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The F-theorem for stochastic models
198
ABSTRACTS
OF PAPERS
TO APPEAR
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to be used is discussed, and the relation to measurement theory is elucidated. The paper summarizes and unities earlier works by Husimi, Arthurs, and Kelly and Braunstein. Caves, and Milburn. A realization in quantum optics is suggested. The
for Stochastic Models. R. F. STREATER. Department of Mathematics, King’s College, Strand, London WCZR 2LS, United Kingdom.
F-Theorem
Heat variables are added to stochastic models, that is, discrete systems obeying a non-linear timeevolution called the discrete Boltzmann ~uation. When. the evolution is rn~i~~ so that the heat variables remain fixed at a given temperature, the H-theorem of the isolated system becomes the F-theorem of the closed system: the free energy F is monotonic decreasing along the orbit. This is used to show that a large family of chemical rate equations converge to equilibrium, including the Ising model with Glauber dynamics. Some changes in certain commonly used equations are suggested, making them compatible with the general scheme. Global
Method CORNELIA
for All S-Matrix GRAMA, N. GRAMA,
Poles I~nt~~cat~on~ AND I. ZAMFIRESCXJ.
New
Classes
of
Poles
and
Resonant
States.
Institute of Atomic Physics, Bucharest, MC-6
Romania. A global method of identifying all the s-matrix poles k = k,(g) in the k-plane for a central potential gV(r) (g E C) is presented. The method involves construction of the Riemann surface Rt’ over the g-plane, on which the function k = k,(g) is single valued and analytic. It implies the division of the Riemann surface Rf’ into zz’ sheets and the construction of the sheet images z0rul in the k-plane. The method is first applied to potentials having Jost functions entire in both,variables g and k. In this case the general properties of the function k = k,(g) and of its Riemann surface Rf) are obtained. The method is then extended to other classes of potentials. The construction of the Riemann surface Rf) as well as the construction of the images of its sheets in the k-plane for some particular potentials is given in detail. The method allows identilication of new classesof resonant state poles. These poles remain in a neighborhood of some special points called “stable-points” as the strength of the potential increases to infinity; i.e., these resonant state poles do not become bound or virtual state poles. Moreover, the wave functions of the resonant states corresponding to the new class of poles situated in the nei~borho~ of the stable-points are almost completely localized outside the potential well. The method provides a quantum number n, characterizing the bound and resonant state S-matrix poles. This quantum number has a topological significance: it is the label of the sheets of the Riemann surface Rz’. The existence of new classes of resonant state poles seems to be a general property of the potentials with barriers used in nuclear scattering theory. Convergence of ~o~i~near Mass&e quanta Fiefd Theory in the Einstein Universe. I. E. SEGAL AND Z. ZHOU. ~as~chu~tts Institute of Technology, Cambridge, Massachusetts 02139.
We treat as a prototype for four-dimensional nonlinear quantum field theories the g(p@theory in the Einstein Universe E = R’ x S3. The underlying free system is defined by the KleinGordon equation in E. We show rigorously, without the intervention of any cutoffs or perturbative renormalizations, that the interaction and total hamiltonians are self-adjoint operators in the free field Hilbert space that depend continuously on g. The boundary condition that the interacting field be asymptotically free in the infinite past is rigorously implemented. and a unitary S-matrix of Yang-Feldman type is given a finite expression. Our formalism agrees with that of conventional relativistic theory within terms of order I/R, where R is the cosmic distance scale (radius of S3) in laboratory units and 2 1040fm. Any mass packet in Minkowski space extends covariantly to the ambient Einstein Universe. The microscopic relevance of cosmic effects is discussed; eg,, Einstein gravity produces an effective cutoK of order 10”’ Gev on the energy of a massive particle.
ABSTRACTS
OF PAPERS
TO APPEAR
IN
FUTURE
ISSUES
to be used is discussed, and the relation to measurement theory is elucidated. The paper summarizes and unities earlier works by Husimi, Arthurs, and Kelly and Braunstein. Caves, and Milburn. A realization in quantum optics is suggested. The
for Stochastic Models. R. F. STREATER. Department of Mathematics, King’s College, Strand, London WCZR 2LS, United Kingdom.
F-Theorem
Heat variables are added to stochastic models, that is, discrete systems obeying a non-linear timeevolution called the discrete Boltzmann ~uation. When. the evolution is rn~i~~ so that the heat variables remain fixed at a given temperature, the H-theorem of the isolated system becomes the F-theorem of the closed system: the free energy F is monotonic decreasing along the orbit. This is used to show that a large family of chemical rate equations converge to equilibrium, including the Ising model with Glauber dynamics. Some changes in certain commonly used equations are suggested, making them compatible with the general scheme. Global
Method CORNELIA
for All S-Matrix GRAMA, N. GRAMA,
Poles I~nt~~cat~on~ AND I. ZAMFIRESCXJ.
New
Classes
of
Poles
and
Resonant
States.
Institute of Atomic Physics, Bucharest, MC-6
Romania. A global method of identifying all the s-matrix poles k = k,(g) in the k-plane for a central potential gV(r) (g E C) is presented. The method involves construction of the Riemann surface Rt’ over the g-plane, on which the function k = k,(g) is single valued and analytic. It implies the division of the Riemann surface Rf’ into zz’ sheets and the construction of the sheet images z0rul in the k-plane. The method is first applied to potentials having Jost functions entire in both,variables g and k. In this case the general properties of the function k = k,(g) and of its Riemann surface Rf) are obtained. The method is then extended to other classes of potentials. The construction of the Riemann surface Rf) as well as the construction of the images of its sheets in the k-plane for some particular potentials is given in detail. The method allows identilication of new classesof resonant state poles. These poles remain in a neighborhood of some special points called “stable-points” as the strength of the potential increases to infinity; i.e., these resonant state poles do not become bound or virtual state poles. Moreover, the wave functions of the resonant states corresponding to the new class of poles situated in the nei~borho~ of the stable-points are almost completely localized outside the potential well. The method provides a quantum number n, characterizing the bound and resonant state S-matrix poles. This quantum number has a topological significance: it is the label of the sheets of the Riemann surface Rz’. The existence of new classes of resonant state poles seems to be a general property of the potentials with barriers used in nuclear scattering theory. Convergence of ~o~i~near Mass&e quanta Fiefd Theory in the Einstein Universe. I. E. SEGAL AND Z. ZHOU. ~as~chu~tts Institute of Technology, Cambridge, Massachusetts 02139.
We treat as a prototype for four-dimensional nonlinear quantum field theories the g(p@theory in the Einstein Universe E = R’ x S3. The underlying free system is defined by the KleinGordon equation in E. We show rigorously, without the intervention of any cutoffs or perturbative renormalizations, that the interaction and total hamiltonians are self-adjoint operators in the free field Hilbert space that depend continuously on g. The boundary condition that the interacting field be asymptotically free in the infinite past is rigorously implemented. and a unitary S-matrix of Yang-Feldman type is given a finite expression. Our formalism agrees with that of conventional relativistic theory within terms of order I/R, where R is the cosmic distance scale (radius of S3) in laboratory units and 2 1040fm. Any mass packet in Minkowski space extends covariantly to the ambient Einstein Universe. The microscopic relevance of cosmic effects is discussed; eg,, Einstein gravity produces an effective cutoK of order 10”’ Gev on the energy of a massive particle.