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Scale transformations for renormalized field operators: Discussion of the soluble model
ABSTRACTS
OF PAPERS
TO APPEAR
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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
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