The structure and dynamics of an n -particle system are described with coupled nonlinear Heisenberg commutator equations where the nonlinear terms are generated by the two-body interaction that excites the reference vacuum via particle–particle and particle–hole excitations. Nonperturbative solutions of the system are obtained with the use of the dynamic linearization approximation and cluster transformation coefficients. The dynamic linearization approximation converts the commutator chain into an eigenvalue problem. The cluster coefficients factorize the matrix elements of the ( n ) particles or particle–hole systems in terms of the matrix elements of the ( n − 1)-systems coupled to particle–particle, particle–hole and hole–hole bosons. Group properties of the particle–particle, particle–hole and hole–hole permutation groups simplify the calculation of these coefficients. The particle–particle vacuum excitations generate superconductive diagrams in the dynamics of three-quark systems. Applications of the model to fermionic and bosonic systems are discussed.
The structure and dynamics of an n -particle system are described with coupled nonlinear Heisenberg commutator equations where the nonlinear terms are generated by the two-body interaction that excites the reference vacuum via particle–particle and particle–hole excitations. Nonperturbative solutions of the system are obtained with the use of the dynamic linearization approximation and cluster transformation coefficients. The dynamic linearization approximation converts the commutator chain into an eigenvalue problem. The cluster coefficients factorize the matrix elements of the ( n ) particles or particle–hole systems in terms of the matrix elements of the ( n − 1)-systems coupled to particle–particle, particle–hole and hole–hole bosons. Group properties of the particle–particle, particle–hole and hole–hole permutation groups simplify the calculation of these coefficients. The particle–particle vacuum excitations generate superconductive diagrams in the dynamics of three-quark systems. Applications of the model to fermionic and bosonic systems are discussed.
The problem of finding the spectrum of the Fock-space Hamiltonian \ifmmode \hat{H}\else \^{H}\fi{} for a system of many fermions is analyzed. The quasiparticle formalism is employed, with ``holes'' and ``particles'' defined with respect to some N-particle determinantal wave function (the model vacuum). The basic idea behind the proposed approach is to perform a similarity transformation of Hamiltonian \ifmmode \hat{H}\else \^{H}\fi{}, such that the resulting effective Hamiltonian \ifmmode \hat{G}\else \^{G}\fi{} is, unlike \ifmmode \hat{H}\else \^{H}\fi{}, a quasiparticle-number--conserving operator. It is shown that eigenvalues of \ifmmode \hat{G}\else \^{G}\fi{}, corresponding to small numbers of quasiparticles (0,1,2) can be easily calculated. This is equivalent to finding eigenvalues of \ifmmode \hat{H}\else \^{H}\fi{} for certain states of N, N\ifmmode\pm\else\textpm\fi{}1, and N\ifmmode\pm\else\textpm\fi{}2 particles. The construction of the operator transforming \ifmmode \hat{H}\else \^{H}\fi{} into \ifmmode \hat{G}\else \^{G}\fi{} (the wave operator) stems from an analysis of the structure of the algebra of linear operators acting in a (finite-dimensional) Fock space. The exponential Ansatz for the wave operator is used, resulting in a generalization of the coupled-cluster (CC) method of Coester [Nucl. Phys. 7, 421 (1958)]. The generalized CC equations determining the wave operator, and equations determining the effective Hamiltonian \ifmmode \hat{G}\else \^{G}\fi{}, are presented in a diagrammatic form. An effort has been made to obtain a concise notation for expressing these equations in an algebraic form. Approximation schemes, necessary for practical applications of the proposed method, are also studied.
In this book, we encounter the hierarchy problem when we apply the equation-of-motion technique to analyze the quantum dynamics of the coupled light–matter system. To obtain systematic approximations, we use the so-called cluster-expansion method where many-body quantities are systematically grouped into cluster classes based on how important they are to the overall quantum dynamics. With increasing complexity, the clusters contain independent single particles (singlets), interacting pairs (doublets), three-particle terms (triplets), coupled four-particle contributions (quadruplets), as well as higher-order correlations. In this context, the N -particle concept is somewhat formal because it refers to generic N -particle expectation values 〈 N 〉, which may consist of an arbitrary mixture of carrier, photon, and phonon operators, as discussed in Section 14.2. In order to truncate the hierarchy problem at a given level, 〈 N 〉 is approximated through a functional structure that includes all clusters up to the predetermined level while all remaining clusters with a higher rank are omitted. It is natural that the corresponding approximations can be systematically improved by increasing the number of clusters included. To the best of our knowledge, the idea of coupled-clusters approaches was first formulated by Fritz Coester and Hermann Kümmel in the 1950s to describe nuclear many-body phenomena. The approach was then modified for the needs of quantum chemistry by Jiri Cizek 1966 to deal with many-body phenomena in atoms and molecules. Currently, it is one of the most accurate methods to compute molecular eigenstates.
For n particles moving in three-dimensional space, each coordinate should carry an index i = 1, 2,3 denoting the vector component and an index l = 1, 2,..., n denoting the particle number. Among the linear point transformations acting on these coordinates there are two commuting subgroups of transformations. The elements of the first one are given by product matrices M 3 × I n where the 3 × 3 matrix M 3 acts on the vector components. The elements of the second one are of the form I 3 × M n where the n × n matrix M n acts on the vectors for the different particles. To the first subgroup there belong the rotations giving rise to the rotation group SO (3, IR). The second subgroup contains permutations and transformations to various types of relative coordinates. In the present section we shall deal with this second subgroup and hence shall drop the reference to the three vector components. Moreover we shall need at present only orthogonal n × n transformation matrices.
Liquids and solids, atoms and molecules, nuclei—all these clearly are interacting many-body systems. Even a nucleon may be regarded as a manyparticle system, not just because it is now known to consist of three quarks interacting via gluons, but because of the possibility in quantum field theory of virtual excitation of many particles from the vacuum. in the table page 56 we list some of the many-particle systems we encounter in the physical world at length scales that range from a few centimeters to a few fermis. Many of these systems exhibit phenomena—superconductivity in solids and fission in nuclei, for example—whose understanding does not follow immediately from knowledge of the constituents of the system and the interactions among the constituents, but requires new concepts and ideas. Many-body physics is the branch of theoretical physics that studies the new phenomena or “emergent properties” that arise from interactions among “elementary” constituents of a many-particle system and provides means and methods for carrying out precise calculations of such characteristic properties of these systems as may be compared with experimental results to verify hypotheses about the nature of the constituents and their interactions.
The nonperturbative Hamiltonian eigenvalue problem for bound states of a quantum field theory is formulated in terms of Dirac's light-front coordinates and then approximated by the exponential operator technique of the many-body coupled-cluster method. This approximation eliminates any need for the usual approximation of Fock-space truncation. Instead, the exponentiated operator is truncated, and the terms retained are determined by a set of nonlinear integral equations. These equations are solved simultaneously with an effective eigenvalue problem in the valence sector, where the number of constituents is small. Matrix elements can be calculated, with extensions of techniques from many-body coupled-cluster theory, to obtain form factors and other observables.
Introduction In many cases in physics, one has to deal simultaneously with collective and single-particle excitations of the system. The collective excitations are usually bosonic in nature while the single-particle excitations are often fermionic. One is therefore led to consider a system which includes bosons and fermions. In this book we discuss applications of a general algebraic theory of mixed Bose- Fermi systems to atomic nuclei. The collective degrees of freedom here can be described in terms of a system of interacting bosons as discussed in a previous book (Iachello and Arima, 1987), henceforth referred to as Volume 1. The single-particle degrees of freedom represent the motion of individual nucleons in the average nuclear field. They are described in terms of a system of interacting fermions. The coupling of fermions and bosons leads to the interacting boson-fermion model which has been used extensively in recent years to discuss the properties of nuclei with an odd number of nucleons. The interacting boson-fermion model was introduced by Arima and one of us in 1975 (Arima and Iachello, 1975). It was subsequently expanded by Iachello and Scholten (1979) and cast into a form more readily amenable to calculations. As in the corresponding case of even-mass systems, the algebra of creation and annihilation operators can be realized in several ways. One of these is the Hoistein-Primakoff realization which leads to a slightly different version of the interacting boson-fermion model called the truncated quadrupole phonon-fermion model (Paar, 1980; Paar and Brant, 1981), based on the boson realization introduced by Janssen, Jolos and Donau in 1974 and discussed in Sect. 1.4.6 of Volume 1.