Perturbation Expansion of Green Functions: Feynman Diagrams and Dyson Equation

2010 ◽  
pp. 477-496
Author(s):  
Carlo Jacoboni
Keyword(s):  
Perturbation Expansion ◽  
Dyson Equation ◽  
Feynman Diagrams ◽  
Green Functions

Target-oriented inversion using the Patched Green′s function method

Geophysics ◽  
2021 ◽  
pp. 1-77
Author(s):  
Danyelle da Silva ◽  
Edwin Fagua Duarte ◽  
Wagner Almeida ◽  
Mauro Ferreira ◽  
Francisco Alirio Moura ◽  
...  
Keyword(s):  
Condensed Matter ◽  
Waveform Inversion ◽  
Time Lapse ◽  
Velocity Model ◽  
Dyson Equation ◽  
Computational Time ◽  
Green Functions ◽  
Target Area ◽  
Computational Domain ◽  
Full Waveform

We have designed a target-oriented methodology to perform Full Waveform Inversion using a frequency-domain wave propagator based on the so-called Patched Green’s Function (PGF) technique. Originally developed in condensed matter physics to describe electronic waves in materials, the PGF technique is easily adaptable to the case of wave propagation in a spatially variable media in general. By dividing the entire computational domain into two sections, namely the target area and the outside target area, we calculate the Green Functions related to each section separately. The calculations related to the section outside the target are performed only once at the beginning of inversion, whereas the calculations in the target area are performed repeatedly for each iteration of the inversion process. With the Green Functions of the separate areas, we calculate the Green Functions of the two systems patched together through the application of a Recursive Dyson equation. By performing 2D and time-lapse experiments on the Marmousi model and a Brazilian Pre-salt velocity model, we demonstrate that the target-oriented PGF reduces the computational time of the inversion without compromising accuracy. In fact, when compared with conventional FWI results, the PGF-based calculations are identical but done in a fraction of the time.

Interacting Fields

2021 ◽  
pp. 237-252
Author(s):  
J. Iliopoulos ◽  
T.N. Tomaras
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Perturbation Expansion ◽  
Field Theories ◽  
Green Functions ◽  
Quantum Field ◽  
Minkowski Space Time ◽  
Dimensional Minkowski Space ◽  
Feynman Rules ◽  
Wightman Axioms

We present a simple form of the Wightman axioms in a four-dimensional Minkowski space-time which are supposed to define a physically interesting interacting quantum field theory. Two important consequences follow from these axioms. The first is the invariance under CPT which implies, in particular, the equality of masses and lifetimes for particles and anti-particles. The second is the connection between spin and statistics. We give examples of interacting field theories and develop the perturbation expansion for Green functions. We derive the Feynman rules, both in configuration and in momentum space, for some simple interacting theories. The rules are unambiguous and allow, in principle, to compute any Green function at any order in perturbation.

The Principles of Renormalisation

2021 ◽  
pp. 304-328
Author(s):  
J. Iliopoulos ◽  
T.N. Tomaras
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Perturbation Expansion ◽  
Power Counting ◽  
Field Theories ◽  
Asymptotic Freedom ◽  
Green Functions ◽  
Quantum Field Theories ◽  
Quantum Field

Loop diagrams often yield ultraviolet divergent integrals. We introduce the concept of one-particle irreducible diagrams and develop the power counting argument which makes possible the classification of quantum field theories into non-renormalisable, renormalisable and super-renormalisable. We describe some regularisation schemes with particular emphasis on dimensional regularisation. The renormalisation programme is described at one loop order for φ‎4 and QED. We argue, without presenting the detailed proof, that the programme can be extended to any finite order in the perturbation expansion for every renormalisable (or super-renormalisable) quantum field theory. We derive the equation of the renormalisation group and explain how it can be used in order to study the asymptotic behaviour of Green functions. This makes it possible to introduce the concept of asymptotic freedom.

Lie Group Calculation of the Green Function of Disordered Systems

1991 ◽  
Vol 253 ◽  
Author(s):  
Christian Brouder
Keyword(s):  
Green Function ◽  
Lie Group ◽  
Disordered Systems ◽  
Perturbation Expansion ◽  
Structural Disorder ◽  
Dyson Equation ◽  
Interatomic Distances ◽  
The Green Function ◽  
Atomic Centre ◽  
Structural Matrix

ABSTRACTWithin the framework of the muffin-tin multiple-scattering theory, the scattering path operators are given by the inverse of a matrix consisting of atomic t-matrices and a structural matrix. The influence of the displacement of an atomic centre on the structural matrix can be described analytically using Lie group techniques. From this analytical expression and the standard perturbation expansion of the Lippmann-Schwinger equation, it is possible to write the Green function of a disordered system as a series of terms whichare averages over configurations. These averages can be calculated analytically from themoments of the interatomic distances. Special terms of this series are then summed up toinfinity using Dyson equation. This formalism is computationally very effective to calculate electronic properties of systems with thermal or structural disorder. In this paper, the theoretical basis of this approach is briefly described and the convergence properties of the expansions are investigated.

scholarly journals Worldline Green functions for arbitrary Feynman diagrams

2007 ◽  
Vol 770 (1-2) ◽  
pp. 107-122 ◽  
Author(s):  
Peng Dai ◽  
Warren Siegel
Keyword(s):  
Feynman Diagrams ◽  
Green Functions

Spurious Structure from Approximations to the Dyson Equation

1972 ◽  
Vol 50 (19) ◽  
pp. 2286-2293 ◽  
Author(s):  
Clas Blomberg ◽  
Birger Bergersen
Keyword(s):  
Spectral Function ◽  
Spectral Properties ◽  
Perturbation Expansion ◽  
Dyson Equation ◽  
The Self ◽  
The Other ◽  
Exact Results ◽  
Electron Level ◽  
Self Energy ◽  
Other Hand

The method of calculating electron spectral properties by substituting perturbation theoretic results for the self-energy into the Dyson equation is investigated for a model of a deep electron level in which exact results are known. The method gives wrong results in the most important range and also predicts spurious structure. On the other hand, if a perturbation expansion is made directly for the spectral function after extracting the appropriate energy shifts no such difficulty arises.

scholarly journals PERTURBATIVE ANALYSIS OF CHERN–SIMONS FIELD THEORY IN THE COULOMB GAUGE

1998 ◽  
Vol 13 (11) ◽  
pp. 1773-1783 ◽  
Author(s):  
FRANCO FERRARI ◽  
IGNAZIO LAZZIZZERA
Keyword(s):  
Perturbative Expansion ◽  
Coulomb Gauge ◽  
Feynman Diagrams ◽  
Field Theories ◽  
Green Functions ◽  
Loop Order ◽  
Yang Mills ◽  
Gauge Fixing ◽  
Infrared Divergences ◽  
Chern Simons

In this paper, we analyse the perturbative aspects of Chern–Simons field theories in the Coulomb gauge. We show that in the perturbative expansion of the Green functions there are neither ultraviolet nor infrared divergences. Moreover, all the radiative corrections are zero at any loop order. Some problems connected with the Coulomb gauge fixing, like the appearance of spurious singularities in the computation of the Feynman diagrams, are discussed and solved. The regularization used here for the spurious singularities can be easily applied also to the Yang–Mills case, which is affected by similar divergences.

Perturbation theory

2021 ◽  
pp. 147-170
Author(s):  
Iosif L. Buchbinder ◽  
Ilya L. Shapiro
Keyword(s):  
Perturbation Theory ◽  
Effective Action ◽  
Momentum Space ◽  
Feynman Diagrams ◽  
Green Functions ◽  
General Description ◽  
Loop Expansion

This chapter provides a general description of perturbation theory in terms of Feynman diagrams. The general prescriptions of constructing Feynman diagrams in momentum space are given, including for an S-matrix. The connected Green functions and the corresponding generation functional are defined with full proofs. After introducing effective action, the chapter addresses loop expansion. The chapter ends with a discussion of Feynman diagrams in fermionic theory.

scholarly journals EUCLIDEAN ASYMPTOTIC EXPANSIONS OF GREEN FUNCTIONS OF QUANTUM FIELDS (II) COMBINATORICS OF THE AS OPERATION

1993 ◽  
Vol 08 (13) ◽  
pp. 2241-2286 ◽  
Author(s):  
G. B. PIVOVAROV ◽  
F. V. TKACHOV
Keyword(s):  
Asymptotic Expansions ◽  
Short Distance ◽  
Feynman Diagrams ◽  
Operator Product ◽  
Green Functions ◽  
Small Parameters ◽  
Combinatorial Technique ◽  
Ms Scheme ◽  
The Masses ◽  
Distance Operator

The results of Ref. 1 are used to obtain full asymptotic expansions of Feynman diagrams renormalized within the MS scheme in the regimes when some of the masses and external momenta are large with respect to the others. The large momenta are Euclidean, and the expanded diagrams are regarded as distributions with respect to them. The small masses may be equal to zero. The As operation for integrals is defined and a simple combinatorial technique is developed to study its exponentiation. The As operation is used to obtain the corresponding expansions of arbitrary Green functions. Such expansions generalize and improve upon the well-known short-distance, operator-product expansions, the decoupling theorem etc.: e.g. the low-energy effective Lagrangians are obtained to all orders of the inverse heavy mass. The obtained expansions possess the property of perfect factorization of large and small parameters, which is essential for meaningful applications to phenomenology. As an auxiliary tool, the inversion of the R operation is constructed. The results are valid for arbitrary QFT models.

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