Experimental Test of a Time-Temperature Formulation of the Uncertainty Principle Via Nanoparticle Fluorescence

No Heading The uncertainty in the measured fluorescence decay lifetimes of 30 nm particles of YAG:Cc was used to evaluate the predictions of a novel form of the Heisenberg uncertainty principle suggested by de Sabbata and Sivaram, T t h/k. The worst-case uncertainty in temperature of 4.5 °K (as derived from the relationship between temperature and lifetime) and the measured uncertainty in decay lifetime, 0.45 ns, yielded an internal estimate of T t = 2.0 × 10^–9 °K s, which is 263 times larger than /k = 7.6 × 10^–12 °K s. An external estimate of T t = 4.5 × 10¹¹ °K s (which is = 6 times /k) is derived from the independently measured uncertainty in the temperature of the sample and the experimentally determined uncertainty in lifetime. These results could be low by a factor of 5.6 if signal averaging must be taken into account. If valid, the findings are consistent with the predictions of this version of the uncertainty principle and they imply the existence of a type of thermal quantum limit.     

The strength of Einstein's equations

The strength of Einstein's empty-space field equations is computed anew and shown to be equal to the amount of initial data required for a local solution of the equations. This same amount of initial data is shown to be precisely that required for a set of 16 unknown first-order differential equations containing 10 field variables and having six identities of second order. The 10 field variables must be functions of second order in the metric coefficients. The 16 field equationsC , = 0 whereC is Weyl's conformal tensor, are shown to have the same properties as those of the unknown equations, suggesting thatC = 0 is a satisfactory local first-order formulation of Einstein's second-order empty-space field equations.     

Ward's identity in critical dynamics

We consider the relaxation of an order-parameter fluctuation of wave numberk in a system undergoing a second-order phase transition. In general, close to the critical point, wherek ^{–1 –1} (the correlation length) the relaxation rate has a linear dependence on/k of the form (k, ) = (k, 0)x(1–a/k). In analogy with the use of Ward's identity in elementary particle physics, we show that the numerical coefficienta is readily calculated by means of a mass insertion. We demonstrate, furthermore, that this initial linear drop is the main feature of the full/k dependence of the scaling functionR ^–x (k,), wherex is the dynamic critical exponent andR=(k²+ ²)^1/2 is the distance variable.     

Inertial range spectrum of a fractal model for turbulence

The present study deals with a model for 3D turbulence which is based on a fractal set ink-space (massM(k)logk, i.e.d _f=0). The energy spectrumE(_k) is calculated by a method similar to the Monte Carlo renormalization technique of Ma [1]. To apply this we decouple the set into next-neighbour pairs of energy shells.E _k is calculated iteratively by simulating the pairs. Their rough scales are forced and their fine scales are damped by an eddy viscosity ^e. We start in the viscous range where ^e is neglected. An intertial range shows up with a spectrumE(k)k^–, =1.68±0.07, and an eddy viscosity ^e(k)k ^-,=1.3±. The possible benefit for the simulation of turbulent flows is pointed out.     

Analyticity properties of eigenfunctions and scattering matrix

For potentialsV=V(x)=O(|x|^–2–) for |x|,x³ we prove that if theS-matrix of (–, –+V) has an analytic extension to a regionO in the lower half-plane, then the family of generalized eigenfunctions of –+V has an analytic extension toO such that for |Imk|<b. Consequently, the resolvent (–+V–z ²)^–1 has an analytic continuation from ⁺ to {kOImk|<b} as an operator from _b ={f=e ^–b|x| g|gL ₂(³)} to _–b . Based on this, we define for potentialsW=o(e ^–2b|x|) resonances of (–+V, –+V+W) as poles of and identify these resonances with poles of the analytically continuedS-matrix of (–+V, –+V+W).The author would like to thank the Institute for Advanced Study for its hospitality and the National Science Foundation for financial support under Grant No. DMS-8610730(1)     

Spectral duality for planar billiards

For a bounded open domain with connected complement inR ² and piecewise smooth boundary, we consider the Dirichlet Laplacian- on and the S-matrix on the complement ^c . We show that the on-shell S-matricesS _k have eigenvalues converging to 1 askk ₀ exactly when-- has an eigenvalue at energyk ₀ ² . This includes multiplicities, and proves a weak form of transparency atk=k ₀. We also show that stronger forms of transparency, such asS _{k 0} having an eigenvalue 1 are not expected to hold in general.     

Current distributions in a two-dimensional random-resistor network

The current and logarithm-of-the-current distributionsn(i) andn(ln i) on bond diluted two-dimensional random-resistor networks at the percolation threshold are studied by a modified transfer matrix method. Thek th moment (–9k8) of n(ln i) i.e., ln i&^k, is found to scale with the linear sizeL as (InL)^(k). The exponents (k) are not inconsistent with the recent theoretical prediction (k)=k, with deviations which may be attributed to severe finitesize effects. For small currents, ln n(y)–, yielding information on the threshold below which the multifractality of (i) breaks down. Our numerical results for the moments of the currents are consistent with other available results.     

The infinite coupling limit and the asymptotics of coefficients of the perturbation expansion

For some functions in quantum field theory and quantum mechanics, presented by the asymptotic expansion in coupling constant g, the method for finding the approximate asymptotics when g is proposed.Such an estimation requires the first exact terms of the perturbation theory, their high-order asymptotics and supposition on the Borel summability. This method for the Gell-Mann-Low function (g) in the (4²/3)g ₍₄₎ ⁴ model gives approximately (g) 1.8 g ².     

Analyticity of effective coupling and propagators in massless models of quantum field theory

For massless models of quantum field theory, some general theorems are proved concerning the analytic continuation of the renormalization group functions as well as the effective coupling and the propagators. Starting points are analytic properties of the effective coupling and the propagators in the momentum variablek ², which can be converted into analyticity of - and -functions in the coupling parameter . It is shown that the -function can have branch point singularities related to stationary points of the effective coupling as a function ofk ². The type of these singularities of () can be determined explicitly. Examples of possible physical interest are extremal values of the effective coupling at space-like points in the momentum variable, as well as complex conjugate stationary points close to the realk ²-axis. The latter may be related to the sudden transition between weak and strong coupling regimes in quantum chromodynamics. Finally, for the effective coupling and for the propagators, the analytic continuation in both variablesk ² and is discussed.On leave from the Max-Planck-Institut für Physik und Astrophysik, D-8000 München, Federal Republic of Germany     

Decay of the Bethe–Salpeter Kernel and Bound States for Lattice Classical Ferromagnetic Spin Systems at High Temperature

We consider lattice classical ferromagnetic spin systems at high temperature (1) with nearest neighbor interactions and even single-spin distributions (ssd). Associated with each system is an imaginary time lattice quantum field theory. It is known that there is a particle of mass m–ln in the energy-momentum spectrum. If s ⁴–3s ²²<0, where s ^k is the kth moment of the ssd, and is sufficiently small, we show that in the two-particle subspace there is no mass spectrum up to 2m. For >0 we show that the only mass spectrum in (m, 2m) is a bound state of mass m _b=2m+ln(1–)+O(), where =(+2s ²²)^–1. A bound on the decay of the kernel of a Bethe–Salpeter equation is obtained and used to prove these results.     

Humidity dependence of the atmospheric absorption coefficient in the transparency windows centered at 0.88 and 0.73 MM

A dependence is investigated of the absorption coefficient of atmospheric water vapor on absolute air humidity in the spectral intervals =11.1+12.0 and 13.2+14.3 cm^–1 by a radiospectrometer with a multipass vacuum cell. Measurements are carried out in the humidity range 3+23 g/m³ at 140 m length of the optical path.Parts of the absorption coefficient linearly and quadratically depending on have been separately defined from the experimental data. In particular, it has been obtained for the centers of transparency windows: (=0.73mm) [dB/km]=1.68 (±0.08) +0.019 (±0.004) ², (=0.88 mm) [dB/km]=0.83 (±0.06) +0.013 (±0.002) ². The given values correspond to the temperature of 306 K and the pressure of dry air of 735 Torr.A comparison is made between our results and theoretical calculations and data of other measurements.     

Surface tension,step free energy,and facets in the equilibrium crystal

Some aspects of the microscopic theory of interfaces in classical lattice systems are developed. The problem of the appearance of facets in the (Wulff) equilibrium crystal shape is discussed, together with its relation to the discontinuities of the derivatives of the surface tension (n) (with respect to the components of the surface normaln) and the role of the step free energy ^step(m) (associated with a step orthogonal tom on a rigid interface). Among the results are, in the case of the Ising model at low enough temperatures, the existence of ^step(m) in the thermodynamic limit, the expression of this quantity by means of a convergent cluster expansion, and the fact that 2^step(m) is equal to the value of the jump of the derivative / (when varies) at the point =0 [withn=(m ₁ sin ,m ₂ sin , cos )]. Finally, using this fact, it is shown that the facet shape is determined by the function ^step(m).     

Phantom scalar fields in five-dimensional Kaluza-Klein theory

The existence of a so-called "phantom" scalar field in some Riemannian spaceV ₄, i.e., a field in which the effective energy momentum tensorT ^(sf) vanishes in the five-dimensional Kaluza-Klein theory, is investigated by means of the integrability conditions for relations of the form _;;=k_,,+bg found in [6]. Phantom fields are found in homogeneous isotropic cosmological models.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 50–56, February, 1996.     

Multiple scattering in random media I. Restricted two-body additive approximation

We present a microscopic theory of the problem of finding the properties of a particle interacting with potentials located at random sites. The sites are governed by a general probability distribution. The starting point is the multiple scattering equations for the amplitude k ₁|T |k ₂ in terms of the individual scattering amplitudes k ₁|T |k ₂. We work with quantitiesA defined by k ₁|T |k ₂=k ₁|T |k ₂exp[i(k ₁–k ₂)R ]. The theory is based on a splitting of the fundamental equation forA into equations for the mean A and the fluctuationsAA . Neglect of the fluctuations yields the quasicrystalline approximation. We rearrange the equation forAA to isolate the collective part of the fluctuations. We then make the simplest microscopic truncation which is thatAA is a restricted two-body additive function of the site positions. With the contribution of the collective fluctuations, this yields results forA that are accurate to ordert ⁴.Work supported in part by the National Science Foundation under Contract No. NSF DMRWork supported in part by the National Science Foundation under Contract No. NSF DMR     

Band-gap structure of the spectrum of periodic Maxwell operators

We investigate the band-gap structure of some second-order differential operators associated with the propagation of waves in periodic two-component media. Particularly, the operator associated with the Maxwell equations with position-dependent dielectric constant (x),xR ³, is considered. The medium is assumed to consist of two components: the background, where (x) = _b , and the embedded component composed of periodically positioned disjoint cubes, where (x) = _a . We show that the spectrum of the relevant operator has gaps provided some reasonable conditions are imposed on the parameters of the medium. Particularly, we show that one can open up at least one gap in the spectrum at any preassigned point provided that the size of cubesL, the distancel=L betwen them, and the contrast = _b / _a are chosen in such a way thatL ^–2, and quantities ^-1-3/2 and ² are small enough. If these conditions are satisfied, the spectrum is located in a vicinity of widthw(^3/2)^-1 of the set {² L ^-2 k ²:kZ³}. This means, in particular, that any finite number of gaps between the elements of this discrete set can be opened simultaneously, and the corresponding bands of the spectrum can be made arbitrarily narrow. The method developed shows that if the embedded component consists of periodically positioned balls or other domains which cannot pack the space without overlapping, one should expect pseudogaps rather than real gaps in the spectrum.     

Convergent perturbation expansions for Euclidean quantum field theory

Mayer perturbation theory is designed to provide computable convergent expansions which permit calculation of Greens functions in Euclidean quantum field theory to arbitrary accuracy, including nonper-turbative contributions from large field fluctuations. Here we describe the expansions at the example of 3-dimensional ⁴-theory (in continuous space). They are not essentially more complicated than standard perturbation theory. Then ^th order term is expressed in terms ofO(n)-dimensional integrals, and is of order ^k if 4k–3n4k.Dedicated to the memory of Kurt Symanzik     

Spatio-temporal Fourier components and soft modes in thermal convection: A neutron scattering study

In a nematic liquid crystal subjected to a vertical temperature difference T in a vertical slab container, we find, using real-time and real-space neutron scattering, that the spatiotemporal pattern can be indexed on a two-dimensional lattice with selection rule (h+k)=2n. Higher index modes are sequentially activated as T increases from the convection threshold T _c . The frequency softening as TT _c and asTT ₀, the super-cooling limit of the nematic, is discussed.     

Differential equations in the spectral parameter

We determine all the potentialsV(x) for the Schrödinger equation (– _x ² +V(x))=k² such that some family of eigenfunctions satisfies a differential equation in the spectral parameterk of the formB(k, _k )ø=(x)ø. For each suchV(x) we determine the algebra of all possible operatorsB and the corresponding functions (x)This research was partially supported by NSF grant DMS 84-03232 and ONR contract NOOO14-84-C-0159     

One-dimensional random walk with self-interaction

A standard random walk on a one-dimensional integer lattice is considered where the probability ofk self-intersections of a path =(0, (1),..., (n) is proportional toe ^–k . It is proven that for <0,n ^–1/3(n) converges to a certain continuous random variable. For >0 the formulas are given for the asymptotic Westerwater velocity of a generic path and for the variance of the fluctuations about the asymptotic motion.     

More surprises in the general theory of lattice systems

I use Israel's methods to prove new theorems of ubiquitous pathology for classical and quantum lattice systems. The main result is the following: Let be any interaction and be any translation-invariant equilibrium state for (extremal or not). Then there exists a sequence { _k } of interactions converging to , having extremal (or even unique) translation-invariant equilibrium states _k , such that { _k } converges to . In certain situations the perturbations _k – can be chosen to lie in a cone of antiferromagnetic pair interactions. I discuss the connection with results of Daniëls and van Enter, and point out an application to the one-dimensional ferromagnetic Ising model with 1/r ² interaction (Thouless effect).