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Physical Review C

Algebraic analysis of physical and spurious states in Dyson boson mapping

P. Park

Published Feb 1, 1987• 24 citations
Publisher

Abstract

Dyson boson mapping of a system of 2n fermions distributed among k single-particle states uses a space generated by two-index ideal bosons. We define a unitary group U(k) to classify this ideal-boson space. The physical subspace is shown to correspond to the antisymmetric irreducible representation [${1}^{2n}$]. This classification enables one to introduce a ``Majorana'' operator S which is a linear combination of one- and two-body Casimir operators of U(k). A zero eigenvalue of S characterizes the physical subspace while positive eigenvalues identify all other irreducible representations that occur. The Hermitian boson image of the Hamiltonian does not connect physical and spurious states because it is a function of the generators of U(k). All the nonphysical eigenvalues and eigenstates of a physical boson Hamiltonian (Dyson boson image of a physical fermion Hamiltonian) which could be Hermitian or non-Hermitian, can be removed without affecting the physical eigenvalues and eigenvectors, by adding a suitable multiple of S to the Hamiltonian.

DOI
10.1103/physrevc.35.807

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Implementation of Dyson boson mapping is discussed in connection with effective Hamiltonians. A feature of the mapping technique, when implemented in an ideal boson basis, is the possible appearance of spurious states. These spurious states typically signal the overcompleteness of the basis. Without truncation, no contamination of the physical states and spectrum takes place. However, in practice one may be required to select from the ideal boson basis the dominant components for a given interaction. It is shown that the correspondence between a perturbative expansion, $\'a\mathit{la}$ Bloch-Horowitz, and Dyson boson mapping allows for the identification of spurious states. The proposed method is applied to the mapping of a bi-fermionic Hamiltonian.

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Comment on ‘‘Algebraic analysis of physical and spurious states in Dyson boson mapping’’

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By stressing the application of boson mapping techniques to the truncated collective branch of nuclear excitations, the usefulness of two recently developed methods for the treatment of spurious states are compared. It is shown why the identification proposed by Geyer et al. is superior to the use of the ``Majorana'' operator S as proposed by Park.

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Dyson boson mapping of a system of 2n fermions distributed among k single-particle states uses a space generated by two-index ideal bosons. We define a unitary group U(k) to classify this ideal-boson space. The physical subspace is shown to correspond to the antisymmetric irreducible representation [12n{1}^{2n}12n]. This classification enables one to introduce a ``Majorana'' operator S which is a linear combination of one- and two-body Casimir operators of U(k). A zero eigenvalue of S characterizes the physical subspace while positive eigenvalues identify all other irreducible representations that occur. The Hermitian boson image of the Hamiltonian does not connect physical and spurious states because it is a function of the generators of U(k). All the nonphysical eigenvalues and eigenstates of a physical boson Hamiltonian (Dyson boson image of a physical fermion Hamiltonian) which could be Hermitian or non-Hermitian, can be removed without affecting the physical eigenvalues and eigenvectors, by adding a suitable multiple of S to the Hamiltonian.
Implementation of Dyson boson mapping is discussed in connection with effective Hamiltonians. A feature of the mapping technique, when implemented in an ideal boson basis, is the possible appearance of spurious states. These spurious states typically signal the overcompleteness of the basis. Without truncation, no contamination of the physical states and spectrum takes place. However, in practice one may be required to select from the ideal boson basis the dominant components for a given interaction. It is shown that the correspondence between a perturbative expansion, aˊla\'a\mathit{la}aˊla Bloch-Horowitz, and Dyson boson mapping allows for the identification of spurious states. The proposed method is applied to the mapping of a bi-fermionic Hamiltonian.
For an even number of nucleons, we consider collective motion based on any dynamical symmetry subalgebra \^A of the valence-shell bifermion algebra; the generators of \^A are coherent linear combinations of bifermion operators of the types ${a}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$${a}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$, ${a}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$a, and aa. A boson algebra is obtained by Dyson mapping the whole valence-shell bifermion algebra, and then ``skeletonizing'' the resulting expressions by deleting all noncollective boson operators from them. This skeleton boson algebra is shown to have the same commutation relations as the collective bifermion algebra, provided that the bifermion algebra is self-conjugate. The resulting boson Hamiltonian is not Hermitian, but does resemble the Hamiltonian of the interacting boson model in containing only one- and two-body terms. Moreover, the skeleton boson algebra (when restricted to a collective boson subspace that is free of spurious states) is shown to be physically equivalent to the collective bifermion algebra. It is shown that by working solely with collective bosons one can identify spurious boson states unambiguously, although the procedure lacks convenience. The similarity transformation that connects conjugate versions of the skeleton boson mapping is discussed, and a general condition for hermiticity of the boson image of a Hermitian operator is given.
The present series of papers deals with various realizations of the dynamical group 𝒮𝓅c(2d,R) of microscopic collective states for an A nucleon system in d dimensions, defined as those A particle states invariant under the orthogonal group O(n) associated with the n=A−1 Jacobi vectors. In the present paper, we derive two boson representations of 𝒮𝓅c(2d,R), namely the Dyson representation and the Holstein–Primakoff (HP) one. Our starting point is a representation of microscopic collective states, as introduced in the first paper of the present series, in a Barut Hilbert space ℱc of analytic functions in ν =(1/2)d(d+1) complex variables. Basis functions in ℱc, classified according to the chain 𝒮𝓅c(2d,R)⊇𝒰c(d), can be put into one-to-one correspondence with basis functions, classified according to the chain 𝔘(ν)⊇𝔘(d), in a Bargmann Hilbert space ℬ of analytic functions in ν complex variables representing ν-dimensional boson states. By equating the complex variables of ℱc and their conjugate momenta with those of ℬ, we get the non-Hermitian Dyson representation of 𝒮𝓅c(d,R). We then go from the latter to the Hermitian HP representation by means of a canonical transformation that restores the Hermiticity properties of the variables and conjugate momenta. The inverse of the HP representation gives the unitary representation in quantum mechanics of the classical canonical transformation relating the oscillator Hamiltonians of the microscopic collective model and the boson macroscopic one. From the ν boson creation and annihilation operators, it is possible to build the generators of a 𝔘(ν) group, which in the physical three-dimensional case reduces to 𝔘(6). The latter is finally compared with the U(6) group appearing in the interacting boson model.
Majorana phase-space representations for fermions map fermionic many-body physics into a distribution over one of the Cartan symmetric spaces of Lie group theory. The representation is in terms of $2M\ifmmode\times\else\texttimes\fi{}2M$ complex antisymmetric matrices, which generate the Gaussian Majorana operators. Here we show how this expansion can be utilized to calculate quantities arising in quantum thermodynamics and quantum information. Purity and the linear entropy are calculated, as well as the quantum fidelity between two general fermionic states, with numerical examples for pure states. We describe the geometrical properties of the phase space, and show that the overlap between two Gaussian Majorana states depends on the product of their antisymmetric matrices. Fermionic phase space is divided up into two orthogonal subspaces of different number parity, whose matrix representations differ by an orthogonal reflection in the phase space.
In the framework of the so-called quasi-Hermitian quantum mechanics of stationary unitary systems, bound states are usually constructed as eigenstates ∣ψn⟩|ψ_n \rangle∣ψn​⟩ of a Hamiltonian operator HHH with real spectrum which is non-Hermitian, H≠H†H \neq H^\daggerH=H†. One of the ways of the standard probabilistic interpretation of such systems consists in a transformation of HHH into its isospectral Hermitian ``twin" h=h†\mathfrak{h}= \mathfrak{h}^\daggerh=h† via one of the so-called Dyson maps Ω:H→hΩ: H \to \mathfrak{h}Ω:H→h. Naturally, the well known ambiguity of these H−H-H−dependent Dyson-map transformations implies also an ambiguity of the physical, Ω−Ω-Ω−dependent probabilistic and experimental interpretation of the system in question. In the present paper, an exhaustive classification of all of the eligible H−H-H−dependent Dyson maps Ω=Ω(H)Ω=Ω(H)Ω=Ω(H) is provided, implying also a systematic framework for a specification of all of the possible probabilistic interpretations of the quantum system characterized by a preselected HHH.