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Mathematical Proceedings of the Cambridge Philosophical Society

Non-linear interactions, causality and the R-matrix

Edwin Albert Power, I. Saavedra

Published Oct 1, 1964• 0 citations
Publisher

Abstract

Abstract The problem of causality in non-relativistic scattering where a bounded region of non-linear interaction occurs is re-examined in the context of a paper by Ebel. In the model considered it is shown that the acausal behaviour cannot be accounted for in the straightforward manner that is possible for an energy-dependent but linear potential which gives rise to a similar R-matrix.

DOI
10.1017/s0305004100038408

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Abstract The problem of causality in non-relativistic scattering where a bounded region of non-linear interaction occurs is re-examined in the context of a paper by Ebel. In the model considered it is shown that the acausal behaviour cannot be accounted for in the straightforward manner that is possible for an energy-dependent but linear potential which gives rise to a similar R-matrix.
We consider exact time-dependent analytic solutions to the Schr\"odinger equation for tunneling in one dimension with cutoff wave initial conditions at t=0.t=0.t=0. We obtain that as soon as $t\ensuremath{\ne}0$ the transmitted probability density at any arbitrary distance rises instantaneously with time in a linear manner. Using a simple model we find that the above nonlocal effect of the time-dependent solution is suppressed by consideration of low-energy relativistic effects. Hence at a distance x0{x}_{0}x0​ from the potential the probability density rises after a time t0=x0/c{t}_{0}{=x}_{0}/ct0​=x0​/c restoring Einstein causality. This implies that the tunneling time of a particle can never be zero.
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.
When multiple quantum particles of different species - or "flavors" - are allowed to interact via pairwise interactions, they may form bound states. If these interactions can be modeled by an attractive contact potential, the strength of the attractions will depend on a single parameter: the coupling g. In 3D, when two or more flavors are present, it is possible for the system to form a bound state if the coupling is above a certain finite threshold g[c]. In contrast, a one-flavor system is just a free particle and can only exist in scattering states (i.e., g[c] = ∞). Although non-integer flavor numbers N[f] have no obvious physical meaning, they are mathematically valid and can be used to examine the analytic relationship between N[f] and g[c]. We used computational methods to vary N[f] in the continuum interval (1,2] in an effort to determine the manner in which g[c] → ∞ as N[f] → 1+. We used Fortran and Python to calculate many-body quantum systems with parameters N[f] and g. Specifically, we used the projection Monte Carlo method to calculate the ground-state energies E[0] of these systems. Plotting E[0] vs. g, we obtained the critical coupling g[c] at which bound state formation occurred for each N[f] sampled. Determining the manner in which g[c] diverges sees applications in theoretical fields such as QCD - where similar problems related to flavor number arise in the study of quarks - as well as in the experimental study of ultracold atoms, where the quantum simulation of increasingly arbitrary systems is becoming a reality.
A new formulation of the causality condition, in terms of the total wave function at a finite distance from the scatterer, is given. This condition is employed to derive all known analytic properties of the scattering amplitude in both variables (energy and scattering angle or momentum transfer) for a scatterer of finite radius. It is shown that all the assumptions are valid in the particular case of scattering by a local potential. The relations between causality, Wigner's inequality, and completeness are discussed.
The macroscopic linear light scattering theory is extended to the semiclassical level starting with a general discussion of spatially nonlocal scattering. An inte- gral relation for the gauge invariant transverse vector potential is given, stressing similarities and differences between the so-called photon-eye and electron-eye picture. The linear constitutive relations and the causal response tensor are key elements in the semiclassical light scattering theory. Elastic scattering processes are discussed from first principles by making use of the transverse Feynman propagator and the causal response tensor. In scattering from mesoscopic me- dia, e.g, the causal response tensor is a two-point function in direct space. In the spatial dispersion subfield this tensor becomes a one-point function, depending only on the vectorial distance between given source and observation points. In the spatial dispersion approximation a simple relation can be obtained between the selfconsistent and causal response tensors in the wave-vector-frequency do- main. Next, we turn to the scattering matrix concept in quantum physics. The treatment describes the transition from an initial to a final object-field tensor product. These states are connected by the S -matrix operator. A general ex- pression for this is obtained in the Interaction Picture. The reduced S -matrix which relates in a direct manner to the scattering between selected initial and final photon states is introduced. The chapter is finished by the establishment and a discussion of the weakly relativistic interaction Hamiltonian, a key ele- ment in the S -matrix operator.