Heuristic Discussion Research Articles

Path integrals, disordered systems, and some ensuing mathematical questions

The path integral describing a particle in a randomly distributed potential is examined. A heuristic discussion is given, which should elucidate the qualitative features as well as the mathematical problems of density-of-states calculations.

Existence of General Equilibrium in Multi-Regional Economies with Public Goods

IN THIS PAPER WE DEMONSTRATE the existence of a general equilibrium in an economy in which the consumers are partitioned into a number of governmental jurisdictions, which provide public goods and collect taxes. The defining nature of a public good is that everyone in the jurisdiction which provides it consumes the total produced. The equilibrium concept used in this essay is due to Ellickson [3], who generalizes a definition due to Foley [4]. The essential features of the equilibrium are that producers maximize profits, consumers maximize utility subject to their after-tax budget constraint, every jurisdiction has a balanced budget, and for no jurisdiction is there an alternative public sector proposal with taxes to pay for it which is unanimously preferred by its members. We shall be dealing with the case where taxes are proportional to wealth. Ellickson offers no existence proof for such a case. Foley provides an existence proof for the special case of one jurisdiction. The central focus of this paper is the presence of regionality (i.e., the existence of more than one region). We provide an existence proof for the multi-jurisdiction case which is very different from Foley's proof for the single jurisdiction case. The techniques used in the proof would appear to have wide applicability to economies involving externalities. In Section 2 we introduce some notation and assumptions. Section 3 discusses the equilibrium concept, Section 4 gives a heuristic discussion of the strategy behind the existence proof, and Section 5 contains the formal existence proof.

Relativistic electron beam trajectories

A heuristic discussion is given of the influence of the self-magnetic field on the trajectory of a high-current relativistic electron beam in different external magnetic fields.

APPLICATIONS OF SPECTRAL ANALYSIS: SOME FURTHER CONSIDERATIONS*

AbstractIncreasing attention has focused in recent years on the use of spectral analytic techniques for the analysis of economic time series data. Sovereign, Nolan and Mandel [26] have recently outlined the basic concepts underlying such analysis and presented examples relating to the management science area including inventory demand, transportation simulation and stock market price behavior. The purpose of this paper is to supplement and extend their tutorial with a discussion of some of the problems involved with the application and interpretation of spectral statistics, emphasizing the problems of stationarity and detrending and data windows. Additionally, the scope of the examples is broadened to include applications in the area of monetary economics.The first part of this paper presents a brief discussion of the objectives of spectral analysis. Since this paper attempts to extend the work of Sovereign, Nolan and Mandel, only a heuristic discussion of spectral techniques of analysis is presented. This is followed by a more detailed description of the problem of stationarity and the effects of detrending upon the interpretation of spectral results. Then we consider the problem of estimating a continuous curve with finite information and the implications of both of these problems by presenting several applied examples in the area of monetary economics. The concluding section of this paper briefly summarizes some additional considerations in harmonic analysis of economic time series data and notes some potential directions for future research.

Changes in Laser Speckle as a Tool for Vibration Analysis

An object that is vibrating in a resonant mode and has a moderately rough surface illuminated by dispersed laser light shows patterns analogous to Chladni (strewn sand) patterns. The laser speckle patterns differ from the rest of the surface by appearing as regions in which the laser speckle has more dark areas interspersed with brighter light areas. These patterns appear at antinodes and are thus complementary to most Chladni patterns. Some examples of this observation along with Chladni patterns at the same resonant frequency are given. A heuristic theoretical discussion is also presented. [This work was performed under the auspices of the U. S. Atomic Energy Commission.]

Relativistic Systems of Particles in Gravitational Interaction

An elementary heuristic discussion of relativistic systems of particles in mutual gravitational interaction is given. Both in the boson and fermion cases, the main features of rigorous treatments are very simply recovered. A conceptual dilemma touching the use of various coordinate systems in such considerations is clarified. Finally, it is shown that in any realistic boson system, the relativistic effects of gravitation are probably masked by the other fundamental interactions, in contradistinction with fermion systems.

Will the Resonances on the Leading Trajectories Become Stable to Strong Decay at High Spin?

A heuristic discussion is given of the decay modes of high-spin particles lying on a linearly rising trajectory. Among other things, it is concluded that for particles with masses \ensuremath{\gtrsim}3500 MeV ($J\ensuremath{\gtrsim}12$), the total decay widths decrease with $J$ and go to zero approximately like ${(J\mathrm{In}J)}^{\ensuremath{-}0.28{J}^{\frac{1}{2}}}$ as $J\ensuremath{\rightarrow}\ensuremath{\infty}$ ($0.28=2{m}_{\ensuremath{\pi}}{b}^{\frac{1}{2}}$, where $b\ensuremath{\simeq}1$ ${\mathrm{GeV}}^{\ensuremath{-}1/2}$ is asymptotic slope of the rising trajectory).

A Development of Tukey's Quick Test of Location

Abstract A simple test for recognising differences in mean is described. It has its origins in a test proposed by Tukey in 1959, and effectively ignores that piece of data which most seriously detracts from the significance of Tukey's statistic. Heuristic discussions and a reference to empirical evidence are given concerning the resultant improvement in power. The test is quick to execute both in formation of the statistic and in deciding on its significance, since in a wide class of situations the 5%, 1% and .1% critical values can be taken as 10, 13 and 17 respectively. Tables of critical values for small samples are however included, as are graphs based on asymptotic considerations.

A Development of Tukey's Quick Test of Location

Abstract A simple test for recognising differences in mean is described. It has its origins in a test proposed by Tukey in 1959, and effectively ignores that piece of data which most seriously detracts from the significance of Tukey's statistic. Heuristic discussions and a reference to empirical evidence are given concerning the resultant improvement in power. The test is quick to execute both in formation of the statistic and in deciding on its significance, since in a wide class of situations the 5%, 1% and .1% critical values can be taken as 10, 13 and 17 respectively. Tables of critical values for small samples are however included, as are graphs based on asymptotic considerations.

Local States

A heuristic discussion is given of the preparation of states in finite space-time regions. Some axioms concerning such states are derived from this discussion. It is shown that states requiring for their preparation a selection of events according to the outcome of a measurement will not be ``strictly local.'' Nonselective states will, however, be ``strictly local.'' The mathematical structure of such states is investigated.

Bound States and Bootstraps in Field Theory

We present a general, model-independent theory of composite particles. Starting from a general local Lagrangian containing an elementary particle, we show how this particle becomes composite (in a precisely defined sense) as its wave-function renormalization constant tends to zero. As this limit is taken, the usual dynamical type of field equation changes its form. We show that the Green's functions of the theory still possess the usual physical interpretation in the limit; in particular, for renormalizable theories, if a certain coupling constant does not tend to zero, the "composite" propagator possesses a pole at the composite mass and with nonzero residue. We show that the operator form of the Green's-function equations describing the composite particle can be manipulated by natural approximations to explain a wide variety of known modeldependent results on composite particles, and in particular to obtain all the usual types of bootstrap equations. We present a preliminary classification of the operator equations, showing that many of them cannot possess physically meaningful solutions. Our results agree qualitatively with earlier, heuristic discussions. Finally we make some remarks about the possibility of "bootstrapped symmetry schemes."

Solution of the Bloch—Nordsieck Model

An operator solution is constructed for the Bloch—Nordsieck model, in the formulation of Bogolubov and Shirkov. The following are some of the features of the solution. (a) The method of solution depends on the following assumption: The renormalized electron current is identical with the free-field current. The resulting solution is consistent with this assumption. (b) The infraparticle structure of the electron is analogous to the infraparticle structure in Schroer's model. (c) The ordering of fields, i.e., the definition of their products, is considered in detail. A modification of Wick ordering (called F-ordering in the text) appears particularly satisfactory. (d) The solution allows a heuristic discussion of the infrared representations of the electromagnetic field. These depend, apparently, on the chosen velocity of the electron.

On the Definition of Scattering in Relativistic Quantum Field Theory

A relativistic interacting theory for neutral bosons and in which there can be no stable bound states is studied. The theory is obtained by embedding two-particle relativistic quantum mechanics in quantum field theory by means of a procedure recently given by Sudarshan. This results in a theory in which there is scattering in the two-particle channel only. An « interacting » field operator ψx is defined for the theory. It is then found that under the usual formulation of axiomatic quantum field theory ψx must be regarded as a « local free-field operator ». However, there is inherent in ψx scattering information. A new asymptotic condition for field operators is then proposed, whereby this scattering information may be extracted. A heuristic discussion of the physical significance of the asymptotic condition is also given.

Stroboscopic-Perturbation Procedure for Treating a Class of Nonlinear Wave Equations

A new perturbation procedure is presented for treating initial-value problems of nonlinear hyperbolic partial differential equations. The characteristic variables of the partial differential equation and the functions of these variables are expanded in powers of ε, and the formal solution is uniformly valid over time intervals O(1/ε). The uniform first-order solution is evaluated for the equationytt=(1+εyx)yxx,subject to the standing-wave initial conditions: y(x, 0) = a sin πx, yt(x, 0) = 0. This equation is the lowest continuum limit of an equation for which numerical computations are available. The uniform zero-order solution breaks down after a time tB = 4/εaπ. A detailed study of the solution is made in the vicinity of the breakdown region of the (x, t) plane, and it demonstrates that the formal solution for yx and yt goes from a single-valued to a triple-valued function while yxx and ytt become singular. To compare the solutions with the available numerical computations, the yx and yt waveforms are decomposed into spatial Fourier modes. The effect of breakdown is manifest in the modal amplitudes ∝ Jn(nT)/nT. The modal amplitudes change their asymptotic behavior, from exponentially decreasing as n → ∞, to algebraically decreasing when t goes from smaller to larger than tB. In the time interval up to breakdown, t < tB, the modal energies are in excellent agreement with the modal energies of the numerical computations, whereas for t > tB they diverge. For t < tB, the total energy calculated from the uniform zero-order solution is conserved and equal to the initial value, EC(0)=2a2π2 ∑ n=1∞[Jn(nT)(nT)]2=12a2π2=EC |t=0,t≤tB. Thus, the lowest-continuum-limit equations describe the dynamics of a discrete model for a finite time. A heuristic discussion is given which suggests that the time of description can be extended beyond tB by including higher spatial derivatives in the continuum model.

Undecidability of Macroscopically Distinguishable States in Quantum Field Theory

A heuristic discussion is presented regarding quantum field theory as a synthesis of the complementary theories of classical mechanics and quantum mechanics. If the states of quantum field theory are partitioned in equivalence classes accordingly as their occupation numbers differ in a finite or an infinite number of places, it is suggested that we define states to be macroscopically distinguishable if they belong to different equivalence classes. It is then proven that there is, in general, no effective procedure for determining whether or not two arbitrarily given states of a quantum system having an infinite number of degrees of freedom are macroscopically distinguishable.

General Considerations in the Analysis of Spectra

The object of this paper is to present a simplified account of the motivation behind the spectral analysis of time-series and to give a heuristic discussion of the statistical problems which arise. It is directed mainly at statisticians with little experience of the theory and applications of spectral analysis. It is of necessity short and omits a great deal of important detail. A much lengthier exposition for non-statisticians is in the course of preparation.

Some Remarks on Oscillators Driven by a Random Force

A brief and partly heuristic discussion of the problem of spacing of zeros of the displacement of an oscillator driven by a random force is given. Although no explicit results have been obtained, the problem is reduced to a study of a certain integral equation and equivalently, of a partial differential equation with a peculiar boundary condition. The boundary condition has previously been conjectured by Uhlenbeck and Wang.