Exact saturation and degeneracy of isospin bounds and zeros trajectories in pion-nucleon scattering

430 Strong

ABSTRACTS

Coupling

Polaron

Theory

OF

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Invariance. EUGENE P. GROSS. Department of Massachusetts 02154.

and Translational

Physics, Brandeis University, Waltham,

We develop a systematic, manifestly translation invariant, strong coupling theory for nonrelativistic Hamiltonians of the polaron type. As in earlier strong coupling theories, the position of the polarization well is a collective coordinate. The field is expanded in a set of basis functions centered about the well with three amplitudes deleted. A particle coordinate relative to the polarization center is introduced. The new coordinates are introduced using a point canonical Hamiltonian leading to a Hermitian Hamiltonian, with properly normalized wavefunctions, and with a Jacobian that is evaluated in closed form. All subsequent approximations to the states are manifestly translation invariant. For the ground state the energy of the recoil terms to leading order depend on the coupling constant g as g-*. The intrinsic part of the Hamiltonian determines the energy terms of order g4 and go. An adiabatic canonical transformation is used to calculate all terms through order g-‘. The coefficients depend on the Green’s function for the electron in a static potential well. We determine the first three terms in the inverse coupling constant expansion of the effective mass. Optimally

Simple

Connection

between

the

Reaction

Matrix

and

the

Observables.

GARY

R.

GOLDSTEIN. Department of Physics, Tufts University, Medford, Massachusetts 02155. MICHAEL J. MORAVCSIK. Institute of Theoretical Science and Department of Physics, University of Oregon, Eugene, Oregon 97403. A class of optimal formalisms is derived to describe in the simplest possible way the relationship between the amplitudes of a arbitrary four-particle reaction and the experimental observables for that reaction. Within the optimal class (which includes none of the currently used formalisms) there are an infinite number of realizations depending on coordinate systems and quantization axes. The matrix connecting the bilinear combinations of amplitudes and the experimental observables (the X-matrix) for the optimal class consists only of small submatrices along the main diagonal, the maximum sizes of which are independent of the values of the spins involved. Two examples are worked out in detail: 4 + 0 + 4 + 0 and 1 + 4 + 0 + 4, and it is shown that the optimal prescription in conjunction with quantization along the normal of the reaction plane gives an unprecedentedly simple X-matrix as well as constraints among the observables imposed by parity conservation, which are also maximally simple. The advantages of the new class of formalisms are enumerated for the purposes of studies of high energy and nuclear reactions. Exact Saturation Scattering. D.

and Degeneracy of Isospin Bounds and Zeros Trajectories in Pion-Nucleon B. ION. Joint Institute for Nuclear Research, Dubna Moscow, P. 0. Box 79,

U.S.S.R. In this paper the exact saturation and degeneracy of the isospin bounds, in terms of the zeros trajectories (ZT) of Im Ni, [N,,-specific bilinear forms] are systematically discussed. The topology of the zeros trajectories and near degeneracy of the isospin bounds in the pion-nucleon scattering are investigated using the CERN-theoretic and CERN experimental solutions for the phase shifts. The interpretation of the experiments in terms of (ZT)-topology as well as certain tests for the possible isospin breaking phenomena are suggested. Representations of the Lorentz Group. SEAN BROWNE. School of Theoretical Physics, Dublin Institute for Advanced Studies, Dublin 4, Ireland AND DJORDJE SIJAEKI. Institute “B. Kidrich” Vin&, Belgrade, Yugoslavia.

On the Zrreducibk

All continuous irreducible representations of the SL(2, C) group (as given by Naimark) are obtained by means of methods developed by Harish-Chandra and Kihlberg. The analysis is done

ABSTRACTS

OF PAPERS

TO APPEAR

IN FUTURE

431

ISSUES

in the SU(2) basis and a single closed expression for the matrix elements of the noncompact generators for an arbitrary irreducible representation of X(2, C) is given. For the unitary irreducible representations, the scalar product for each irreducible Hilbert space is found explicitly. The connection between the unitary irreducible representations of X(2, C) and those of T3 o ScI(2) is discussed by means of Inonii and Wigner contraction procedure and the Gel]-Mann formula. Finally, due to physical interest, the addition of a four-vector operator to X(2, C) unitary irreducible representations in a minimal way is considered; and all group extensions of the parity and time reversal operators by SL(2, C) are explicitly obtained and some aspects of their representations are treated. Scale

Transformations

for

Renormalired

Field

Operators:

Discussion

of

the

Soluble

Model.

J. LUKIERSKI. International Centre for Theoretical Physics, Trieste, Italy, AND A. OGIELSKI. Institute of Theoretical Physics, University of Wroclaw, Poland. We discuss the operator formulation of the Zachariasen-Thirring model, describing the chain approximation to the propagator (the sum of three-particle massless bubbles) in massless X@ theory. Such a model is formally scale-invariant and explicitly soluble. All intermediate steps of conventional renormalization procedure: regularization, introduction of appropriate counterterms and cut-off free limit, are explicitly performed. In every step the scaling properties are discussed and respective dilatation currents are written down. After the proper choice of scale transformations for the renormalized field operator, we obtain the nonlocal dilatation current, defining the renormalized dilatation generator DnR(t), In the cut-off free limit il ---f -u the ET commutator of DnR(t) with renormalized field operators reproduces the CallanSymanzik modification of “naive” canonical scale transformations. The renormalized scale transformations coincide in the cut-off free limit with renormalized dimensional transformations and define the exact symmetry of the renormalized theory. “Clusrers”

in the Ising

Model,

Metastable

States,

and Essential

Sitlgularity.

richtung 11. I, Theoret. Physik, Universitat des Saarlandes, 66 Saarbriicken

K. BINDER. Fach11, West Germany.

Various possibilities for the definition of “clusters” which are used in theories of critical phenomena and nucleation are discussed for the case of nearest-neighbor lsing models. Using a two coordinate description in terms of a contour (of “size” s) around I reversed spins, it is shown that scaling assumptions for the cluster concentration g(l, s) imply that the critical behavior cannot be attributed to fully “ramified” clusters as suggested by Domb. Monte Carlo results for p(l, s) are also presented and shown to be consistent with scaling. For large I a crossover to geometric behavior is found and again interpreted in terms of scaling. Relating the “clusters” to fluctuations of a coarse-grained order parameter, the arguments of Andreev in favor of an essential singularity at the coexistence curve below the critical temperature are recovered. The stability limit of the metastable states, which can thus be defined in terms of dynamic considerations only, is obtained for the whole temperature range from computer simulations. Recoil

Corrections

to Elastic

Electron

Scattering

in the Breit

Approximation.

J. L. FRIAR. Depart-

ment of Physics, Brown University, Providence, Rhode Island 02912. Recoil corrections to the cross sections for elastic electron scattering from spin-0 nuclei are investigated in the Breit approximation. The form of the scattering amplitude in first- and secondBorn approximation is investigated in detail using time-dependent perturbation theory, and it is found that the center-of-mass (CM) frame is particularly convenient to work in. Transformation equations relating the lab and CM frames are developed. Those parts of the second-Born amplitude