A general system composed of many weakly interacting bosons and/or fermions is considered. The object is to develop a procedure for renormalizing the single-particle creation operators, so as to remove the interactions to successively higher orders of perturbation. The basic idea is that creation operators ${{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$ must satisfy the Hamiltonian commutator equations $[\mathcal{H},{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}]={\ensuremath{\omega}}_{i}{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$, where
H is the Hamiltonian and ${\ensuremath{\omega}}_{i}$ are the particle energies. For the many-weakly-interacting-particle system, the zeroth-order boson and fermion creation operators satisfy these equations to zeroth order. It is shown that if the creation operators are renormalized so as to satisfy the Hamiltonian commutator equations to order
m, and also to satisfy the appropriate boson commutators and fermion anticommutators to order
m, then the problem is solved to order
m. In particular, the vectors formed by operating on the ground state with the renormalized creation operators, according to the usual boson and fermion occupation-number representation, are eigenvectors of
H and are orthonormal, all to order
m. A procedure is otained for finding the (
m+1)-order contributions to the particle creation energies, in terms of the
m-order operators. Explicit first-order calculations of these general results are provided for a system of bosons and a system of fermions, and these first-order results are shown to include similar results of Rayleigh-Schr\"odinger perturbation theory, the random-phase approximation, and the method of thermodynamic Green's functions. The problem of anharmonic lattice dynamics is studied in detail, and a method of undetermined coefficients is used to renormalize the phonon creation operators to first order. The phonon energies are calculated to second order, and this calculation shows that the interactions between renormalized phonons cannot be removed in second order. Statistical averages of the phonon energies give the energy shifts and lifetimes which have been calculated previously by various propagator techniques. In addition, the renormalized energies are used to calculate the temperature-dependent part of the Helmholtz free energy correct to second order. As a final example, electron-phonon interactions in a normal metal are studied. The electron and phonon creation operators are renormalized to first order, the particle energies are calculated to second order, and statistical averages of the particle energies recover the usual thermodynamic Green's-function results for energy shifts and lifetimes. These examples show the simplicity by which the renormalization procedure obtains a great amount of detailed information about the single-particle nature of a many-particle system.
We introduce a non perturbative general approximation scheme (NGAS) that can handle interactions of any strength in quantum theory. This approach starts with an input Hamiltonian that can be solved exactly. The interaction effects are then built into this Hamiltonian through nonlinear feedback enforced by self consistency conditions. While the method itself is nonperturbative it can be systematically improved using a new perturbation method called 'mean field perturbation theory' which does not involve power series expansion in any small parameter. We put this scheme to the test on one dimensional anharmonic interactions using the harmonic approximation. The results are consistently accurate across various cases including quartic, sextic, and octic anharmonic oscillators, as well as the quartic double well oscillator (QDWO) even when the coupling strength varies widely. The flexibility of the method is demonstrated when we swap the input Hamiltonian for that of an infinite square well and still achieve comparable accuracy. When applied to the λϕ4 quantum field theory this approach aligns with the Gaussian effective potential method under the harmonic approximation. Beyond that it reveals the condensate structure of the effective vacuum and highlights the instability of the perturbative ground state. Notably, our ground-state energy results for the QDWO stand in stark contrast to those from standard perturbation theory where Borel summation fails regardless of coupling strength.
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.
The Schrodinger equation with a two-dimensional delta-function potential is a simple example of an asymptotically free theory that undergoes dimensional transmutation. Renormalization requires the introduction of a mass scale, which can be lowered perturbatively until an infrared cutoff produced by non-perturbative effects such as bound state formation is encountered. We outline the effective field theory and similarity renormalization group techniques for producing renormalized cutoff hamiltonians, and illustrate the control of logarithmic and inverse-power-law errors both techniques provide.
I review the formalism, Feynman rules, and combinatorics that constrain a field to propagate ``classically", strictly in tree diagrams, either by itself, or interacting with other, purely quantum fields.
The perturbation theory is reorganized by virtue of the linear terms that introduce the constraints via Lagrange multipliers, generalizing and giving results that cannot be obtained with the standard procedures which start at the quadratic terms.
I apply the formalism to a theory of an
O(N)-symmetric quantum field interacting with a ``classical" scalar field via cubic interactions in six spacetime dimensions. Using the renormalization group, I examine the effective potential, symmetry breaking with radiative corrections, the fixed points in
d=6−ε dimensions, and compare with other works.
Other possible generalizations and applications of the formalism are also discussed.
We present a simple renormalization-group method of investigating ground-state properties of interacting bosonic systems. Our method reduces the number of particles in a system, which makes numerical calculations possible for large systems. It is conceptually simple and easy to implement, and allows investigation of properties unavailable through mean-field approximations, such as one- and two-particle reduced density matrices of the ground state. As an example, we model a weakly interacting one-dimensional Bose gas in a harmonic trap. Compared to the mean-field Gross-Pitaevskii approximation, our method provides a more accurate description of the ground-state one-particle density matrix. We have also obtained the Hall-Post lower bounds for the ground-state energy of the gas. All results have been obtained by straightforward numerical diagonalization of the Hamiltonian matrix.