Large-Order Behaviour of Perturbation Theory

✍ Scribed by J.C. LE GUILLOU and J. ZINN-JUSTIN (Eds.)

• Publisher: North Holland
• Year: 1990
• Tongue: English
• Leaves: 585
• Series: Current Physics–Sources and Comments 7
• Category: Library

✦ Synopsis

This volume is concerned with the determination of the behaviour of perturbation theory at large orders in quantum mechanics and quantum field theory, and its application to the problem of summation of perturbation series.

Perturbation series in quantum field theory and in many quantum mechanics models are only asymptotic and thus diverge for all values of the expansion parameter. Their behaviour at large orders provides information about whether they define the theory uniquely (the problem of Borel summability). It suggests methods to extract numerical information from the series when the expansion parameter is not small.

The articles reprinted here deal with the explicit evaluation of large-order behaviour in many quantum mechanics and field theory models. The large-order behaviour is related to barrier penetration effects for unphysical values of the expansion parameter, which can be calculated by WKB or instanton methods. The calculation of critical exponents of &fgr;⁴ field theory is presented as a practical application.

✦ Table of Contents

Content:
Current Physics — Sources and Comments
Page ii

Front Matter
Page iii

Copyright page
Page iv

Preface
Pages v-vi
J.-C. Le Guillou, J. Zinn-Justin

Acknowledgements
Pages vii-ix

1 - Methods and Simple Examples
Pages 1-12

2 - More Complex Systems
Pages 13-18

3 - Field Theories with Fermions
Pages 19-21

4 - Large-Order Behaviour: Difficulties, Applications
Pages 23-25

5 - Additional references
Page 27

6 - General articles
Pages 29-31

Divergence of Perturbation Theory in Quantum Electrodynamics
Pages 32-33
F.J. DYSON

Theory of Bound States in a Random Potential
Pages 34-40
J. ZITTARTZ, J.S. LANGER

Anharmonic Oscillator. II. A Study of Perturbation Theory in Large Order
Pages 41-57
Carl M. Bender, Tai Tsun Wu

Physical Review Letters: Statistical Analysis of Feynman Diagrams
Pages 58-61
Carl M. Bender, Tai Tsun Wu

Behavior of Very-High-Order Perturbation Diagrams
Pages 62-78
C.S. LAM

Divergence of the perturbation-theory series and pseudoparticles
Pages 79-82
L.N. Lipatov

Divergence of the perturbation-theory series and the quasi-classical theory
Pages 83-90
L.N. Lipatov

Perturbation theory at large order. I. The Φ^2N interaction
Pages 91-104
E. Brézin, J.C. Le Guillou, J. Zinn-Justin

Perturbation theory at large order. II. Role of the vacuum instability
Pages 105-111
E. Brézin, J.-C. Le Guillou, J. Zinn-Justin

Asymptotic Estimates in Perturbation Theory
Pages 112-114
G. PARISI

Calculation of the Green Functions by the Coupling Constant Dispersion Relations
Pages 115-116
E.B. BOGOMOLNY

Asymptotic Estimates of Feynman Diagrams
Pages 117-120
G. PARISI

Large Order Expansion in Perturbation Theory
Pages 121-146
JOHN C. COLLINS, DAVISON E. SOPER

On the Perturbation Theory of the Anharmonic Oscillator at Large Orders
Pages 147-169
G. AUBERSON, G. MENNESSIER, G. MAHOUX

Perturbation theory at large orders for a potential with degenerate minima
Pages 170-174
E. Brézin, G. Parisi, J. Zinn-Justin

Calculation of Instanton–Anti-Instanton Contributions in Quantum Mechanics
Pages 175-179
E.B. BOGOMOLNY

Expansion around instantons in quantum mechanics
Pages 180-189
J. Zinn-Justin

Multi-Instanton Contributions in Quantum Mechanics
Pages 190-205
J. ZINN-JUSTIN

Multi-Instanton Contributions in Quantum Mechanics (II)
Pages 206-221
J. ZINN-JUSTIN

Instantons in quantum mechanics: Numerical evidence for a conjecture
Pages 222-228
J. Zinn-Justin

Large-order perturbation theory for the O(2) anharmonic oscillator with negative anharmonicity and for the double-well potential
Pages 229-239
R Damburg, R Propint, V Martyshchenko

Borel Summability: Application to the Anharmonic Oscillator
Pages 240-243
S. GRAFFI, V. GRECCHI, B. SIMON

The Zeeman Effect Revisited
Pages 244-246
J. AVRON, I. HERBST, B. SIMON

Bender-Wu Formula, the SO(4,2) Dynamical Group, and the Zeeman Effect in Hydrogen
Pages 247-249
J.E. Avron, B.G. Adams, J. Čížek, M. Clay, M.L. Glasser, P. Otto, J. Paldus, E. Vrscay

Dispersion Relations for the Zeeman Effect in the Hydrogen Atom
Pages 250-255
S.C. KANAVI, S.H. PATIL

Bender-Wu formulas for degenerate eigenvalues
Pages 256-258
B.G. Adams, J.E. Avron, J. Čížek, P. Otto, J. Paldus, R.K. Moats, H.J. Silverstone

The Hydrogen Atom in Strong Magnetic Fields: Summation of the Weak Field Series Expansion
Pages 259-286
J.C. LE GUILLOU, J. ZINN-JUSTIN

Physical Review Letters: Stark Effect Revisited
Pages 287-289
I.W. Herbst, B. Simon

Bender-Wu Formula and the Stark Effect in Hydrogen
Pages 290-293
L. Benassi, V. Grecchi, E. Harrell, B. Simon

Errata
Page 294

Expansion of the H₂⁺ ground state energy in inverse powers of the distance between the two protons
Pages 295-296
E. Brézin, J. Zinn-Justin

1/R Expansion for H₂⁺: Analyticity, Summability, Asymptotics, and Calculation of Exponentially Small Terms
Pages 297-300
Robert J. Damburg, Rafail Kh. Propin, Sandro Graffi, Vincenzo Grecchi, Evans M. Harrell II, Jiří Čížek, Josef Paldus, Harris J. Silverstone

Asymptotic Estimates in Scalar Electrodynamics
Pages 301-305
C. Itzykson, G. Parisi, J.B. Zuber

High Orders of the Perturbation Theory in Scalar Electrodynamics
Pages 306-308
A.P. BUCHVOSTOV, L.N. LIPATOV

Asymptotic estimates of high order perturbation theory approximations in scalar electrodynamics
Pages 309-317
A.P. Bukhvostov, L.N. Lipatov

Large Order Calculations in Gauge Theories
Pages 318-321
E.B. BOGOMOLNY, V.A. FATEYEV

Large-order estimates for perturbation theory of a Yang-Mills field coupled to a scalar field
Pages 322-331
L.N. Lipatov, A.P. Bukhvostov, E.I. Malkov

High-order behavior in Φ³ field theories and the percolation problem
Pages 332-340
A. Houghton, J.S. Reeve, D.J. Wallace

Large-order behaviour of the 1/N expansion in zero and one dimensions
Pages 341-352
S Hikami, E Brézin

The Non-Linear Sigma Model in the 1/N Expansion and the Inverse Scattering Transformation in the Angular Momentum
Pages 353-357
H.J. de VEGA

Classical solutions by inverse scattering transformation in any number of dimensions. I. The gap equation and the effective action
Pages 358-370
J. Avan, H.J. de Vega

Classical solutions by inverse scattering transformation in any number of dimensions. II. Instantons and large orders of the 1/N series for the theory in v dimensions (1 ≤ v ≤ 4)
Pages 371-382
J. Avan, H.J. de Vega

Asymptotic Estimates in Perturbation Theory with Fermions
Pages 383-385
G. PARISI

Asymptotic estimates in quantum electrodynamics
Pages 386-403
C. Itzykson, G. Parisi, J-B. Zuber

Asymptotic estimates in quantums electrodynamics. II
Pages 404-415
R. Balian, C. Itzykson, J.B. Zuber, G. Parisi

The Dyson Instability and Asymptotics of the Perturbation Series in QED
Pages 416-418
E.B. BOGOMOLNY, V.A. FATEYEV

Asymptotic estimates for graphs with a fixed number of fermion loops in quantum electrodynamics. The choice of the form of the steepest-descent solutions
Pages 419-424
E.B. Bogomol'ny, Yu.A. Kubyshin

Asymptotic estimates for diagrams with a fixed number of fermion loops in quantum electrodynamics. The extremal configurations with the symmetry group O(2) ⊗ O(3)
Pages 425-430
E.B. Bogomol'nyi, Yu. A. Kubyshin

Eighth-Order Anomalous Magnetic Moment of the Electron
Pages 431-434
T. Kinoshita, W.B. Lindquist

Vacuum Energy Density in Large Orders of Perturbation Theory for the Scalar Yukawa₂ Field Theory
Pages 435-437
M.P. FRY

On High Order Estimates in QED
Pages 438-440
B. LAUTRUP

On Borel Singularities in Quantum Field Theory
Pages 441-444
S. CHADHA, P. OLESEN

On Vacuum Instability in Quantum Field Theory
Pages 445-447
P. OLESEN

Singularities of the Borel Transform in Renormalizable Theories
Pages 448-449
G. PARISI

Ambiguities of Renormalized Φ₄⁴ Field Theory and the Singularities of its Borel Transform
Pages 450-454
M.C. BERGÈRE, F. DAVID

On Infrared Divergences
Pages 455-464
G. PARISI

Non-Perturbative Effects and Infrared Renormalons within the 1/N Expansion of the O(N) Non-Linear Sigma Model
Pages 465-492
F. DAVID

On the Ambiguity of Composite Operators, ir Renormalons and the Status of the Operator Product Expansion
Pages 493-507
F. DAVID

The Operator Product Expansion and Renormalons: A Comment
Pages 508-519
F. DAVID

Physical Review Letters: Ising-Model Critical Indices in Three Dimensions from the Callan-Symanzik Equation
Pages 520-523
George A. Baker Jr., Bernie G. Nickel, Melville S. Green, Daniel I. Meiron

Workshop on Padé approximants
Page 524

Transformation of an asymptotic series in a convergent one
Pages 525-526
J.J. LOEFFEL

Critical Exponents for the n-Vector Model in Three Dimensions from Field Theory
Pages 527-530
J.C. Le Guillou, J. Zinn-Justin

Critical exponents from field theory
Pages 531-553
J.C. Le Guillou, J. Zinn-Justin

Accurate critical exponents from the ε-expansion
Pages 554-558
J.C. Le Guillou, J. Zinn-Justin

Accurate critical exponents for Ising like systems in non-integer dimensions
Pages 559-564
J.C. Le Guillou, J. Zinn Justin

Le Journal de Physique: Accurate critical exponents from field theory
Pages 565-570
J.C. Le Guillou, J. Zinn-Justin

Essentially perturbative quantities in the Kondo model
Pages 571-580
A. Berkovich, J.H. Lowenstein