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Journal of Mathematical Physics

On the propagation of the operator average in truncated space

Masao Nomura

Published May 1, 1985• 12 citations
Publisher

Abstract

The propagation of the operator average is described in truncated space with some quantum number(s) being fixed. It is first shown that the propagation coefficient satisfies an analog of the Chapman–Kolmogorov equation. Next, particle-hole symmetry is incorporated into the propagation of the operator average. It yields an expression that facilitates evaluation of many-body trace. Fermion and boson systems are treated alike.

DOI
10.1063/1.526557

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The propagation of the operator average is described in truncated space with some quantum number(s) being fixed. It is first shown that the propagation coefficient satisfies an analog of the Chapman–Kolmogorov equation. Next, particle-hole symmetry is incorporated into the propagation of the operator average. It yields an expression that facilitates evaluation of many-body trace. Fermion and boson systems are treated alike.
The particle-hole symmetry (equivalence) of the full shell-model Hilbert space is straightforward and routinely used in practical calculations. In this work I show that this symmetry is preserved in the subspace truncated up to a certain generalized seniority and give the explicit transformation between the states in the two types (particle and hole) of representations. Based on the results, I study particle-hole symmetry in popular theories that could be regarded as further truncations on top of the generalized seniority, including the microscopic interacting boson (fermion) model, the nucleon-pair approximation, and other models.
The authors derive the equation of evolution in the configuration space for a system of semiclassical bosons or fermions starting from the recently derived nonlinear Kramers-Chandrasekhar equation for such particles. The latter equation does not contract to the corresponding Fokker-Planck equation in the velocity space; however, it is shown that in the context of a Chapman-Enskog approximation the contraction to the physical space leads to the diffusion equation with the classical relationship between the self-diffusion coefficient and the friction coefficient maintained.
Abstract Although highly successful, the truncated Wigner approximation (TWA) does not account for genuine many-body (MB) quantum interference between different solutions of the mean-field equations of a bosonic MB system. This renders the TWA essentially classical, where a large number of particles formally takes the role of the inverse of Planck’s constant ℏ . The failure to describe genuine interference phenomena, such as localization and scarring in Fock space, can be seen as a virtue of this quasiclassical method, which thereby allows one to identify genuine quantum effects when being compared with ‘exact’ quantum calculations that do not involve any a priori approximation. A rather prominent cause for such quantum effects that are not accounted for by the TWA is the constructive interference between the contributions of symmetry-related trajectories, which would occur in the presence of discrete symmetries provided the phase-space distribution of the initial state and the observable to be evaluated feature a strong localization about the corresponding symmetry subspaces. Here we show how one can conceive an augmented version of the TWA which can account for this particular effect. This augmented TWA effectively amounts to complementing conventional TWA calculations by separate truncated Wigner simulations that are restricted to symmetric subspaces and involve weight factors that account for the dynamical stability of sampling trajectories with respect to perpendicular deviations from those subspaces. We illustrate the validity of this method at pre- as well as post-Ehrenfest time scales in prototypical Bose–Hubbard systems displaying chaotic classical dynamics, where it also reveals the existence of additional MB interference effects.
Starting from a general, microscopic fermion-pair-to-boson mapping of a complete fermion space that preserves Hermitian conjugation, we show that the resulting infinite and non-convergent boson Hamilitonian can be factored into a finite (e.g., a 1 + 2-body fermion Hamiltonian is mapped to a 1 + 2-body boson Hamiltonian) image Hamilitonian times the norm operator, and it is the norm operator that is infinite and non-convergent. We then truncate to a collective boson space and we give conditions under which the exact boson images of finite fermion operators are also finite in the truncated basis.
Starting from a general, microscopic fermion-to-boson mapping that preserves Hermitian conjugation, we discuss truncations of the boson Fock space basis. We give conditions under which the exact boson images of finite fermion operators are also finite [e.g., a (1+2)-body fermion Hamiltonian is mapped to a (1+2)-body boson Hamiltonian] in the truncated basis. For the most general case, where the image is not necessarily exactly finite, we discuss how to make practical and controlled approximations.