##
**Geometry of differential equations, secondary differential calculus and quantum field theory.**
*(English.
Russian original)*
Zbl 0616.58009

Sov. Math. 30, No. 1, 14-25 (1986); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1986, No. 1, 13-21 (1986).

The author explains motives why he isn’t in accord with a statement ascribed to J. von Neumann that the language of quantum field theory (QFT) is functional analysis. The author considers the problem of what kind of mathematics is needed for quantum field theory and tries to show that the natural concepts and constructions of the current geometric theory of sets of nonlinear partial differential equations enable one at any rate to raise this problem to firm basis. The author gives arguments stating that the special mathematics necessary for quantum field theory represents differential calculus on the secondary or second-quantized level - secondary differential calculus (SDC) since in field theory the process of quantization starts from partial differential equations (PDE). It is necessary to construct objects whose characteristics are partial differential equations, represented as a category of differential equations (DE), is an instrument which enables one effectively to construct the elements of SDC.

The author begins the construction of SDC from the notion of the secondary vector field. Understanding the manifold M, in which the vector field X is specified, as a phase space of some classical mechanical system, then proceeding to field theory, it is necessary to replace M by a unit of the type Sol Y, where Y is a set of partial differential equations (generally nonlinear) which describes the field we are considering, and Sol Y signifies a ”manifold” of all the local solutions of this system. The solutions of system Y must be considered in all possible areas, in order for the theory to be localizable. Thus it is necessary to determine the vector field concept in a ”manifold” of the Sol Y type. If X is the vector field in Sol Y, then the current it generates is a symmetry of system Y, which is combined with the localization operation. Therefore X is an infinitesimal symmetry of system Y. So the problem reduces to construction a theory of localizable infinitesimal symmetries for arbitrary PDE.

Consider a smooth stratification \(\pi\) : \(E\to M\), dim M\(=n\), dim E\(=m+n\) and \(J^ k=J^ k\) is the manifold of the kth order jets of this stratification. Suppose \(\pi_ k: J^ k \to M\), \(\pi_{k,1}: J^ k \to J^ 1\). The infinite-dimensional manifold \(J^{\infty}\) has a canonical contact infinite-order structure, which is the n-dimensional distribution \(\theta \to C_ 0\), \(\theta \in J^{\infty}\), \(C_ 0\subset T_ 0(J^{\infty})\). The distribution \(C: \theta\to C_ 0\) in \(J^{\infty}\) is a Frobenius one in the sense that it satisfies the prerequisites of the classical Frobenius theorem. Geometrically the set of equations determines in \(J^{\infty}\) the submanifold \(Y_{\infty}\). The distribution C in \(J^{\infty}\), being bounded by the submanifold \(Y_{\infty}\), determines in it the n-dimensional Frobenius distribution C(Y). The manifold (generally infinite-dimensional), which is provided by the finite-dimensional Frobenius distribution and is locally isomorphic to the pair \((Y_{\infty},C(Y))\) is called a diffeotop and by a set of differential partial derivative equations we mean a diffeotop.

Diffeotops are considered as the objects of the DE category. As morphisms in this category it is convenient to consider the smooth mappings of diffeotops for which the corresponding structural Frobenius distributions map each other, as morphisms. The morphisms so determined are locally nonlinear differential operators. Further, the Lie algebra of the localizable vector fields in Sol Y or, equivalently, the Lie algebra of all the localizable infinitesimal symmetries of the system Y is determined. In this connection the objects of secondary calculus in \(Y_{\infty}\) are the corresponding ”primary” objects in \(Y_{\infty}\) which ”project” to Sol Y using the virtual stratification \(Y_{\infty}\to Sol Y\). Using this principle, it is possible to construct secondary differential operators of arbitrary order k. Further, the author defines the notion of secondary functions in Sol Y as elements of the \(\bar H{}^ n(Y_{\infty})\), where \(\bar H{}^ n(Y_{\infty})\) is the horizontal de Rham complex. The secondary operators affect the secondary functions. The notions of a scalar differential operator as a mapping of the form \(\Delta\) : \(\bar H{}^ n\to \bar H^ n\) and a secondary analogue of the de Rham cohomologies as the first term of the \(\sigma\)-spectral sequence are indicated by the author. The approach discussed by the author enables him to obtain secondary analogues of objects such as jets, Spencer complexes, Hamiltonian formalism, etc.

The author’s conclusion is that the considerations presented in his paper (and his other papers also) enable one to consider it highly probable that the conversion of quantum field theory into theory completeness to classical mechanics is possible only in the language of secondary differential calculus, but undoubtedly to achieve this goal it takes time and the work of many researchers.

The author begins the construction of SDC from the notion of the secondary vector field. Understanding the manifold M, in which the vector field X is specified, as a phase space of some classical mechanical system, then proceeding to field theory, it is necessary to replace M by a unit of the type Sol Y, where Y is a set of partial differential equations (generally nonlinear) which describes the field we are considering, and Sol Y signifies a ”manifold” of all the local solutions of this system. The solutions of system Y must be considered in all possible areas, in order for the theory to be localizable. Thus it is necessary to determine the vector field concept in a ”manifold” of the Sol Y type. If X is the vector field in Sol Y, then the current it generates is a symmetry of system Y, which is combined with the localization operation. Therefore X is an infinitesimal symmetry of system Y. So the problem reduces to construction a theory of localizable infinitesimal symmetries for arbitrary PDE.

Consider a smooth stratification \(\pi\) : \(E\to M\), dim M\(=n\), dim E\(=m+n\) and \(J^ k=J^ k\) is the manifold of the kth order jets of this stratification. Suppose \(\pi_ k: J^ k \to M\), \(\pi_{k,1}: J^ k \to J^ 1\). The infinite-dimensional manifold \(J^{\infty}\) has a canonical contact infinite-order structure, which is the n-dimensional distribution \(\theta \to C_ 0\), \(\theta \in J^{\infty}\), \(C_ 0\subset T_ 0(J^{\infty})\). The distribution \(C: \theta\to C_ 0\) in \(J^{\infty}\) is a Frobenius one in the sense that it satisfies the prerequisites of the classical Frobenius theorem. Geometrically the set of equations determines in \(J^{\infty}\) the submanifold \(Y_{\infty}\). The distribution C in \(J^{\infty}\), being bounded by the submanifold \(Y_{\infty}\), determines in it the n-dimensional Frobenius distribution C(Y). The manifold (generally infinite-dimensional), which is provided by the finite-dimensional Frobenius distribution and is locally isomorphic to the pair \((Y_{\infty},C(Y))\) is called a diffeotop and by a set of differential partial derivative equations we mean a diffeotop.

Diffeotops are considered as the objects of the DE category. As morphisms in this category it is convenient to consider the smooth mappings of diffeotops for which the corresponding structural Frobenius distributions map each other, as morphisms. The morphisms so determined are locally nonlinear differential operators. Further, the Lie algebra of the localizable vector fields in Sol Y or, equivalently, the Lie algebra of all the localizable infinitesimal symmetries of the system Y is determined. In this connection the objects of secondary calculus in \(Y_{\infty}\) are the corresponding ”primary” objects in \(Y_{\infty}\) which ”project” to Sol Y using the virtual stratification \(Y_{\infty}\to Sol Y\). Using this principle, it is possible to construct secondary differential operators of arbitrary order k. Further, the author defines the notion of secondary functions in Sol Y as elements of the \(\bar H{}^ n(Y_{\infty})\), where \(\bar H{}^ n(Y_{\infty})\) is the horizontal de Rham complex. The secondary operators affect the secondary functions. The notions of a scalar differential operator as a mapping of the form \(\Delta\) : \(\bar H{}^ n\to \bar H^ n\) and a secondary analogue of the de Rham cohomologies as the first term of the \(\sigma\)-spectral sequence are indicated by the author. The approach discussed by the author enables him to obtain secondary analogues of objects such as jets, Spencer complexes, Hamiltonian formalism, etc.

The author’s conclusion is that the considerations presented in his paper (and his other papers also) enable one to consider it highly probable that the conversion of quantum field theory into theory completeness to classical mechanics is possible only in the language of secondary differential calculus, but undoubtedly to achieve this goal it takes time and the work of many researchers.

Reviewer: H.Kilp

### MSC:

58C99 | Calculus on manifolds; nonlinear operators |

81Q99 | General mathematical topics and methods in quantum theory |

53C80 | Applications of global differential geometry to the sciences |

35A99 | General topics in partial differential equations |