On the Topological Structure of Nonlocal Continuum Field Theories

Abstract

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

Directive Properties of Radiating Source Systems in Massive Electromagnetism

“…If the medium's response parameters also depend on k, the wavevector, then the medium becomes nonlocal (or spatially dispersive if the material is also homogeneous) [27]- [30]. Spatial dispersion often leads to unusual physical behaviour like negative group velocity [28], [31], nonreciprocity [32], [33], topological materials [34], and energy localization [35].…”

“…From (10), it is clear that because of the explicit dependence on k, the Proca MTM is nonlocal. In fact, since the medium possess continuous spatiotemporal translation symmetry (homogeneous and time-invariant), the domain is spatially dispersive [28], [30]. It can be shown that the normalized mass m is related to the nonlocality radius of the Proca of MTM [8].…”

“…It is will known that one can establish from general considerations alone that reflection and transmission of waves at material interfaces where one medium is nonlocal (spatially dispersive) leads to the production of new "additional waves" that cannot be predicted by the usual paradigm of electromagnetic boundary-value problems in mathematical physics. Indeed, while the latter paradigm of boundary-value problem in mathematical physics has been successfully applied so far mainly to local electromagnetics [9], [43], [44], it cannot be extended in a straightforward manner to deal with nonlocal electromagnetic problems [20], [26], [28], [30], [45]. The problem of additional waves, whose discovery is essentially due to Pekar [36], [37], makes the computational or even the theoretical analysis of the problem in Fig.…”

“…Since the Proca radiation formula (25) differs from the corresponding classical expression (32) by the phase factor ζ 2 − 1, it is imperative to compare the corresponding behaviour of the two radiating source array types at different values of the normalized frequency ζ. The total radiated power of a 6-element linear uniform Proca array can be computed by means of ( 25) and (30). In general, the spectral dependence of the source current excitation amplitude |J 0 | is important to ensure convergence at high frequencies.…”

Artificial General Intelligence, Noncomputability, and Dynamical Systems: A Critical Reexamination

“…Securing the existence of nonlocal predictive processes is inherently beyond the means of classical computing machines, which are often based on classical (Kolmogorov) probability and classical physics [39], [40]. Nonlocality in intelligent agent networks requires the establishment of regimes of longspatiotemporal correlations allowing distinct sub-components of the system to exchange information among each other [41]- [44] and, more importantly, to evaluate and assess each other [45], [46], leading then to at least the possibility of creating a novel behaviour that surpasses the capacities of each sub-component when taken on its own [47]. Thus, while nonlocality and novelty are ontologically distinct concepts, they are closely related with each other [48].…”