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Progress of Theoretical Physics

Time-Dependent Variational Approach to (1 + 1)-Dimensional Scalar-Field Solitons in Terms of Squeezed Coherent States

Y. Tsue, Y. Fujiwara

Published Aug 1, 1991• 12 citations
Publisher

Abstract

With the view of investigating quantum effects in (1 + 1)-dimensional scalar-field solitons, coupled equations for a soliton profile function and a background boson field are derived by the use of a squeezed coherent state and the time-dependent variational principle. These equations take the form of canonical equations of motion with the aid of canonicity conditions developed in the time-dependent Hartree-Fock theory. They also incorporate self-consistent higher order quantum corrections in the sense of the mean field theory. A method of heat-kernel expansion is developed to deal with the background field of static solitons. At the semi-classical level, this method can reproduce the mass corrections obtained by WKB method with respect to the sine-Gordon system and φ4-model.

DOI
10.1143/ptp.86.469

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With the view of investigating quantum effects in (1 + 1)-dimensional scalar-field solitons, coupled equations for a soliton profile function and a background boson field are derived by the use of a squeezed coherent state and the time-dependent variational principle. These equations take the form of canonical equations of motion with the aid of canonicity conditions developed in the time-dependent Hartree-Fock theory. They also incorporate self-consistent higher order quantum corrections in the sense of the mean field theory. A method of heat-kernel expansion is developed to deal with the background field of static solitons. At the semi-classical level, this method can reproduce the mass corrections obtained by WKB method with respect to the sine-Gordon system and φ4-model.
A general framework for time-dependent variational approach in terms of squeezed coherent states is constructed with the aim of describing quantal systems by means of classical mechanics including higher order quantal effects with the aid of canonicity conditions developed in the time-dependent Hartree-Fock theory. The Maslov phase occurring in a semi-classical quantization rule is investigated in this framework. In the limit of a semi-classical approximation in this approach, it is definitely shown that the Maslov phase has a geometric nature analogous to the Berry phase. It is also indicated that this squeezed coherent state approach is a possible way to go beyond the usual WKB approximation.
A general framework for a time-dependent variational approach in terms of squeezed coherent states is given with the aid of canonicity conditions developed in the time-dependent Hartree-Fock theory. By using a formula given by Balian and Brezin for canonical decompositions of generalized Bogoliubov transformations, we show that this approach is entirely equivalent to that given by Jackiw and Kerman in terms of the most general Gaussian functions, which is originally motivated for the variational definition of the effective action in the quantum field theory. The resultant equations of motion have structure of classical Hamiltonian systems consisting of displacement degree of freedom and squeezing degree of freedom in the configuration space. The initial conditions for solving these equations, particularly for the squeezing degree of freedom, are discussed in connection with the minimum-uncertainty relations of the squeezed coherent state. For a simple one-dimensional quantum-mechanical Hamiltonian with potentials of quadratic terms or less, the present formulation gives an exact result for the time-evolution of the Gaussian wave packet.
This paper introduces a novel Self-Consistent WKB-Variational Theory designed to investigate the complex dynamics of non-equilibrium quantum fields. Traditional approaches often struggle with strongly correlated systems or those far from thermal equilibrium, necessitating methods that capture both semiclassical evolution and quantum fluctuations. Our proposed framework unifies the Wentzel-Kramers-Brillouin (WKB) approximation, which provides a semiclassical description of the background field, with a time-dependent variational principle applied to quantum fluctuations around this background. The core of the theory lies in the self-consistent determination of the classical background field dynamics, influenced by quantum backreaction, and the evolution of the quantum fluctuations. This intricate interplay allows for a comprehensive description of non-perturbative phenomena, particle production, and phase transitions in dynamic settings. We elaborate on the theoretical derivation, outlining the coupled equations of motion for the background field and the variational parameters that characterize the quantum state. The advantages of this approach include its ability to address strong coupling regimes and provide a physically intuitive understanding of the interplay between classical and quantum degrees of freedom. Potential applications span diverse areas, from early universe cosmology to heavy-ion collisions and quantum quenches in condensed matter systems.
Starting from the Gaussian-variational approximation the state of the single sine-Gordon soliton is constructed using the coupled cluster method. The quantum contributions to the soliton mass and those to the space dependence of the soliton are computed and found to be non-negligible. If the coupling constant ${\ensuremath{\beta}}^{2}$ approaches the critical value $4\ensuremath{\pi}$ the mass becomes classical, but in other respects the soliton remains a quantum object.
Abstract This paper employs the Cauchy matrix approach to investigate the (1+1)-dimensional Gross-Neveu (GN) model. We establish a theoretical framework using Sylvester-type equations and master function matrices, and reveal the connection between the model’s time-dependent Hartree–Fock (TDHF) potential and the sine-Gordon equation. Through variable transformations, exact analytical N-soliton solutions are derived. The localized propagation characteristics of the soliton fundamental components and key physical quantities (condensate function and fermion density for N = 1 and N = 2) are validated via spatiotemporal evolution analysis and static slice visualizations. The results confirm the model’s full dynamical solvability and its parameter dependency. This work builds a concrete bridge between quantum field theory and classical physics, offering analytical evidence for the quantum particle behavior in the semiclassical GN model.
Toward the classical description of quantum fluctuations in quantal systems, the Maslov phase occurring in a semi-classical quantization rule is investigated in a framework of the time-dependent variational principle with squeezed coherent states. In the limit of a semi-classical approximation in this approach, it is definitely shown that the Maslov phase has a geometric nature analogous to Berry's phases or canonical phases including dynamical effects. The Maslov phase appears as a winding number in this framework. It is also indicated that this squeezed coherent state approach is a possible way to go beyond the usual WKB approximation.