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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Many-body scattering processes in a one-dimensional boson system

H. B. Thacker

Published Dec 15, 1976• 27 citations
Publisher

Abstract

The one-dimensional many-body problem described by the nonrelativistic ${|\ensuremath{\varphi}|}^{4}$ theory of a complex scalar boson field (also known as the $\ensuremath{\delta}$-function model) is studied. By induction, it is shown that the set of $n\mathrm{th}$-order perturbation theory graphs for the $N$-particle in-state wave function can be combined and reduced to an equivalent set of factorized graphs in which the particle lines are numbered according to their ordering in momentum space at time $t=\ensuremath{-}\ensuremath{\infty}$. By carrying out loop integrations, each factorized graph is reduced to one of a finite number of skeleton graphs multiplied by a dressing function. The dressing functions are related to the multiparticle phase shifts which appear in Bethe's form of the $N$-body wave function. The skeleton graphs are shown to be associated with the ordering of particles in configuration space which characterizes Bethe's hypothesis. The analogy of a classical system of billiard balls is found to be helpful in interpreting the form of Bethe's hypothesis and the physical significance of the skeleton graphs. The use of factorized graphs in the scattering theory of statistical mechanics is demonstrated by a graphical calculation of the second virial coefficient.

DOI
10.1103/physrevd.14.3508

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The quantum field theory in one space + one time dimension described by the Lagrangian $\mathfrak{L} = (\frac{i}{2}){\ensuremath{\varphi}}^{*}{\stackrel{}{\ensuremath{\partial}}}_{0}\ensuremath{\varphi}\ensuremath{-}({\ensuremath{\partial}}_{1}{\ensuremath{\varphi}}^{*})({\ensuremath{\partial}}_{1}\ensuremath{\varphi})\ensuremath{-}c{|\ensuremath{\varphi}|}^{4}$ is studied for systems with finite temperature and particle density. Using momentum-space techniques previously developed, a graphical procedure is obtained for calculating inner products of many-particle scattering state wave functions. The unitarity of the wave operator $U(0,\ensuremath{-}\ensuremath{\infty})$ is demonstrated as a pattern of graphical cancellations. An operator formulation of statistical mechanics is derived in which partition functions are given in terms of matrix elements having the form of diagonal (forward) inner products. The importance of noncommutativity of the forward limit and the $i\ensuremath{\epsilon}\ensuremath{\rightarrow}0$ limit is noted and traced to the presence of forward singular graphs in the inner product. Combining this observation with wave-operator unitarity, we obtain a graphical recipe for calculating $N$-body partition functions. The thermodynamics first described by Yang and Yang is obtained by summation of the fugacity series for the grand partition function.

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arXiv:0811.3853•Physical Review A•Feb 4, 2009•43 citations
The one-dimensional many-body problem described by the nonrelativistic ∣\ensuremathφ∣4{|\ensuremath{\varphi}|}^{4}∣\ensuremathφ∣4 theory of a complex scalar boson field (also known as the \ensuremathδ\ensuremath{\delta}\ensuremathδ-function model) is studied. By induction, it is shown that the set of nthn\mathrm{th}nth-order perturbation theory graphs for the NNN-particle in-state wave function can be combined and reduced to an equivalent set of factorized graphs in which the particle lines are numbered according to their ordering in momentum space at time t=\ensuremath−\ensuremath∞t=\ensuremath{-}\ensuremath{\infty}t=\ensuremath−\ensuremath∞. By carrying out loop integrations, each factorized graph is reduced to one of a finite number of skeleton graphs multiplied by a dressing function. The dressing functions are related to the multiparticle phase shifts which appear in Bethe's form of the NNN-body wave function. The skeleton graphs are shown to be associated with the ordering of particles in configuration space which characterizes Bethe's hypothesis. The analogy of a classical system of billiard balls is found to be helpful in interpreting the form of Bethe's hypothesis and the physical significance of the skeleton graphs. The use of factorized graphs in the scattering theory of statistical mechanics is demonstrated by a graphical calculation of the second virial coefficient.
The quantum field theory in one space + one time dimension described by the Lagrangian $\mathfrak{L} = (\frac{i}{2}){\ensuremath{\varphi}}^{*}{\stackrel{}{\ensuremath{\partial}}}_{0}\ensuremath{\varphi}\ensuremath{-}({\ensuremath{\partial}}_{1}{\ensuremath{\varphi}}^{*})({\ensuremath{\partial}}_{1}\ensuremath{\varphi})\ensuremath{-}c{|\ensuremath{\varphi}|}^{4}$ is studied for systems with finite temperature and particle density. Using momentum-space techniques previously developed, a graphical procedure is obtained for calculating inner products of many-particle scattering state wave functions. The unitarity of the wave operator $U(0,\ensuremath{-}\ensuremath{\infty})$ is demonstrated as a pattern of graphical cancellations. An operator formulation of statistical mechanics is derived in which partition functions are given in terms of matrix elements having the form of diagonal (forward) inner products. The importance of noncommutativity of the forward limit and the $i\ensuremath{\epsilon}\ensuremath{\rightarrow}0$ limit is noted and traced to the presence of forward singular graphs in the inner product. Combining this observation with wave-operator unitarity, we obtain a graphical recipe for calculating NNN-body partition functions. The thermodynamics first described by Yang and Yang is obtained by summation of the fugacity series for the grand partition function.
This chapter explores multi-body scattering in quantum integrable models, emphasizing the emergence of Factorized Scattering Theory. The presence of an extensive set of conservation laws ensures the non-diffractive nature of scattering, leading to a distinctive form of the asymptotic wave function – known as the Bethe wave function – as a linear superposition of plane waves. Elementary scattering events, such as reflection and transmission, are described by Yang’s scattering operators and the two-body S -matrix. The multi-body S -matrix is then expressed as an ordered product of two-body S -matrices, with order-independence guaranteed by the Yang–Baxter equation. To illustrate these ideas, the Lieb–Liniger model for distinguishable particles is introduced. The chapter also examines general symmetry constraints on the Bethe wave function imposed by quantum statistics.
Identical Particles Second Quantization Independent-Particle Model for Fermions in Finite Systems Two-Particle States and Interactions Noninteracting Bosons and Fermions Propagators in One-Particle Quantum Mechanics Single-Particle Propagator in Many-Particle Systems Perturbation Expansion of the Single-Particle Propagator Dyson Equation and Self-consistent Green's Functions Mean-Field or Hartree-Fock Approximation Beyond the Mean-Field Approximation Interacting Boson Systems Excited States in Finite Systems Excited States in Infinite Systems Excited States in N 2 Systems and In-Medium Scattering Dynamical Treatment of the Self-energy in Infinite Systems Dynamical Treatment of the Self-energy in Finite Systems Bogoliubov Perturbation Expansion for the Bose Gas Boson Perturbation Theory Applied to Physical Systems In-Medium Interaction and Scattering of Dressed Particles Conserving Approximations and Excited States Pairing Phenomena Propagator Framework for Phenomenological Description of Many-Body Systems Finite Temperature Formalism and Applications.
A method of solution of a number of quantum field theory and statistical mechanics models in two space-time dimensions is presented in this Part. This method was first suggested by H. Bethe in 1931 and is traditionally called the Bethe Ansatz. Later on the method was developed by Hulthen, Yang and Yang, Lieb, Sutherland, Baxter, Gaudin and others (see, and). We begin the presentation with the coordinate Bethe Ansatz not only due to historical reasons but also because of its simplicity and clarity. The multi-particle scattering matrix appears to be equal to the product of two-particle matrices for integrable models. This property of two-particle reducibility is of primary importance when constructing the Bethe wave function. The important feature of integrable models is that there is no mass-shell multiple particle production. This property is closely connected to the existence of an infinite number of conservation laws in such models; this will be clear from Part II. Four main models, namely the one-dimensional Bose gas, the Heisenberg magnet, the massive Thirring model and the Hubbard model, are considered in Part I. Eigenfunctions of the Hamiltonians of these models are constructed. Imposing periodic boundary conditions leads to a system of equations for the permitted values of momenta. These are known as the Bethe equations. This system can also be derived from a certain variational principle, the corresponding action being called the Yang-Yang action.
We derive a multiconfigurational time-dependent Hartree theory for systems with particle conversion. In such systems particles of one kind can convert to another kind and the total number of particles varies in time. The theory thus extends the scope of the available and successful multiconfigurational time-dependent Hartree methods---which were solely formulated for and applied to systems with a fixed number of particles---to a broader class of physical systems and problems. As a guiding example we treat explicitly a system where bosonic atoms can combine to form bosonic molecules and vice versa. In the theory for particle conversion, the time-dependent many-particle wave function is written as a sum of configurations made of a different number of particles, and assembled from sets of atomic and molecular orbitals. Both the expansion coefficients and the orbitals forming the configurations are time-dependent quantities that are fully determined according to the Dirac-Frenkel time-dependent variational principle. By employing the Lagrangian formulation of the Dirac-Frenkel variational principle we arrive at two sets of coupled equations of motion, one for the atomic and molecular orbitals and one for the expansion coefficients. The first set is comprised of first-order differential equations in time and nonlinear integrodifferential equations in position space, whereas the second set consists of first-order differential equations with coefficients forming a time-dependent Hermitian matrix. Particular attention is paid to the reduced density matrices of the many-particle wave function that appear in the theory and enter the equations of motion. There are two kinds of reduced density matrices: particle-conserving reduced density matrices which directly only couple configurations with the same number of atoms and molecules, and particle nonconserving reduced density matrices which couple configurations with a different number of atoms and molecules. Closed-form and compact equations of motion are derived for contact as well as general two-body interactions, and their properties are analyzed and discussed.