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arXiv:1611.09725 · Journal of Physics A Mathematical and Theoretical

Coherent functional expansions in quantum field theory

P. D. Drummond

Published Sep 19, 2017• 1 citation
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Abstract

Abstract A new formalism is introduced to treat problems in quantum field theory, using coherent functional expansions rather than path integrals. The basic results and identities of this approach are developed. In the case of a Bose gas with point-contact interactions, this leads to a soluble functional equation in the weak interaction limit, where the perturbing term is part of the kinetic energy. This approach has the potential to prevent the Dyson problem of divergence in perturbation theory.

DOI
10.1088/1751-8121/aa8d8c

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Abstract A new formalism is introduced to treat problems in quantum field theory, using coherent functional expansions rather than path integrals. The basic results and identities of this approach are developed. In the case of a Bose gas with point-contact interactions, this leads to a soluble functional equation in the weak interaction limit, where the perturbing term is part of the kinetic energy. This approach has the potential to prevent the Dyson problem of divergence in perturbation theory.
We construct a quantum kinetic theory of a weakly interacting critical boson gas using the expectation values of products of Heisenberg field operators in the grand canonical ensemble. Using a functional representation for the Wick theorem for time-ordered products, we construct a perturbation theory for the generating functional of these time-dependent Green’s functions at a finite temperature. We note some problems of the functional-integral representation and discuss unusual apparent divergences of the perturbation expansion. We propose a regularization of these divergences using attenuating propagators. Using a linear transformation to variables with well-defined scaling dimensions, we construct an infrared effective field theory. We show that the structure of the regularized model is restored by renormalization. We propose a multiplicatively renormalizable infrared effective model of the quantum dynamics of a boson gas.
We study perturbation theory in certain quantum mechanics problems in one dimension in which the perturbing potential diverges at some points, even though the energy eigenvalues are smooth functions of the coefficient of the potential. We discuss some of the unusual techniques which are required to obtain perturbative expansions of the energies in such cases. These include a point-splitting prescription with a δ′′ function for expanding around the Dirichlet (fermionic) limit of the δ function potential, and performing a similarity transformation to a non-Hermitian potential in the Calogero–Sutherland model. As an application of the first technique, we study the ground state of the δ function Bose gas near the fermionic limit.
The path-integral is a formulation of quantum mechanics equivalent to the standard one, offering a new point oh view of the subject which is more intuitive than the usual approach. In this work we firstly present the Dirac orginal idea and then we describe how Feynman developed it. Therefore we apply the path-integral approach to the case of a scalar field theory, introducing the concept of generating functional. Finally we illustrate two alternative representations of the generating functional, dual to the Schwinger formalism, discussing how they lead to a realation with the Hermite polynomials. We also introduce the Schwinger-Dyson equation and express it through the representations introduced.
Strongly correlated quantum field theories present a formidable challenge to conventional theoretical methods, particularly those rooted in perturbative expansions. The breakdown of perturbation theory in regimes where interaction strengths are comparable to or exceed kinetic energies necessitates the development of non-perturbative approaches. This paper introduces and elaborates on a novel framework for diagrammatic renormalization specifically designed for strongly correlated quantum fields, moving beyond the limitations inherent in iterative or weak-coupling expansions. We explore how a systematic resummation of infinite classes of diagrams, guided by a functional renormalization group perspective, can capture the complex interplay of quantum fluctuations and emergent phenomena characteristic of these systems. Our approach leverages a self-consistent treatment of Green's functions and self-energies, allowing for the faithful description of spectral functions, phase transitions, and exotic ground states in a manner that traditional perturbation theory cannot. We discuss the theoretical underpinnings, computational strategies, and potential applications across diverse domains, including condensed matter physics, quantum chromodynamics, and ultracold atomic systems. The methodology aims to provide a robust, diagrammatically transparent pathway to understanding the non-trivial fixed points and scaling behaviors that govern strongly correlated phenomena, offering a powerful alternative where standard approximations fail.
In this thesis the path integral formalism is applied to the calculation of the dynamics of dissipative quantum systems. The time evolution of a system of bilinearly coupled bosonic modes is treated using the real-time path integral technique in coherent-state representation. This method is applied to a damped harmonic oscillator within the Caldeira-Leggett model. In order to get the stationary trajectories the corresponding Lagrangian function is diagonalized and then the path integrals are evaluated by means of the stationary-phase method. The time evolution of the reduced density matrix in the basis of coherent states is given in simple analytic form for weak system-bath coupling, i.e. the so-called rotating-wave terms can be evaluated exactly but the non-rotating-wave terms only in a perturbative manner. The validity range of the rotating-wave approximation is discussed from the viewpoint of spectral equations. In addition, it is shown that systems without initial system-bath correlations can exhibit initial jumps in the population dynamics even for rather weak dissipation. Only with initial correlations the classical trajectories for the system coordinate can be recovered. The path integral formalism in a combined phase-space and coherent-state representation is applied to the problem of curve-crossing dynamics. The system of interest is described by two coupled one-dimensional harmonic potential energy surfaces interacting with a heat bath. The mapping approach is used to rewrite the Lagrangian function of the electronic part of the system. Using the Feynman-Vernon influence-functional method the bath is eliminated whereas the non-Gaussian part of the path integral is treated using the perturbation theory in the small coordinate shift between potential energy surfaces. The vibrational and the population dynamics is considered in a lowest order of the perturbation. The dynamics of a Gaussian wave packet is analyzed along a one-dimensional reaction coordinate. Also the damping rate of coherence in the electronic part of the relevant system is evaluated within the ordinary and variational perturbation theory. The analytic expressions for the rate functions are obtained in the low and high temperature regimes.