On the pointwise behavior of semi-classical measures

In this paper we concern ourselves with the small asymptotics of the inner products of the eigenfunctions of a Schrödinger-type operator with a coherent state. More precisely, let _j and E _j denote the eigenfunctions and eigenvalues of a Schrödinger-type operator H with discrete spectrum. Let _(x,) be a coherent state centered at the point (x, ) in phase space. We estimate as 0 the averages of the squares of the inner products ( _a ^(x,) , _j ) over an energy interval of size around a fixed energy, E. This follows from asymptotic expansions of the form for certain test function and Schwartz amplitudes a of the coherent state. We compute the leading coefficient in the expansion, which depends on whether the classical trajectory through (x, ) is periodic or not. In the periodic case the iterates of the trajectory contribute to the leading coefficient. We also discuss the case of the Laplacian on a compact Riemannian manifold.Research supported in part by NSF grant DMS-9303778     

Transquantum Dynamics

Segal proposed transquantum commutation relations with two transquantum constants , besides Planck's quantum constant and with a variable i. The Heisenberg quantum algebra is a contraction—in a more general sense than that of Inönü and Wigner—of the Segal transquantum algebra. The usual constant i arises as a vacuum order-parameter in the quantum limit ,0. One physical consequence is a discrete spectrum for canonical variables and space-time coordinates. Another is an interconversion of time and energy accompanying space-time meltdown (disorder), with a fundamental conversion factor of some kilograms of energy per second.     

The Schrödinger equation and canonical perturbation theory

LetT ₀(, )+V be the Schrödinger operator corresponding to the classical HamiltonianH ₀()+V, whereH ₀() is thed-dimensional harmonic oscillator with non-resonant frequencies =(₁, ... , _d ) and the potentialV(q ₁, ... ,q _d) is an entire function of order (d+1)^–1. We prove that the algorithm of classical, canonical perturbation theory can be applied to the Schrödinger equation in the Bargmann representation. As a consequence, each term of the Rayleigh-Schrödinger series near any eigenvalue ofT ₀(, ) admits a convergent expansion in powers of of initial point the corresponding term of the classical Birkhoff expansion. Moreover ifV is an even polynomial, the above result and the KAM theorem show that all eigenvalues _n (, ) ofT ₀+V such thatn coincides with a KAM torus are given, up to order , by a quantization formula which reduces to the Bohr-Sommerfeld one up to first order terms in .     

Some remarks on braided group reconstruction and braided doubles

The cross coproduct braided group AutC)B is obtained by Tannaka-Krein reconstruction from C ^B C for a braided group B in braided category C. We apply this construction to obtain partial solutions to two problems in braided group theory, namely the tensor problem and the braided double. We obtain AutC) Aut(C) Aut(C) Aut(C) and higher braided group spin chains. The example B(R) B(R) ... B(R) is described explicitly by R-matrix relations. We also obtain Aut(C) Aut(C)* as a dual quasitriangular codouble braided group by reconstruction from the dual category C° C. General braided double crossproducts B C are also considered.     

Short-time propagator on curved spaces

It is shown that the standard WKB-approximation (the approximation of the first order in) to the propagator is not sufficient for the construction of the short-time propagator on curved spaces. The proper short-time propagator can be obtained by means of the second order (in) WKB-approximation and then no subtraction of a quantum correction proportional to ² from the original Lagrangian is necessary.The authors are indebted to J. Tolar for valuable critical comments and advices.     

On the failure of the time-energy uncertainty principle

We establish, for the quantum system made up of a single free particle, the formula E t(v/c) , where E is the precision to whichE can be ascertained in time t. The measurement can be carried out with zero disturbance inE itself.     

Notes on the classical and the nonrelativistic limits in quantum mechanics

A method is presented which can be used to discuss both the classical and also the nonrelativistic limit of quantum mechanics. A one-to-one correspondence may be established between the asymptotic convergence of the resolvent and that of the timedependent solution. In so far as the question of dynamics is concerned we investigate the relation between families of nonrelativistic Hamiltonians and the corresponding Dirac-Hamiltonians when c± or when c±0. The nonrelativistic free theory formally shows the same pattern when ±0 (the classical limit) or when ±. The investigation finally shows how the asymptotic convergence of the relativistic theory can take place under some fairly general conditions of the radiation field.     

Stochastic gravity

We investigate stochastic gravity as a potentially fruitful avenue for studying quantum effects in gravity. Following the approach of stochastic electrodynamics (sed), as a representation of the quantum gravity vacuum we construct a classical state of isotropic random gravitational radiation, expressed as a spin-2 field,h _µ (x), composed of plane waves of random phase on a flat spacetime manifold. Requiring Lorentz invariance leads to the result that the spectral composition function of the gravitational radiation,h(), must be proportional to 1/ ². The proportionality constant is determined by the Planck condition that the energy density consist of/2 per normal mode, and this condition sets the amplitude scale of the random gravitational radiation at the order of the Planck length, giving a spectral composition functionh() =16c ²L_p/². As an application of stochastic gravity, we investigate the Davies-Unruh effect. We calculate the two-point correlation function (R _iojo(O–/2)R _kolo(O,+/2)) of the measureable geodesic deviation tensor field,R _iojo, for two situations: (i) at a point detector uniformly accelerating through the random gravitational radiation, and (ii) at an inertial detector in a heat bath of the random radiation at a finite temperature. We find that the two correlation functions agree to first order ina/c provided that the temperature and acceleration satisfy the relationkT=a/2c.     

Electronic structure and optical spectra of YBa2Cu3O7−δ thin films in the infrared and UV region

Optical reflectance of YBa₂Cu₃O_7- thin films grown by laser ablation is measured within photon energies of 0.1 eV to 3.5 eV at room temperature. The spectra can be fit congruently with the anisotropic dielectric constants which take account of the intraband free carrier transition and interband transition. The anisotropic plasma frequencies are simulated to be _pl=2.18 eV and _ch=2.80 eV contributed from free carriers on the plane and in the chain, respectively. The interband transition occurs near 2.5 eV and is pertinent to a rather broad line width.     

Some results on electronic two- and one photon raman scattering in CdS

Two photon Raman scattering (TPRS) via virtually excited biexcitons is observed in CdS over a rather large spectral region in a scattering configuration which favours stimulated emission. We observe either a pure longitudinal exciton or-for the first time—a bound exciton (I ₂) as final state particles. Furthermore, the anomaly in the relation between _exc and _R at _exc= E_blex is observed for the first time in a II–VI compound, indicating an energy of the ₁ biexciton level of 5.098 eV in agreement with two photon absorption measurements. With an applied magnetic fieldB, the corresponding shift of the exciton eigenenergies can be observed. For the longitudinal exciton, the diamagnetic shift is 0.35 meV atB=10T forBc in agreement with theoretical predictions. In this configuration also a stimulated one photon spin flip Raman scattering is observed, yielding the well known electronicg-value of 1.78.     

Small ħ asymptotics for quantum partition functions associated to particles in external Yang-Mills potentials

To a gauge field on a principalG-bundlePM is associated a sequence of quantum mechanical Hamiltonians, as Planck's constant 0 and a sequence of representations _n ofG is taken. This paper studies the associated quantum partition functions, trace exp (–tH _n ), and produces a complete asymptotic expansion, as 0, =1/n, of which the principal term, proportional to the classical partition function, is the familiar classical limit.Research supported in part by Deutsche Forschungsgemeinschaft and NSF grant NoPhy 81-09011A-01On leave of absence from Freie Universität, BerlinResearch supported by NSF grant MCS 820176A01     

Smoothed Wigner functions: a tool to resolve semiclassical structures

The Wigner and Husimi distributions are the usual phase space representations of a quantum state. The Wigner distribution has structures of order 2. On the other hand, the Husimi distribution is a Gaussian smearing of the Wigner function on an area of size and then, it only displays structures of size . We have developed a phase space representation which results a Gaussian smearing of the Wigner function on an area of size , with 1. Within this representation, the Husimi and Wigner functions are recovered when =1 and respectively. We treat the application of this intermediate representation to explore the semiclassical limit of quantum mechanics. In particular we show how this representation uncover semiclassical hyperbolic structures of chaotic eigenstates.     

Derivation of the Evans Wave Equation from the Lagrangian and <Emphasis Type="Italic">Action</Emphasis>: Origin of the Planck Constant in General Relativity

No Heading The Evans wave equation is derived from the appropriate Lagrangian and action, identifying the origin of the Planck constant in general relativity. The classical Fermat principle of least time, and the classical Hamilton principle of least action, are expressed in terms of a tetrad multiplied by a phase factor exp(iS/), where S is the action in general relativity. Wave (or quantum) mechanics emerges from these classical principles of general relativity for all matter and radiation fields, giving a unified theory of quantum mechanics based on differential geometry and general relativity. The phase factor exp(iS/) is an eigenfunction of the Evans wave equation and is the origin in general relativity and geometry of topological phase effects in physics, including the Aharonov-Bohm class of effects, the Berry phase, the Sagnac effect, related interferometric effects, and all physical optical effects through the Evans spin field B⁽³⁾ and the Stokes theorem in differential geometry. The Planck constant is thus identified as the least amount possible of action or angular momentum or spin in the universe. This is also the origin of the fundamental Evans spin field B⁽³⁾, which is always observed in any physical optical effect. It originates in torsion, spin and the second (or spin) Casimir invariant of the Einstein group. Mass originates in the first Casimir invariant of the Einstein group. These two invariants define any particle.     

Heat equation on phase space and the classical limit of quantum mechanical expectation values

The expectation value of a quantum mechanical operator, taken in coherent states and suitably rescaled, is the solution of an initial value problem for the heat equation on phase space, in which plays the role of time, and the classical observable is the distribution of temperature at =0.     

On the stability of spin waves at finite temperatures

The temperature dependent exchange stiffnessD, defined for long wavelength spin waves by=Dq ², is calculated for very weak itinerant ferromagnets. It is found that for a general single band of d-electrons,D reduces to a very simple form givingD(T)=D(0)/ ₀ in the limit ₀0. The stability condition for spin waves at finite temperatures is thenD(0) > 0 and/ ₀ > 0, where is the relative occupation ± spin sub-bands and ₀ its value at 0°K.The authors are grateful to E. P. Wohlfarth for helpful discussions.     

Apparent wave function collapse caused by scattering

Some experimental implications of the recent progress on wave function collapse are calculated. Exact results are derived for the center-of-mass wave function collapse caused by random scatterings and applied to a range of specific examples. The results show that recently proposed experiments to measure the GRW effect are likely to fail, since the effect of naturally occurring scatterings is of the same form as the GRW effect but generally much stronger. The same goes for attempts to measure the collapse caused by quantum gravity as suggested by Hawking and others. The results also indicate that macroscopic systems tend to be found in states with xp =/ , but microscopic systems in highly tiltedly squeezed states with xp ».     

Twisted Lie Group C*-Algebras as Strict Quantization

A nonzero 2-cocycle Z2(g, R) on the Lie algebra g of a compact Lie group G defines a twisted version of the Lie–Poisson structure on the dual Lie algebra g*, leading to a Poisson algebra C (g*()). Similarly, a multiplier c Z2(G, U(1)) on G which is smooth near the identity defines a twist in the convolution product on G, encoded by the twisted group C-algebra C*(G,c). Further to some superficial yet enlightening analogies between C (g*()) and C*(G,c), it is shown that the latter is a strict quantization of the former, where Plancks constant assumes values in (Z\{0})-1. This means that there exists a continuous field of C*-algebras, indexed by 0 (Z\{0})-1, for which A0= C0(g*) and A=C*(G,c) for 0, along with a cross-section of the field satisfying Diracs condition asymptotically relating the commutator in A to the Poisson bracket on C(g*()). Note that the quantization of does not occur for =0.     

Isospectral transformations in relativistic and nonrelativistic mechanics

The modified Korteweg de-Vries hierarchy of partial differential equations generating transformations of the one-dimensional Dirac equation, is shown to reduce in the limitc to the Korteweg de-Vries hierarchy, generating isospectral transformations of the Schrödinger equation. The former hierarchy reduces into relativistic and the latter into nonrelativistic isoperiodic transformation in the limit0.     

Correspondence-principle condition on the Reissner-Nordström solution for charged leptons

Current experimental evidence supports the picture of three charged leptons (e ^±, ^±, ^±), a triplet with mass splitting that bears a resemblance to the Gell-Mann-Okubo form. To elucidate the overall mass scale, a charged lepton is viewed as a mass point that engenders a local Reissner-Nordström spacetime geometry, and the Einstein-Maxwell action is evaluated through an invariant space-time region associated with the particle's radiation reaction interval 2e ²/3m. What emerge for the electron, muon, and tau are values of the Einstein-Maxwell action in the neighborhood of. The mean value of the three action integrals is =(140.93)e ²=(1.0284), and thus the apparent quantum condition sets the mass scale for the three charged leptons.     

Higher approximations for quasiclassical trajectory-coherent states of a nonrelativistic particle

A scheme is developed for constructing higher approximations for quasiclassical trajectory-coherent states of a nonrelativistic particle. The basic property of these states is that the quantum-mechanical mean of the coordinate and momentum operators in these states are accurate solutions of the Hamiltonian system in the limit as 0. The quantum-mechanical means of the energy, coordinate and momentum operators and the mean square deviations from the classical trajectory are calculated with an accuracy of order 0(²). Higher approximations are obtained by the Maslov complexgrowth method.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 14–18, October, 1987.