958).     

Gravitational and Electroweak Unification

Einstein suggested that a unified field theorybe constructed by replacing the diffeomorphisms (thecoordinate transformations of general relativity) withsome larger group. We have constructed a theory that unifies the gravitational and electroweakfields by replacing the diffeomorphisms with the largestgroup of coordinate transformations under whichconservation laws are covariant statements. Thisreplacement leads to a theory with field equations whichimply the validity of the Einstein equations of generalrelativity, with a stress-energy tensor that is justwhat one expects for the electroweak field andassociated currents. The electroweak field appears as aconsequence of the field equations (rather than as a"compensating field" introduced to secure gaugeinvariance). There is no need for symmetry breaking toaccommodate mass, because the U(1) × SU(2) gaugesymmetry is approximate from the outset. Thegravitational field is described by the space-timemetric, as in general relativity. The electroweak fieldis described by the "mixed symmetry" part of the Riccirotation coefficients. The gauge symmetry-breakingquantity is a vector formed by contracting theLevi-Civita symbol with the totally antisymmetric partof the Ricci rotation coefficients.     

Necessary and sufficient conditions for the existence of a Lagrangian in field theory. III. Generalized analytic representations of tensorial field equations

In this paper we first study the equivalence transformations of class C², regular, tensorial, quasi-linear systems of field equations which (a) preserve the continuity, regularity, and quasi-linear structure of the systems; and (b) occur within a fixed system of Minkowski coordinates and field components. We identify, among the transformations of this class, those which either induce or preserve a self-adjoint structure of the field equations and we term them genotopic and isotopic transformations, respectively. We then give the necessary and sufficient conditions for an equivalence transformation of the above type to be either genotopic or isotopic. By using this methodology, we then extend the theorem on the necessary and sufficient condition for the existence of ordered direct analytic representations introduced in the preceding paper to the case of ordered indirect analytic representations in terms of the conventional Lagrange equations; we introduce a method for the construction of a Lagrangian, when it exists, in this broader context; and we explore some implications of the underlying methodology for the problem of the structure of the Lagrangian capable of representing interactions within the framework of the indirect analytic representations. Some of the several aspects which demand an inspection prior to the use of this analytic approach in actual models are pointed out. In particular, we indicate a possible deep impact in the symmetries and conservation laws of the system generated by the use of the concept of indirect analytic representation. As a preparatory step prior to the analysis of these problems, we study some methodological aspects which underlie the generalized Lagrange equations postulated in the first paper of this series for the case when they are regular, namely, when they are simple equivalence transformations of the conventional Lagrange equations. We first introduce a generalization of the action principle capable of inducing the generalized as well as the conventional equations. In this way we establish that the former equations are “bona fide” analytic equations. Finally, as our most general analytic framework for the case of unconstrained field equations, we work out the necessary and sufficient condition for the existence of ordered direct analytic representations of quasi-linear systems in terms of the generalized analytic equations and study their relationship to the conventional representations.     

GROUP OF TRANSFORMATIONS WITH RESPECT TO THE COUNTERPART OF RAPIDITY AND RELATED FIELD EQUATIONS

The Lorentz-group of transformations usually consists of linear transformations of the coordinates, keeping as invariant the norm of the four-vector in (Minkowski) space-time. Besides those linear transformations, one may construct different forms of nonlinear transformations of the coordinates keeping unchanged that respective invariant. In this paper we explore nonlinear transformations of second-order which have a natural interpretation within the framework of Yamaleev's concept of the counterpart of rapidity (co-rapidity). The purpose of developed concept is to show that the formulae for energy and momentum of the relativistic particle become regular near the zero-mass and speed of light states. Furthermore, in a covariant formulation, the co-rapidity is presented as a four-vector which admits an extension of the Lorentz-group of transformations. In this paper we additionally show, that in the same way as the rapidity is related to the electromagnetic field, the co-rapidity is related to the field of strengths, which are given by a four-vector. The corresponding equations of such a field are also constructed.     

Complex symmetries of electrodynamics

We prove that a set of nonsingular free solutions of Maxwell's equations forms a representation of the group obtained by analytic continuation of the Poincaré group to complex values of the group parameters, and that a set of singular solutions forms a representation of the group obtained by analytic continuation of the conformal group to complex values of the group parameters. These results are obtained by constructing a theory governing 2 × 2 complex matrix fields defined for complex values of position and time; the equations of this theory are invarient with respect to complex Poincaré transformations and complex conformal transformations, but the set of nonsingular solutions is in one-to-one correspondence with a set of nonsingular solutions of Maxwell's equations, and a similar correspondence exists for the singular solutions. Certain collections of solutions of Maxwell's equations for the field of a current form representations of these complex groups if both magnetic and electric currents are permitted, in which case complex transformations provide a natural connection between electric and magnetic charge. A class of complex transformations also yield natural relations between sources moving slower than light and sources moving faster than light.     

一类带限定变换的二阶耦合线性微分方程组的解析解

用函数和方程变换将二阶耦合线性微分方程组转化为一阶非线性类椭圆方程，并给出了一次和二次限定变换下方程组的Jacobi椭圆函数解析解，所得结果修正了文献中超导特例的近似解，进一步肯定了超导边界层电场的存在性. 关键词： 微分方程 Jacobi椭圆函数 解析解 超导     

The group of external transformations of electromagnetic field variables

A quaternion formulation of Maxwell equations in a medium is given. Adding gradient terms to these equations, they become invariant under the transformations of field variables forming a representation of the group S0(3, 3) — the group of external transformations of electromagnetic field variables. A discussion of possible physical applications of the generalized equations of macroscopic electrodynamics is given.     

Effect of radiation and chemical reaction on natural convective MHD flow through a porous medium with double diffusion

The influence of radiation and chemical reaction on a natural convective MHD flow through a porous medium bounded by a vertical infinite surface in the presence of transverse magnetic field is studied. The basic equations governing the flow, heat and mass transfer are reduced to a set of ordinary differential equations by appropriate transformations. Governing equations are solved by perturbation technique for velocity, temperature and concentration, and that has been presented graphically for different values of involved parameters. It is observed that effects of magnetic parameter and radiation parameter in the flow field affect the flow significantly.     

String field equations from the generalized sigma model II

We improve and extend a method introduced in an earlier paper for deriving string field equations. The idea is to impose conformal invariance on a generalized sigma model, using a background field method that ensures covariance under very general non-local coordinate transformations. The method is used to derive the free string equations, as well as the interacting equations for the graviton-dilaton system. The full interacting string field equations derived by this method should be manifestly background independent.     

A complete action for the spinning string

We present an action for the Neveu-Schwarz-Ramond model from which follow both the field equations and the gauge and supergauge constraints. This is done by coupling the free-field action to two-dimensional supergravity in a geometrically clear way. The constraints arise as the supergravity field equations, the supergravity fields playing the role of Lagrange multipliers. The action is invariant under local supersymmetry transformations and, as a consequence, the field equations and the constraints are consistent. The commutator structure of the local supersymmetry algebra is exhibited. It is also shown that there exists a special gauge in which the action, the field equations and the constraints take the free-field from of the usual formulation of the Neveu-Schwarz-Ramond model.     

Mean motion of Dirac electrons in a gravitational field

A sequence of Foldy-Wouthuysen transformations applied to the Dirac equation coupled to a background gravitational field is used to obtain evolution equations for the mean position, mean momentum, and mean spin operators. These equations are compared with the analogous Papapetrou equations for a classical gravitationally coupled pole-dipole particle.     

A space-time functional formalism for classical field equations

The equations derived by R.L. Levis and R.H. Kraichnan for a space-time functional for turbulence are further investigated in the general case of classical field equations. The compact form of the equations obtained allows for their different transformations and enables to discuss the effect of initial conditions on the final results.     

非中心力场中经典粒子的轨道参数方程与对称性

把非中心力场中经典粒子运动微分方程写成Ermakov方程的形式，得到Ermakov不变量.用改变时间坐标标度的方法得到用能量H和Ermakov不变量表示的轨道参数方程，并研究两守恒量（能量和Ermakov不变量）相应的无限小变换的Noether对称性、Lie对称性和形式不变性.研究结果表明：与两守恒量相应的无限小变换既具有Noether对称性，也具有Lie对称性和形式不变性. 关键词： 非中心力场 轨道参数方程 守恒量 对称性