Why Differential Forms? Functional Substitution and PullBack

Cartan's theory of exterior differential forms is NOT just another notational system of fancy. The importance of the Cartan concept resides with the fact that differential forms are well defined objects with respect to Functional Substitution and the PullBack (FS_PB for short) relative to C1 differentiable maps from an initial variety of variables of dimension M to a final state or variety of variables of dimension N. The map does not need to have an inverse (or an invertible Jacobian as the dimensions of the varieties are different) and yet the forms and their behavior are well defined in functional format with regards to FS_PB. In particular, if a differential form is zero on the final state, it is zero on the initial state.

This situation is different from the constraints used to develop the theory of tensor analysis, where the map and its inverse, as well as the differential map and its inverse are required. Such maps are called diffeomorphisms; they are what the physicist usually calls a coordinate transformation. A tensor which is zero on the final state is also zero on the initial state, but for tensors the states must be connected by a diffeomorphism. Differential Forms are well behaved in a retrodictive, pullback sense with regards to maps that are NOT diffeomorphisms. Hence Differential Forms transcend tensor calculus, but include tensor calculus as a special case. In particular, if a differential form is zero on the final state, it is zero on the initial state, relative to maps which are differentiable but without inverse. This result will be used to construct invariant properties that are associated with deformations, where the geometrical properties of size and shape are not preserved. Such properties are called topological properties.

Given a differential form in terms of the variables and their differentials (say x and dx) on the final state (target of the map), the differential form is well defined functionally in terms of the variables ( say y and dy) on the initial state (domain of the map). A differential form is a "scalar invariant or a density invariant" with respect to diffeomorphisms. It is a product of covariant or covariant density coefficients and contravariant differentials. A differential form is not a vector, but a column vector of differential forms can be constructed (called a vector valued differential form, where each component of the vector is a differential form). Matrices of differential forms also prove to be useful.

Much of theoretical physics is governed by a set of partial differential equations. The PDE,s consist of dependent functions of independent variables. When a PDE is specified, the independent variables are fixed, hence transformations of the independent variables does not necessarily preserve the formulation of the PDE's. It is known that all PDE,s can be written in terms of an exterior system of differential forms (Pfaffian 1-forms set equal to zero). When written in this manner, both the coefficient functions of independent variables and the independent variables can be transformed into other variables in an invariant manner. The exterior differential system imposes topological (not necessarily geometrical) constraints on the system. This technique may be used to demonstrate that the theory of electromagnetism is a topological theory independent from metric (or connection), and is valid in all coordinate systems (Lorentzian, Galilean, or any other diffeomorphically constrained systems).

As P. Libermann points out, higher order differential processes (jets) on the tangent space (of vectors like velocity and acceleration) are not always linear (hence do not form a vector bundle), but on the co-tangent space (of gradient fields or differential forms with covariant coefficients) everything is well behaved. It is only when constraints are imposed to make the higher order contravariant jets into vector bundles that the wave - particle dogma of Copenhagen quantum mechanics is strictly valid. Otherwise the concept and behavior of a wave (think differential forms with covariant coefficients on the cotangent space) are distinct from the concept and behavior of a particle (think contravariant fields on the tangent space). The behavioral differences are experimentally distinct with regard to processes that are thermodynamically irreversible (where topology changes) -- and therefor are not Hamiltonian processes.

The point is that without metric, without the constraint of a group based connection, without restriction to diffeomorphisms or even homeomorphisms, differential forms are well defined quantities in a FS_PB sense. They can be used to describe situations that involve topological evolution, while the usual contra-variant tensor fields of particle mechanics are restricted (usually) to a reversible evolution that preserves topological features.

The coefficients of differential forms are either anti-symmetric co-variant, tensors or tensor densities, the most useful of field structures used to describe physical systems. To appreciate these extraordinary features of differential forms, consider C1 maps from a space (variety) of M dimensions to a space of N dimensions. Construct a differential form on (meaning in terms of the variables of x, dx) the final state, and then by functional substitution and use of the Jacobian map construct the well defined functional form of the differential form on the initial state (in terms of y and dy). This is what is meant by Retrodiction ( Retrodictive Determinism ). The process works even though the inverse map and/or inverse Jacobian does not exist, hence can be applied to maps between spaces that are not of the same dimension. Remember the Jacobian is a N by M matrix and is without an inverse.

The reciprocal process (Predictive determinism) fails, where given the differential form on the initial state in terms of y and dy, it is impossible to predict (define) the differential form on the final state in terms of x and dx, unless the inverse map and the inverse Jacobian exist. Such a constraint can be true only when the dimension of the initial and the final state are the same. It may be thought that contravariant fields are predictive, but that is not necessarily true. For given a contravariant tensor V(y) in terms of M component functions of y on the initial state, the Jacobian map may be used to push forward the M components to the N components of the final state, but the N functional forms on the final state are functions of y, not x.

The bottom line is that differential forms carry topological content, and can be used to study irreversible phenomena, in a deterministic, but retrodictive, manner.

 Tensors, Jacobians and Metrics

 Differential forms Notes 1

 Classical Field theory Notes 1

 As I have time I will replace this under construction symbol with a link to a more detailed pdf or tex file download.

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Last update 07/19/2001