arXiv subjects

L. I. Petrova

Publications and source records attributed to L. I. Petrova.

At least 19 recordsLinked to original sources

arXiv2024

Quantum Properties of Mathematical Physics Equations. Generation of Quantum Structures

It is shown with the help of skew-symmetric forms that the mathematical physics equations, on which no additional conditions are imposed, have quantum properties. And this is due to the integrability properties of differential equations, which depends on the consistency of derivatives with respect to different variables and the consistency of equations, if the mathematical physics equations are a system of equations. It was found that such equations on the original tangent space turn out to be non-integrable. Their derivatives do not form a differential. The integrability of such equations is realized only on the structures of a cotangent integrable manifold. This happens using a degenerate, non-differential-preserving transformation that has quantum properties. When implementing degenerate transformations, mini structures (quanta) arise, from which integrable structures are formed. Such properties of integrability of mathematical physics equations and features of degenerate transformations reveal the quantum properties of mathematical physics equations and their ability to generate quantum structures.

math.GM↗

arXiv2023

Internal connection between the field theory equations. Fundamentals of the field theory

It is shown that there is a correspondence between field theory equations such as the Dirac, Shrődinger, Maxwell, Einstein equations and closed exterior forms of a certain degree. In this case, the Dirac and Shrődinger equations for the wave function correspond to closed exterior forms of zero degree. The Shrődinger equation for the state functional corresponds to closed exterior forms of the first degree. The Maxwell's equations based on exterior forms of second degree. Einstein's equation for the gravitational field consists of covariant tensors, which correspond to closed exterior forms of the second degree. However, the covariant tensors of the Einstein equation are derived from the covariant tensors, which correspond to closed exterior forms of third degree. Such a correspondence between the field theory equations and closed exterior forms of a certain degree reveals the internal connection between the field theory equations. At the same time, it was shown that closed exterior forms, on which the field theory equations for physical fields are based, are associated with the equations of mathematical physics for material media, such as thermodynamic, gas-dynamic, electromagnetic, cosmological equations, etc. This indicates the connection between the field theory equations and the mathematical physics equations and reveals the foundations of field theory.

math.GM↗

arXiv2022

Hidden unique possibilities of mathematical physics equations (Formalism of skew-symmetric forms)

It is shown that mathematical physics differential equations have properties that allow describing processes such as the structures emergence, discrete transitions, quantum jumps. The peculiarity is that such properties are hidden. They do not follow directly from the mathematical physics equations but are realized discretely in the solving process. This is due to the mathematical physics equations integrability, which, as shown, can be realized only discretely in the presence of any degrees of freedom. In this case, a transition occurs from the original coordinate space with a solution that is not a function (the solution derivatives do not compose a differential) to integrable structures with a solution that is a discrete function. It is the double solutions and spatial transitions that can describe the processes of the emergence of any structures or phenomena. Due to hidden properties, the mathematical physics equations have unique possibilities in describing physical processes and phenomena that cannot be described in the framework of other mathematical formalisms. Such results were obtained using skew-symmetric differential forms.

math.GM↗

arXiv2010

Role of skew-symmetric differential forms in mathematics

Skew-symmetric forms possess unique capabilities. The properties of closed exterior and dual forms, namely, invariance, covariance, conjugacy and duality, either explicitly or implicitly appear in all invariant mathematical formalisms. This enables one to see an internal connection between various branches of mathematics. However, the theory of closed exterior forms cannot be completed without an answer to a question of how the closed exterior forms emerge. In the present paper we discus essentially new skew-symmetric forms, which generate closed exterior forms. Such skew-symmetric forms, which are evolutionary ones, are derived from differential equations, and, in contrast to exterior forms, they are defined on nonintegrable manifolds.

math.GM↗

arXiv2010

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.

physics.gen-ph↗

arXiv2009

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.

math-ph↗

arXiv2009

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.

math-ph↗

arXiv2009

Skew-symmetric differential forms. Invariants. Realization of invariant structures

Skew-symmetric differential forms play an unique role in mathematics and mathematical physics. This relates to the fact that closed exterior skew-symmetric differential forms are invariants. The concept of "Exterior differential forms" was introduced by E.Cartan for a notation of integrand expressions, which can create the integral invariants.(The existence of integral invariants was recognized by A.Poincare while studying the general equations of dynamics.) All invariant mathematical formalisms are based on invariant properties of closed exterior forms. The invariant properties of closed exterior forms explicitly or implicitly manifest themselves essentially in all formalisms of field theory, such as the Hamilton formalism, tensor approaches, group methods, quantum mechanics equations, the Yang-Mills theory and others. They lie at the basis of field theory. However, in this case the question of how the closed exterior forms are obtained arises. In present work it is shown that closed exterior forms, which possess the invariant properties, are obtained from skew-symmetric differential forms, which, as contrasted to exterior forms, are defined on nonintegrable manifolds. The process of generating closed exterior forms describes the mechanism of realization of invariants and invariant structures.

math.GM↗

arXiv2008

Two types of conservation laws. Connection of physical fields with material systems. Peculiarities of field theories

Historically it happen so that in branches of physics connected with field theory and of physics of material systems (continuous media) the concept of "conservation laws" has a different meaning. In field theory "conservation laws" are those that claim the existence of conservative physical quantities or objects. These are conservation laws for physical fields. In contrast to that in physics (and mechanics) of material systems the concept of "conservation laws" relates to conservation laws for energy, linear momentum, angular momentum, and mass that establish the balance between the change of physical quantities and external action. In the paper presented it is proved that there exist a connection between of conservation laws for physical fields and those for material systems. This points to the fact that physical fields are connected with material systems. Such results has an unique significance for field theories. This enables one to substantiate many basic principles of field theories, such as, for example, the unity of existing field theories and the causality. The specific feature of field theory equations, namely, their connection to the equations for material systems, is elicited. Such results have been obtained by using skew-symmetric differential forms, which reflect the properties of conservation laws.

physics.gen-ph↗

arXiv2007

The connection between field-theory and the equations for material sistems

The existing field theories are based on the properties of closed exterior forms, which correspond to conservation laws for physical fields. 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 sistems (material media). The process of obtaining closed exterior forms demonstrates the connection between field-theory equations and the equations for material sistems and points to the fact that the foundations of field theories must be conditioned by the properties of equations conservation laws for material sistems.

physics.gen-ph↗

arXiv2007

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.

math-ph↗

arXiv2007

Specific features of differential equations of mathematical physics

Three types of equations of mathematical physics, namely, the equations, which describe any physical processes, the equations of mechanics and physics of continuous media, and field-theory equations are studied in this paper. In the first and second case the investigation is reduced to the analysis of the nonidentical relations of the skew-symmetric differential forms that are obtained from differential equations. It is shown that the integrability of equations and the properties of their solutions depend on the realization of the conditions of degenerate transformations under which the identical relations are obtained from the nonidentical relation. The field-theory equations, in contrast to the equations of first two types, are the relations made up by skew-symmetric differential forms or their analogs (differential or integral ones). This is due to the fact that the field-theory equations have to describe physical structures (to which closed exterior forms correspond) rather than physical quantities. The equations that correspond to field theories are obtained from the equations that describe the conservation laws (of energy, linear momentum, angular momentum, and mass) of material systems (of continuous media). This disclose a connection between field theories and the equations for material systems (and points to that material media generate physical fields).

math-ph↗

arXiv2006

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.

physics.gen-ph↗

arXiv2005

Analysis of the equations of mathematical physics and foundations of field theories with the help of skew-symmetric differential forms

In the paper it is shown that, even without a knowledge of the concrete form of the equations of mathematical physics and field theories, with the help of skew-symmetric differential forms one can see specific features of the equations of mathematical physics, the relation between mathematical physics and field theory, to understand the mechanism of evolutionary processes that develop in material media and lead to emergency of physical structures forming physical fields. This discloses a physical meaning of such concepts like "conservation laws", "postulates" and "causality" and gives answers to many principal questions of mathematical physics and general field theory. In present paper, beside the exterior forms, the skew-symmetric differential forms, whose basis (in contrast to the exterior forms) are deforming manifolds, are used. Mathematical apparatus of such differential forms(which were named evolutionary ones) includes nontraditional elements like nonidentical relations and degenerate transformations and this enables one to describe discrete transitions, quantum steps, evolutionary processes, and generation of various structures.

math-ph↗

arXiv2005

Role of exterior and evolutionary skew-symmetric differential forms in mathematical physics

A role of skew-symmetric differential forms in mathematical physics relates to the fact that they reflect the properties of conservation laws. The closed exterior forms correspond to the conservation laws for physical fields, whereas the evolutionary forms correspond to the conservation laws for material media. Skew-symmetric differential forms can describe a conjugacy of any objects (that correspond to the conservation laws). The closed exterior forms describe conjugated objects. And the evolutionary forms, whose basis are deforming manifolds, describe the process of conjugating objects and obtaining conjugated objects. From the evolutionary forms the closed exterior forms are obtained. This shows that material media generate physical fields. The relation between evolutionary and closed exterior forms discloses the relation between the equations of mathematical physics and field theories. This explains the field theory postulates. Conjugacy is possible if there is symmetry. Symmetries of closed exterior forms, which are conditions of fulfilment of the conservation laws for physical fields, are interior symmetries of field theories. And symmetries of dual forms (due to the degrees of freedom of material media) are external symmetries of the equations of field theories. This shows connection between internal and external symmetries of field theories.

math-ph↗

arXiv2005

Evolutionary forms: The generation of differential-geometrical structures. (Symmetries and Conservation laws.)

Evolutionary forms, as well as exterior forms, are skew-symmetric differential forms. But in contrast to the exterior forms, the basis of evolutionary forms is deforming manifolds (with unclosed metric forms). Such forms possess a peculiarity, namely, the closed inexact exterior forms are obtained from that. The closure conditions of inexact exterior form (vanishing the differentials of exterior and dual forms) point out to the fact that the closed inexact exterior form is a quantity conserved on pseudostructure having the dual form as the metric form. We obtain that the closed inexact exterior form and corresponding dual form made up a conservative object, i.e. a quantity conserved on pseudostructure. Such conservative object corresponds to the conservation law and is a differential-geometrical structure. Transition from the evolutionary form to the closed inexact exterior form describes the process of generating the differential-geometrical structures. This transition is possible only as a degenerate transformation, the condition of which is a realization of a certain symmetry. Physical structures that made up physical fields are such differential-geometrical structures. And they are generated by material systems (medias). Relevant symmetries are caused by the degrees of freedom of material system.

math.DG↗

arXiv2005

Evolutionary forms: Conservation laws and causality

Evolutionary forms are skew-symmetric differential forms the basis of which, as opposed to exterior forms, are deforming manifolds (with unclosed metric forms). Such differential forms arise when describing physical processes. A specific feature of evolutionary forms is the fact that from the evolutionary forms, which correspond to the conservation laws for material media, the closed exterior forms, which correspond to the conservation laws for physical fields, are obtained. This shows that material media generate physical fields. And by this the determinacy of physical processes and phenomena is revealed. In this paper we obtain the mathematic apparatus that allows to describe discrete transitions and quantum jumps. This relates to the fact that the mathematic apparatus of exterior and evolutionary forms, which basis involves nonidentical relations and degenerate transformations, can describe transitions from nonconjugate operators to conjugate ones. None of mathematic formalisms contains such possibilities. The physical results that disclose a mechanism of evolutionary processes in material media and a generation of physical fields are obtained. These results explain many actual processes.

math-ph↗

arXiv2005

Qualitative investigation of Hamiltonian systems by application of skew-symmetric differential forms

A great number of works is devoted to qualitative investigation of Hamiltonian systems. One of tools of such investigation is the method of skew-symmetric differential forms. In present work, under investigation Hamiltonian systems in addition to skew-symmetric exterior differential forms, skew-symmetric differential forms, which differ in their properties from exterior forms, are used. These are skew-symmetric differential forms defined on manifolds that are nondifferentiable ones. Such manifolds result, for example, under describing physical processes by differential equations. This approach to investigation of Hamiltonian systems enables one to understand a connection between Hamiltonian systems and partial differential equations, which describe physical processes, and to see peculiarities of Hamiltonian systems and relevant phase spaces connected with this fact.

math-ph↗