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General expansions of a scattering amplitude
ANNALSOFPHYSICS:
Abstracts
55, 392-393(1969)
of Papers
to
Appear
in Future
Issues
General Expansions of a Scattering Amplitude, G. FELDMAN AND P. T. MATHEWS. A general theory is developed of the expansions of a two-particle scattering amplitude based on its Poincare invariance properties for particles of arbitrary mass and spin. The theory is then applied to (i) the conventional little group expansions, including the 0(3, 1) expansion for equal mass, forward scattering, (ii) the auxiliary group expansions in terms of 0(3, 1) functions for the general case, and (iii) the double variable expansions of Smorodinsky et al. The implications of poles in these various amplitudes is discussed in connection with high energy behaviour and analyticity at zero momentum transfer. The Generalized Angular Momentum Crossing Relation. D. B. FAIRLIE AND M. T. NOGA. In low-energy meson baryon scattering a specific partial wave amplitude in the direct channel will contain contributions from the exchange of resonances in all partial waves in the crossed channel. These contributions, first considered by Carruthers for pion-nucleon scattering, are evaluated in the limit where the baryon kinetic energy is small compared to its mass, and expressed succinctly in terms of a 9j symbol. Consequences for bootstrap theory are noted.
On Violation of the Superselection Rules. ELIHU LUBKIN. Dept. of Physics, Univ. of Wisconsin. If Q = EYE’=,Qi, and Q2 = 1, @ a.. @ lie1 @ qi Q I,,, @ *a* 0 1, (Q, Qi form an additive system of quantities), if [Q, P] = 0 (superselection rule), and if the reduced density matrix pi is obtained from the density matrix P by tracing out over the j spaces, j # i, then [qi , pi] = 0. This shows that a superselection rule for an additive quantity propagates to subsystems. It is argued that superselection rules are, therefore, universal for additive quantities, unless a process of relativization with respect to a Q reservoir is employed. Since such relativization is required to produce the coherent superpositions of momentum eigenstates needed for the notion of localization in position, it is argued that analogous processes of relativization can be used to break other superselection rules.
Spin Wave Theory of the Heisenberg Model for Large Spin and the Classical Limit. P. D. LOLY. The conventional results of spin wave theory are now known to be valid only for T < TJS. This paper deals with the results that obtain when that condition does not hold. Using the appropriate expansion of the Planck function and a boson model for the spin wave system we correct and extend recently published results of Mattis and of Vaks, Larkin, and Pikin. Those authors appear to have overlooked the possibility of using this form of spin wave theory to investigate the low temperature phase of the classical Heisenberg model. Simplifications occurring on taking this limit are contrasted with the problems of the finite spin case. The early work of Heller and Kramers for this classical case is extended herein to include the effects of spin wave interactions. For both the large spin and classical cases detailed results are given up to second order interaction effects and a conjecture is made about the nth order behaviour and its significance in the classical case. 392
Abstracts
55, 392-393(1969)
of Papers
to
Appear
in Future
Issues
General Expansions of a Scattering Amplitude, G. FELDMAN AND P. T. MATHEWS. A general theory is developed of the expansions of a two-particle scattering amplitude based on its Poincare invariance properties for particles of arbitrary mass and spin. The theory is then applied to (i) the conventional little group expansions, including the 0(3, 1) expansion for equal mass, forward scattering, (ii) the auxiliary group expansions in terms of 0(3, 1) functions for the general case, and (iii) the double variable expansions of Smorodinsky et al. The implications of poles in these various amplitudes is discussed in connection with high energy behaviour and analyticity at zero momentum transfer. The Generalized Angular Momentum Crossing Relation. D. B. FAIRLIE AND M. T. NOGA. In low-energy meson baryon scattering a specific partial wave amplitude in the direct channel will contain contributions from the exchange of resonances in all partial waves in the crossed channel. These contributions, first considered by Carruthers for pion-nucleon scattering, are evaluated in the limit where the baryon kinetic energy is small compared to its mass, and expressed succinctly in terms of a 9j symbol. Consequences for bootstrap theory are noted.
On Violation of the Superselection Rules. ELIHU LUBKIN. Dept. of Physics, Univ. of Wisconsin. If Q = EYE’=,Qi, and Q2 = 1, @ a.. @ lie1 @ qi Q I,,, @ *a* 0 1, (Q, Qi form an additive system of quantities), if [Q, P] = 0 (superselection rule), and if the reduced density matrix pi is obtained from the density matrix P by tracing out over the j spaces, j # i, then [qi , pi] = 0. This shows that a superselection rule for an additive quantity propagates to subsystems. It is argued that superselection rules are, therefore, universal for additive quantities, unless a process of relativization with respect to a Q reservoir is employed. Since such relativization is required to produce the coherent superpositions of momentum eigenstates needed for the notion of localization in position, it is argued that analogous processes of relativization can be used to break other superselection rules.
Spin Wave Theory of the Heisenberg Model for Large Spin and the Classical Limit. P. D. LOLY. The conventional results of spin wave theory are now known to be valid only for T < TJS. This paper deals with the results that obtain when that condition does not hold. Using the appropriate expansion of the Planck function and a boson model for the spin wave system we correct and extend recently published results of Mattis and of Vaks, Larkin, and Pikin. Those authors appear to have overlooked the possibility of using this form of spin wave theory to investigate the low temperature phase of the classical Heisenberg model. Simplifications occurring on taking this limit are contrasted with the problems of the finite spin case. The early work of Heller and Kramers for this classical case is extended herein to include the effects of spin wave interactions. For both the large spin and classical cases detailed results are given up to second order interaction effects and a conjecture is made about the nth order behaviour and its significance in the classical case. 392