Perturbative Expansion in Field Theory

2016 ◽  
pp. 61-66
Keyword(s):  
Field Theory ◽  
Perturbative Expansion

Hyper-asymptotic expansions and instantons

2019 ◽  
pp. 471-492
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Quantum Mechanics ◽  
Asymptotic Expansions ◽  
Perturbative Expansion ◽  
Quantum Field ◽  
Wkb Method ◽  
Additional Information ◽  
Spectral Equation ◽  
Double Well Potential

Chapter 24 examines the topic of hyper–asymptotic expansions and instantons. A number of quantum mechanics and quantum field theory (QFT) examples exhibit degenerate classical minima connected by quantum barrier penetration effects. The analysis of the large order behaviour, based on instanton calculus, shows that the perturbative expansion is not Borel summable, and does not define unique functions. An important issue is then what kind of additional information is required to determine the exact expanded functions. While the QFT examples are complicated, and their study is still at the preliminary stage, in quantum mechanics, in the case of some analytic potentials that have degenerate minima (like the quartic double–well potential), the problem has been completely solved. Some examples are described in Chapter 24. There, the perturbative, complete, hyper–asymptotic expansion exhibits the resurgence property. The perturbative expansion can be related to the calculation of the spectral equation via the complex WKB method.

Field theory: Perturbative expansion and summation methods

2019 ◽  
pp. 451-470
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Renormalization Group ◽  
Conformal Mapping ◽  
Critical Exponents ◽  
Perturbative Expansion ◽  
Divergent Series ◽  
Quantum Field ◽  
Summation Methods ◽  
Borel Transformation

Chapter 23 examines perturbative expansion and summation methods in field theory. In quantum field theory, all perturbative expansions are divergent series in the mathematical sense. This leads to a difficulty when the expansion parameter is not small. In the case of Borel summable series, using the knowledge of the large order behaviour, a number of summation techniques have been developed to derive convergent sequences from divergent series. Some methods apply directly on the series like Padé approximants or order–dependent mapping (the ODM method). Others involve first a Borel transformation, like the Padé–Borel method. The method of Borel transformation, suitably modified, followed by a conformal mapping, has been applied to renormalization group (RG) functions of the phi4 3 field theory and has led to precise values of critical exponents.

scholarly journals SOME ISSUES IN CONFORMAL FIELD THEORY WITH BOUNDARIES AND CROSSCAPS

2004 ◽  
Vol 19 (supp02) ◽  
pp. 174-178
Author(s):  
B. GATO-RIVERA
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
String Theory ◽  
Open String ◽  
Perturbative Expansion ◽  
Conformal Field ◽  
The Subject

This is a brief introduction to the subject of Conformal Field Theory on surfaces with boundaries and crosscaps, which describes the perturbative expansion of open string theory.

Dimensional continuation, regularization, minimal subtraction (MS). Renormalization group (RG) functions

2021 ◽  
pp. 220-239
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Field Theory ◽  
Dimensional Regularization ◽  
Perturbative Expansion ◽  
Feynman Diagrams ◽  
Essential Property ◽  
Quantum Field ◽  
Regularization Technique ◽  
Four Dimensions ◽  
Lattice Regularization ◽  
Quantum Operators

In this chapter, the notions of dimensional continuation and dimensional regularization are introduced, by defining a continuation of Feynman diagrams to analytic functions of the space dimension. Dimensional continuation, which is essential for generating Wilson–Fisher's famous ϵexpansion in the theory of critical phenomena, and dimensional regularization seem to have no meaning outside the perturbative expansion of quantum field theory (QFT). Dimensional regularization is a powerful regularization technique, which is often used, when applicable because it leads to much simpler perturbative calculations. Dimensional regularization performs a partial renormalization, cancelling what would show up as power-law divergences in momentum or lattice regularization. In particular it cancels the commutator of quantum operators in local QFTs. These cancellations may be convenient but may also, occasionally, remove divergences that have an important physical meaning. It is not applicable when some essential property of the field theory is specific to the initial dimension. For example, in even space dimensions, the relation between γS (identical to γ5 in four dimensions) and the other γ matrices involving the completely antisymmetric tensor ϵμ1···μd, may be needed in theories violating parity symmetry. Its use requires some care in massless theories because its rules may lead to unwanted cancellations between ultraviolet and infrared logarithmic divergences. Explicit calculations at two-loop order in a scalar QFT with a general four-field interaction are performed.

scholarly journals Open giant magnons on LLM geometries

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
David Berenstein ◽  
Adolfo Holguin
Keyword(s):  
Field Theory ◽  
Angular Momentum ◽  
Dispersion Relation ◽  
Spin Chain ◽  
Radial Direction ◽  
Sigma Model ◽  
Perturbative Expansion ◽  
Additional Information ◽  
Open Strings ◽  
Special Case

Abstract We compute sigma model solutions for rigidly rotating open strings suspended between giant gravitons in general LLM geometries. These solutions are confined to the LLM plane. These all have a dispersion relation for ∆ − J that is consistent with saturation of a BPS bound of the centrally extended spin chain. For the special case of circularly symmetric LLM geometries, we can further evaluate the amount of angular momentum J carried by these strings. This quantity diverges for string configurations that try to move between different “coloring regions” in the LLM plane. All of these quantities have a perturbative expansion in the t’Hooft coupling. For the strings suspended between AdS giants, we can compute in field theory the leading result of J carried by the string via an analytic continuation of the SU(2) result, with the help of the Bethe Ansatz for the SL(2) sector. We thus provide additional information on how the radial direction of AdS arises from (open) spin chain calculations.

Critical exponents and equation of state from series summation

2021 ◽  
pp. 975-991
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Field Theory ◽  
Phase Transition ◽  
Superfluid Helium ◽  
Critical Exponents ◽  
Summation Method ◽  
Perturbative Expansion ◽  
Field Theories ◽  
Summation Methods ◽  
Universal Properties ◽  
Borel Transformation

Universal quantities near the phase transition of O(N) symmetric vector models, can be determined, in the framework of the (f2 )2 field theory, and the corresponding renormalization group (RG), in the form of perturbative series. The O(N) symmetric field theories describe, in particular for N = 0, the universal properties of the statistics of long polymers, for N = 1, the liquid–vapour transition, for N = 2, superfluid helium transition, and so on. Universal quantities have been calculated within two different schemes, the Wilson-Fisher ϵ = 4 − d expansion, and perturbative expansion at fixed dimensions 2 and 3 (as suggested by Parisi). In both cases, the series are divergent, and the expansion parameters are not small. In fixed dimensions smaller than 4, the series are proven to be Borel summable. For the ϵ expansion, there are reasons that the property is equally true, but a proof is lacking. With this assumption, in both cases, although the series are divergent, they define unique functions. Since the expansion parameters are not small, summation methods are then required to determine these functions. A specific summation method, based on a parametric Borel transformation and mapping, in which the knowledge of the large order behaviour has been incorporated, has been successfully applied to the series, and has led to a precise evaluation of critical exponents and other universal quantities.

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