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arXiv:1601.07169 · WORLD SCIENTIFIC (EUROPE) eBooks

Bound States as Fundamental Quantum Structures

Ruth E. Kastner

Published Nov 14, 2016• 1 citation
arXivHTMLPDF

Abstract

Bound states arise in many interactions among elementary field states, and are represented by poles in the scattering matrix. The emergent nature of bound states suggests that they play a perhaps under-appreciated role in specifying the ontologically relevant degrees of freedom pertaining to composite systems. The basics of this ontology are presented, and it is discussed in light of an example of Arsenijevi\' c et al.

DOI
10.1142/9781786341419_0013

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Bound states arise in many interactions among elementary field states, and are represented by poles in the scattering matrix. The emergent nature of bound states suggests that they play a perhaps under-appreciated role in specifying the ontologically relevant degrees of freedom pertaining to composite systems. The basics of this ontology are presented, and it is discussed in light of an example of Arsenijevi' c et al.
Chapters 5–7 have shown us why and how states play such an essential role in quantum theory. States are distinguished through their labeling, which consists of a set of quantum numbers specific to the state. In general, each degree of freedom gives rise to a quantum number, where, in the present context, the phrase “degrees of freedom” refers to the number of spatial coordinates characterizing a given quantum system. The 1-D systems employed in the two chapters illustrating the postulates were thus single-degree-of-freedom systems: all the eigenstates encountered there were labeled by a single quantum number ( n for bound states and k for continuum states) related to the energy of the system. Having studied systems with a single degree of freedom, it might seem that our next step would be to consider systems with more than one degree of freedom, for example, two 1-D particles or a particle in 3-D subject to a potential, etc. Among the goals of such studies would be the assigning of labels to the eigenstates. If the assignment of quantum numbers were based solely on spatial degrees of freedom, then the preceding endeavor would indeed be a way to proceed with our development of quantum theory. However, spatial degrees of freedom are not the only source of the labels which specify eigenstates. An additional source is the sets of eigenvalues of the operators that commute with Ĥ and with each other. Such operators are the main subject of this chapter.
Abstract We proceed in this chapter to discuss scattering theory and related aspects of the bound-state problem in quaternionic quantum mechanics. We begin with a detailed discussion of a simple one-dimensional example, the delta function potential model. From this model will emerge the result (Adler, 1988) that will be the principal theme of this and the succeeding two chapters: with the standard ray representative choice of Sec. 4.2, for short-range potentials the asymptotic scattering states and the S-matrix in quaternionic quantum mechanics are always complex ℂ(1, i).
Abstract The generalized hydrodynamic (GHD) approach has been extremely successful in describing the out-of-equilibrium properties of a great variety of integrable many-body quantum systems. It naturally extracts the large-scale dynamical degrees of freedom of the system, and is thus a particularly good probe for emergent phenomena. One such phenomenon is the presence of unstable particles, traditionally seen via special analytic structures of the scattering matrix. Because of their finite lifetime and energy threshold, these are especially hard to study. In this paper we apply the GHD approach to a model possessing both unstable excitations and quantum integrability. The largest family of relativistic integrable quantum field theories known to have these features are the homogeneous sine-Gordon models. We consider the simplest non-trivial example of such theories and investigate the effect of an unstable excitation on various physical quantities, both at equilibrium and in the non-equilibrium state arising from the partitioning protocol. The hydrodynamic approach sheds new light onto the physics of the unstable particle, going much beyond its definition via the analytic structure of the scattering matrix, and clarifies its effects both on the equilibrium and out-of-equilibrium properties of the theory. Crucially, within this dynamical perspective, we identify unstable particles as finitely-lived bound states of co-propagating stable particles of different types, and observe how stable populations of unstable particles emerge in large-temperature thermal baths.
Time-independent scattering methods are widely employed to analyze transport in non-Hermitian systems. Their application, however, rests on a critical yet often overlooked assumption: that an incident wave is a pure superposition of scattering states. In practice, any physically realistic, spatially localized wave packet will generally have a nonzero overlap with the system's bound states, thereby violating this premise. While this violation is inconsequential in Hermitian systems, it can invalidate the conventional scattering picture in their non-Hermitian counterparts. The underlying cause is the emergence of time-growing bound states, which manifest as poles of the scattering matrix (SSS matrix) in the first quadrant of the complex wave-number plane. Any initial overlap with these states becomes exponentially amplified, eventually dominating the long-time dynamics. Consequently, the actual evolution of a wave packet diverges dramatically from the conventional scattering picture, rendering the transmission and reflection coefficients derived from time-independent scattering methods unphysical. Using tight-binding models with non-Hermiticity introduced via imaginary on-site potentials or asymmetric hopping, we demonstrate that parameter regimes supporting such growing states are common. We therefore conclude that an analysis of SSS-matrix poles is an indispensable step to confirm the physical relevance of time-independent scattering calculations in non-Hermitian systems.