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Continuum limit of QED2 on a lattice
ANNALS
OF PHYSICS
122, 465-466 (1979)
Abstracts
of Papers
to Appear
in Future
Issues
Number of Bound States of Three-Body Systems and Efimov’s E&et. Yu. N. OUCHINNIKOV. L. D. Landau Institute of Theoretical Physics, Moscow, USSR. I. M. SIGAL. Department of Mathematics, Princeton University, Princeton, New Jersey 08540. Using the variational approach we obtain lower bounds for number of eigenvalues of threeparticle Hamiltonians with short-range potentials close to critical ones and construct corresponding trial functions. A new proof of the infiniteness of the number of eigenvahtes for critical potentials (Efimov’s effect) is given. Semiclassical Approximations GRAMMATICOS AND A. VOROS.
Nuclear Hamiltonians. I. Spin-Independent Potentials. B. Centre d’Etudes Nucleaires de Saclay, 91190 Gif-sur-Yvette, France.
for
A systematic procedure for calculating semiclassical expansions of physically interesting quantities is presented. The method is based on the Wigner transform of operators. It is applied specifically to the case of fermions governed by a one-body Hamiltonian. Expansions up to fourth order in fi of the density matrix for various spin-independent potentials are derived. Expressions of the kinetic energy density in terms of the matter density are given, which are of particular interest for nuclear structure calculations. On the Algebraic Formulation of Collective Models. I. The Mass Quadrupole Collective Model. G. ROSENSTEEL.Department of Physics, Tulane University, New Orleans, Louisiana 70118. D. J. ROWE. Department of Physics, University of Toronto, Toronto, Ontario MSS lA7 Canada. This paper is the first in a series of three which together present a microscopic formulation of the Bohr-Mottelson (BM) collective model of the nucleus. In this article the mass quadrupole collective (MQC) model is defined and shown to be a generalization of the BM model. The MQC model eliminates the small oscillation assumption of BM and also yields the rotational and CM(3) submodels by holonomic constraints on the MQC configuration space. In addition, the MQC model is demonstrated to be an algebraic model, so that the state space of the MQC model carries an irrep of a Lie algebra of microscopic observables, the MQC algebra. An infinite class of new collective models is then given by the various inequivalent irreps of this algebra. A microscopic embedding of the BM model is achieved by decomposing the representation of the MQC algebra on manyparticle state space into its irreducible components. In the second paper this decomposition is studied in detail. The third paper presents the symplectic model, which provides the realization of the collective model in the harmonic oscillator shell model. Continuum Limit of QED2 on a Lattice. D. H. WEINGARTEN AND J. L. CHALLIFOUR. Physics Department, Indiana University, Bloomington, Indiana 47405. A path integral is defined for the vacuum expectation values of Euclidean QED, on a periodic lattice. Wilson’s expression is used for the coupling between fermion and gauge fields. The action for the gauge field by itself is assumed to be a quadratic in place of Wilson’s periodic action. The integral over the fermion field is carried out explicitly to obtain a Matthews-Salam formula for vacuum expectation values. For a combination of gauge and fermion fields 3 on a lattice with spacing proportional to N-l, NE E, the Matthews-Salam formula for the vacuum expectation <97>~ has the form <3)~ = s dpWN(S, f) where dp is an N-independent measure on a random electromagnetic 465
0003-4916/79/110465-02$05.00/0 Copyright
0 1979 by Academic
Press,
Inc.
466
ABSTRACTS
OF
PAPERS
TO
APPEAR
IN
FUTURE’
ISSUES
fieldfand W,(S,f) is an N-dependent function offdetermined by g. For a class of 9 we prove that as N -+ 00, W,(V,ff> has a limit W(S,f) except possibly for a set of p measure zero. In subsequent articles it will be shown that l dpW(S,f) exists and limN,m~ = l dpW(Y,f). The Meson
Mass
Spectrum
and
Unitarity.
NILS
A. T~RNQVIST.
Research Institute for Theoretical
Physics, University of Helsinki, Finland. Using general arguments we discuss corrections from unitarity and analyticity to the meson masses and mixing angles corresponding to the lowest qq states of the quark model. Although the input S wave qq states are nearly degenerate and the input couplings obey exact flavor symmetry and the Okubo-Zweig-Iizuka rule, the output unitarized masses and mixing angles are close to the observed ones. The much larger unitarity effects for the pseudoscalars compared to the vector mesons are accounted for by the extra spin-counting factor the P - PV - P and PV - V - P loops compared to the V - PP - V loops.
OF PHYSICS
122, 465-466 (1979)
Abstracts
of Papers
to Appear
in Future
Issues
Number of Bound States of Three-Body Systems and Efimov’s E&et. Yu. N. OUCHINNIKOV. L. D. Landau Institute of Theoretical Physics, Moscow, USSR. I. M. SIGAL. Department of Mathematics, Princeton University, Princeton, New Jersey 08540. Using the variational approach we obtain lower bounds for number of eigenvalues of threeparticle Hamiltonians with short-range potentials close to critical ones and construct corresponding trial functions. A new proof of the infiniteness of the number of eigenvahtes for critical potentials (Efimov’s effect) is given. Semiclassical Approximations GRAMMATICOS AND A. VOROS.
Nuclear Hamiltonians. I. Spin-Independent Potentials. B. Centre d’Etudes Nucleaires de Saclay, 91190 Gif-sur-Yvette, France.
for
A systematic procedure for calculating semiclassical expansions of physically interesting quantities is presented. The method is based on the Wigner transform of operators. It is applied specifically to the case of fermions governed by a one-body Hamiltonian. Expansions up to fourth order in fi of the density matrix for various spin-independent potentials are derived. Expressions of the kinetic energy density in terms of the matter density are given, which are of particular interest for nuclear structure calculations. On the Algebraic Formulation of Collective Models. I. The Mass Quadrupole Collective Model. G. ROSENSTEEL.Department of Physics, Tulane University, New Orleans, Louisiana 70118. D. J. ROWE. Department of Physics, University of Toronto, Toronto, Ontario MSS lA7 Canada. This paper is the first in a series of three which together present a microscopic formulation of the Bohr-Mottelson (BM) collective model of the nucleus. In this article the mass quadrupole collective (MQC) model is defined and shown to be a generalization of the BM model. The MQC model eliminates the small oscillation assumption of BM and also yields the rotational and CM(3) submodels by holonomic constraints on the MQC configuration space. In addition, the MQC model is demonstrated to be an algebraic model, so that the state space of the MQC model carries an irrep of a Lie algebra of microscopic observables, the MQC algebra. An infinite class of new collective models is then given by the various inequivalent irreps of this algebra. A microscopic embedding of the BM model is achieved by decomposing the representation of the MQC algebra on manyparticle state space into its irreducible components. In the second paper this decomposition is studied in detail. The third paper presents the symplectic model, which provides the realization of the collective model in the harmonic oscillator shell model. Continuum Limit of QED2 on a Lattice. D. H. WEINGARTEN AND J. L. CHALLIFOUR. Physics Department, Indiana University, Bloomington, Indiana 47405. A path integral is defined for the vacuum expectation values of Euclidean QED, on a periodic lattice. Wilson’s expression is used for the coupling between fermion and gauge fields. The action for the gauge field by itself is assumed to be a quadratic in place of Wilson’s periodic action. The integral over the fermion field is carried out explicitly to obtain a Matthews-Salam formula for vacuum expectation values. For a combination of gauge and fermion fields 3 on a lattice with spacing proportional to N-l, NE E, the Matthews-Salam formula for the vacuum expectation <97>~ has the form <3)~ = s dpWN(S, f) where dp is an N-independent measure on a random electromagnetic 465
0003-4916/79/110465-02$05.00/0 Copyright
0 1979 by Academic
Press,
Inc.
466
ABSTRACTS
OF
PAPERS
TO
APPEAR
IN
FUTURE’
ISSUES
fieldfand W,(S,f) is an N-dependent function offdetermined by g. For a class of 9 we prove that as N -+ 00, W,(V,ff> has a limit W(S,f) except possibly for a set of p measure zero. In subsequent articles it will be shown that l dpW(S,f) exists and limN,m
Mass
Spectrum
and
Unitarity.
NILS
A. T~RNQVIST.
Research Institute for Theoretical
Physics, University of Helsinki, Finland. Using general arguments we discuss corrections from unitarity and analyticity to the meson masses and mixing angles corresponding to the lowest qq states of the quark model. Although the input S wave qq states are nearly degenerate and the input couplings obey exact flavor symmetry and the Okubo-Zweig-Iizuka rule, the output unitarized masses and mixing angles are close to the observed ones. The much larger unitarity effects for the pseudoscalars compared to the vector mesons are accounted for by the extra spin-counting factor the P - PV - P and PV - V - P loops compared to the V - PP - V loops.