currentsformalism

Currentsformalism is a framework in differential geometry and mathematical physics that uses the theory of currents to model distributions of physical quantities and their flows. In this approach, physical densities and fluxes are encoded as generalized currents, enabling a unified treatment of smooth and singular sources within a single formalism.

Mathematically, an r-current is a continuous linear functional on the space of smooth, compactly supported differential r-forms. Currents can be added, scaled, and subjected to a boundary operator ∂, yielding the (r−1)-current that represents the boundary or flux of the original object. The mass or mass norm provides a notion of size, and rectifiable currents describe objects like oriented curves or surfaces with well-defined geometric measure.

Currentsformalism builds on geometric measure theory, notably the theory of currents developed by Federer and Fleming. It has been used to recast conservation laws and source terms in electromagnetism, fluid dynamics, and general relativity as statements about currents and their boundaries, often handling singular sources such as point charges, line defects, or branes.

Typical examples include point-like sources represented by 0-currents (Dirac measures), line currents by 1-currents along a curve, and membrane currents by 2-currents. In variational problems, currentsformalism provides a natural language for describing limits of minimizing sequences and for formulating weak solutions.

Advantages include the ability to model singular sources and to work with weak convergence in a geometric setting. Limitations include mathematical sophistication and technical prerequisites, and the framework is most fully developed for smooth or rectifiable ambient spaces. Currentsformalism is related to differential forms, distribution theory, and conservation principles guided by Noether's theorem.