On the correspondence between fermion and boson states and operators

Mapping of shell-model (fermion) Hamiltonians onto boson Hamiltonians which underly the interaction boson model ^1–5) is investigated. A simple correspondence is defined and a sufficient condition given for shell-model Hamiltonians to simply correspond to finite hermitian boson Hamiltonians. A special case is discussed where diagonalization of a shell-model Hamiltonian for valence protons and neutrons can be exactly carried out in an equivalent (but different) boson space. If, however, the proton Hamiltonian and neutron Hamiltonian are diagonal in the seniority scheme, mapping of fermion states onto orthogonal boson states cannot be a simple correspondence. In that case the boson quadrupole operators equivalent to fermion guadrupole operators cannot be single-boson operators but must be more complicated, ones.     

Fermions and bosons in a two-dimensional world

We derive the formal equivalence of a free massless two-dimensional theory and a free massless two-dimensional boson theory constructed from the bilinear products of the self-same fermion theory. The sense of this equivalence is investigated. Using a box normalization, it is found that the fermion states are Glauber coherent states of bosons, where the boson vacuum is the ground state of the charge sector corresponding to the given fermion state. The massless boson is the Goldstone boson and the degenerate vacua are the ground states of the various charge sectors. A complete operator identity between fermion and boson operators can be obtained, but to do this an additional boson operator must be introduced which cannot be defined in terms of bilinear products of the fermion operators. Doing this makes the charge spectrum continuous.     

A schematic model for monopole and quadrupole pairing in nuclei

A schematic Hamiltonian with a pairing interaction plus a quadrupole-quadrupole interaction between nucleons is presented. It is shown that all the states of the fermion system can be classified according to the number of nucleons u not coupled to coherent monopole or quadrupole pairs. The states with u = 0 are shown to have a one-to-one correspondence to the states of the interacting boson model. The spectra for these states are derived analytically for various limits of the pairing strength and the quadrupole strength. Analytical forms for the matrix elements of operators are derived for these limits. The operators in fermion space are mapped onto boson operators. The matrix elements of operators in the fermion space are shown to be equal to matrix elements of the boson operators multiplied by analytical Pauli factors which are state dependent. The two-nucleon transfer strength is calculated in two limits and is compared to experimental values.     

Microscopic calculation of IBM parameters by using intrinsic states

The overlaps between intrinsic fermionic and bosonic wave functions are required to be the same. This provides relations between fermion and boson variables. These relations are used in conjunction with an OAI procedure for intrinsic states to map the shell-model space operators onto their equivalent boson space operators. As an example, a QQ interaction is mapped.     

A unification of boson expansion theories: (I). Functional representations of fermion states

The method for obtaining boson expansion by representing the fermion states as holomorphic functions of many complex variables is presented. Such functional representation is explicitly constructed for each space which is the carrier space of an irreducible representation of a semisimple compact Lie group. This is achieved by proving the unity resolution in terms of holomorphically parametrized Perelomov's generalized coherent states. The functional images of fermion states are polynomials of complex variables, while those of fermion operators are differential operators of finite order with polynomial coefficients.     

Boson description of collective states: (II). Description of odd vibrational nuclei

By addition of the so-called ideal quasiparticle to the boson space one can represent the odd fermion states in that product space. In such a way one finds various representations of the fermion operators in terms of the boson operators and ideal quasiparticles. From these boson expansions of the fermion operators a finite one is selected by considering non-unitary transformations. Thus, the direct generalization, of the Dyson representation for even systems is given for the case of odd systems. The Hamiltonian can be divided into three parts: the boson term which describes the vibrational motion of the even core, the unperturbed motion of the quasiparticle, and the interaction between the quasiparticle and the bosons. This interaction consists of two terms, one of which agrees with the term used by Kisslinger and Sorensen ²), which is usually called the dynamical interaction, and the additional term is due to the antisymmetrization between the extra particle and the even core. The latter term can be identified as kinematical interaction which is responsible for the anomalous coupling states. For example, it is demonstrated that this term produces qualitatively the same splitting of the one-phonon multiplet as was obtained by Kuriyama et al. ³) for the j-shell. Furthermore, it is shown for the more complicated case of ¹¹⁷Sn that the effect of this additional interaction between phonons and quasiparticle is important when many shells to the states in the odd nucleus are taken into account.     

Infinite-Dimensional Grade Star Representations of the osp (2,l) and spl(2,l) Superalgebras

A grade adjoint operation for the boson and fermion operators is considered. A graded bosonfermion algebra is constructed. Explicit expressions for the generators of the osp(2,l) and spl(2,l) superalgebras in terms of suitable pairs of graded boson and fermion operators are given. Infinite-dimensional grade star representations are obtained.     

Fermions and associated bosons of one-dimensional model

The representation of the canonical commutation relations involved in the construction of boson operators from fermion operators according to the recipe of the neutrino theory of light is studied. Starting from a cyclic Fock-representation for the massless fermions the boson operators are reduced by the spectral projectors of two charge-operators and form an infinite direct sum of cyclic Fock-representations. Kronig's identity expressing the fermion kinetic energy in terms of the boson kinetic energy and the squares of the charge operators is verified as an identity for strictly selfadjoint operators. It provides the key to the solubility ofLuttinger's model. A simple sufficient condition is given for the unitary equivalence of the representations linked by the canonical transformation which diagonalizes the total Hamiltonian.Work supported by the National Science Foundation.     

Tomographic probability representation for quantum fermion fields

Tomographic probability representation is introduced for fermion fields. The states of the fermions are mapped onto the probability distribution of discrete random variables (spin projections). The operators acting on the fermion states are described by fermionic tomographic symbols. The product of the operators acting on the fermion states is mapped onto the star-product of the fermionic symbols. The kernel of the star-product is obtained. The antisymmetry of the fermion states is formulated as a specific symmetry property of the tomographic joint probability distribution associated with the states.     

Large-mass expansions or one-loop effective actions and fermion currents

The large-mass expansion of the functional determinants for second-order elliptic operators and general Dirac operators is calculated for four-dimensional flat euclidean space using zeta function regularisation and heat kernel methods. The results are applicable to one-loop boson and fermion effective actions. In addition the expansions of covariantly regularised fermion currents are derived. It is also possible for the corresponding Pauli-Villars regularised forms to be then simply obtained and the modified currents then reproduce the usual Bardeen anomaly. Although covariant methods are used it is shown how to derive the expansion for the phase of the fermion determinant, which is non-covariant and produces the anomaly, in terms of a representation as a five-dimensional integral which is related to the spectral asymmetry for a suitable spinor hamiltonian. This relation is essentially exact and is demonstrated by considering the variation of the phase with respect to the Dirac operator.     

Shape transitions in even Mo and Sm isotopes: Study in a new microscopic interacting boson model scheme

A simple dynamic procedure, based on the deformed Hartree-Fock solution of a nucleus, is presented to construct the IBM operators in microscopic basis. The parameters of these operators are evaluated by establishing a Marumori mapping from the truncated shell model space onto the boson space. The transitions from spherical to axial-rotor shape observed in the low-lying levels ofeven ^96–108Mo and^146–154Sm isotopes are reproduced qualitatively by applying this procedure with a fixed set of fermion input parameters to each chain. Variation of a few parameters in fermion space leads to quantitative agreement.     

A Relation of the Noncommutative Parameters in Generalized Noncommutative Phase Space

We introduce the deformed boson operators which satisfy a deformed boson algebra in some special types of generalized noncommutative phase space.Based on the deformed boson algebra,we construct coherent state representations.We calculate the variances of the coordinate operators on the coherent states and investigate the corresponding Heisenberg uncertainty relations.It is found that there are some restriction relations of the noncommutative parameters in these special types of noncommutative phase space.     

Boson expansions and quantization of time-dependent self-consistent fields (I). Particle-hole excitations

Two concrete methods are presented for quantizing the time-dependent Hartree equations in terms of boson operators. The first is the well-known infinite boson expansion analogous to the Holstein-Primakoff representation of angular momentum operators. The second, a new development, consists of finite boson quadratic forms, and is analogous to the Schwinger representation of angular momenta. In each case, a physical boson subspace can easily be constructed within which the full fermion dynamics is exactly duplicated. It therefore follows that quantization of the time-dependent Hartree equations, including all degrees of freedom, retrieves the exact many-body problem. The discussion in this paper is limited to particle-hole excitations of an N-particle system. A generalization to one-nucleon transfer processes on the N-particle system is also given in terms of ideal odd nucleons, but this brings in infinite expansions.     

Construction of microscopic boson states and their relevance for IBM-2

We investigate four methods for the construction of collective shell model states which may be mapped onto boson states of the IBM-2. These methods use, as building blocks for the wave functions, particle-particle pair operators, particle-hole operators, pair operators with seniority projection and energy-weighted quadrupole operators. It is demonstrated that one obtains stronger collectivity with the energy-weighted quadrupole operator than with the other methods.On the basis of a comparison of calculated and empirical IBM-2 interaction parameters we can rule out the seniority projection method. This implies that particle-particle and particle-hole approaches difler.The ratios between quadrupole matrix elements of the microscopic boson states appear to be similar to the IBM predictions. For states corresponding to those with two d-bosons coupled to J = 0 there is a smaller quadrupole matrix element when subshells with small angular momenta dominate near the Fermi level. Especially for this type of states the collective quadrupole space will be larger than represented in the IBM, however, which may compensate the smaller proton-neutron quadrupole coupling.The calculated bare quadrupole interaction between like bosons is found to be weak.     

Schwinger boson mapping of SO(8) fermion model

The Schwinger representation of the SO(8) fermion pair algebra in terms ofd and quasispin vector (u, s, v) bosons is used in deriving a microscopic boson coherent state having both particle-hole and pair excitations. The coherent state is the exact boson image of the HFB variational solution. We can study the shape phase transition and pairing behaviour of the nuclear ground states using the coherent states.     

Boson-fermion relations in several dimensions

Zero and positive temperature fermion field operators in several dimensions are constructed as stochastic integrals of certain reflection valued processes with respect to the corresponding boson field operator processes. The authors felicitate Prof. D S Kothari on his eightieth birthday and dedicate this paper to him on this occasion.     

Nature of correlations in the atomic limit of the boson fermion model

Using the equation of motion technique for Green's functions we derive the exact solution of the boson fermion model in the atomic limit. Both (fermion and boson) subsystems are characterised by the effective three level excitation spectra. We compute the spectral weights of these states and analyse them in detail with respect to all possible parameters. Although in the atomic limit there is no true phase transition, we notice that upon decreasing temperature some pairing correlations start to appear. Their intensity is found to be proportional to the depleted amount of the fermion nonbonding state. We notice that pairing correlations behave in a fashion observed for the optimally doped and underdoped high T_c superconductors. We try to identify which parameter of the boson fermion model can possibly correspond to the actual doping level. This study clarifies the origin of pairing correlations within the boson fermion model and may elucidate how to apply it for interpretation of experimental data. Received 31 January 2003 / Received in final form 18 March 2003 Published online 23 May 2003 RID="a" ID="a"e-mail: doman@kft.umcs.lublin.pl     

An application of boson second quantization techniques to the calculation of electron scattering form factors

It is suggested that boson second quantization, in terms of harmonic oscillator bosons, may be much more useful than its fermion counterpart for shell-model calculations in an LS coupled basis. The bosons carry the fundamental representation of SU(3). The combined set of boson creation and annihilation operators also carry the fundamental representation of Sp(3, R). Boson second quantization therefore provides a mechanism for expressing operators in terms of SU(3) and Sp(3, R) irreducible tensors. This is of major importance for shell-model calculations in an SU(3) basis and for the development of the symplectic shell model. Applications are made to the calculation of electron scattering form factors and it is shown how major simplifications arise when the space is restricted to a single major shell. For example, longitudinal form factors for 0 → 2 transitions in the sd-shell are shown to depend on just three parameters while the corresponding transverse form factors are uniquely determined up to an overall multiplicative constant. Further simplifications result on restriction to a single SU(3) irreducible representation.