The existing field theories are based on the properties of closed

1. The connection of field-theory equations with the equations for material systemsL.I. Petrova (Moscow State University, Russia) 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 systems (material media). • The process of obtaining closed exterior forms demonstrates the connection between field-theory equations and the equations for material systems and points to the fact that the foundations of field theories must be conditioned by the properties of equations conservation laws for material systems.

2. 2. Closed exterior form:conservation laws for physical fields The closure condition for exterior form: describes conservative quantity. This correspond toconservation law The closure condition for exterior form defined on pseudostructureand closure condition for dual form also correspond to conservation laws - a conservative object (physical structures) (dual form describes a pseudostructure) It isthe conservation laws for physical fields. The existing field theories are based on the properties of closed exterior forms, which correspond to conservation laws for physical fields. Closed exterior forms corresponding to field theories are obtained from the equations modelling conservation (balance) laws for material systems (material media).

3. 3. Peculiarities of differential equations for material systems • Equations for material systems are differential (or integral) equations that describe the conservation laws (balance) for energy, linear momentum, angular momentum and mass. In the accompanying system of coordinates(connected with the manifold built by the trajectories of the material system elements) equations of energy and linear momentumis now written in the form - the functional of the state(action functional, entropy, 1) wave functioncan be regarded as examples of the functional ) • -coordinate along thetrajectory 2) • - coordinates normal to the trajectory • Eqs. (1), (2) can be convoluted into the relation

4. 4. This relation can be written as where - the differential form of thefirst degree. In the general case (for energy, linear momentum, angular momentum and mass) this relation will be the form Since the balance conservation laws are evolutionary ones, this relation is also an evolutionary relation The evolutionary relation obtained from equations of balance conservation laws for material systems (continuous media) carries not only mathematical but also large physical loading. This is due to the fact that the evolutionary relation possesses the duality. On the one hand, this relation corresponds to material system, and on other, as it will be shown below, describes the mechanism of generating physical structures (from that it is obtained closed inexact exterior form). This discloses the properties and peculiarities of the field-theory equations and their connection with the equations of balance conservation laws for material systems.

5. 5. Properties ofevolutionary relation Evolutionary relation isanonidentical relation as it involves an unclosed skew-symmetric evolutionary differential form : Example:( ) as the commutator of the form is nonzero: • (The coefficients of the form • depend on energetic action and • on force action which have different nature. And so the commutator of the form constructed fromderivatives of such coefficients is nonzero.) • Nonidentical evolutionary relation possesses an unique peculiarity, namely, this relation can generate closed exterior forms, which correspond to conservation laws for physical fields and disclose physical structures.

6. 6. Peculiaritiesof evolutionary relation The evolutionary nonidentical relation is a selfvarying one(it is an evolutionary relation and it contains two objects one of which appears to be unmeasurable and cannot be compared with another one, and thereforethe process of mutual variation cannot terminate). (The selfvariation of thenonequilibrium state of the material system takes place). The selfvarying evolutionary relation leads torealization of the conditions of degenerate transform.Under degenerate transformation from unclosed evolutionary form (which differential is nonzero) it is obtained closed exterior form (which differential equals zero) that corresponds to conservation law for physical fields. In this case from nonidentical relation the relation that is identical on pseudostructure is obtained.

7. Obtaining closed (inexact) exterior formand identical relation Under degenerate transform it is realized the transition That is, it is realized apseudostructure(equations pseudostructure ) , on which the differential evolutionary formsvanish:. It is obtained closed (inexact) exterior form . On the pseudostructure evolutionary relation transforms intoan identical relation(closed exterior form is differential). It is shown that the closed exterior forms are obtained from evolutionaryforms. This disclose a relation between differential equations for material media and the invariant field theories.

8. 8. Specific features of field-theory equations • Nonidentical evolutionary relation discloses the features of field- theory equationsand their distinction from equations for material media. Equations for material media are partial differential equations for desired functions like a velocity of particles (elements), temperature, pressure and density, which correspond to physical quantities of material systems. Such functions describe the character of varying physical quantities of material system. The functionals (and state- functions) like wave-function, action functional, entropy and others, which specify the state of material systems, and corresponding relations (nonidentical evolutionary relation) are used only for analysisof integrability of these equations. And in field theories such relations play a role of equations. It can be shown that all equations of existing field theories are nonidentical evolutionary relation or their analogs (differential or tensor ones)

9. 9. The connection the field theory equations with skew-symmetricforms of appropriate degrees • The Hamilton formalismis based on the properties of closed exterior and dual forms of the first degree, quantum mechanics – zero degree, the electromagnetic field equations- second degree. The third degree correspond to the gravitational field. • The identical relations of the field theories are the identical relations for exterior forms. In general form identical relation can be written as ( - closed exterior forms). • The examples: Einstein equation,the Poincare invariant,vector and tensor identical relations, the Cauchi-Riemann conditions, canonical relations, the thermodynamic relations, the eikonal relations and so on.

10. References • Petrova L.I. Evolutionary forms: Conservation laws and causality, http://arXiv.org/abs/math-ph/0506018, 2005 2. Petrova L.I. Evolutionary forms: The generation of differential-geometrical structures.(Symmetries and Conservation laws.), http://arXiv.org/abs/math.DG/0510142, 2005 3. Petrova L.I. The connection between field-theory and the equations for material systems, http://arXiv.org/abs/0705.0222, 2007 4. Petrova L.~I. The quantum character of physical fields. Foundations of field theories, Electronic Journal of Theoretical Physics, v.3, 10 (2006), 89-107p. 5. Петрова Л.И. Кососимметричные дифференциальные формы: Законы сохранения. Основы теории поля. Москва, URSS, 2007