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Analysis of Prigogine's theory of subdynamics
ANNALS
OF PHYSICS
150. 552-553
Abstracts
(1983)
of Papers to Appear
Supersymmetric Quantum Mechanics. M. DE CROMBRUGGHE Institut. Universitiit Bonn. Bonn, West Germany.
in Future
AND
Issues
V. RITTENBERG.
Physikalisches
A general construction for supersymmetric Hamiltonians in quantum mechanics is given. It is found that N-extended supersymmetry imposes very strong constraints, and for N > 4 the Hamiltonian is integrable. A variety of examples, for one-particle and for many-particle systems, in different numbers of dimensions are given.
High- and Low-Frequency Behaviour of Response Functions in a Bose Liquid: One-Loop Approximalion. E. TALBOT AND A. GRIFFIN. Department of Physics and Scarborough College, University of Toronto. Toronto. Ontario M5S lA7, Canada. The field-theoretic analysis of Wong and Gould for a Bose fluid is used to study response functions at arbitrary temperatures. Several rigorous identities are derived for density and momentum current response functions in the high- and low-frequency limits. These involve the condensate fraction n, and the superfluid density ps. By explicit calculations using the one-loop diagrammatic approximation for the proper irreducible functions, these rigorous results are verified at finite temperatures. In particular, the zero-frequency Gavoret-Nozieres relation, the high-frequency Wong-Gould f-sum rule. and the impulse approximation at high momentum transfers are all satisfied in this simple microscopic calculation. On the other hand. the superfluidfsum rule is found not to be satisfied.
Decoupling Physik
of Heall,’ Quarks in Quantum Chromodynamics. der Universitat Heidelberg, D-6900 Heidelberg,
W. BERNREUTHER. West Germany.
Institut
fiir Theoretische
Decoupling of heavy quarks in Quantum Chromodynamics (QCD) defined by mass-independent renormalization is investigated. The structure of the relations between the parameters offllavour QCD and the parameters of the effective/I flavour QCD below a heavy quark threshold is discussed to all orders in the loop expansion, and the relations are computed to two-loop approximation for the minimal subtraction schemes (MS) and to one-loop approximation for some Weinberg schemes. These matching relations can be used to systematically determine the renormalization group (RG)-invariant parameters of the effective theory in terms of the m-invariant parameters of the theory which includes the heavy quark. or vice versa. For the MS scheme the connection between A,- i and A, to two and three loops and the two-loop connection between the RG-invariant mass parameters of thefI andfflavour theory are given. The effect of heavy quarks on the evolution of the QCD coupling is of significance for present QCD phenomenology based on next-to-leading order perturbation theory. This is illustrated with a few examples within the MS scheme.
Ana&sis of Prigogine’s Group. University
Theory of Subdvnamics. C. H. OBCEMEA AND of Uppsala, Box 518, 75 120 Uppsala, Sweden.
The basic results of the theory of subdynamics of the projective components of the resolvent
are discussed in full detail. in the theory are analyzed,
552 0003.4916/83 Copyright All rights
$7.50
‘B 1983 by Academic Press. Inc. of reproduction in any form reserved.
E. BRANDAS.
Quantum
The regularity and a recent
Chemistry
requirements mathematical
ABSTRACTS
OF PAPERS
TO APPEAR
IN FUTURE
553
ISSUES
result of M. Courbage regarding the impossibility of simultaneous analytic continuation of the collision and creation operators is corrected. Further, some critiques on the interpretation of the various results of the subdynamics formalism are offered.
Variational Problems.
Helsinki,
Principles
JOUKO Finland.
and
Linked-Cluster
ARPONEN.
Research
e.upS
Expansions
Institute
for
for
Theoretical
Static
Physics,
and
Dynamic
University
Many-Bad)
of Helsinki,
The expS formalism for the ground state of a many-body system is derived from a variational principle. An energy functional is constructed using certain n-body linked-cluster amplitudes with respect to which the functional is required to be stationary. By using two different sets of amplitudes one either recovers the normal expS method or obtains a new scheme called the extended expS method. The same functional can also be used to obtain the average values of any operators as well as the linear response to static perturbations. The theory is extended to treat dynamical phenomena by introducing time depen dence to the cluster amplitudes. This allows the calculation both of non-linear dynamical behaviour and of dynamical linear response and Green’s functions. Practical approximation schemes are considered. In a SUBn approximation the m-body amplitudes are restricted to the order m < n and the energy functional is a finite-order multinomial in the amplitudes to be variationally determined. It is shown that the solution corresponds to summing well-defined subsets of Goldstone diagrams. These subsets are conveniently specified in terms of tree structures, the normal or extended g.t.0. trees. The extended expS method is in the SUBn approximation able to sum. in addition to the normal SUBn diagrams, a set which contains m-body cluster amplitudes of arbitrarily high order (m > n) in the ordinary sense. The article also discusses how the SUBn truncation schemes must be modified to be able to treat a system with a strong repulsive core in the two-body interaction. The method is formulated for the general cases of Bose and Fermi systems which may or may not conserve total particle number. It is shown that the simplest approximation, SUBl. in the extended expS method agrees with the mean field theory. which is the coherent-state approximation in the boson case or the Hartree-Fock approximation in the fermion case. It is argued that the extended expS method already in low-order approximations can realistically treat a great variety of diverse many-body problems, including even systems which may undergo groundstate phase transitions. A few applications are described in more detail. The Bose liquid is treated in the extended SUB2 approximation. It is shown that the ground-state results in the uniform limit are exact and agree with the hypernetted-chain approximation. The modifications due to hard-core interactions and the non-linear equations of motion are also discussed in this case. For Fermi systems it is shown that the superconductive phase transition of the BCS model Hamiltonian and the deformation phase transrtion of the Lipkin model are properly obtained by the extended expS method in a low-order approximation.
Eirtstein
Gravity
Washington,
Emerging
Seattle,
from
Washington
Quantum
B’qr’l Gravity.
A. ZEE. Department
of Physics,
University
of
98195.
A conformal invariant world described by the sum of the Weyl, Dirac, and Yang-Mills actions is advocated. Quantum fluctuations bring back Einstein gravity so that the long-distance phenomenology is as observed. Formulas for the induced Newton’s constant and Eddington’s constant are derived in quan tized Weyl gravity. The analogue of the trace anomaly for the Weyl action is shown to be structurally similar to that for the Yang-Mills action.
OF PHYSICS
150. 552-553
Abstracts
(1983)
of Papers to Appear
Supersymmetric Quantum Mechanics. M. DE CROMBRUGGHE Institut. Universitiit Bonn. Bonn, West Germany.
in Future
AND
Issues
V. RITTENBERG.
Physikalisches
A general construction for supersymmetric Hamiltonians in quantum mechanics is given. It is found that N-extended supersymmetry imposes very strong constraints, and for N > 4 the Hamiltonian is integrable. A variety of examples, for one-particle and for many-particle systems, in different numbers of dimensions are given.
High- and Low-Frequency Behaviour of Response Functions in a Bose Liquid: One-Loop Approximalion. E. TALBOT AND A. GRIFFIN. Department of Physics and Scarborough College, University of Toronto. Toronto. Ontario M5S lA7, Canada. The field-theoretic analysis of Wong and Gould for a Bose fluid is used to study response functions at arbitrary temperatures. Several rigorous identities are derived for density and momentum current response functions in the high- and low-frequency limits. These involve the condensate fraction n, and the superfluid density ps. By explicit calculations using the one-loop diagrammatic approximation for the proper irreducible functions, these rigorous results are verified at finite temperatures. In particular, the zero-frequency Gavoret-Nozieres relation, the high-frequency Wong-Gould f-sum rule. and the impulse approximation at high momentum transfers are all satisfied in this simple microscopic calculation. On the other hand. the superfluidfsum rule is found not to be satisfied.
Decoupling Physik
of Heall,’ Quarks in Quantum Chromodynamics. der Universitat Heidelberg, D-6900 Heidelberg,
W. BERNREUTHER. West Germany.
Institut
fiir Theoretische
Decoupling of heavy quarks in Quantum Chromodynamics (QCD) defined by mass-independent renormalization is investigated. The structure of the relations between the parameters offllavour QCD and the parameters of the effective/I flavour QCD below a heavy quark threshold is discussed to all orders in the loop expansion, and the relations are computed to two-loop approximation for the minimal subtraction schemes (MS) and to one-loop approximation for some Weinberg schemes. These matching relations can be used to systematically determine the renormalization group (RG)-invariant parameters of the effective theory in terms of the m-invariant parameters of the theory which includes the heavy quark. or vice versa. For the MS scheme the connection between A,- i and A, to two and three loops and the two-loop connection between the RG-invariant mass parameters of thefI andfflavour theory are given. The effect of heavy quarks on the evolution of the QCD coupling is of significance for present QCD phenomenology based on next-to-leading order perturbation theory. This is illustrated with a few examples within the MS scheme.
Ana&sis of Prigogine’s Group. University
Theory of Subdvnamics. C. H. OBCEMEA AND of Uppsala, Box 518, 75 120 Uppsala, Sweden.
The basic results of the theory of subdynamics of the projective components of the resolvent
are discussed in full detail. in the theory are analyzed,
552 0003.4916/83 Copyright All rights
$7.50
‘B 1983 by Academic Press. Inc. of reproduction in any form reserved.
E. BRANDAS.
Quantum
The regularity and a recent
Chemistry
requirements mathematical
ABSTRACTS
OF PAPERS
TO APPEAR
IN FUTURE
553
ISSUES
result of M. Courbage regarding the impossibility of simultaneous analytic continuation of the collision and creation operators is corrected. Further, some critiques on the interpretation of the various results of the subdynamics formalism are offered.
Variational Problems.
Helsinki,
Principles
JOUKO Finland.
and
Linked-Cluster
ARPONEN.
Research
e.upS
Expansions
Institute
for
for
Theoretical
Static
Physics,
and
Dynamic
University
Many-Bad)
of Helsinki,
The expS formalism for the ground state of a many-body system is derived from a variational principle. An energy functional is constructed using certain n-body linked-cluster amplitudes with respect to which the functional is required to be stationary. By using two different sets of amplitudes one either recovers the normal expS method or obtains a new scheme called the extended expS method. The same functional can also be used to obtain the average values of any operators as well as the linear response to static perturbations. The theory is extended to treat dynamical phenomena by introducing time depen dence to the cluster amplitudes. This allows the calculation both of non-linear dynamical behaviour and of dynamical linear response and Green’s functions. Practical approximation schemes are considered. In a SUBn approximation the m-body amplitudes are restricted to the order m < n and the energy functional is a finite-order multinomial in the amplitudes to be variationally determined. It is shown that the solution corresponds to summing well-defined subsets of Goldstone diagrams. These subsets are conveniently specified in terms of tree structures, the normal or extended g.t.0. trees. The extended expS method is in the SUBn approximation able to sum. in addition to the normal SUBn diagrams, a set which contains m-body cluster amplitudes of arbitrarily high order (m > n) in the ordinary sense. The article also discusses how the SUBn truncation schemes must be modified to be able to treat a system with a strong repulsive core in the two-body interaction. The method is formulated for the general cases of Bose and Fermi systems which may or may not conserve total particle number. It is shown that the simplest approximation, SUBl. in the extended expS method agrees with the mean field theory. which is the coherent-state approximation in the boson case or the Hartree-Fock approximation in the fermion case. It is argued that the extended expS method already in low-order approximations can realistically treat a great variety of diverse many-body problems, including even systems which may undergo groundstate phase transitions. A few applications are described in more detail. The Bose liquid is treated in the extended SUB2 approximation. It is shown that the ground-state results in the uniform limit are exact and agree with the hypernetted-chain approximation. The modifications due to hard-core interactions and the non-linear equations of motion are also discussed in this case. For Fermi systems it is shown that the superconductive phase transition of the BCS model Hamiltonian and the deformation phase transrtion of the Lipkin model are properly obtained by the extended expS method in a low-order approximation.
Eirtstein
Gravity
Washington,
Emerging
Seattle,
from
Washington
Quantum
B’qr’l Gravity.
A. ZEE. Department
of Physics,
University
of
98195.
A conformal invariant world described by the sum of the Weyl, Dirac, and Yang-Mills actions is advocated. Quantum fluctuations bring back Einstein gravity so that the long-distance phenomenology is as observed. Formulas for the induced Newton’s constant and Eddington’s constant are derived in quan tized Weyl gravity. The analogue of the trace anomaly for the Weyl action is shown to be structurally similar to that for the Yang-Mills action.