On the Topological Structure of Nonlocal Continuum Field Theories

An alternative to conventional spacetime is proposed and rigorously formulated for nonlocal continuum field theories through the deployment of a fiber bundle-based superspace extension method. We develop, in increasing complexity, concept nonlocality starting from general considerations, going spatial dispersion, ending up with broad formulation that unveils link between topology generic material media. It shown naturally leads Banach (vector) bundle structure serving as an enlarged space (superspace) inside which physical processes, such electromagnetic ones, take place. The added structures, essentially fibered spaces, model topological microdomains physics-based provide fine-grained geometrical picture field–matter interactions metamaterials. utilize standard techniques theory smooth manifolds construct by paying careful attention relevant physics. response tensor then reformulated homomorphism various tools needed proceed local global domains are developed. For concreteness simplicity, our presentations both fundamental examples given illustrate mathematics all emphasize case theory, but formalism developed here quite can be easily extended other types theories. application given, consists utilizing metamaterials order explain why materials often require additional boundary conditions or extra input microscopic relative electromagnetism, where latter not needed. Real-life studies quantitatively illustrating microdomain semiconductors provided. Moreover, series connected appendices, we outline new view emerging electromagnetism domains, which, together main introduced text, may considered unified introduction physics methods