Exterior Differential Forms in Field Theory

A role of the exterior differential forms in field theory is connected with a fact that they reflect properties of the conservation laws. In field theory a role of the closed exterior forms is well known. A condition of closure of the form means that the closed form is the conservative quantity, and this corresponds to the conservation laws for physical fields. In the present work a role in field theory of the exterior forms, which correspond to the conservation laws for the material systems is clarified. These forms are defined on the accompanying nondifferentiable manifolds, and hense, they are not closed. Transition from the forms, which correspond to the conservation laws for the material systems, to those, which correspond to the conservation laws for physical fields (it is possible under the degenerate transform), describe a mechanism of origin of the physical structures that format physical fields. In the work it is shown that the physical structures are generated by the material systems in the evolutionary process. In Appendices we give an analysis of the principles of thermodinamics and equations of the electromagnetic field. A role of the conservation laws in formation of the pseudometric and metric spaces is also shown.Comment: 42 pages, Latex 2.0

The quantum character of physical fields. Foundations of field theories

The existing field theories are based on the properties of closed exterior forms, which are invariant ones and correspond to conservation laws for physical fields. Hence, to understand the foundations of field theories and their unity, one has to know how such closed exterior forms are obtained. In the present paper it is shown that closed exterior forms corresponding to field theories are obtained from the equations modelling conservation (balance)laws for material media. It has been developed the evolutionary method that enables one to describe the process of obtaining closed exterior forms. The process of obtaining closed exterior forms discloses the mechanism of evolutionary processes in material media and shows that material media generate, discretely, the physical structures, from which the physical fields are formed. This justifies the quantum character of field theories. On the other hand, this process demonstrates the connection between field theories and the equations for material media and points to the fact that the foundations of field theories must be conditioned by the properties of material media. It is shown that the external and internal symmetries of field theories are conditioned by the degrees of freedom of material media. The classification parameter of physical fields and interactions, that is, the parameter of the unified field theory, is connected with the number of noncommutative balance conservation laws for material media.Comment: 23 page

Physical structures. Forming physical fields and manifolds. (Properties of skew-symmetric differential forms)

It is shown that physical fields are formed by physical structures, which in their properties are differential-geometrical structures. These results have been obtained due to using the mathematical apparatus of skew-symmetric differential forms. This apparatus discloses the controlling role of the conservation laws in evolutionary processes, which proceed in material media and lead to origination of physical structures and forming physical fields and manifolds.Comment: 17 page

Invariant and evolutionary properties of the skew-symmetric differential forms

The present work pursues the aim to draw attention to unique possibilities of the skew-symmetric differential forms. At present the theory of skew-symmetric exterior differential forms that possess invariant properties has been developed. In the work the readers are introduced to the skew-symmetric differential forms that were called evolutionary ones because they possess evolutionary properties. The combined mathematical apparatus of exterior and evolutionary skew-symmetric differential forms in essence is a new mathematical language. This apparatus can describe transitions from nonconjugated operators to conjugated ones. There are no such possibilities in any mathematical formalism. In the present work it has been shown that the properties of exterior or evolutionary forms are explicitly or implicitly accounted for in all mathematical (and physical) formalisms.Comment: LaTex, 36 page

Conservation laws. Generation of physical fields. Principles of field theories

In the paper the role of conservation laws in evolutionary processes, which proceed in material systems (in material media) and lead to generation of physical fields, is shown using skew-symmetric differential forms. In present paper the skew-symmetric differential forms on deforming (nondifferentiable) manifolds were used in addition to exterior forms, which have differentiable manifolds as a basis. Such skew-symmetric forms (which were named evolutionary ones since they possess evolutionary properties), as well as the closed exterior forms, describe the conservation laws. But in contrast to exterior forms, which describe conservation laws for physical fields, the evolutionary forms correspond to conservation laws for material systems. The evolutionary forms possess an unique peculiarity, namely, the closed exterior forms are obtained from these forms. It is just this that enables one to describe the process of generation of physical fields, to disclose connection between physical fields and material systems and to resolve many problems of existing field theories.Comment: 14 page

Skew-symmetric forms: On integrability of equations of mathematical physics

The study of integrability of the mathematical physics equations showed that the differential equations describing real processes are not integrable without additional conditions. This follows from the functional relation that is derived from these equations. Such a relation connects the differential of state functional and the skew-symmetric form. This relation proves to be nonidentical, and this fact points to the nonintegrability of the equations. In this case a solution to the equations is a functional, which depends on the commutator of skew-symmetric form that appears to be unclosed. However, under realization of the conditions of degenerate transformations, from the nonidentical relation it follows the identical one on some structure. This points out to the local integrability and realization of a generalized solution. In doing so, in addition to the exterior forms, the skew-symmetric forms, which, in contrast to exterior forms, are defined on nonintegrable manifolds (such as tangent manifolds of differential equations, Lagrangian manifolds and so on), were used. In the present paper, the partial differential equations, which describe any processes, the systems of differential equations of mechanics and physics of continuous medium and field theory equations are analyzed.Comment: 8 pages, LaTex 2

Physical meaning and a duality of concepts of wave function, action functional, entropy, the Pointing vector, the Einstein tensor

Physical meaning and a duality of concepts of wave function, action functional, entropy, the Pointing vector, the Einstein tensor and so on can be disclosed by investigating the state of material systems such as thermodynamic and gas dynamic systems, systems of charged particles, cosmologic systems and others. These concepts play a same role in mathematical physics. They are quantities that specify a state of material systems and also characteristics of physical fields. The duality of these concepts reveals in the fact that they can at once be both functionals and state functions or potentials. As functionals they are defined on nonintegrable manifold (for example, on tangent one), and as a state function they are defined on integrable manifold (for example, on cotangent one). The transition from functionals to state functions dicribes the mechanism of physical structure origination. The properties of these concepts can be studied by the example of entropy and action. The role of these concepts in mathematical physics and field theory will be demonstrated. Such results have been obtained by using skew-symmetric forms. In addition to exterior forms, the skew-symmetric forms, which are obtained from differential equations and, in distinction to exterior forms, are evolutionary ones and are defined on nonintegrable manifolds, were used.Comment: 17 page

Identical and Nonidentical Relations. Nondegenerate and Degenerate Transformations. (Properties of skew-symmetric differential forms)

Identical relations occur in various branches of mathematics and mathematical physics. The Cauchy-Riemann relations, characteristical and canonical relations, the Bianchi identities and others are examples of identical relations. It can be shown that all these relations express either the conditions of closure of exterior (skew-symmetric) differential forms and corresponding dual forms or the properties of closed exterior forms. Since the closed differential forms are invariant under all transformations, which conserve the differential (these are gauge transformations: unitary, canonical, gradient and others), from this it follows that identical relations are a mathematical representation of relevant invariant and covariant objects. The theory of exterior differential forms cannot answer the question of how do invariant objects appear and what does these objects generate? The answer to this question can be obtained using the skew-symmetric differential forms, which possess the evolutionary properties. The mathematical apparatus of such evolutionary forms contains nonidentical relations, from which the identical relations corresponding to invariant objects are obtained with the help of degenerate transformations. Due to such potentialities, the mathematical apparatus of skew-symmetrical differential forms enables one to describe discrete transitions, evolutionary processes and generation of various structures.Comment: 15 page

Qualitative investigation of the solutions to differential equations. (Application of the skew-symmetric differential forms)

The presented method of investigating the solutions to differential equations is not new. Such an approach was developed by Cartan in his analysis of the integrability of differential equations. Here this approach is outlined to demonstrate the role of skew-symmetric differential forms. The role of skew-symmetric differential forms in a qualitative investigation of the solutions to differential equations is conditioned by the fact that the mathematical apparatus of these forms enables one to determine the conditions of consistency for various elements of differential equations or for the system of differential equations. This enables one, for example, to define the consistency of the partial derivatives in the partial differential equations, the consistency of the differential equations in the system of differential equations, the conjugacy of the function derivatives and of the initial data derivatives in ordinary differential equations and so on. The functional properties of the solutions to differential equations are just depend on whether or not the conjugacy conditions are satisfied.Comment: 7 page

The noncommutativity of the conservation laws: Mechanism of origination of vorticity and turbulence

From the equations of conservation laws for energy, linear momentum, angular momentum and mass the evolutionary relation in differential forms follows. This relation connects the differential of entropy and the skew-symmetric form, whose coefficients depend on the characteristics of gas-dynamic system and the external actions. The evolutionary relation turns out to be nonidentical that is explained by the noncommutativity of conservation laws. The properties of such nonidentical relation (selfvariation, degenerate transformation) enable one to disclose the mechanism of evolutionary processes in gas-dynamic system that are accompanied by origination of vorticity and turbulence. In this case the intensity of vorticity and turbulence is defined by the commutator on unclosed skew-symmetric form in the nonidentical evolutionary relation.Comment: 12 page