Generalized Correspondence Principle Research Articles

Solving superconducting quantum circuits in Dirac's constraint analysis framework**The paper is dedicated to the memory of Professor Roman Jackiw, who apart from working in diverse branches of theoretical physics, has contributed significantly in quantization of constraint systems.

Abstract In this work we exploit Dirac's Constraint Analysis (DCA) in Hamiltonian formalism to study different types of Superconducting Quantum Circuits (SQC) in a unified way. The Lagrangian of a SQC reveals the constraints, that are classified in a Hamiltonian framework, such that redundant variables can be removed to isolate the canonical degrees of freedom for subsequent quantization of the Dirac Brackets via a generalized Correspondence Principle. This purely algebraic approach makes the application of concepts such as graph theory, null vector, loop charge, etc that are in vogue, (each for a specific type of circuit), completely redundant. The universal validity of DCA scheme in SQC, proposed by us, is demonstrated by correctly re-deriving existing results for different SQCs, obtained previously exploiting different formalisms each applicable for a specific SQC. Furthermore, we have also analysed and predicted new results for a generic form of SQC - it will be interesting to see its validation in an explicit circuit implementation.

Anisotropic Continuum-Molecular Models: A Unified Framework Based on Pair Potentials for Elasticity, Fracture and Diffusion-Type Problems

This paper presents a unified framework for continuum-molecular modeling of anisotropic elasticity, fracture and diffusion-based problems within a generalized two-dimensional peridynamic theory. A variational procedure is proposed to derive the governing equations of the model, that postulates oriented material points interacting through pair potentials from which pairwise generalized actions are computed as energy conjugates to properly defined pairwise measures of primary field variables. While mass is considered as continuous function of volume, we define constitutive laws for long-range interactions such that the overall anisotropic behavior of the material is the result of the assigned elastic, conductive and failure micro-interaction properties. The non-central force assumption in elasticity, together with the definition of specific orientation-dependent micromoduli functions respecting material symmetries, allow to obtain a fully anisotropic non-local continuum using a purely pairwise description of deformation and constitutive properties. A general and consistent micro-macro moduli correspondence principle is also established, based on the formal analogy with the classic elastic and conductivity tensors. The main concepts presented in this work can be used for further developments of anisotropic continuum-molecular formulations to include other mechanical behaviors and coupled phenomena involving different physics.

Topological lattice defects by groupoid methods and Kasparov’s KK-theory* *This work was supported by the National Science Foundation through the Grant DMR-1823800.

The bulk-boundary and a new bulk-defect correspondence principles are formulated using groupoid algebras. The new strategy relies on the observation that the groupoids of lattices with boundaries or defects display spaces of units with invariant accumulation manifolds, hence they can be naturally split into disjoint unions of open and closed invariant sub-sets. This leads to standard exact sequences of groupoid C*-algebras that can be used to associate a Kasparov element to a lattice defect and to formulate an extremely general bulk-defect correspondence principle. As an application, we establish a correspondence between topological defects of a two-dimensional square lattice and Kasparov’s group . Numerical examples of non-trivial bulk-defect correspondences are supplied.

Uniqueness of a Furstenberg system

Given a countable amenable group $G$, a Folner sequence $(F_N) \subseteq G$, and a set $E \subseteq G$ with $\bar{d}_{(F_N)}(E)=\limsup_{N \to \infty} \frac{|E \cap F_N|}{|F_N|}>0$, Furstenberg's correspondence principle associates with the pair $(E,(F_N))$ a measure preserving system $(X,\mathcal{B},\mu,(T_g)_{g \in G})$ and a set $A \in \mathcal{B}$ with $\mu(A)=\bar{d}_{(F_N)}(E)$, in such a way that for all $r \in \mathbb{N}$ and all $g_1,\dots,g_r \in G$ one has $\bar{d}_{(F_N)}(g_1^{-1}E \cap \dots \cap g_r^{-1}E)\geq\mu((T_{g_1})^{-1}A \cap \dots \cap (T_{g_r})^{-1}A)$. We show that under some natural assumptions, the system $(X,\mathcal{B},\mu,(T_g)_{g \in G})$ is unique up to a measurable isomorphism. We also establish variants of this uniqueness result for non-countable discrete amenable semigroups as well as for a generalized correspondence principle which deals with a finite family of bounded functions $f_1,\dots,f_{\ell}: G \rightarrow \mathbb{C}$.

Nota na temat pojmowania uogólnionej zasady korespondencji

The author, referring to the text of Jan Woleński published on the pages of Prace Komisji Historii Nauki PAU in 2014, discusses the understanding of the generalized correspondence principle in the context of the following concepts: cumulativism (C.G. Hempel, P. Oppenhaim, E. Nagel), extreme anticumulativism (P. Feyerabend, T.S. Kuhn), dialectical cumulativism (W. Krajewski) and the hypothetico‑ deductive method of correspondence‑oriented thinking as well as Copernicus’s methodology (M. Kokowski).

Complex Covariance

According to some generalized correspondence principle the classical limit of a non-Hermitian quantum theory describing quantum degrees of freedom is expected to be the well known classical mechanics of classical degrees of freedom in the complex phase space, i.e., some phase space spanned by complex-valued space and momentum coordinates. As special relativity was developed by Einstein merely for real-valued space-time and four-momentum, we will try to understand how special relativity and covariance can be extended to complex-valued space-time and four-momentum. Our considerations will lead us not only to some unconventional derivation of Lorentz transformations for complex-valued velocities, but also to the non-Hermitian Klein-Gordon and Dirac equations, which are to lay the foundations of a non-Hermitian quantum theory.

Biomechanics of Articular Cartilage and Determination of Material Properties

Descriptions of the mechanical behaviors of articular cartilage and their correlations with collagen, proteoglycan, water, and ions are summarized, with particular emphasis on understanding the osmotic effect inside the tissue. First, a descriptive explanation is presented of the biphasic theory required to understand how interstitial water contributes toward the viscoelastic behavior of any hydrated soft tissue. Then, the famous osmotic effect in charged, hydrated soft tissue is interpreted in light of the triphasic mixture theory framework. In the introduction of mechanical testing methods, our emphasis is on the popular indentation technique, which can determine the material properties of cartilage in situ or in vivo. The widely accepted indentation analysis solutions in cartilage biomechanics history are summarized and evaluated. At the end of this paper, a new generalized correspondence principle between charged, hydrated soft tissue and linear, isotropic, elastic material (i.e., elasticity theory) is introduced. This principle makes the employment of triphasic theory as straightforward as using an elasticity theory to solve any equilibrium problem where the elasticity theory can be used to model the material. By using this generalized correspondence principle, the fixed charge density of bovine cartilage has been simply and conveniently calculated from the indentation testing data. The results of proteoglycan content from this mechanical test are remarkably consistent with those from standard biochemical assay. This new correspondence principle significantly improves the power of indentation tests in the determination of mechanoelectrochemical properties of articular cartilage.

The generalized triphasic correspondence principle for simultaneous determination of the mechanical properties and proteoglycan content of articular cartilage by indentation

The triphasic mixture theory has been used to describe the mechanical and physicochemical behaviors of articular cartilage under some specialized loading conditions. However, the mathematical complexities of this theory have limited its applications for theoretical analyses of experimental studies and models for predicting cartilage and other biological tissues’ deformational behaviors. A generalized correspondence principle has been established in the present study, and this principle shows that the equilibrium deformational behavior of a charged-hydrated material under loading is identical to that of an elastic medium without charge. A set of explicit formulas has been derived to correlate the mechanical properties of an equivalent material with the intrinsic elastic moduli, fixed charge density and free-ion concentration within the cartilage tissue. The validity of these formulas is independent of the deformation state of the elastic solid matrix under an infinitesimal strain. Therefore they can be employed for any loading conditions, such as confined or unconfined compression, tension, and indentation tests, etc. In the current study, the fixed charge density of bovine cartilage is determined from the indentation creep data using this generalized correspondence principle. The proteoglycan content results were then compared with those from biochemical assay, yielding a linear regression slope of 1.034. Additionally a correspondence principle within a framework of cubic symmetry and a bilinear response in tension-compression (the conewise linear elasticity model) has also been developed to demonstrate the potential application of current methodology for inhomogeneous, anisotropic and nonlinear situations.

A general correspondence principle for time-domain electromagnetic wave and diffusion fields

SUMMARY A general correspondence principle is presented that relates any time-domain electromagnetic diffusion field to an electromagnetic wavefield in a ‘corresponding’ configuration. The principle applies to arbitrarily inhomogeneous and anisotropic media and arbitrary transmitters and receivers. For the correspondence between the two types of electromagnetic fields to hold, the electric conductivity in the diffusive case and the permittivity in the wavefield case should have the same spatial variation, while the permeability distributions in space in the two cases are to be identical. Essential steps in the derivation of the correspondence principle are the use of the time Laplace transformation of causal signals, taken at real, positive values of the transform parameter, the Schouten-Van der Pol theorem in the theory of the Laplace transformation, and the reliance upon Lerch’s theorem of the uniqueness of the interrelation between causal field quantities and their time-Laplace-transform representations at real, positive values of the transform parameter. Correspondence is then established between the tensorial Green’s functions in the two cases, where the Green’s functions are the point-receiver responses (either electric or magnetic field) to point-transmitter excitations (either electric- or magnetic-current source). Through the correspondence principle, all transient electromagnetic wavefields (where losses are neglected) have as a counterpart a transient diffusive electromagnetic field (where the electric displacement current is neglected). The interrelation yields the tool to compare quantitatively the potentialities of the two types of fields in transient electromagnetic geophysical prospecting. Finally, a general medium-parameter scaling law for time-domain electromagnetic wavefields is presented.

Ionization of the excited states of hydrogen by proton impact.

Ionization by impact of protons on hydrogen targets in different initial states, with principal quantum number n=1--5, is studied at collision energies from 10 up to ${10}^{3}$ keV/amu. Total cross sections are calculated employing the first-order-Born approximation and the continuum-distorted-wave--eikonal-initial-state model. Scaling laws predicted by the generalized correspondence principle are analyzed.

Heuristics and the Generalized Correspondence Principle

AbstractSeveral philosophers of science have claimed that the correspondence principle can be generalized from quantum physics to all of (particularly physical) science and that in fact it constitu...

Correspondence principles: the relationship between classical trajectories and quantum spectral

The general correspondence principle specifies a relationship between wave functions and families of trajectories. For bounded systems with regular trajectories, this general correspondence principle leads to the principle of quantization of action. The general principle also leads to a correspondence principle for finite-resolution spectra: oscillation in the average density of states, or in the average oscillator strength as a function of energy, are connected to periodic or to closed orbits. This principle holds independently or whether classical trajectories are orderly or chaotic.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Theoretical support for the generalized correspondence principle

Although most people associate the correspondence principle with quantum mechanics, it has been used more generally for many years in the development of theories in essentially all branches of physics. Based upon simple assumptions which would seem to be necessary for any exact science, a more precise statement of the generalized correspondence principle is provided, with due regard to the accuracy of theories. Term correspondence is presented as a corollary to the principle. Evidently, the principle has as strong a basis as other scientific theories. Its value in theory development as well as in furthering the understanding of the history of science is discussed.

Solution of the problems of a conical rod, a plane wedge, and a porous tube for nonlinear viscoelastic materials using the generalized correspondence principle

Solution of the problems of a conical rod, a plane wedge, and a porous tube for nonlinear viscoelastic materials using the generalized correspondence principle

Quantum Theory of Electron-Cavity Interactions

The quantum-theoretic method developed in one of our earlier papers is applied to the interaction of an electron with a multicavity system. This analysis is extended to the two-cavity oscillator, linear accelerator (linac), and frequency mixer. It is found that the experimental observations done so far in the above systems do not depend on the quantum state of the cavity field. A comparison of the oscillator theory and the linac given here indicates that the weak-field limit should be taken as the correct classical limit according to the generalized correspondence principle. Finally, a possible refinement of linac design, as a result of the present calculation, is pointed out.

Excitation of highly excited hydrogenic ions and atoms by charged particles III

For pt. II see ibid, Vol 4, 918-31. Heisenberg's form of the correspondence principle has been used to calculate collisional excitation cross sections for transitions between two highly excited levels. An impact parameter formulation of the sudden approximation and first order perturbation theory has been used. The range of incident energies for which the results are given 2M/me mod ZZ' mod n<E/Ryd<M/meZ'2 is determined by the condition that these approximations should have a common region of validity. Where possible the authors have checked these approximations using a more general correspondence principle. Results for the hydrogen atom (Z=1) have been previously reported. The cross-section is obtained in analytic form and is at worst in error by about 20%.

A generalized correspondence principle and proton-hydrogen collisions

The Monte Carlo method is used to obtain total classical cross sections for ionization and charge transfer of hydrogen atoms in any level n by protons in the energy range 38/n2 kev to 218/n2 in the laboratory frame. The cross sections have normal statistical errors of about ±10% for ionization and ±9% or more for charge transfer. For n = 1 the classical ionization cross section agrees with experiment to within about 10 or 15 per cent over the entire range of energies, showing that the agreement between earlier classical theories of ionization and experiment is not simply the result of further dynamical approximations. The classical charge transfer cross section is somewhat higher than experiment. A generalized correspondence principle is derived which states that, for a non-relativistic quantal system of charged particles, the classically scale-invariant quantities tend to their classical values when the linear dimensions tend to infinity and other quantities are also scaled. The above cross sections are thus correct in the limit n -> [infinity]. Validity and application of the results are discussed.

The Classical Theories of Radiation Reaction

Classical theories of radiation reaction are critically reviewed. The renormalized Dirac theory is discussed in particular detail. it is shown that all regularization prescriptions for this theory may be derived from a general dynamical correspondence principle. It is also shown that the physical content of the theory is severely limited by a general radiation condition. Various non- local and extended electron theories are classified with the help of the Herglotz- Wildermuth runaway theorem. The existence of electrodynamic collective modes are discussed; in particular the radiationless and self-excited states of extended charge structures. A possible connection with the existence of the mu meson is pointed out. In the concluding section various problems raised by the phenomenological radiation reaction theory of Ginzburg and Eidman are considered.

Instability of a continuously inhomogeneous viscoelastic half-space under initial stress

Surface folding due to instability is analyzed for continuously inhomogeneous half-space of viscoelastic properties. The medium is under the combined action of a horizontal compression and gravity. The material is of uniform density. Its viscoelastic properties vary exponentially with depth so that it degenerates into a fluid at large depth. It is assumed that the deformations verify the thermodynamic theory of irreversible processes. Folding wavelengths are evaluated. The general correspondence principle as formulated earlier by the author shows the results to be immediately applicable to either viscoelastic or purely elastic media. A numerical application to the geophysical scale indicates that a continuous viscosity gradient resulting from the temperature variation with depth cannot account for orogenesis by the mechanism of instability in compression.

An Extension of Bohr's Correspondence Principle to Apply to Small Quantum Numbers

Generalized correspondence principle for the Bohr atom.---It is shown that when an electron spirals in from an orbit of quantum number ${n}_{2}$ to an orbit of quantum number ${n}_{1}$, the frequency of the radiation emitted is equal to an integer (${n}_{2}\ensuremath{-}{n}_{1}$) times the mean frequency of rotations in its orbit during the spiraling, assuming that the number of rotations made for a change of quantum level $\mathrm{dn}$ is proportional to $\mathrm{dn}$. That is, the mean is defined as the ratio to (${n}_{2}\ensuremath{-}{n}_{1}$) of the integral from ${n}_{1}$ to ${n}_{2}$ of ${\ensuremath{\nu}}_{\mathrm{c}} \mathrm{dn}$ where ${\ensuremath{\nu}}_{c}$ is the classical frequency of rotation corresponding to any value of $n$. No explanation is offered as to why radiation begins and stops at integral values of $n$, nor why the radiation emitted in the manner assumed is monochromatic.