Secondary calculus and cohomological physics
Encyclopedia
In mathematics
, secondary calculus is a proposed expansion of classical differential calculus
on manifold
s, to the "space" of solutions of a (nonlinear) partial differential equation
. It is a sophisticated theory at the level of jet spaces and employing algebraic methods.
s (usually non-linear equations). When the number of independent variables is zero, i.e. the equations are algebraic ones, secondary calculus reduces to classical differential calculus
.
All objects in secondary calculus are cohomology classes of differential complexes growing on diffieties
. The latter are, in the framework of secondary calculus, the analog of smooth manifolds.
class. It is not by chance that formulas of this kind, such as the well known Stokes formula
, though being a natural part of classical differential calculus, have entered in modern mathematics from physics.
s on differentiable manifolds. The Euler operator, which associates to each variational problem
the corresponding Euler-Lagrange equation
, is the analog of the classical differential associating to a function on a variety its differential. The Euler operator is a secondary differential operator of first order, even if, according to its expression in local coordinates, it looks like one of infinite order. More generally, the analog of differential form
s in secondary calculus are the elements of the first term of the so-called C-spectral sequence, and so on.
The simplest diffieties are infinite prolongations of partial differential equations, which are sub varieties of infinite jet spaces
. The latter are infinite dimensional varieties that can not be studied by means of standard functional analysis
. On the contrary, the most natural language in which to study these objects is differential calculus over commutative algebras
. Therefore, the latter must be regarded as a fundamental tool of secondary calculus. On the other hand, differential calculus over commutative algebras gives the possibility to develop algebraic geometry as if it were differential geometry.
, based on quantum field theories and its generalizations, have led to understand the deep cohomological nature of the quantities describing both classical and quantum fields. The turning point was the discovery of the famous BRST transformation. For instance, it was understood that observables in field theory are classes in horizontal de Rham cohomology which are invariant under the corresponding gauge group and so on. This current in modern theoretical physics is actually growing and it is called Cohomological Physics.
It is relevant that secondary calculus and cohomological physics, which developed for twenty years independently from each other, arrived at the same results. Their confluence took place at the international conference Secondary Calculus and Cohomological Physics (Moscow, August 24–30, 1997).
and algebraic geometry
, homological algebra
and differential topology
, Lie group
and Lie algebra
theory, differential geometry, etc.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, secondary calculus is a proposed expansion of classical differential calculus
Differential calculus
In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus....
on manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
s, to the "space" of solutions of a (nonlinear) partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
. It is a sophisticated theory at the level of jet spaces and employing algebraic methods.
Secondary calculus
Secondary calculus acts on the space of solutions of a system of partial differential equationPartial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
s (usually non-linear equations). When the number of independent variables is zero, i.e. the equations are algebraic ones, secondary calculus reduces to classical differential calculus
Differential calculus
In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus....
.
All objects in secondary calculus are cohomology classes of differential complexes growing on diffieties
Diffiety
In mathematics a diffiety, is a geometrical object introduced by playing the same role in the modern theory of partial differential equations as algebraic varieties play for algebraic equations....
. The latter are, in the framework of secondary calculus, the analog of smooth manifolds.
Cohomological physics
Cohomological physics was born with Gauss's theorem, describing the electric charge contained inside a given surface in terms of the flux of the electric field through the surface itself. Flux is the integral of a differential form and, consequently, a de Rham cohomologyDe Rham cohomology
In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes...
class. It is not by chance that formulas of this kind, such as the well known Stokes formula
Stokes' theorem
In differential geometry, Stokes' theorem is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. Lord Kelvin first discovered the result and communicated it to George Stokes in July 1850...
, though being a natural part of classical differential calculus, have entered in modern mathematics from physics.
Classical analogues
All the constructions in classical differential calculus have an analog in secondary calculus. For instance, higher symmetries of a system of partial differential equations are the analog of vector fieldVector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
s on differentiable manifolds. The Euler operator, which associates to each variational problem
Calculus of variations
Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
the corresponding Euler-Lagrange equation
Euler-Lagrange equation
In calculus of variations, the Euler–Lagrange equation, Euler's equation, or Lagrange's equation, is a differential equation whose solutions are the functions for which a given functional is stationary...
, is the analog of the classical differential associating to a function on a variety its differential. The Euler operator is a secondary differential operator of first order, even if, according to its expression in local coordinates, it looks like one of infinite order. More generally, the analog of differential form
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
s in secondary calculus are the elements of the first term of the so-called C-spectral sequence, and so on.
The simplest diffieties are infinite prolongations of partial differential equations, which are sub varieties of infinite jet spaces
Jet bundle
In differential geometry, the jet bundle is a certain construction which makes a new smooth fiber bundle out of a given smooth fiber bundle. It makes it possible to write differential equations on sections of a fiber bundle in an invariant form...
. The latter are infinite dimensional varieties that can not be studied by means of standard functional analysis
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
. On the contrary, the most natural language in which to study these objects is differential calculus over commutative algebras
Differential calculus over commutative algebras
In mathematics the differential calculus over commutative algebras is a part of commutative algebra based on the observation that most concepts known from classical differential calculus can be formulated in purely algebraic terms...
. Therefore, the latter must be regarded as a fundamental tool of secondary calculus. On the other hand, differential calculus over commutative algebras gives the possibility to develop algebraic geometry as if it were differential geometry.
Theoretical physics
Recent developments of particle physicsParticle physics
Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...
, based on quantum field theories and its generalizations, have led to understand the deep cohomological nature of the quantities describing both classical and quantum fields. The turning point was the discovery of the famous BRST transformation. For instance, it was understood that observables in field theory are classes in horizontal de Rham cohomology which are invariant under the corresponding gauge group and so on. This current in modern theoretical physics is actually growing and it is called Cohomological Physics.
It is relevant that secondary calculus and cohomological physics, which developed for twenty years independently from each other, arrived at the same results. Their confluence took place at the international conference Secondary Calculus and Cohomological Physics (Moscow, August 24–30, 1997).
Prospects
A large number of modern mathematical theories harmoniously converges in the framework of secondary calculus, for instance: commutative algebraCommutative algebra
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
and algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, homological algebra
Homological algebra
Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...
and differential topology
Differential topology
In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...
, Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
and Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
theory, differential geometry, etc.