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arXiv:2310.10940

A BBGKY-like Hierarchy for Quantum Field Theories

Michael Updike, J. W. Burby

Published Oct 17, 2023• 0 citations
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Abstract

We present a Hamiltonian method of constructing BBGKY-like hierarchies for quantum field theories. With suitable choices, our method creates a hierarchical system of evolution equations for the k-th order reduced density matrices. These equations can be closed at finite order using methods developed for the classical BBGKY hierarchy to give non-perturbative approximations for the full quantum equations of motion. Classical observables can then be numerically computed from these approximate equations, providing an analytically tractable method of modeling high-energy environments where quantum effects play a pronounced role.

DOI
10.48550/arxiv.2310.10940

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We present a Hamiltonian method of constructing BBGKY-like hierarchies for quantum field theories. With suitable choices, our method creates a hierarchical system of evolution equations for the k-th order reduced density matrices. These equations can be closed at finite order using methods developed for the classical BBGKY hierarchy to give non-perturbative approximations for the full quantum equations of motion. Classical observables can then be numerically computed from these approximate equations, providing an analytically tractable method of modeling high-energy environments where quantum effects play a pronounced role.
We establish a theoretical framework for the exploration of the quantum dynamics of finite ultracold bosonic ensembles based on the Born-Bogoliubov-Green-Kirkwood-Yvon (BBGKY) hierarchy for equations of motion for few-particle reduced density matrices (RDMs). The theory applies to zero as well as low temperatures and is formulated in a highly efficient way by utilizing dynamically optimized single-particle basis states and representing the RDMs in terms of permanents with respect to those. An energy, RDM compatibility and, if applicable, symmetry conserving closure approximation is developed on the basis of a cluster expansion for these finite systems, using suitably normalized RDMs and symmetrization operators. The underlying cluster expansion is formulated recursively by means of super-operators, which allows for going to large truncation orders. In order to enforce necessary representability conditions, two novel, minimal-invasive and energy-conserving correction algorithms are proposed, involving the dynamical purification of the solution of the truncated BBGKY hierarchy and the correction of the equations of motion themselves, respectively. For gaining conceptual insights, the impact of two-particle correlations on the dynamical quantum depletion is studied analytically. The theoretical framework provided here is extensively applied and evaluated in our immediate follow-up work [1], involving truncation orders as large as twelve.
In the paper we discuss possible approaches to the problem of the rigorous derivation of quantum kinetic equations from underlying many-particle dynamics. For the description of a many-particle evolution we construct solutions of the Cauchy problems of the BBGKY hierarchy and the dual BBGKY hierarchy in suitable Banach spaces. In the framework of the conventional approach to the description of kinetic evolution the mean-field asymptotics of the quantum BBGKY hierarchy solution is constructed. We develop also alternative approaches. One method is based on the construction of the solution asymptotics of the initial-value problem of the quantum dual BBGKY hierarchy. One more approach is based on the generalized quantum kinetic equation that is a consequence of the equivalence of the Cauchy problems of such evolution equation and the BBGKY hierarchy with initial data determined by the one-particle density operator.
We establish a theoretical framework for exploring the quantum dynamics of finite ultracold bosonic ensembles based on the Born-Bogoliubov-Green-Kirkwood-Yvon (BBGKY) hierarchy of equations of motion for few-particle reduced density matrices (RDMs). The theory applies to zero as well as low temperatures and is formulated in a highly efficient way by utilizing dynamically optimized single-particle basis states and representing the RDMs in terms of permanents with respect to those. An energy, RDM compatibility and symmetry conserving closure approximation is developed on the basis of a recursively formulated cluster expansion for these finite systems. In order to enforce necessary representability conditions, two novel, minimal-invasive and energy-conserving correction algorithms are proposed, involving the dynamical purification of the solution of the truncated BBGKY hierarchy and the correction of the equations of motion themselves, respectively. For gaining conceptual insights, the impact of two-particle correlations on the dynamical quantum depletion is studied analytically. We apply this theoretical framework to both a tunneling and an interaction-quench scenario. Due to our efficient formulation of the theory, we can reach truncation orders as large as twelve and thereby systematically study the impact of the truncation order on the results. While the short-time dynamics is found to be excellently described with controllable accuracy, significant deviations occur on a longer time-scale in sufficiently far off-equilibrium situations. Theses deviations are accompanied by exponential-like instabilities leading to unphysical results. The phenomenology of these instabilities is investigated in detail and we show that the minimal-invasive correction algorithm of the equation of motion can indeed stabilize the BBGKY hierarchy truncated at the second order.
Time-resolved measurement techniques are opening a window on nonequilibrium quantum phenomena that is radically different from the traditional picture in the frequency domain. The simulation and interpretation of nonequilibrium dynamics are conspicuous challenges for theory. This paper presents an approach to quantum many-body dynamics that is based on a Hamiltonian formulation of the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy of equations of motion for reduced density matrices. These equations have an underlying symplectic structure, and we write them in the form of the classical Hamilton equations for canonically conjugate variables. Applying canonical perturbation theory or the Krylov-Bogoliubov averaging method to the resulting equations yields a systematic approximation scheme. The possibility of using memory-dependent functional approximations to close the Hamilton equations at a given level of the hierarchy is discussed. The geometric structure of the equations gives rise to reduced geometric phases that are observable even for noncyclic evolutions of the many-body state. The approach is applied to a finite Hubbard chain which undergoes a quench in on-site interaction energy UUU. Canonical perturbation theory, carried out to second order, fully captures the nontrivial real-time dynamics of the model, including resonance phenomena and the coupling of fast and slow variables.
We introduce a hierarchical system of approximations for summing both conventional perturbation theory and large N vector expansions of models in quantum field theory and condensed matter physics. Each stage of the hierarchy consists of a closed set of nonlinear equations for one particle irreducible correlation functions with no more than K points and captures the perturbation expansion of each of those functions up to some finite order. As K goes to infinity the full Schwinger Dyson hierarchy is approached. We present an argument that for ordinary finite dimensional integrals, the procedure converges to the exact answer in this limit, despite the fact that no finite order of the hierarchy sees effects from non-leading saddle points of the integral. Some potential applications to the calculation of critical exponents, to low dimensional condensed matter systems, and to the phase structure of the homogeneous electron gas are outlined, but no detailed calculations are presented.
In these notes, we review some recent mathematical results concerning the derivation of effective evolution equations from many body quantum mechanics. In particular, we discuss the emergence of the Hartree equation in the so-called mean field regime (for example, for systems of gravitating bosons), and we show that the Gross-Pitaevskii equation approximates the dynamics of initially trapped Bose-Einstein condensates. We explain how effective evolution equations can be derived, on the one hand, by analyzing the so called BBGKY hierarchy, describing the time-evolution of reduced density matrices, and, on the other hand, by studying the dynamics of coherent initial states in a Fock-space representation of the many body system.