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Physical Review Letters

Sum-Rule Dynamics

T. A. Osborn, Richard Froese, S. F. Howes

Published Dec 22, 1980• 4 citations
Publisher

Abstract

A new class of sum rules in nonrelativistic scattering theory is obtained. By extending the concept of the Fredholm determinant to a local form defined at each space point $x$, a function with analytic properties analogous to the Fredholm determinant, but containing more information, is found. These rules assume the form of the power-moment problem and are valid in both quantum and classical mechanics.

DOI
10.1103/physrevlett.45.1987

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A new class of sum rules in nonrelativistic scattering theory is obtained. By extending the concept of the Fredholm determinant to a local form defined at each space point xxx, a function with analytic properties analogous to the Fredholm determinant, but containing more information, is found. These rules assume the form of the power-moment problem and are valid in both quantum and classical mechanics.
The connection between causality and analyticity in scattering theory is formulated in terms of Hilbertspace concepts. The usual rules of nonrelativistic quantum mechanics are assumed to hold for the "in" and "out" states of the scattering system. We show that there is a (physically verifiable) causality condition which implies that each diagonal SSS-matrix element ${S}_{\ensuremath{\alpha}\ensuremath{\alpha}}(E)$ must be the limit of an analytic function of the energy EEE, regular in ImE>0\mathrm{Im}E>0ImE>0. The implications for partial-wave amplitudes and for the forward scattering amplitude in elastic two-body collisions are discussed.
The quadratic density response function and the quadratic dynamical structure function (the Fourier transform of the equilibrium three-point density correlations) contain important information about a many body system; they are also ingredients for an improved dynamical mean field theory for strongly coupled Fermi systems. We examine the analytic properties of the density response function and establish new single frequency and double frequency moment sum rules. We relate the sum rule coefficients to the high frequency expansion of the response function. Next we invoke the quadratic fluctuation-dissipation theorem to relate these frequency moments to weighted frequency moments of the dynamical structure function. These latter reduce to straight frequency moments in the high temperature classical and zero temperature degenerate limits. \textcopyright{} 1996 The American Physical Society.
Using a classical phase space integral to approximate certain quantum mechanical operations, the classical limit of the Fredholm determinant for a general multichannel scattering system is derived. This statistical form of the classical limit of scattering is found to give exactly the same ``classical S matrix'' for the case of potential scattering as does a dynamical treatment of the classical limit. For inelastic scattering, however, the classical S matrix which results from this Fredholm determinant is only an approximation to that obtained by exact classical dynamics. Reasons for this failure of the statistical version of the classical limit for inelastic scattering are discussed.
We study finite temperature dynamical correlation functions of the magnetization operator in the one-dimensional Ising quantum field theory. Our approach is based on a finite temperature form factor series and on a Fredholm determinant representation of the correlators. While for space-like separations the Fredholm determinant can be efficiently evaluated numerically, for the time-like region it has convergence issues inherited from the form factor series. We develop a method to compute the correlation functions at time-like separations based on the analytic continuation of the space-time coordinates to complex values. Using this numerical technique, we explore all space-time and temperature regimes in both the ordered and disordered phases including short, large, and near-light-cone separations at low and high temperatures. We confirm the existing analytic predictions for the asymptotic behavior of the correlations except in the case of space-like correlations in the paramagnetic phase. For this case we derive a new closed form expression for the correlation length that has some unusual properties: it is a non-analytic function of both the space-time direction and the temperature, and its temperature dependence is non-monotonic.