Статті в журналах: "Quantum-classical correspondence principle"

1

WÓJCIK, ANTONI, and RAVINDRA W. CHHAJLANY. "QUANTUM-CLASSICAL CORRESPONDENCE IN THE ORACLE MODEL OF COMPUTATION." International Journal of Quantum Information 04, no. 04 (August 2006): 633–40. http://dx.doi.org/10.1142/s0219749906002109.

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The oracle model of computation is believed to allow a rigorous proof of quantum over classical computational superiority. Since quantum and classical oracles are essentially different, a correspondence principle is commonly implicitly used as a platform for comparison of oracle complexity. Here, we question the grounds on which this correspondence is based. Obviously, results on quantum speed-up depend on the chosen correspondence. So, we introduce the notion of genuine quantum speed-up which can serve as a tool for reliable comparison of quantum versus classical complexity, independent of the chosen correspondence principle.

2

KAZAKOV, KIRILL A. "CLASSICAL SCALE OF QUANTUM GRAVITY." International Journal of Modern Physics D 12, no. 09 (October 2003): 1715–19. http://dx.doi.org/10.1142/s0218271803004110.

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Characteristic length scale of the post-Newtonian corrections to the gravitational field of a body is given by its gravitational radius r g . The role of this scale in quantum domain is discussed in the context of the low-energy effective theory. The question of whether quantum gravity effects appear already at r g leads to the question of correspondence between classical and quantum theories, which in turn can be unambiguously resolved by considering the issue of general covariance. The O(ℏ0) loop contributions turn out to violate the principle of general covariance, thus revealing their essentially quantum nature. The violation is O(1/N), where N is the number of particles in the body. This leads naturally to a macroscopic formulation of the correspondence principle.

3

Chen, Jin-Fu, Tian Qiu, and Hai-Tao Quan. "Quantum–Classical Correspondence Principle for Heat Distribution in Quantum Brownian Motion." Entropy 23, no. 12 (November 29, 2021): 1602. http://dx.doi.org/10.3390/e23121602.

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Quantum Brownian motion, described by the Caldeira–Leggett model, brings insights to the understanding of phenomena and essence of quantum thermodynamics, especially the quantum work and heat associated with their classical counterparts. By employing the phase-space formulation approach, we study the heat distribution of a relaxation process in the quantum Brownian motion model. The analytical result of the characteristic function of heat is obtained at any relaxation time with an arbitrary friction coefficient. By taking the classical limit, such a result approaches the heat distribution of the classical Brownian motion described by the Langevin equation, indicating the quantum–classical correspondence principle for heat distribution. We also demonstrate that the fluctuating heat at any relaxation time satisfies the exchange fluctuation theorem of heat and its long-time limit reflects the complete thermalization of the system. Our research study justifies the definition of the quantum fluctuating heat via two-point measurements.

4

Liu, Q. H., and B. Hu. "The hydrogen atom's quantum-to-classical correspondence in Heisenberg's correspondence principle." Journal of Physics A: Mathematical and General 34, no. 28 (July 6, 2001): 5713–19. http://dx.doi.org/10.1088/0305-4470/34/28/307.

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5

Lu, Jun, and Xue Mei Wang. "Quantum Spectra and Classical Orbits in Nano-Microstructure." Advanced Materials Research 160-162 (November 2010): 625–29. http://dx.doi.org/10.4028/www.scientific.net/amr.160-162.625.

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A kind of new classical-quantum correspondence principle is proposed using the idea of closed-orbit theory. The quantum spectrum function is introduced by means of the eigenvalues and the eigenfunctions in the system of one-dimensional nano-microstructure. The Fourier transformation of the quantum spectrum function is found corresponding with the classical orbits in the system. These results give new evidence about the classical-quantum correspondence. All the methods and results can be used in a lot of other systems, including some two-dimensional and three-dimensional systems. The researches about these systems are very important in the field of applied science, for example, molecular reaction dynamics and quantum information.

6

TZENOV, STEPHAN I. "IRROTATIONAL MOMENTUM FLUCTUATIONS CONDITIONING THE QUANTUM NATURE OF PHYSICAL PROCESSES." International Journal of Modern Physics A 21, no. 26 (October 20, 2006): 5299–316. http://dx.doi.org/10.1142/s0217751x06033866.

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Starting from a simple classical framework and employing some stochastic concepts, the basic ingredients of the quantum formalism are recovered. It has been shown that the traditional axiomatic structure of quantum mechanics can be rebuilt, so that the quantum mechanical framework resembles to a large extent that of the classical statistical mechanics and hydrodynamics. The main assumption used here is the existence of a random irrotational component in the classical momentum. Various basic elements of the quantum formalism (calculation of expectation values, the Heisenberg uncertainty principle, the correspondence principle) are recovered by applying traditional techniques, borrowed from classical statistical mechanics.

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Mauro, M. Di, A. Drago, and A. Naddeo. "Understanding the relation between classical and quantum mechanics: prospects for undergraduate teaching." Journal of Physics: Conference Series 2727, no. 1 (March 1, 2024): 012013. http://dx.doi.org/10.1088/1742-6596/2727/1/012013.

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Abstract Classical and quantum mechanics are two very different theories, each describing the world within its own range of validity. It is often stated that classical mechanics emerges from quantum mechanics in a certain limit. This is known as the correspondence principle. According to Planck’s version of the correspondence principle, classical mechanics is recovered when the limit in which a dimensionless parameter containing Planck’s constant h goes to zero is taken, while Bohr’s version entails taking the limit of large quantum numbers. However, despite what is usually stated in textbooks, the relation between the two theories is much more complex to state and understand. Here we deal with this issue by analysing some key examples, in some of which also the analogously subtle relation between wave and geometric optics is considered. Implications for quantum mechanics teaching at undergraduate level are carefully discussed.

8

Bonnar, James D., and Jeffrey R. Schmidt. "Classical orbits from the wave function in the large-quantum-number limit." Canadian Journal of Physics 81, no. 7 (July 1, 2003): 929–39. http://dx.doi.org/10.1139/p03-065.

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Classical trajectories for the Coulomb potential are obtained from the large principle quantum-number limit of solutions to the nonrelativistic Schrödinger equation, by use of integral equations satisfied by the radial probability density function. These trajectories are found to be in excellent agreement with those computed directly from classical mechanics, in accordance with a statement of the Bohr Correspondence principle, except in a region very close to the center of force. PACS No.: 05.45.Mt

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Manjavidze, J., and A. Sissakian. "Symmetries, variational principles, and quantum dynamics." Discrete Dynamics in Nature and Society 2004, no. 1 (2004): 205–12. http://dx.doi.org/10.1155/s1026022604310022.

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We describe the role of symmetries in formation of quantum dynamics. A quantum version of d'Alembert's principle is proposed to take into account the symmetry constrains more exact. It is argued that the time reversibility of quantum process, as the quantum analogy of d'Alembert's principle, makes the measure of the corresponding path integralδ-like. The argument of thisδ-function is the sum of all classical forces of the problem under consideration plus the random force of quantum excitations. Such measure establishes the one-to-one correspondence with classical mechanics and, for this reason, allows a free choice of the useful dynamical variables. The analysis shows that choosing the action-angle variables, one may get to the free-from-divergences quantum field theory. Moreover, one can try to get an independence from necessity to extract the degrees of freedom constrained by the symmetry. These properties of new quantization scheme are vitally essential for such theories as the non-Abelian Yang-Mills gauge theory and quantum gravity.

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Astapenko, Valery, and Timur Bergaliyev. "Comparison of Harmonic Oscillator Model in Classical and Quantum Theories of Light-Matter Interaction." Foundations 3, no. 3 (September 4, 2023): 549–59. http://dx.doi.org/10.3390/foundations3030031.

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A brief review of the classical and quantum description of the interaction of electromagnetic radiation with matter based on the model of a harmonic oscillator is presented. This review includes the generalized Bohr correspondence principle, the excitation of a quantum oscillator by electromagnetic pulses including saturation effect, the harmonic limit of the Bloch equations, and a phenomenological account of the damping of the quantum oscillator. In all cases, at the mathematical level, the relationship between the classical and quantum descriptions of the electromagnetic interaction is established and the conditions for such compliance are identified.

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CAVAGLIÀ, MARCO, and VITTORIO DE ALFARO. "ON A QUANTUM MINIUNIVERSE FILLED WITH YANG-MILLS RADIATION." Modern Physics Letters A 09, no. 07 (March 7, 1994): 569–77. http://dx.doi.org/10.1142/s0217732394003804.

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The Wheeler-DeWitt equation for a Robertson-Walker closed universe filled with an SU(2) Yang-Mills spatially uniform field is separable; we are then led to impose on the wave function the usual rules for a quantum bound system: square integrability and orthogonality. We investigate the correspondence principle for large gravitational quantum numbers. The classical probability depends on the gauge chosen for the time, and we examine lapse functions of the form N=N(a) (having in mind cosmic time and conformal time). The correspondence principle holds for these gauges if the differential representation of the momentum is gauge-dependent. This gives a prescription for the ordering of operators in the Wheeler-DeWitt equation. We also discuss the measure in the minisuperspace needed to ensure invariance of matrix elements of gauge-invariant operators.

12

Lippolis, Domenico, and Akira Shudo. "Towards the Resolution of a Quantized Chaotic Phase-Space: The Interplay of Dynamics with Noise." Entropy 25, no. 3 (February 24, 2023): 411. http://dx.doi.org/10.3390/e25030411.

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We outline formal and physical similarities between the quantum dynamics of open systems and the mesoscopic description of classical systems affected by weak noise. The main tool of our interest is the dissipative Wigner equation, which, for suitable timescales, becomes analogous to the Fokker–Planck equation describing classical advection and diffusion. This correspondence allows, in principle, to surmise a finite resolution, other than the Planck scale, for the quantized state space of the open system, particularly meaningful when the latter underlies chaotic classical dynamics. We provide representative examples of the quantum-stochastic parallel with noisy Hopf cycles and Van der Pol-type oscillators.

13

Boreham, B. W., H. Hora, and H. Hora. "Energy spectra of electrons emitted from laser irradiated low-density gas and the correspondence principle of electromagnetic interaction." Laser and Particle Beams 13, no. 1 (March 1995): 71–81. http://dx.doi.org/10.1017/s0263034600008855.

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It is expected (Hora & Handel 1987) that the energy spectra of electrons emitted from laserirradiated atoms in low-density gases would be fundamentally different for laser intensities above and below the threshold of a correspondence principle. Below such a threshold, the emission is a quantum mechanical interaction while, in contrast, above the threshold it is a classical process. Both of our earlier experiments (Boreham & Hora 1979; Boreham & Luther-Davies 1979) and those of several others, including some very recent results (Monot et al. 1993) confirm—after some controversy—the existence of such a correspondence principle. Details are discussed.

14

NOVELLO, M., V. M. C. PEREIRA, and N. PINTO-NETO. "NONMINIMAL COUPLING AND QUANTUM COSMOLOGY." International Journal of Modern Physics D 04, no. 05 (October 1995): 673–84. http://dx.doi.org/10.1142/s0218271895000454.

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We consider a minisuperspace cosmological model generated by coupling nonminimally a vector field with the gravitational field. The classical solutions are divided into three sets: singular solutions, eternal universes with an expansion phase and flat spacetime. We apply quantum cosmological arguments to investigate which of them is the most probable classical solution. The semiclassical cosmological wave functions satisfy the correspondence principle, and examples are shown in which the three sets of classical solutions can be predicted from different classes of WKB wave functions. As a particular example we have considered the no-boundary wave function, which predicts flat spacetime. No solutions can be obtained from the tunneling boundary condition.

15

FLOYD, EDWARD R. "CLASSICAL LIMIT OF THE TRAJECTORY REPRESENTATION OF QUANTUM MECHANICS, LOSS OF INFORMATION AND RESIDUAL INDETERMINACY." International Journal of Modern Physics A 15, no. 09 (April 10, 2000): 1363–78. http://dx.doi.org/10.1142/s0217751x00000604.

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The trajectory representation in the classical limit (ℏ→0) manifests a residual indeterminacy. We show that the trajectory representation in the classical limit goes to neither classical mechanics (Planck's correspondence principle) nor statistical mechanics. This residual indeterminacy is contrasted to Heisenberg uncertainty. We discuss the relationship between residual indeterminacy and 't Hooft's information loss and equivalence classes.

16

Neumaier, Arnold. "Ensembles and Experiments in Classical and Quantum Physics." International Journal of Modern Physics B 17, no. 16 (June 30, 2003): 2937–80. http://dx.doi.org/10.1142/s0217979203018338.

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A philosophically consistent axiomatic approach to classical and quantum mechanics is given. The approach realizes a strong formal implementation of Bohr's correspondence principle. In all instances, classical and quantum concepts are fully parallel: the same general theory has a classical realization and a quantum realization. Extending the ''probability via expectation'' approach of Whittle to noncommuting quantities, this paper defines quantities, ensembles, and experiments as mathematical concepts and shows how to model complementarity, uncertainty, probability, nonlocality and dynamics in these terms. The approach carries no connotation of unlimited repeatability; hence it can be applied to unique systems such as the universe. Consistent experiments provide an elegant solution to the reality problem, confirming the insistence of the orthodox Copenhagen interpretation on that there is nothing but ensembles, while avoiding its elusive reality picture. The weak law of large numbers explains the emergence of classical properties for macroscopic systems.

17

Sanz, Ángel S. "Quantum–Classical Entropy Analysis for Nonlinearly-Coupled Continuous-Variable Bipartite Systems." Entropy 24, no. 2 (January 27, 2022): 190. http://dx.doi.org/10.3390/e24020190.

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The correspondence principle plays a fundamental role in quantum mechanics, which naturally leads us to inquire whether it is possible to find or determine close classical analogs of quantum states in phase space—a common meeting point to both classical and quantum density statistical descriptors. Here, this issue is tackled by investigating the behavior of classical analogs arising upon the removal of all interference traits displayed by the Wigner distribution functions associated with a given pure quantum state. Accordingly, the dynamical evolution of the linear and von Neumann entropies is numerically computed for a continuous-variable bipartite system, and compared with the corresponding classical counterparts, in the case of two quartic oscillators nonlinearly coupled under regular and chaos conditions. Three quantum states for the full system are considered: a Gaussian state, a cat state, and a Bell-type state. By comparing the quantum and classical entropy values, and particularly their trends, it is shown that, instead of entanglement production, such entropies rather provide us with information on the system (either quantum or classical) delocalization. This gradual loss of information translates into an increase in both the quantum and the classical realms, directly connected to the increase in the correlations between both parties’ degrees of freedom which, in the quantum case, is commonly related to the production of entanglement.

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SERGEENKO, M. N. "QUASICLASSICAL ANALYSIS OF THREE-DIMENSIONAL SCHRÖDINGER'S EQUATION AND ITS SOLUTION." Modern Physics Letters A 15, no. 02 (January 20, 2000): 83–100. http://dx.doi.org/10.1142/s0217732300000104.

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Three-dimensional Schrödinger's equation is analyzed with the help of the correspondence principle between classical and quantum-mechanical quantities. Separation is performed after reduction of the original equation to the form of the classical Hamilton–Jacobi equation. Each one-dimensional equation obtained after separation is solved by the conventional WKB method. Quasiclassical solution of the angular equation results in the integral of motion [Formula: see text] and the existence of nontrivial solution for the angular quantum number l = 0. Generalization of the WKB method for multi-turning-point problems is given. Exact eigenvalues for solvable and some "insoluble" spherically symmetric potentials are obtained. Quasiclassical eigenfunctions are written in terms of elementary functions in the form of a standing wave.

19

Boorstein, J. L., and T. Uzer. "Dynamical barriers to transport in pulsed spin systems: quantum–classical correspondence in the kicked top." Canadian Journal of Chemistry 70, no. 2 (February 1, 1992): 488–96. http://dx.doi.org/10.1139/v92-070.

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Classical phase space is not homogeneous, but contains barriers (such as KAM tori) that, whether intact or broken, affect the temporal evolution of dynamical systems. In this article, we study the quantal manifestations of these classical-mechanical structures. Here, the particular system is the kicked top, which consists of an angular momentum vector precessing about one direction and experiencing periodic sudden kicks around another direction. We find that a suitably defined probability distribution function (constructed from the time-dependent state vector) shows transitions from classically allowed regions to inaccessible regions at the locations where the classical dynamics places the KAM tori. Keywords: bottlenecks, phase space, transport, spin systems, Correspondence Principle.

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SITE, L. DELLE. "ON THE CORRESPONDENCE BETWEEN CLASSICAL AND QUANTUM MECHANICS IN DEFINING ATOMS AND MOLECULES IN CONDENSED SYSTEMS." International Journal of Modern Physics B 15, no. 18 (July 20, 2001): 2485–90. http://dx.doi.org/10.1142/s0217979201006586.

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We analyze and discuss the concept of "proper quantum subsystem" (PQS). In particular we focus the attention on the theory of "quantum mechanics of a subspace" developed by R. F. W. Bader and coworkers whose definition of a PQS is based on the Schwinger's Quantum Action Principle; we illustrate some properties of this definition and propose a problem whose solution could be relevant in formulating a correct physical interpretation of the concept above.

21

Wu, Jiu Hui, Lin Zhang, and Ke jiang Zhou. "A novel kind of equations linking the quantum dynamics and the classical wave motions based on the catastrophe theory." Europhysics Letters 136, no. 4 (November 1, 2021): 40004. http://dx.doi.org/10.1209/0295-5075/ac2b5b.

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Abstract Considering that the catastrophe theory could describe quantitatively any phase transition process, we adopt the folding and cusp catastrophe types as the potential functions in the Schrödinger equation to attempt to link the quantum dynamics and the classical wave motions. Thus, through the dimensionless analysis a novel kind of partial differential equations is derived out. When the scaling parameter of the novel equation is equal to the Planck's constant, this equation becomes a detailed time-independent Schrödinger equation, from which Bohr correspondence principle can be found. On the other hand, when the scaling parameter tends to zero, this equation could degenerate to the classical Helmholtz equation. Therefore, this novel kind of equations could describe quantitatively the variation process of the wave functions from the macroscopic level to the quantum size.

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Medel-Portugal, Carlos, Juan Manuel Solano-Altamirano, and José Luis E. Carrillo-Estrada. "Classical and Quantum H-Theorem Revisited: Variational Entropy and Relaxation Processes." Entropy 23, no. 3 (March 19, 2021): 366. http://dx.doi.org/10.3390/e23030366.

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We propose a novel framework to describe the time-evolution of dilute classical and quantum gases, initially out of equilibrium and with spatial inhomogeneities, towards equilibrium. Briefly, we divide the system into small cells and consider the local equilibrium hypothesis. We subsequently define a global functional that is the sum of cell H-functionals. Each cell functional recovers the corresponding Maxwell–Boltzmann, Fermi–Dirac, or Bose–Einstein distribution function, depending on the classical or quantum nature of the gas. The time-evolution of the system is described by the relationship dH/dt≤0, and the equality condition occurs if the system is in the equilibrium state. Via the variational method, proof of the previous relationship, which might be an extension of the H-theorem for inhomogeneous systems, is presented for both classical and quantum gases. Furthermore, the H-functionals are in agreement with the correspondence principle. We discuss how the H-functionals can be identified with the system’s entropy and analyze the relaxation processes of out-of-equilibrium systems.

23

AREF’EVA, I. YA, B. DRAGOVICH, P. H. FRAMPTON, and I. V. VOLOVICH. "THE WAVE FUNCTION OF THE UNIVERSE AND p-ADIC GRAVITY." International Journal of Modern Physics A 06, no. 24 (October 10, 1991): 4341–58. http://dx.doi.org/10.1142/s0217751x91002094.

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A new approach to the wave function of the universe is suggested. The key idea is to take into account fluctuating number fields and present the wave function in the form of a Euler product. For this purpose we define a p-adic generalization of both classical and quantum gravitational theory. Elements of p-adic differential geometry are described. The action and gravitation field equations over the p-adic number field are investigated. p-adic analogs of some known solutions to the Einstein equations are presented. It follows that in quantum cosmology one should consider summation only over algebraic manifolds. The correspondence principle with the standard approach is considered.

24

Henner, V., A. Klots, A. Nepomnyashchy, and T. Belozerova. "The Correspondence Principle for Spin Systems: Simulations of Free Induction Decay with Classical and Quantum Spins." Applied Magnetic Resonance 52, no. 7 (May 26, 2021): 859–66. http://dx.doi.org/10.1007/s00723-021-01351-0.

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25

Canero, Armando TomÃ¡s. "Sound as a transverse wave." JOURNAL OF ADVANCES IN PHYSICS 13, no. 1 (February 28, 2017): 4522–34. http://dx.doi.org/10.24297/jap.v13i1.5670.

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This paper presents sound propagation based on a transverse wave model which does not collide with the interpretation of physical events based on the longitudinal wave model, but responds to the correspondence principle and allows interpreting a significant number of scientific experiments that do not follow the longitudinal wave model. Among the problems that are solved are: the interpretation of the location of nodes and antinodes in a Kundt tube of classical mechanics, the traslation of phonons in the vacuum interparticle of quantum mechanics and gravitational waves in relativistic mechanics.

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Bonnemain, Thibault, Benjamin Doyon, and Gennady El. "Generalized hydrodynamics of the KdV soliton gas." Journal of Physics A: Mathematical and Theoretical 55, no. 37 (August 19, 2022): 374004. http://dx.doi.org/10.1088/1751-8121/ac8253.

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Abstract We establish the explicit correspondence between the theory of soliton gases in classical integrable dispersive hydrodynamics, and generalized hydrodynamics (GHD), the hydrodynamic theory for many-body quantum and classical integrable systems. This is done by constructing the GHD description of the soliton gas for the Korteweg–de Vries equation. We further predict the exact form of the free energy density and flux, and of the static correlation matrices of conserved charges and currents, for the soliton gas. For this purpose, we identify the solitons’ statistics with that of classical particles, and confirm the resulting GHD static correlation matrices by numerical simulations of the soliton gas. Finally, we express conjectured dynamical correlation functions for the soliton gas by simply borrowing the GHD results. In principle, other conjectures are also immediately available, such as diffusion and large-deviation functions for fluctuations of soliton transport.

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Gomez, Ignacio, and Mario Castagnino. "On the classical limit of quantum mechanics, fundamental graininess and chaos: Compatibility of chaos with the correspondence principle." Chaos, Solitons & Fractals 68 (November 2014): 98–113. http://dx.doi.org/10.1016/j.chaos.2014.07.008.

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Paik, Biplab. "A UV complete picture of black hole conforming to low energy effective field theory." International Journal of Modern Physics A 33, no. 36 (December 30, 2018): 1850219. http://dx.doi.org/10.1142/s0217751x18502196.

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In this paper, we propose a UV complete, quantum improved picture of a black hole geometry that conforms to the IR gravity of effective field theory. Our work builds on identifying an effective space-distributed notion of black hole fluid in quantum improved regular Einstein gravity and its theoretical correspondence with a cosmology inspired power law fluctuation of matter. Hence, we make use of phenomenological asymptotic scales of matter fluctuation in static space to consequently derive a UV complete line-element of black hole space–time. In this appraisal, it gets explicit how principle of causality is preserved even while there is an effective spread of black hole fluid across horizon(s). Gravity changes from its conventional classical geometry-state to a quantum masked profile across a hypersurface of characteristic radius [Formula: see text]. We make analyses that probe the newly proposed quantum improved gravity in the contexts of regularity of Einstein fields, complete predictability of Hawking radiation process, and first law of black hole thermodynamics. It emerges that quantum black hole geometry self-regulates a regular timelike core that is abide by every quantum theoretical constraint while being flat around its center.

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Porter, Max D., and Ilon Joseph. "Observability of fidelity decay at the Lyapunov rate in few-qubit quantum simulations." Quantum 6 (September 8, 2022): 799. http://dx.doi.org/10.22331/q-2022-09-08-799.

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In certain regimes, the fidelity of quantum states will decay at a rate set by the classical Lyapunov exponent. This serves both as one of the most important examples of the quantum-classical correspondence principle and as an accurate test for the presence of chaos. While detecting this phenomenon is one of the first useful calculations that noisy quantum computers without error correction can perform [G. Benenti et al., Phys. Rev. E 65, 066205 (2001)], a thorough study of the quantum sawtooth map reveals that observing the Lyapunov regime is just beyond the reach of present-day devices. We prove that there are three bounds on the ability of any device to observe the Lyapunov regime and give the first quantitatively accurate description of these bounds: (1) the Fermi golden rule decay rate must be larger than the Lyapunov rate, (2) the quantum dynamics must be diffusive rather than localized, and (3) the initial decay rate must be slow enough for Lyapunov decay to be observable. This last bound, which has not been recognized previously, places a limit on the maximum amount of noise that can be tolerated. The theory implies that an absolute minimum of 6 qubits is required. Recent experiments on IBM-Q and IonQ imply that some combination of a noise reduction by up to 100× per gate and large increases in connectivity and gate parallelization are also necessary. Finally, scaling arguments are given that quantify the ability of future devices to observe the Lyapunov regime based on trade-offs between hardware architecture and performance.

30

MANDAL, SWAPAN. "ANALYTICAL SOLUTIONS OF ARBITRARY ORDERS TO THE CLASSICAL AND QUANTUM OSCILLATORS WITH VELOCITY-DEPENDENT QUARTIC ANHARMONICITIES." Modern Physics Letters B 18, no. 28n29 (December 20, 2004): 1453–66. http://dx.doi.org/10.1142/s021798490400792x.

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The classical oscillator with velocity-dependent anharmonicity (COVDA) arises when the velocity of the oscillator is reasonably high. By using an intuitive approach, we obtain an approximate analytical solution of arbitrary order to the problem of a COVDA. In addition to the third harmonic generation manifested by the nonlinear interaction, it is found that the solution contains the secular terms since the intuitive approach basically depends on the perturbation method. By assuming the small anharmonic constant, the secular terms are summed up for all orders and we obtain the renormalization of the frequency. The frequency of the oscillator decreases with the increase of the anharmonic constant. Interestingly, the magnitude of the shifts of the frequency of the oscillator with velocity-dependent quartic anharmonicity is identical with those of the oscillator with q-dependent quartic anharmonicity. However, the sign of the shifts for those two types of anharmonic oscillator is opposite in nature. These results indicate that the frequency shifts of the oscillator are actually the resultant effects (shifts) due to the q-dependent and the velocity-dependent anharmonicities. Finally, with the help of the correspondence principle, the solution of a quantum oscillator with velocity-dependent quartic anharmonicity (QOVDA) is obtained from the knowledge of the solution of its classical counterpart.

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GHANEH, T., F. DARABI, and H. MOTAVALLI. "SIGNATURE CHANGE BY GUP." International Journal of Modern Physics D 22, no. 05 (April 2013): 1350026. http://dx.doi.org/10.1142/s0218271813500260.

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We revisit the issue of continuous signature transition from Euclidean to Lorentzian metrics in a cosmological model described by Friedmann–Robertson–Walker (FRW) metric minimally coupled with a self-interacting massive scalar field. Then, using a noncommutative (NC) phase space of dynamical variables deformed by generalized uncertainty principle (GUP), we show that the signature transition occurs even for a model described by the FRW metric minimally coupled with a free massless scalar field accompanied by a cosmological constant. This indicates that the continuous signature transition might have been easily occurred at early universe just by a free massless scalar field, a cosmological constant and a NC phase space deformed by GUP, without resorting to a massive scalar field having an ad hoc complicate potential. We also study the quantum cosmology of the model and obtain a solution of Wheeler–DeWitt (WD) equation which shows a good correspondence with the classical path.

32

Bojowald, Martin. "Black-Hole Models in Loop Quantum Gravity." Universe 6, no. 8 (August 14, 2020): 125. http://dx.doi.org/10.3390/universe6080125.

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Dynamical black-hole scenarios have been developed in loop quantum gravity in various ways, combining results from mini and midisuperspace models. In the past, the underlying geometry of space-time has often been expressed in terms of line elements with metric components that differ from the classical solutions of general relativity, motivated by modified equations of motion and constraints. However, recent results have shown by explicit calculations that most of these constructions violate general covariance and slicing independence. The proposed line elements and black-hole models are therefore ruled out. The only known possibility to escape this sentence is to derive not only modified metric components but also a new space-time structure which is covariant in a generalized sense. Formally, such a derivation is made available by an analysis of the constraints of canonical gravity, which generate deformations of hypersurfaces in space-time, or generalized versions if the constraints are consistently modified. A generic consequence of consistent modifications in effective theories suggested by loop quantum gravity is signature change at high density. Signature change is an important ingredient in long-term models of black holes that aim to determine what might happen after a black hole has evaporated. Because this effect changes the causal structure of space-time, it has crucial implications for black-hole models that have been missed in several older constructions, for instance in models based on bouncing black-hole interiors. Such models are ruled out by signature change even if their underlying space-times are made consistent using generalized covariance. The causal nature of signature change brings in a new internal consistency condition, given by the requirement of deterministic behavior at low curvature. Even a causally disconnected interior transition, opening back up into the former exterior as some kind of astrophysical white hole, is then ruled out. New versions consistent with both generalized covariance and low-curvature determinism are introduced here, showing a remarkable similarity with models developed in other approaches, such as the final-state proposal or the no-transition principle obtained from the gauge-gravity correspondence.

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Aulin, Viktor, Andrey Grinkiv, Serhii Lysenko, Oleksandr Livitskyi, Andrii Chernai, Dmytro Holub, and Artem Holovatyi. "Theoretical Substantiation of Management of Functioning of Technical and Transport Systems on the Basis of Methods of the System Theory of Information." Central Ukrainian Scientific Bulletin. Technical Sciences, no. 4(35) (2021): 178–89. http://dx.doi.org/10.32515/2664-262x.2021.4(35).178-189.

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In the work from the theoretical point of view the management of functioning of transport and operation of technical systems on the basis of a method of the system theory of information is substantiated. The state vector and matrix on the basis of diagnostic information and sensitivity of influence of diagnostic parameters on a condition of transport or technical systems are offered. The properties of sensitivity (sensitivity) are reflected on the basis of classical and generalized information formulas of A. Kharkevich, K. Shannon, R. Hartley. The coefficients of quantum (emergence) of information according to the expressions of Shannon-Kharkevich and Hartley are used, the level of complexity of the changed states of the control system is taken into account. The scheme of connections of classical and generalized formulas which reflect functions of density of the diagnostic information on a condition of technical and transport systems is constructed. A universal numerical method and tools for it have been developed, which allows to perform information calculations on the basis of diagnostic data of the state of the studied objects. Numerical calculation refers to the sensitivity of the state (sensitivity factor) of technical and transport systems. The level of system organization of transport and technical systems and their management as active objects is considered. It is noted that the principle of correspondence is fulfilled, which is mandatory for a more general information theory. The system modification is implemented using the Hartley formula, and the modification in the classical case is the Kharkevich formula. It is determined that the semantic information model allows for small samples and comparison of partial criteria to well substantiate the interpretations of object state recognition, developed by the vectors of their classes. It is revealed that the semantic information model combines the advantages of meaningful and statistical models created for the implementation of the automated system of cognitive analysis of transport and technical systems. The behavior of systems is revealed when they are completely deterministic and completely random, which is important in predicting their state.

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Tamburini, Fabrizio, and Ignazio Licata. "Majorana quanta, string scattering, curved spacetimes and the Riemann Hypothesis." Physica Scripta 96, no. 12 (December 1, 2021): 125276. http://dx.doi.org/10.1088/1402-4896/ac4553.

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Abstract The Riemann Hypothesis states that the Riemann zeta function ζ(z) admits a set of ‘non-trivial’ zeros that are complex numbers supposed to have real part 1/2. Their distribution on the complex plane is thought to be the key to determine the number of prime numbers before a given number. Hilbert and Pólya suggested that the Riemann Hypothesis could be solved through the mathematical tools of physics, finding a suitable Hermitian or unitary operator that describe classical or quantum systems, whose eigenvalues distribute like the zeros of ζ(z). A different approach is that of finding a correspondence between the distribution of the ζ(z) zeros and the poles of the scattering matrix S of a physical system. Our contribution is articulated in two parts: in the first we apply the infinite-components Majorana equation in a Rindler spacetime and compare the results with those obtained with a Dirac particle following the Hilbert-Pólya approach showing that the Majorana solution has a behavior similar to that of massless Dirac particles and finding a relationship between the zeros of zeta end the energy states. Then, we focus on the S-matrix approach describing the bosonic open string scattering for tachyonic states with the Majorana equation. Here we find that, thanks to the relationship between the angular momentum and energy/mass eigenvalues of the Majorana solution, one can explain the still unclear point for which the poles and zeros of the S-matrix of an ideal system that can satisfy the Riemann Hypothesis, exist always in pairs and are related via complex conjugation. As claimed in the literature, if this occurs and the claim is correct, then the Riemann Hypothesis could be in principle satisfied, tracing a route to a proof.

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Ruiz, D. E., and I. Y. Dodin. "On the correspondence between quantum and classical variational principles." Physics Letters A 379, no. 40-41 (October 2015): 2623–30. http://dx.doi.org/10.1016/j.physleta.2015.06.014.

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36

Czachor, Marek. "Cosmic-Time Quantum Mechanics and the Passage-of-Time Problem." Universe 9, no. 4 (April 16, 2023): 188. http://dx.doi.org/10.3390/universe9040188.

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A new dynamical paradigm merging quantum dynamics with cosmology is discussed. We distinguish between a universe and its background space-time. The universe here is the subset of space-time defined by Ψτ(x)≠0, where Ψτ(x) is a solution of a Schrödinger equation, x is a point in n-dimensional Minkowski space, and τ≥0 is a dimensionless ‘cosmic-time’ evolution parameter. We derive the form of the Schrödinger equation and show that an empty universe is described by a Ψτ(x) that propagates towards the future inside some future-cone V+. The resulting dynamical semigroup is unitary, i.e., ∫V+d4x|Ψτ(x)|2=1 for τ≥0. The initial condition Ψ0(x) is not localized at x=0. Rather, it satisfies the boundary condition Ψ0(x)=0 for x∉V+. For n=1+3 the support of Ψτ(x) is bounded from the past by the ‘gap hyperboloid’ ℓ2τ=c2t2−x2, where ℓ is a fundamental length. Consequently, the points located between the hyperboloid and the light cone c2t2−x2=0 satisfy Ψτ(x)=0, and thus do not belong to the universe. As τ grows, the gap between the support of Ψτ(x) and the light cone increases. The past thus literally disappears. Unitarity of the dynamical semigroup implies that the universe becomes localized in a finite-thickness future-neighbourhood of ℓ2τ=c2t2−x2, simultaneously spreading along the hyperboloid. Effectively, for large τ the subset occupied by the universe resembles a part of the gap hyperboloid itself, but its thickness Δτ is non-zero for finite τ. Finite Δτ implies that the three-dimensional volume of the universe is finite as well. An approximate radius of the universe, rτ, grows with τ due to Δτrτ3=Δ0r03 and Δτ→0. The propagation of Ψτ(x) through space-time matches an intuitive picture of the passage of time. What we regard as the Minkowski-space classical time can be identified with ctτ=∫d4xx0|Ψτ(x)|2, so tτ grows with τ as a consequence of the Ehrenfest theorem, and its present uncertainty can be identified with the Planck time. Assuming that at present values of τ (corresponding to 13–14 billion years) Δτ and rτ are of the order of the Planck length and the Hubble radius, we estimate that the analogous thickness Δ0 of the support of Ψ0(x) is of the order of 1 AU, and r03∼(ctH)3×10−44. The estimates imply that the initial volume of the universe was finite and its uncertainty in time was several minutes. Next, we generalize the formalism in a way that incorporates interactions with matter. We are guided by the correspondence principle with quantum mechanics, which should be asymptotically reconstructed for the present values of τ. We argue that Hamiltonians corresponding to the present values of τ approximately describe quantum mechanics in a conformally Minkowskian space-time. The conformal factor is directly related to |Ψτ(x)|2. As a by-product of the construction, we arrive at a new formulation of conformal invariance of m≠0 fields.

37

Delos, J. B., and M. L. Du. "Correspondence principles: the relationship between classical trajectories and quantum spectral." IEEE Journal of Quantum Electronics 24, no. 7 (July 1988): 1445–52. http://dx.doi.org/10.1109/3.983.

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38

Yukalov, Vyacheslav I. "Evolutionary Processes in Quantum Decision Theory." Entropy 22, no. 6 (June 18, 2020): 681. http://dx.doi.org/10.3390/e22060681.

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The review presents the basics of quantum decision theory, with an emphasis on temporary processes in decision making. The aim is to explain the principal points of the theory. How an operationally-testable, rational choice between alternatives differs from a choice decorated by irrational feelings is elucidated. Quantum-classical correspondence is emphasized. A model of quantum intelligence network is described. Dynamic inconsistencies are shown to be resolved in the frame of the quantum decision theory.

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MARMO, G., A. SIMONI, and F. VENTRIGLIA. "QUANTUM SYSTEMS AND ALTERNATIVE UNITARY DESCRIPTIONS." International Journal of Modern Physics A 19, no. 15 (June 20, 2004): 2561–78. http://dx.doi.org/10.1142/s0217751x04018385.

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Motivated by the existence of bi-Hamiltonian classical systems and the correspondence principle, in this paper we analyze the problem of finding Hermitian scalar products which turn a given flow on a Hilbert space into a unitary one. We show how different invariant Hermitian scalar products give rise to different descriptions of a quantumsystem in the Ehrenfest and Heisenberg picture.

40

Horvat, Sebastian, and Borivoje Dakić. "Interference as an information-theoretic game." Quantum 5 (March 8, 2021): 404. http://dx.doi.org/10.22331/q-2021-03-08-404.

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The double slit experiment provides a clear demarcation between classical and quantum theory, while multi-slit experiments demarcate quantum and higher-order interference theories. In this work we show that these experiments pertain to a broader class of processes, which can be formulated as information-processing tasks, providing a clear cut between classical, quantum and higher-order theories. The tasks involve two parties and communication between them with the goal of winning certain parity games. We show that the order of interference is in one-to-one correspondence with the parity order of these games. Furthermore, we prove the order of interference to be additive under composition of systems both in classical and quantum theory. The latter result can be used as a (semi)device-independent witness of the number of particles in the quantum setting. Finally, we extend our game formulation within the generalized probabilistic framework and prove that tomographic locality implies the additivity of the order of interference under composition. These results shed light on the operational meaning of the order of interference and can be important for the identification of the information-theoretic principles behind second-order interference in quantum theory.

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WEN, KAI, and GUI LU LONG. "ONE-PARTY QUANTUM-ERROR-CORRECTING CODES FOR UNBALANCED ERRORS: PRINCIPLES AND APPLICATION TO QUANTUM DENSE CODING AND QUANTUM SECURE DIRECT COMMUNICATION." International Journal of Quantum Information 08, no. 04 (June 2010): 697–719. http://dx.doi.org/10.1142/s0219749910006289.

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In this article, we present unbalanced-quantum-error-correcting codes (one-party QECCs) — a novel idea for correcting unbalanced quantum errors. In some quantum communication tasks using entangled pairs, the error distributions between two parts of the pairs are unbalanced, and one party holds the whole entangled pairs at the final stage, and he or she is able to perform joint measurements on the pairs. In this situation the proposed one-party QECCs can improve error correction by allowing a higher-tolerated error rate. We have established the general correspondence between linear classical codes and the one-party QECCs, and we have given the general definition for these types of quantum-error-correcting codes. It has been shown that the one-party QECCs can correct errors as long as the error threshold is not larger than 0.5. They work even for fidelity less than 0.5 as long as it is larger than 0.25. We give several concrete examples of the one-party QECCs. We provide the applications of the one-party QECCs in quantum dense coding, so that it can function in noisy channels. As a result, a large number of quantum secure direct communication protocols based on dense coding are also able to be protected by this new type of one-party QECCs.

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Oneto, Alessandro, and Andrea Petracci. "On the quantum periods of del Pezzo surfaces with ⅓ (1, 1) singularities." Advances in Geometry 18, no. 3 (July 26, 2018): 303–36. http://dx.doi.org/10.1515/advgeom-2017-0048.

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AbstractIn earlier joint work with collaborators we gave a conjectural classification of a broad class of orbifold del Pezzo surfaces, using Mirror Symmetry. We proposed that del Pezzo surfaces X with isolated cyclic quotient singularities such that X admits a ℚ-Gorenstein toric degeneration correspond via Mirror Symmetry to maximally mutable Laurent polynomials f in two variables, and that the quantum period of such a surface X, which is a generating function for Gromov–Witten invariants of X, coincides with the classical period of its mirror partner f.In this paper we give strong evidence for this conjecture. Contingent on conjectural generalisations of the Quantum Lefschetz theorem and the Abelian/non-Abelian correspondence, we compute many quantum periods for del Pezzo surfaces with $\begin{array}{} \frac{1}{3} \end{array} $(1, 1) singularities. Our computations also give strong evidence for the extension of these two principles to the orbifold setting.

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Jarzynski, Christopher, H. T. Quan, and Saar Rahav. "Quantum-Classical Correspondence Principle for Work Distributions." Physical Review X 5, no. 3 (September 17, 2015). http://dx.doi.org/10.1103/physrevx.5.031038.

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44

Wang, F., M. T. Greenaway, A. G. Balanov, and T. M. Fromhold. "Non-KAM classical chaos topology for electrons in superlattice minibands determines the inter-well quantum transition rates." Scientific Reports 14, no. 1 (March 4, 2024). http://dx.doi.org/10.1038/s41598-024-52351-6.

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AbstractWe investigate the quantum-classical correspondence for a particle tunnelling through a periodic superlattice structure with an applied bias voltage and an additional tilted harmonic oscillator potential. We show that the quantum mechanical tunnelling rate between neighbouring quantum wells of the superlattice is determined by the topology of the phase trajectories of the analogous classical system. This result also enables us to estimate, with high accuracy, the tunnelling rate between two spatially displaced simple harmonic oscillator states using a classical model, and thus gain new insight into this generic quantum phenomenon. This finding opens new directions for exploring and understanding the quantum-classical correspondence principle and quantum jumps between displaced harmonic oscillators, which are important in many branches of natural science.

45

Lorenz, Heribert, Sigmund Kohler, Anton Parafilo, Mikhail Kiselev, and Stefan Ludwig. "Classical analogue to driven quantum bits based on macroscopic pendula." Scientific Reports 13, no. 1 (October 26, 2023). http://dx.doi.org/10.1038/s41598-023-45118-y.

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AbstractQuantum mechanics increasingly penetrates modern technologies but, due to its non-deterministic nature seemingly contradicting our classical everyday world, our comprehension often stays elusive. Arguing along the correspondence principle, classical mechanics is often seen as a theory for large systems where quantum coherence is completely averaged out. Surprisingly, it is still possible to reconstruct the coherent dynamics of a quantum bit (qubit) by using a classical model system. This classical-to-quantum analogue is based on wave mechanics, which applies to both, the classical and the quantum world. In this spirit we investigate the dynamics of macroscopic physical pendula with a modulated coupling. As a proof of principle, we demonstrate full control of our one-to-one analogue to a qubit by realizing Rabi oscillations, Landau-Zener transitions and Landau-Zener-Stückelberg-Majorana interferometry. Our classical qubit demonstrator can help comprehending and developing useful quantum technologies.

46

Zhu, Long, Zongping Gong, Biao Wu, and H. T. Quan. "Quantum-classical correspondence principle for work distributions in a chaotic system." Physical Review E 93, no. 6 (June 3, 2016). http://dx.doi.org/10.1103/physreve.93.062108.

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Paul, Susobhan, Satadal Bhattacharyya, and Joydip Mitra. "Study of correspondence principle in $1$-dimension through examples." European Journal of Physics, July 6, 2023. http://dx.doi.org/10.1088/1361-6404/ace502.

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Abstract We present several examples both for non-perturbative and perturbative systems to illustrate the Bohr's correspondence principle which are not usually treated in common text-books. In particular, the importance of the limit $n\to \infty$ and $h \to 0$ in understanding of the classical limit of quantum theory has been discussed explicitly in each case.

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Luna Hernández, Edgar Luis, Jorge Alejandro Bernal Arroyo, and Luis Enrique Ramón Pedrero. "The free fall in three Physics theories." Revista Mexicana de Física E 21, no. 1 Jan-Jun (January 19, 2024). http://dx.doi.org/10.31349/revmexfise.21.010214.

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This paper explores the interplay between Classical Mechanics, Relativistic Mechanics, and Quantum Mechanics through an analysis of the free fall phenomenon. We investigate the probability density functions and corresponding plots in each theory, alongside calculating the expected values of position and momentum. By observing the behavior of these results as they approach the classical limit, we confirm the hypothesis that these theories can be connected through their probability density functions. Furthermore, we discuss the validity of the correspondence principle in Quantum Mechanics, while also examining, in a non-rigorous manner, the validity of the weak equivalence principle within each of the aforementioned theories.

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Lin, De-Hone, Jee-Gong Chang, and Chi-Chuan Hwang. "Aharonov-Bohm effect in quantum-to-classical correspondence of the Heisenberg principle." Physical Review A 67, no. 4 (April 22, 2003). http://dx.doi.org/10.1103/physreva.67.042109.

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50

Berman, Robert J., Tristan C. Collins, and Daniel Persson. "Emergent Sasaki-Einstein geometry and AdS/CFT." Nature Communications 13, no. 1 (January 18, 2022). http://dx.doi.org/10.1038/s41467-021-27951-9.

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AbstractA central problem in any quantum theory of gravity is to explain the emergence of the classical spacetime geometry in some limit of a more fundamental, microscopic description of nature. The gauge/gravity-correspondence provides a framework in which this problem can, in principle, be addressed. This is a holographic correspondence which relates a supergravity theory in five-dimensional Anti-deSitter space to a strongly coupled superconformal gauge theory on its 4-dimensional flat Minkowski boundary. In particular, the classical geometry should therefore emerge from some quantum state of the dual gauge theory. Here we confirm this by showing how the classical metric emerges from a canonical state in the dual gauge theory. In particular, we obtain approximations to the Sasaki-Einstein metric underlying the supergravity geometry, in terms of an explicit integral formula involving the canonical quantum state in question. In the special case of toric quiver gauge theories we show that our results can be computationally simplified through a process of tropicalization.

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