Reciprocity Invariance Research Articles

Restoring the reciprocity invariance in nonlinear systems with broken mirror symmetry

Circumventing the reciprocity invariance has posed an interesting challenge in the design of modern devices for wave engineering. In passive devices, operating the device in the nonlinear response regime is a common means for realizing nonreciprocity. Because mirror-symmetric systems are trivially reciprocal, breaking the mirror symmetry is a necessary requirement for nonreciprocal dynamics to exist in nonlinear systems. However, the response of an asymmetric nonlinear system is not necessarily nonreciprocal. In this work, we report on the existence of stable, steady-state nonlinear reciprocal dynamics in coupled asymmetric systems subject to external harmonic excitation. We restore reciprocity in the asymmetric system by tuning two symmetry-breaking parameters simultaneously under certain operating conditions. Specifically, we identify response regimes in the vicinity of the primary resonances of the system where the steady-state left-to-right transmission characteristics are identical to the right-to-left characteristics in terms of frequency, amplitude and phase. We interpret these regimes of reciprocal dynamics in the context of phase nonreciprocity, wherein incident waves undergo a nonreciprocal phase shift depending on their direction of travel. We hope these findings help design devices with new functionalities for controlling and steering of elastic waves.

Computation of nonreciprocal dynamics in nonlinear materials

<p style='text-indent:20px;'>The reciprocity theorem in elastic materials states that the response of a linear, time-invariant system to an external load remains invariant with respect to interchanging the locations of the input and output. In the presence of nonlinear forces within a material, circumventing the reciprocity invariance requires breaking the mirror symmetry of the medium, thus allowing different wave propagation characteristics in opposite directions along the same transmission path. This work highlights the application of numerical continuation methods for exploring the steady-state nonreciprocal dynamics of nonlinear periodic materials in response to external harmonic drive. Using the archetypal example of coupled oscillators, we apply continuation methods to analyze the influence of nonlinearity and symmetry on the reciprocity invariance. We present symmetry-breaking bifurcations for systems with and without mirror symmetry, and discuss their influence on the nonreciprocal dynamics. Direct computation of the reciprocity bias allows the identification of response regimes in which nonreciprocity manifests itself as a phase shift in the output displacements. Various operating regimes, bifurcations and manifestations of nonreciprocity are identified and discussed throughout the work.</p>

Time and energy operators in the canonical quantization of special relativity

Based on Lorentz invariance and Born reciprocity invariance, the canonical quantization of special relativity is shown to provide a unified origin for (i) the complex vector space formulation of quantum mechanics; (ii) the momentum and space commutation relations and the corresponding representations; (iii) the Dirac Hamiltonian in the formulation of relativistic quantum mechanics; (iv) the existence of a self adjoint time operator that circumvents Pauli’s objection.

On Madelung systems in nonlinear optics: A reciprocal invariance

The role of the de Broglie-Bohm potential, originally established as central to Bohmian quantum mechanics, is examined for two canonical Madelung systems in nonlinear optics. In a seminal case, a Madelung system derived by Wagner et al. via the paraxial approximation and in which the de Broglie-Bohm potential is present is shown to admit a multi-parameter class of what are here introduced as “q-gaussons.” In the limit, as the Tsallis parameter q → 1, the q-gaussons are shown to lead to standard gausson solitons, as admitted by the logarithmic nonlinear Schrödinger equation encapsulating the Madelung system. The q-gaussons are obtained for optical media with dual power-law refractive index. In the second case, a Madelung system originally derived via an eikonal approximation in the context of laser beam propagation and in which the de Broglie Bohm term is neglected is shown to admit invariance under a novel class of two-parameter class of reciprocal transformations. Model optical laws analogous to the celebrated Kármán-Tsien law of classical gas dynamics are introduced.

Reciprocal Class of Jump Processes

Processes having the same bridges as a given reference Markov process constitute its reciprocal class. In this paper we study the reciprocal class of compound Poisson processes whose jumps belong to a finite set $$\mathcal {A}\subset \mathbb {R}^{d}$$ . We propose a characterization of the reciprocal class as the unique set of probability measures on which a family of time and space transformations induces the same density, expressed in terms of the reciprocal invariants. The geometry of $$\mathcal {A}$$ plays a crucial role in the design of the transformations, and we use tools from discrete geometry to obtain an optimal characterization. We deduce explicit conditions for two Markov jump processes to belong to the same class. Finally, we provide a natural interpretation of the invariants as short-time asymptotics for the probability that the reference process makes a cycle around its current state.

Bridges of Markov counting processes. Reciprocal classes and duality formulas

Processes having the same bridges are said to belong to the same reciprocal class. In this article we analyze reciprocal classes of Markov counting processes by identifying their reciprocal invariants and we characterize them as the set of counting processes satisfying some duality formula.

On a discrete symmetry of the Bremsstrahlung function in $ \mathcal{N} $ = 4 SYM

We consider the quark anti-quark potential on the three sphere in planar N = 4 SYM and the associated vacuum potential in the near BPS limit with L units of R-charge. The associated Bremsstrahlung function BL has been recently computed analytically by means of the Thermodynamical Bethe Ansatz. We discuss it at strong coupling by computing it at large but nite L. We provide strong support to a special symmetry of the Bremsstrahlung function under the formal discrete Z2 symmetry L ! 1 L. In this context, it is the counterpart of the reciprocity invariance discovered in the past in the spectrum of various gauge invariant composite operators. The Z2 symmetry has remarkable consequences in the scaling limit where L is taken to be large with xed ratio to the 't Hooft coupling. This limit organizes in inverse powers of the coupling and resembles the semiclassical expansion of the dual string theory which is indeed known to capture the leading classical term. We show that the various higher-order contributions to the Bremsstrahlung function obey several constraints and, in particular, the next-to-leading term, formally associated with the string one-loop correction, is completely determined by the classical contribution. The large L limit at strong coupling is also discussed.

Three loop anomalous dimensions of twist-3 gauge operators in 𝒩 = 4 SYM

We propose a closed expression for the three loop anomalous dimension of a class of twist-3 operators built with gauge fields and covariant derivatives. To this aim, we solve the long-range Bethe Ansatz equations at finite spin and provide a consistent analytical formula obtained assuming maximal transcendentality violation as suggested by the known one-loop anomalous dimension. The final result reproduces the universal cusp anomalous dimension and obeys recursion relations inspired by the principle of reciprocity invariance.

Curvature Effects in Ocean Surface Scattering

Curvature effects in EM scattering from ocean surface are described using a generalized curvature expansion of the fields at an elevated nonperfect conducting surface. The new expansion formalism allows us to describe analytically and in general, without separating into different scales, the scattering of EM waves from an undulated ocean surface. The model is exact to first order in curvature for nonshadowing imaging geometry, and obeys the law of reciprocity and tilt invariance. Explicit expressions for EM fields at the surface, including both the projection and the self induced fields, are derived up to first order in surface curvature. Analytic closed form expressions for the scattered fields are derived from the surface field solutions, and applied to the case of backscattering, providing a general expression for the normalized radar cross section. The analytic expression for the normalized radar cross section is implemented for a linear surface model using both the Eulerian and the Lagrangian frame of reference. The results show that the model is capable of describing the expected dependency on polarization, incidence angle, and wind field with minimal restrictions in terms of range of validity. Comparison of polarization ratio shows good agreement between the model and measurements from the Envisat ASAR instrument

Invariant description of solutions of hydrodynamic-type systems in hodograph space: hydrodynamic surfaces

Hydrodynamic surfaces are solutions of hydrodynamic type systems viewed as non-parametrized submanifolds of the hodograph space. We propose an invariant differential-geometric characterization of hydrodynamic surfaces by expressing the curvature form of the characteristic web in terms of the reciprocal invariants.

Lie sphere geometry and integrable systems

Two basic Lie-invariant forms uniquely defining a generic (hyper)surface in Lie sphere geometry are introduced. Particularly interesting classes of surfaces associated with these invariants are considered. These are the diagonally cyclidic surfaces and the Lie-minimal surfaces, the latter being the extremals of the simplest Lie-invariant functional generalizing the Willmore functional in conformal geometry. Equations of motion of a special Lie sphere frame are derived, providing a convenient unified treatment of surfaces in Lie sphere geometry. In particular, for diagonally cyclidic surfaces this approach immediately implies the stationary modified Veselov-Novikov equation, while the case of Lie-minimal surfaces reduces in a certain limit to the integrable coupled Tzitzeica system. In the framework of the canonical correspondence between Hamiltonian systms of hydrodynamic type and hypersurfaces in Lie sphere geometry, it is pointed out that invariants of Lie-geometric hypersurfaces coincide with the reciprocal invariants of hydrodynamic type systems. Integrable evolutions of surfaces in Lie sphere geometry are introduced. This provides an interpretation of the simplest Lie-invariant functional as the first local conservation law of the (2+1)-dimensional modified Veselov-Novikov hierarchy. Parallels between Lie sphere geometry and projective differential geometry of surfaces are drawn in the conclusion.

Binormal motion of curves of constant curvature and torsion. Generation of soliton surfaces

The purely binormal motion of curves of constant curvature or torsion, respectively, is shown to lead to integrable extensions of the Dym and classical sine–Gordon equations. In the case of the extended Dym equation, a reciprocal invariance is used to establish the existence of novel dual–soliton surfaces associated with a given soliton surface. A cc–ideal formulation is adduced to obtain a matrix Darboux invariance for the extended Dym and reciprocally linked m 2 KdV equations. A Backlund transformation with a classical constant–length property is thereby constructed which allows the generation of associated soliton surfaces. Analogues of both Backlund9s and Bianchi9s classical transformations are derived for the extended sine–Gordon system.

Reciprocal Transformations of Two-Component Hyperbolic Systems and Their Invariants

Criteria for the equivalence under reciprocal transformations of two-component hyperbolic systems are established in terms of reciprocal invariants. These results are applied to a 1+1-dimensional isentropic gasdynamics system to determine reciprocal equivalence to the canonical system with a polytropic gas law. The four-parameter model pressure-density relations for which this reciprocal equivalence holds are parameterized in terms of the Lagrangian signal speed.

Perceiving Persistence Under Change and Over Structure

We explored the hypothesis that to perceive an event is to detect an invariant specific to the style of change and an invariant specific to the object undergoing the style of change-the reciprocal invariants of transformation and structure, respectively. Events were defined by seven geometric transformations that produced a variety of styles of change over different objects. In Experiment 1, we examined the generative and abstract aspects of transformational invariants. An event was presented as a sequence of static samples and examined in interpolation and extrapolation tasks. The transformational invariant proved sufficiently generative and abstract to specify a unique style of change. In Experiment 2, we found that types of change and types of objects interacted, which implies that certain styles of change are more compatible with certain object structures. In Experiment 3, we found that the structural invariant was reliably detectable but that the degree to which structural properties were preserved depended on the kind and amount of transformation applied. Taken together, the experiments provide support for the hypothesis that perceiving an event is detecting the two reciprocal invariants of transformation and structure that together specify the event.

Gaussian reciprocal diffusions and their associated conservation laws

Krener and collaborators have shown that Gaussian Reciprocal Diffusions (GRDs) x(t) must satisfy on [0,T] a 2nd order stochastic differential equation of the form , where ξ(t) is a generalized noise process. Krener has also shown that such GRDs have conditional moments satisfying a system of conservation laws similar to those of continuum mechanics In this paper we further investigate the relationship between GRDs. their conditional moments and their associated conservation laws. We show that a GRD is determined by its lifetime, its reciprocal invariants and the initial values of its conditional moments. A characterization is then given of those functions that are allowable as the initial values for the conditional moments of a GRD having a given lifetime and given reciprocal invariants. We show further that the conservation laws associated with a GRD form a hyperbolic system for either all or none of the process lifetime, and we give criteria for determining which of these alternatives holds

Reciprocally invariant electrodynamics on quantum space-time

According to Born’s reciprocity principle, a relativistic theory is reciprocally invariant if it displays symmetry under the exchange of space-time and 4-momentum labels. It is shown that recently proposed relativistic field theories on quantum space-time display reciprocal invariance for the electromagnetic as well as for the massive fields if the proper wave functions of the extended particles satisfy the Born-Lande equation in addition to the Klein-Gordon equation. The basic features of the reciprocally invariant free electromagnetic field are examined and the reciprocally invariant electrodynamics on quantum space-time (EQS) is formulated. It is shown that the Feynman rules for EQS are totally analogous to those for quantum electrodynamics (QED), but that EQS displays none of the ultraviolet divergences of QED except in the limita → + 0, wherea is the proper radius of the massive particles. In that limit, EQS formally merges into QED.

Functional Approach to the Critical Dimension in Veneziano's Model

We give an explicit derivation of the critical dimension of space-time associated with the Veneziano model starting from the path-integral expression based on a Dirichlet-type boundary-value problem. It is shown that our expression for four-leg amplitudes happens to reduce to the beta-function formula of Veneziano only when the effective number of transverse dimensions is 24. Up to the goal, the transformation properties are not selfevident since our representation has a batch of the theta ·functions in the integrand_ Our next concern is therefore with a discussion how the Mobius invariance or the reciprocal invariance is connected with the critical dim~nsionality.

On the inelastic deuteron form factor

The similarity that exists between the deuteron form factor and the off- the-mass-shell neutron-proton scattering amplitude when final state neutron- proton interactions are accounted for in the impulse approximation for the inelastic scattering of a particle by a deuteron is discussed. The lappmann- Schwinger integral equation for the two-nucleon wave function in the final state and the reciprocity and space reflection invariance properties of Green's function are used in the derivations.