A unified formulation of the construction of variational principles

Gerjuoy, E.; Rau, A. R. P.; Spruch, Larry

doi:10.1103/revmodphys.55.725

Cited by 119 publications

(78 citation statements)

References 140 publications

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“…There exists a systematic procedure for constructing variational principles, and hence variational approximations, especially designed to evaluate some quantity of interest [1,2]. Let us briefly recall this method.…”

Section: Variational Principle For the Dynamics Of The Generating Funmentioning

confidence: 99%

“…whereρ ≡ ρ (0) +ρ (1) is itself expanded up to first-order in the ζ's (rather than in the sources as in sect. 4).…”

Section: Extension To Linear Responsesmentioning

confidence: 99%

“…Following the general scheme for building variational principles [1,2], we shall treat the equation (1.8) for A(t) as a set of constraints, with which we associate Lagrangian multipliers (sect. 3).…”

Section: Heisenberg Equation Suffers From Two Defects (I) For a Timementioning

confidence: 99%

“…We shall rely on a general method [1,2] that allows for a systematic construction of variational principles specifically suited to the optimization of a desired quantity. A prerequisite is the knowledge of some simple set of equations characterizing this desired quantity, which is here ϕ.…”

Section: (17)mentioning

confidence: 99%

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Variational approach to multi-time correlation functions

Balian¹,

Vénéroni²

1993

Nuclear Physics B

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URL: http://www-spht.cea.fr/articles/T93/030International audienceA variational method is proposed to evaluate the generating functional $ \varphi $ which gives the multi-time correlation functions in equilibrium and non-equilibrium statistical mechanics or field theories. Its definition, $ \varphi = {\rm ln\ Tr} \ D\ A, $ involves the density operator $ D $ describing the initial state and an operator $ A $ depending on the observables of interest (in the Heisenberg picture), on the associated time-dependent sources and on the initial time. Regarded as function of this time, $ A $ is specified by a simple differential equation with the time running backward. Lagrangian multipliers are introduced to account both for this equation and for the one which characterizes $ D. $ Through this procedure $ \varphi $ is obtained as the stationary value of a functional depending on state-like and observable-like trial operators, with a complex time. Within a trial subspace, the resulting approximation optimizes both the initial state and the dynamics. The example of interacting fermions in many-body physics is worked out by restricting the trial objects to exponentials of single-particle operators. This leads to an extended mean-field approximation for the generating functional associated with any set of observables. Expansion in powers of the sources provides a variational approximation for the two-time causal or response functions. The result incorporates both the static and dynamic Hartree-Fock equations as well as the associated RPA equations. It is free of several inconsistencies occurring in the conventional mean-field approximations. ----- Les auteurs proposent une méthode variationnelle très souple pour évaluer la fonction génératrice des fonctions de corrélations d'observables arbitraires dans un système de mécanique quantique général, où l'état initial est une matrice densité arbitraire. L'un des aspects originaux est d'envisager la dépendance des quantités dans la variable temporelle où sont identifiées les représentations de Schrödinger et de Heisenberg. Les données et les équations du mouvement sont traitées comme des contraintes dans ce formalisme en introduisant des multiplicateurs de Lagrange appropriés. Les auteurs présentent à titre d'application l'exemple d'un système de fermions en interaction et trouvent des généralisations des diverses méthodes de champ moyen, statique ou dynamique (approximation de Hartree-Fock stationnaire ou dépendant du temps, méthode des phases aléatoires...). Ils montrent que leur nouvelle théorie est plus cohérente que celles envisagées jusqu'ici. \vskip 20mm \hfill{C. Itzykson

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Section: Variational Principle For the Dynamics Of The Generating Funmentioning

confidence: 99%

“…whereρ ≡ ρ (0) +ρ (1) is itself expanded up to first-order in the ζ's (rather than in the sources as in sect. 4).…”

Section: Extension To Linear Responsesmentioning

confidence: 99%

Section: Heisenberg Equation Suffers From Two Defects (I) For a Timementioning

confidence: 99%

Section: (17)mentioning

confidence: 99%

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Variational approach to multi-time correlation functions

Balian¹,

Vénéroni²

1993

Nuclear Physics B

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“…This principle is an illustration of the general method developped in reference [13] to construct a variational principle suited to the evaluation of a desired quantity. The basic idea is to consider the equation (2.3) which defines A(J, t 0 ) as a constraint and to introduce a lagrangian multiplier D(t 0 ) associated to this constraint.…”

Section: Variational Evaluation Of the Generating Functionalmentioning

confidence: 99%

Variational Approximations for Correlation Functions in Quantum Field Theories

Martin

1999

Annals of Physics

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Applying the time-dependent variational principle of Balian and Vénéroni, we derive variational approximations for multi-time correlation functions in Φ 4 field theory. We assume first that the initial state is given and characterized by a density operator equal to a Gaussian density matrix. Then, we study the more realistic situation where only a few expectation values are given at the initial time and we perform an optimization with respect to the initial state. We calculate explicitly the two-time correlation functions with two and four field operators at equilibrium in the symmetric phase. *

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On the Absence of Anomalous Singularities in the S‐Matrix Version of the Kohn Variational Principle for Quantum Scattering

Gaucher

Miller

1989

Israel Journal of Chemistry

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Recent calculations by Apagyi, B.; Levay, P.; Ladanyi, K. (Phys. Rev. A, 1988,37: 4577) have shown that anomalous (i.e., spurious, unphysical) singularities arise in e -+ H( Is) scattering calculations using the Schwinger variational principle.It has been known for a long time that such singularities can exist in the K-matrix version of the Kohn variational principle, but this is the first time (to our knowledge) that they have been seen in the Schwinger variational principle. This paper reports the results of calculations for this same problem using the S-matrix version of the Kohn variational principle (Zang, J.Z.H.; Chu, S.l.; Miller, W.H. J. Chern. Phys., 1988, 88: 6233), showing that there are no such anomalous singularities in this approach.

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A unified formulation of the construction of variational principles

Cited by 119 publications

References 140 publications

Variational approach to multi-time correlation functions

Variational approach to multi-time correlation functions

Variational Approximations for Correlation Functions in Quantum Field Theories

On the Absence of Anomalous Singularities in the S‐Matrix Version of the Kohn Variational Principle for Quantum Scattering

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