Critical behaviour at the displacive limit of structural phase transitions

For a special critical point at zero temperature,T _c =0, which is called the displacive limit of a classical or of a quantum-mechanical model showing displacive phase transitions, we derive a set of static critical exponents in the large-n limit. Due to zero-point motions and quantum fluctuations at low temperatures, the exponents of the quantum model are different from those of the classical model. Moreover, we report results on scaling functions, corrections to scaling, and logarithmic factors which appear ford=2 in the classical case and ford=3 in the quantum-mechanical case. Zero-point motions cause a decrease of the critical temperature of the quantum model with respect to the classicalT _c , which implies a difference between the classical and the quantum displacive limit. However, finite critical temperatures are found in both cases ford>2, while critical fluctuations still occur atT _c =0 for 0<d≦2 in the classical case and for 1 <d≦2 in the quantum model. Further we derive the slope of the critical curve at the classical displacive limit exactly. The absence of 1/n-corrections to the exponents of the classical model is also discussed.     

The classical limit for quantum mechanical correlation functions

For quantum systems of finitely many particles as well as for boson quantum field theories, the classical limit of the expectation values of products of Weyl operators, translated in time by the quantum mechanical Hamiltonian and taken in coherent states centered inx- andp-space around? ^?1/2 (coordinates of a point in classical phase space) are shown to become the exponentials of coordinate functions of the classical orbit in phase space. In the same sense,? ^?1/2 [(quantum operator) (t) — (classical function) (t)] converges to the solution of the linear quantum mechanical system, which is obtained by linearizing the non-linear Heisenberg equations of motion around the classical orbit.     

QCD2 and the classical correspondence in the large-N limit

It is shown that the large-N limit of quantum chromodynamics in twodimensions is determined by classical equations with boundary conditions. The nonperturbative quantum spectrum of mesonic bound states is obtained from a classical equation with a simple N-dependent boundary condition on the local charge density. The simplicity of the classical correspondence is shown to be directly tied to the simplicity of the space of gauge invariant operators of the theory. Implications for other large-N models are discussed.     

A Smooth Path between the Classical Realm and the Quantum Realm

A simple example of classical physics may be defined as classical variables, p and q, and quantum physics may be defined as quantum operators, P and Q. The classical world of p&q, as it is currently understood, is truly disconnected from the quantum world, as it is currently understood. The process of quantization, for which there are several procedures, aims to promote a classical issue into a related quantum issue. In order to retain their physical connection, it becomes critical as to how to promote specific classical variables to associated specific quantum variables. This paper, which also serves as a review paper, leads the reader toward specific, but natural, procedures that promise to ensure that the classical and quantum choices are guaranteed a proper physical connection. Moreover, parallel procedures for fields, and even gravity, that connect classical and quantum physical regimes, will be introduced.     

Normal forms for classical and boson systems

The group of Bogoliubov transformations of annihilation and creation operators is a subgroup of U(n,n) where n is the number of distinct pairs of annihilation and creation operators. Here, it is established that this subgroup of U(n,n) is isomorphic to Sp(2n,$\text{R}$), which appears in classical dynamics as the group of linear canonical transformations on a 2n-dimensional phase space. Well-known results in classical dynamics are then to used to deduce the full set of normal forms for Boson Hamiltonians. These are classified using a para-eigenvalue notation applicable to both classical and Bose field systems. A simple sufficient condition is given for the non-removability of pairs of creation operators. Explicit normal forms have not previously been given for Hamiltonians with this pathology, which may occur even when the corresponding classical Hamiltonian can be diagonalized.     

Quantum Communication Complexity

Can quantum communication be more efficient than its classical counterpart? Holevo's theorem rules out the possibility of communicating more than n bits of classical information by the transmission of n quantum bits—unless the two parties are entangled, in which case twice as many classical bits can be communicated but no more. In apparent contradiction, there are distributed computational tasks for which quantum communication cannot be simulated efficiently by classical means. In some cases, the effect of transmitting quantum bits cannot be achieved classically short of transmitting an exponentially larger number of bits. In a similar vein, can entanglement be used to save on classical communication? It is well known that entanglement on its own is useless for the transmission of information. Yet, there are distributed tasks that cannot be accomplished at all in a classical world when communication is not allowed, but that become possible if the non-communicating parties share prior entanglement. This leads to the question of how expensive it is, in terms of classical communication, to provide an exact simulation of the spooky power of entanglement.     

Thermodynamics of the quantum and classical Ising models with skew magnetic field

The thermodynamics of the unitary (normalized spin) quantum and classical Ising models with skew magnetic field, for |J|β?0.9, is derived for the ferromagnetic and antiferromagnetic cases. The high-temperature expansion (β-expansion) of the Helmholtz free energy is calculated up to order β⁷ for the quantum version (spin S≥1/2) and up to order β¹⁹ for the classical version. In contrast to the S=1/2 case, the thermodynamics of the transverse Ising and that of the XY model for S>1/2 are not equivalent. Moreover, the critical line of the T=0 classical antiferromagnetic Ising model with skew magnetic field is absent from this classical model, at least in the temperature range of |J|β?0.9.     

Evolution of Wigner's phase-space density under a nonintegrable quantum map

A nonintegrable area-preserving map for a system with one freedom is quantized, and the evolution of Wigner's function W(q,p) illustrated by contour plots of W in the paase plane. In the classical limit, propagation is governed by Liouville's equation and the contours of W rapidly develop an intricate structure of whorls and tendrils. When Planck's constant ? is not zero, the quantum map smooths out classical detail in phase-space areas smaller than ?. The quantum-mechanical distributions spread more slowly than their classical counterparts.     

Two-dimensional electron fluid at high temperature

The two-dimensional and one-component plasma (OCP) model withr ^?1 interactions is investigated in the high-temperature limit, where the thermal wavelength gets larger than the classical distance of closest approach. Nonnegligible diffraction effects are rigorously taken care of (up toe ²) through a temperature-dependent effective interaction. Debye thermodynamics, analyzed in terms of a classical plasma parameter Λ, is shown to diverge as Λ Inh, whenh→0. There is no classical limit. A result at variance with the corresponding one in three dimensions.     

Induced superradiation for a collection of three-level atoms

N three-level atoms interact simultaneously with classical and quantum fields, which are in quasiresonance with various atomic transitions. The classical and quantum fields exchange photons by means of the atoms. It is shown that under certain conditions this process is collective. The number of photons in a quantized mode oscillates, and the amplitude of these oscillations is proportional to N ². The frequency of the oscillations is determined by the frequencies of the classical and external fields.     

Colored,spinning classical particle in an external non-abelian gauge field

We derive classical non-relativistic equations of motion for a colored, spinning point-like particle in an external SU(2) gauge field from Dirac equation. We find that in addition to the classical spin and color spin vector, S, I, it is necessary to introduce a new classical dynamical variable [J^ab], a, b = 1, 2, 3, describing a mixing of the spin and color. The constraint relations between [J^ab], S, I are also found.     

Classical string dynamics with non-trivial topology

The possibility of branching processes for classical strings is investigated on the basis of the Nambu-Goto action. We parametrize the world sheet by a Riemann surface M and introduce a degenerate, semi-Riemannian metric η on M. Well-known results about the conformal group Diff(S¹) × Diff(S¹) are generalized to the case of (M, η). We provide an infinite dimensional Hamiltonian setting for branching processes of strings. Finally, the classical background for the theory of quantum strings as developed by Krichever and Novikov is discussed within this classical framework.     

Internal Structure of the Heisenberg and Robertson-Schrödinger Uncertainty Relations

It is known that the Heisenberg and Robertson-Schrödinger uncertainty relations can be replaced by sharper uncertainty relations in which the “classical” (depending on the gradient of the phase of the wave function) and “quantum” (depending on the gradient of the envelope of the wave function) parts of the variances 〈(Δx)²〉 and 〈(Δp)²〉 are separated. In this paper, three types of uncertainty relations for a different number of classical parts (2, 1 or 0) with different time behaviour of their left-hand and right-hand sides are discussed. For the Gaussian wave packet and two classical parts, the left-hand side of the corresponding relations increases for t→∞ as t ² and is much larger than ? ²/4. For one classical part, the left-hand side of the corresponding relation goes to the right-hand side equal to ? ²/4. For no classical part, both the right-hand and left-hand sides of the corresponding relation go quickly to zero. Therefore, the well-known limitations following from the usual uncertainty relations can be overcome in the corresponding measurements.     

Quantum effects in the Gowdy T3 cosmology

The Gowdy T³ Cosmology is an exact solution to the vacuum Einstein equations interpreted to be a single polarization of gravitational waves propagating in an anisotropic, spatially inhomogeneous background. The classical behavior is reviewed and related to standard cosmological parameters. Canonical quantization of the dynamical degrees of freedom is reviewed. An adiabatic vacuum state is constructed. Adiabatic regularization is used to obtain non-divergent stress-energy tensor vacuum expectation values. Casimir energy terms due to T³ imposed discrete modes are evaluated. The vacuum expectation values are analyzed in early and late time limits and evaluated numerically. The regularized expectation value is used as a source for the classical background spacetime in the spirit of semi-classical gravity. An entirely vacuum expectation value source term produces essentially the time reverse of the classical evolution. Classical stress-energy added to the source restores the classical behavior at late times only. The combined system collapses from infinite to small but non-zero volume and reexpands. The classical singularity is replaced by a symmetric bounce.     

Correspondence between classical and quantummechanical Boltzmann equations

To compare quantummechanical and classical Boltzmann equations for molecular gases, a correspondence is proposed for functions of angular momentum. Equivalence between irreducible tensors of both kinds is prescribed in a unique way by demanding that trace-averages of binary operator products be equal to solid-angle averages of products of the classical equivalents. Application to the linearized Waldmann-Snider equation for rigid linear molecules leads to an equivalent system for a set of functions φ_j(r, υ, ω, t), j = 0, 1, 2,…. If the quantum number j is approximated by a continuous variable, the system goes over into a single classical equation.     

Classical and quantum oscillators of quartic anharmonicities: second-order solution

An analytical solution up to the second order in the coupling constant λ is obtained for a classical quartic anharmonic oscillator by using Taylor series method. Our solution yields, as a special instance, the corresponding results obtained by using Laplace transform. With the help of correspondence principle, the classical solution is used to obtain the solution corresponding to a quantum quartic anharmonic oscillator. In the weak coupling regime (i.e., anharmonic constant λ⪡1), the so-called secular terms in classical and quantum solutions are tucked in (summed up) to avoid the nonconvergence. Both the classical and quantum solutions are used to obtain the frequency shifts of the quartic oscillators. It is found that these frequency shifts coincide exactly with those of the earlier results obtained by other methods. From the quantum field theoretic point of view, our solution exhibits the so-called Lamb shift. As an application of the solution for the quantum oscillator, we examine the possibility of getting squeezed states out of the input coherent light interacting with a nonlinear medium of inversion symmetry.     

Classical and quantum motion in an inverse square potential

Classical motion in an inverse square potential is shown to be equivalent to free motion on a hyperbola. The existence of a classical splitting between the q>0 and q<0 regions of motion is demonstrated. We show that this last property may be regarded as the classical counterpart of the superselection rule occurring in the corresponding quantum problem. We solve the quantum problem in momentum space finding that there is no way of quantizing its energy but that the eigenfunctions suffice to describe the single renormalized bound state of the system. The dynamical symmetry of the classical problem is found to be O(1,1). Both this symmetry and the symmetry of inversion through the origin are found to be broken.     

Classical and quantum turbulence

The reconnection of two singularities in 2D, 3D, and 4D classical and quantum turbulence is examined. Singularity reconnection plays an essential role in the dissipation of the incompressible part of kinetic energy. A reconnection condition 2(d_s+1)≥d+1 is derived, which crucially depends on the dimension d_s of the singular structure in relation to the spatial dimension d of the system. The feasibility of this condition is examined using direct numerical simulations of the Navier-Stokes and Gross-Pitaevskii equations for the classical and quantum turbulence, respectively. We observed that the condition was satisfied for d=3 and 4, in agreement with the occurrence of energy cascades in both classical and quantum turbulence in those dimensions.     

Can classical physics give g-factors > 1?

It has long been held that in flat space-time a classical slowly moving body with a charge density proportional to its mass density has g = 1. In this communication we demonstrate that classical special relativistic physics is sufficient to give g-factors which may exceed unity for such systems.