Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Variational approximations for correlation functions in quantum field theories

Cécile Martin

Published Jan 1, 1999• 4 citations
OpenAlex

Abstract

Applying the time-dependent variational principle of Balian and Vénéroni, we derive variational approximations for multi-time correlation functions in 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.

Related papers

Variational approximation for two-time correlation functions in Φ4 theory: Optimization of the dynamics

Cécile Martin

We apply the time-dependent variational principle of Balian and V\'en\'eroni to ${\mathrm{\ensuremath{\Phi}}}^{4}$ theory. An appropriate parametrization for the variational objects allows us to write coupled dynamical equations from which we derive approximations for the two-time correlation functions involving two, three, or four field operators. \textcopyright{} 1995 The American Physical Society.

arXiv:hep-th/9505031•Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Dec 15, 1995•17 citations

Variational Multi-Time Green’s Functions for Nonequilibrium Quantum Fields

Mohamed Benarous

The time-dependent variational principle proposed by Balian and Vénéroni is used to provide the best approximation to the generating functional for multi-time Green’s functions of a set of (bosonic) observables Qµ. By suitably restricting the trial spaces, the computation of the two-time Green’s function, obtained by a second order expansion in the sources, is considerably simplified. This leads to a tractable formalism suited to quantum fields out of equilibrium. We propose an illustration on the finite temperature Φ4-theory in curved space and coupled to gravity.

•Jan 1, 2008•5 citations

A perturbation theory for multi-time correlation functions in open quantum systems

Piotr Szańkowski

Dynamical maps are the principal subject of the open system theory. Formally, the dynamical map of a given open quantum system is a density matrix transformation that takes any initial state and sends it to the state at a later time. Physically, it encapsulates the system’s evolution due to coupling with its environment. Hence, the theory provides a flexible and accurate framework for computing expectation values of open system observables. However, expectation values-or more generally, single-time correlation functions-capture only the simplest aspects of a quantum system’s dynamics. A complete characterization of the dynamics requires access to multi-time correlation functions as well: phenomena like detailed balance, linear and non-linear response, non-equilibrium transport in general, or even sequential measurements of system observables are all described in terms of multi-time correlations. For closed systems, such correlations are well-defined, even though knowledge of the system’s state alone is insufficient to determine them fully. In contrast, the standard dynamical map formalism for open systems does not account for multi-time correlations, as it is fundamentally limited to describing state evolution. Here, we extend the scope of open quantum system theory by developing a systematic perturbation theory for computing multi-time correlation functions.

arXiv:2502.19137•SciPost Physics•Sep 3, 2025•1 citation

Time-dependent variational principle for quantum dynamics in the Schrodinger-picture

Fred Cooper, So-Young Pi

Starting with the exact variational formulation of quantum mechanics (or quantum field theory) in the Schroedinger picture, and using Gaussian trail wave functions, an exact functional Schroedinger equation is obtained. The time-dependent Hartree-Fock approximation is obtained and extended to many oscillators and to field theory. The problem of quantum roll, in which an initial Gaussian wave packet starting at an unstable classical equilibrium point evolves in time, is treated using this formalism. Quantum tunnelling is also studied. (LEW)

•Jan 1, 1986•1 citation

7 On the Poisson Structure of the Time-Dependent Mean-Field Equations For Systems of Bosons out of Equilibrium

Mohamed Benarous

We analyze the Poisson structure of the time-dependent mean-field equations for bosons and construct the Lie-Poisson bracket associated to these equations. The latter follow from the time-dependent variational principle of Balian and Veneroni when a gaussian Ansatz is chosen for the density operator. We perform a stability analysis of both the full and the linearized equations. We also search for the canonically conjugate variables. In certain cases, the evolution equations can indeed be cast in a Hamiltonian form.

•Jan 1, 1998•6 citations

Gaussian time-dependent variational principle for the Bose-Hubbard model

Tommaso Guaita, Lucas Hackl, Tao Shi +3 more

We systematically extend Bogoliubov theory beyond the mean-field approximation of the Bose-Hubbard model in the superfluid phase. Our approach is based on the time-dependent variational principle applied to the family of all Gaussian states (i.e., Gaussian TDVP). First, we find the best ground-state approximation within our variational class using imaginary time evolution in 1D, 2D, and 3D. We benchmark our results by comparing to Bogoliubov theory and DMRG in 1D. Second, we compute the approximate one- and two-particle excitation spectrum as eigenvalues of the linearized projected equations of motion (linearized TDVP). We find the gapless Goldstone mode, a continuum of two-particle excitations and a doublon mode. We discuss the relation of the gap between Goldstone mode and two-particle continuum to the excitation energy of the Higgs mode. Third, we compute linear response functions for perturbations describing density variation and lattice modulation and discuss their relations to experiment. Our methods can be applied to any perturbations that are linear or quadratic in creation/annihilation operators. Finally, we provide a comprehensive overview how our results are related to well-known methods, such as traditional Bogoliubov theory and random phase approximation.

arXiv:1907.04837•Physical review. B./Physical review. B•Sep 24, 2019•45 citations
Applying the time-dependent variational principle of Balian and Vénéroni, we derive variational approximations for multi-time correlation functions in 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.
The time-dependent variational principle proposed by Balian and Vénéroni is used to provide the best approximation to the generating functional for multi-time Green’s functions of a set of (bosonic) observables Qµ. By suitably restricting the trial spaces, the computation of the two-time Green’s function, obtained by a second order expansion in the sources, is considerably simplified. This leads to a tractable formalism suited to quantum fields out of equilibrium. We propose an illustration on the finite temperature Φ4-theory in curved space and coupled to gravity.
Dynamical maps are the principal subject of the open system theory. Formally, the dynamical map of a given open quantum system is a density matrix transformation that takes any initial state and sends it to the state at a later time. Physically, it encapsulates the system’s evolution due to coupling with its environment. Hence, the theory provides a flexible and accurate framework for computing expectation values of open system observables. However, expectation values-or more generally, single-time correlation functions-capture only the simplest aspects of a quantum system’s dynamics. A complete characterization of the dynamics requires access to multi-time correlation functions as well: phenomena like detailed balance, linear and non-linear response, non-equilibrium transport in general, or even sequential measurements of system observables are all described in terms of multi-time correlations. For closed systems, such correlations are well-defined, even though knowledge of the system’s state alone is insufficient to determine them fully. In contrast, the standard dynamical map formalism for open systems does not account for multi-time correlations, as it is fundamentally limited to describing state evolution. Here, we extend the scope of open quantum system theory by developing a systematic perturbation theory for computing multi-time correlation functions.
Starting with the exact variational formulation of quantum mechanics (or quantum field theory) in the Schroedinger picture, and using Gaussian trail wave functions, an exact functional Schroedinger equation is obtained. The time-dependent Hartree-Fock approximation is obtained and extended to many oscillators and to field theory. The problem of quantum roll, in which an initial Gaussian wave packet starting at an unstable classical equilibrium point evolves in time, is treated using this formalism. Quantum tunnelling is also studied. (LEW)
We analyze the Poisson structure of the time-dependent mean-field equations for bosons and construct the Lie-Poisson bracket associated to these equations. The latter follow from the time-dependent variational principle of Balian and Veneroni when a gaussian Ansatz is chosen for the density operator. We perform a stability analysis of both the full and the linearized equations. We also search for the canonically conjugate variables. In certain cases, the evolution equations can indeed be cast in a Hamiltonian form.
We systematically extend Bogoliubov theory beyond the mean-field approximation of the Bose-Hubbard model in the superfluid phase. Our approach is based on the time-dependent variational principle applied to the family of all Gaussian states (i.e., Gaussian TDVP). First, we find the best ground-state approximation within our variational class using imaginary time evolution in 1D, 2D, and 3D. We benchmark our results by comparing to Bogoliubov theory and DMRG in 1D. Second, we compute the approximate one- and two-particle excitation spectrum as eigenvalues of the linearized projected equations of motion (linearized TDVP). We find the gapless Goldstone mode, a continuum of two-particle excitations and a doublon mode. We discuss the relation of the gap between Goldstone mode and two-particle continuum to the excitation energy of the Higgs mode. Third, we compute linear response functions for perturbations describing density variation and lattice modulation and discuss their relations to experiment. Our methods can be applied to any perturbations that are linear or quadratic in creation/annihilation operators. Finally, we provide a comprehensive overview how our results are related to well-known methods, such as traditional Bogoliubov theory and random phase approximation.