On causality of wave motion in nonlocal theories of elasticity: a Kramers–Kronig relations study

In this paper, wave dispersion properties in nonlocal theories of elasticity models are critically examined. Both gradient type as well as integral-type non-locality within a setting of the rod and beam are considered. The mathematical framework here involves the Fourier frequency analysis that leads to the frequency spectrum relation (FSR) and system transfer function. Utilizing the FSR, wave modes and group speeds are examined. One main difference that arises between the two nonlocal model types is in the number of wave modes that are possible from the FSR. In gradient-type non-locality, the number of modes is finite and equal to the order of the displacement gradient in the governing equation of motion. However, in an integral-type non-locality, wave modes are infinite in number. Further, in contrast to classical theories there exist nonclassical wave modes showing the existence of nonphysical features, such as either exponential instabilities or undefined wavenumbers, negative group speeds, infinitesimally small, and infinite group speed values. Existence of such features, then, naturally raises the aspect of physically realizable wave motion and causality in these models. In literature, there exist dispersion relations or Kramers–Kronig (K-K) relations as an aid to examine wave motion in a linear, passive, and causal system. In this paper, K-K relations are utilized in order to further examine the wave dispersion properties. Discrepancies are seen between FSR predictions and K-K predictions, especially at frequencies with nonphysical features. An example is also presented that contradicts the general concept that all time-domain formulations agree well with the K-K relations.KeywordsNonlocal elasticityKramers–Kronig relationsCausalityWave dispersion

A parameter retrieval algorithm based on the causality principle and Kramers—Kronig (KK) relations is employed to calculate the effective parameters of three-dimensional (3D) metamaterials. Using KK relations, the branch selecting problem, which is the challenge of effective parameter retrieval method, can be removed. To reveal the validity of the proposed algorithm, the constitutive refractive index of a homogeneous polymide cube is extracted. The result is in excellent agreement with the intrinsic refractive index of the polymide. Finally, the two terahertz metamaterials with 3D structures are designed and their effective parameters are then retrieved using the proposed algorithm. Numerical simulations are performed using the full-wave electromagnetic solver, CST Microwave Studio.

II: Dispersion Relations and Phase Retrieval in Optical Spectroscopy

This Chapter provides a brief overview of the interconnections between the various causality and locality concepts in algebraic quantum field theory such as causal dynamics, primitive causality, local primitive causality, no-signaling, selective and nonselective measurements, local determinism, stochastic Einstein locality.

Four experimental situations are analyzed from the viewpoint of fulfilling the causality principle: instantaneous collapse of the quantum state vector of a system of entangled particles, a quantum eraser, the quantum Zeno paradox, and light transformation by a nonlinear beam splitter. The last is a planar interface of two transparent dielectrics, at least one of which possesses Kerr nonlinearity, that is, its refractive index depends on the intensity of radiation that penetrates it. It has been shown that the causality principle in the first two cases can be violated only in the sense of the instantaneousness of the manifestation of the effect with respect to the cause. For a nonlinear beam splitter, in addition to the directly opposite predictions given by the quantum and classical theories of its description for the behavior of phase fluctuations of the radiation transformed by the splitter, the causality principle is violated in the most general sense: the effect of the following event on the previous one. The quantum Zeno paradox occupies an intermediate position of democratic involvement of the cause and effect in the general cascade of two subsequent events, i.e., the first event can prevent the second event, just as the second event can prevent the first one.

The parameter retrieval method based on the causality principle and Kramers- Kronig relations, which has been proposed for extracting effective parameters of metamateri- als (MTMs) is extended to extract effective parameters of chiral metamaterials (CMMs) as well. Using Kramers-Kroning relations, the branch selecting problem which is the challenge of effective media parameters retrieval methods could be simply removed. To demonstrate the validity of the method, first, the constitutive parameters of a homogeneous chiral media are retrieved in the excellent agreement with the supposed parameters. Second validation is achieved by extracting the effective parameters of two well-known CMM structures whose effective parameters have been reported in the literature. The results are in excellent agree- ment with the reports of the previous works. Finally, a new type of CMMs called windmill structure is suggested, and its effective parameters are then extracted using the proposed method. V C 2012 Wiley Periodicals, Inc. Int J RF and Microwave CAE 00:000-000, 2012.

Physics is an empirical science in the sense that its theories and the basic principles on which they are founded are always open to discussion, reformulation, and even to rejection – if the facts require it. Surprisingly, some of these important principles have withstood all storms of time and also the exponential increase both in data taken and in knowledge gained about nature. Some of the sturdiest ones are conservation of energy, conservation of charge and the causality principle. In this paper, we will try to give simple, not too technical, arguments why the claim that, in a number of experiments with microwaves encountering and passing a barrier in a waveguide, the principle of causality has been violated, is not acceptable. These measurements are contained within the domain of classical electromagnetism such that there is no need for applying the quantum mechanical concept of tunneling. Some of the difficulties which occured during the debate concerning the interpretation of the experiments, seem to be due to an overstretching of the concepts of “particle” and “field”, respectively. These concepts, in classical physics, are less intertwined than in quantum theory.

Finite conformal transformations for quantized fields are obtained. The apparent conflict with Einstein's causality principle is resolved. It is shown that in quantum field theory one is in general dealing with representations of the universal covering group of the conformal group.

Causality study and numerical response of the magnetic permeability as a function of the frequency of ferrites using Kramers–Kronig relations

The photon is modeled as a monochromatic solution of Maxwell's equations confined as a soliton wave by the principle of causality of special relativity. The soliton travels rectilinearly at the speed of light. The solution can represent any of the known polarization (spin) states of the photon. For circularly polarized states the soliton's envelope is a circular ellipsoid whose length is the observed wavelength (A), and whose diameter is λ/π; this envelope contains the electromagnetic energy of the wave (hv = hc/λ). The predicted size and shape is confirmed by experimental measurements: of the sub-picosecond time delay of the photo-electric effect, of the attenuation of undiffracted transmission through slits narrower than the soliton's diameter of λ/π, and by the threshold intensity required for the onset of multiphoton absorption in focussed laser beams. Inside the envelope the wave's amplitude increases linearly with the radial distance from the axis of propagation, being zero on the axis. Outside the envelope the wave is evanescent with an amplitude that decreases inversely with the radial distance from the axis. The evanescent wave is responsible for the observed double-slit interference phenomenon.

It is well known that the causality principle which applies during the acoustic propagation of a wave packet inside any material should be described by the universal Kramers‐Kronig (K‐K) relationships, enabling to formally link dispersion to attenuation curves. The application of the K‐K formalism to porous networks has been studied lately, but unfortunately, the metrology of dispersion features at very low frequency is intricate. The use of parametric arrays of nonlinear acoustics enables to get at the same time, some precise information on dispersion and absorption on various porous media, for instance in the frame of the “equivalent fluid” model. In the present work, we describe some very recent findings obtained along these ideas for a truly relevant application of the full KK formalism in various fibrous materials. We have checked numerical predictions of the absorption versus frequency computed from the dispersion curves with the K‐K modeling, as well as confrontation of these numerical predictions with experimental data on absorption obtained with various set‐ups, on different fibrous materials.

A rotating universe with violation of causality

In this paper, we present an exact analytical solution for the inverse scattering problem formulated by using the coupled-mode theory in the microwave range, assuming single-mode operation. The solution is a series that gives the required coupling coefficient for the microwave device as a function of the target frequency response. A clear physical meaning for the terms of the series is discussed, and several useful properties for the synthesis are given, thus providing a deep understanding of the synthesis mechanics. The method allows for the design of microwave devices with arbitrary frequency response only constrained by principles of causality, passivity, and stability. The resulting device is, in general, a nonuniform waveguide that features a continuously varying profile, e.g., a nonuniform microstrip line with continuously varying strip width. As an example of the potential of the synthesis method, a matched filter for ultra-wideband applications is finally designed in microstrip technology and successfully tested.

Quantum mechanics is not the unique no-signaling theory which is endowed with stronger-than-classical correlations, and there exists a broad class of no-signaling theories allowing even stronger-than-quantum correlations. The principle of information causality has been suggested to distinguish quantum theory from these nonphysical theories, together with an elegant information-theoretic proof of the quantum bound of two-particle correlations. In this work, we extend this to genuine $N$-particle correlations that cannot be reduced to mixtures of states in which a smaller number of particles are entangled. We first express Svetlichny's inequality in terms of multipartite no-signaling boxes, then prove that the strongest genuine multipartite correlations lead to the maximal violation of information causality. The maximal genuine multipartite correlations under the constraint of information causality is found to be equal to the quantum mechanical bound. This result consolidates information causality as a physical principle defining the possible correlations allowed by nature, and provides intriguing insights into the limits of genuine multipartite correlations in quantum theory.

The time-domain studies of the modal fields in a lossy waveguide are executed. The waveguide has a perfectly conducting surface. Its cross section domain is bounded by a singly-connected contour of rather arbitrary but enough smooth form. Possible waveguide losses are modeled by a conductive medium which fills the waveguide volume. Standard formulation of the boundary-value problem for the system of Maxwell’s equations with time derivative is given and rearranged to the transverse-longitudinal decompositions. Hilbert space of the real-valued functions of coordinates and time is chosen as a space of solutions. Complete set of the TE-time-domain modal waves is established and studied in detail. A continuity equation for the conserved energetic quantities for the time-domain modal waves propagating in the lossy waveguide is established. Instant velocity of transportation of the modal flux energy is found out as a function of time for any waveguide cross section. Fundamental solution to the problem is obtained in accordance with the causality principle. Exact explicit solutions are obtained and illustrated by graphical examples.