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.
Highlights
Numerous research studies point toward a basic fact: topology and physics are destined to come closer to each other in the following decades [1,2,3,4]
While the essential idea of the superspace formalism introduced here will be valid for a generic nonlocal continuum field theory, it is much easier sometimes to work with a concrete example, especially in explaining what nonlocality is for someone who is coming to the subject for the first time
This was accomplished by building a bundle homomorphism to replace the well-known, but inadequate, material tensor linear operators commonly utilized in local continuum field theories, for instance, conventional electromagnetism
Summary
Numerous research studies point toward a basic fact: topology and physics are destined to come closer to each other in the following decades [1,2,3,4]. The principal conceptual and philosophical message behind this work is that spacetime (or space–frequency) is not adequate for formulating nonlocal continuum field theories, and that a more appropriate natural approach is the superspace formalism proposed below, which, in our case, is based on a specific fiber bundle construction taking into account the intricate physics-based microdomain structure of the generic nonlocal continuum It is the hope of the author that by helping scientists generate new insights into their physics and models, this formalism may provide a rigorous approach complementing some of the exciting theories and researches currently addressing various topics in continuum field theories, nonlocal metamaterials, and topological materials, while possibly stimulating the creation of novel algorithms for the computation of suitable topological invariant characterizing complex material domains. Due to the wide scope and complexity of this work, we first provide in Section 2 a relatively lengthy overview on the our contribution, where high-level information about this work, in addition to a guide to the literature and how to read the present paper, are outlined before moving to the more technical treatments of the subsequent sections and appendices
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