Table Of ContentLecture Notes in Physics
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Mikio Namiki
Stochastic Quantization
In Collaboration with
Ichiro Ohba, Keisuke Okano, Yoshiya Yamanaka,
Ashok K. Kapoor, Hiromichi Nakazato,
and Satoshi Tanaka
Springer-Verlag
Berlin Heidelberg NewYork
London Paris Tokyo
HongKong Barcelona
Budapest
Author
MikioNamiki
WasedaUniversity, DepartmentofPhysics
3-4-1'pkubo,Shinjuku-ku,Tokyo 169,Japan
ISBN 3-540-55563-3Springer-Verlag Berlin Heidelberg NewYork
ISBN0-387-55563-3 Springer-Verlag NewYork Berlin Heidelberg
Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartof
the material is concerned, specifically the rights oftranslation, reprinting, re-use of
illustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherway,
andstorageindatabanks.Duplicationofthispublicationorpartsthereofispermitted
onlyundertheprovisionsofthe GermanCopyrightLawofSeptember9, 1965,inits
currentversion,andpermissionforusemustalwaysbeobtainedfromSpringer-Verlag.
ViolationsareliableforprosecutionundertheGermanCopyrightLaw.
©Springer-VerlagBerlinHeidelberg 1992
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Preface
This is a textbook on stochastic quantization which was originally proposed by G. Parisi and
Y.S. Wu in 1981 and then developed by many workers. I assume that the reader has finished a
standard course in quantum field theory. The Parisi-Wu stochastic quantization method gives
quantum mechanics as the thermal-equilibrium limit of a hypothetical stochastic process with
respect to some fictitious time other than ordinary time. We can consider this to be a third
method ofquantization; remarkably different from the conventional theories, i.e, the canonical
and path-integral ones. Over the past ten years, we have seen the technical merits of this
method in quantizing gauge fields and in performing large numerical simulations, which have
never been obtained by the other methods. I believe that the stochastic quantization method
has the potential to extend the territory ofquantum mechanics and ofquantumfield theory.
However, I should remark that stochastic quantization is still under development through
many mathematical improvements and physical applications, and also that the fictitious time of
the theory is only a mathematical tool, for which we do not yet know its origin in the physical
background. For these reasons, in this book, I attempt to describe its theoretical formulation in
detail as well as practical achievements.
I have organized the book by arranging and partially rewriting the following contributions:
by Mikio Namiki to Chapters I and Chapter II (supplemented by Satoshi Tanaka in Section
4), and Sections 1, 2 and 6 of Chapter III; by !chiro Ohba to Sections 4 and 5 of Chapter III,
Section 3ofChapter IV, Subsection 1.1 ofChapter V, and Section 2ofChapter VI; by Keisuke
Okano to Subsection 1.2 and Section 2 of Chapter V, Section 1of Chapter VI, Sections 3 and
4ofChapter VIII, and Section 3 ofChapter XII; by Yoshiya Yamanaka to Sections 1 and 2of
Chapter IV, Section 2 of Chapter VIII, and Chapter IX; by Ashok K. Kapoor to Section 2 of
Chapter VI, Chapter VII, and Section 1 of Chapter VIII; by Hiromichi Nakazato to Section 1
(supplementedbyKeisukeOkano) andSubsection2.1ofChapterXII; SatoshiTanakatoSections
3and 7(supplemented by YoshiyaYamanaka) ofChapter III, Section 3ofChapter VI, Chapter
X, Chapter XI, Subsection 2.2 ofChapter XII, and Appendices A and B,
Iwouldlike toexpressmanythanks to all the contributorsfor theireffort and helpin making
this book. I am very much indebted to Dr. Satoshi Tanaka for his painstaking assistance in
preparing the manuscript, and to Dr. Andrew Berkin for his valuable advice in improving the
English style ofthe original manuscript.
August 1991 Mikio Namiki
Contents
Preface ii
Chapter I Background Ideas 1
Chapter II Elements ofthe Theory ofStochastic Processes 5
1. Brownian motion 5
2. Langevin equation and Fokker-Planck equation 7
3. Eigenvalue problemofthe Fokker-Planckoperator 12
4. Path-integral representation and randomization condition 15
5. Operator formalism 20
6. Perturbation theory 25
7. Generatingfunctional and Green's function 27
Chapter III General Prescription ofStochastic Quantization 31
1. Basic ideas ofSQM 31
2. Simple examples 35
2.1 Harmonic oscillator 35
2.2 Free neutral scalar field 37
2.3 Anharmonic oscillator and interactingfield 39
3. Fermion field 41
4. Abelian gaugefield 45
5. Finite temperature problem 47
6. Five-dimensional "stochastic" field theory for SQM 50
6.1 "Stochastic-canonical" field theory- "classical" formalism 51
6.2 "Stochastic-canonical" field theory- "operator" formalism 53
7. Generalized path-integral formulation 57
Chapter IV Perturbative Approach to Scalar Field Theory 62
1. Stochastic diagramsfrom Langevin equation 62
2. Stochastic diagrams from operator formalism 68
3. Reduction supersymmetry 74
Chapter V Perturbative Approach to Gauge Fields 78
1. Stochastic quantization without gaugefixing 78
1.1 Vacuum polarization tensor ofQED 78
VIII
1.2 Gluon self-energy in non-Abeliangauge theory 81
2. Stochastic quantization withgaugefixing 87
2.1 Stochastic gauge fixing 87
2.2 Perturbation theory ofnon-Abelian gauge field with stochastic gauge fixing 89
2.3 Discussion on the Gribov problem 91
Chapter VI Stochastic Quantization ofConstrained Systems 95
1. Stochastic quantization ofconstrainedsystems 95
2. Constrained Hamiltonian systems 100
2.1 Stochastic quantizationin phase space 100
2.2 Systems withfirst class constraints 102
3. Stochastic quantization ofcompact gauge field 106
Chapter VII Superfield Formulation 108
1. Superfield formulation ofstochastic quantization 108
2. Supersymmetry and Ward-Takahashi identities 110
3. Dimensional reduction 111
4. Connection with operator formalism 113
Chapter VIII Renormalization Scheme in Stochastic Quantization 117
1. General discussion 117
2. Power counting approach to renormalization 118
3. Superspace approach to renormalization 127
3.1 Superspace formulation ofstochastic quantization 127
3.2 Renormalizability ofthestochastic dynamics 129
3.3 Renormalization scheme and Wardidentities
- Scalar theory in 4-dimension 131
3.4 Problem ofthe boundarycondition- twisted boundary condition 133
3.4.1 Superspace Feynman rules and boundary conditions 134
3.4.2 Determinant matching and boundary conditions 136
3.5 Higher order calculations 138
3.5.1 First order results 139
3.5.2 Second order contributions 140
4. Gauge theory 144
4.1 Generatingfunctional andstochastic Ward identity 144
4.2 Gauge Ward identity and restricted gauge invariance 146
4.3 The background field method 147
IX
4.3.1 The background gauge invariant stochastic generating functional 148
Chapter IX New Regularizations in Stochastic Quantization 154
1. General approach to regularization and fictitious-time-smearing regularization 154
2. Fictitious-time-smearing regularization II 158
3. Continuumregularization 160
Chapter X Generalized Langevin Equation and Anomaly 164
1. Generalized Langevin equation 164
1.1 Basic ideas ofgeneralized Langevin equation 164
1.2 SU(N) lattice gauge theory 166
1.3 Fermion field theory 167
2. Anomaly 169
2.1 Chiral anomaly 169
2.2 Conformal anomaly 173
Chapter XI Application to Numerical Simulations 176
1. Basic procedure ofLangevin simulation 176
2. Langevin source method 177
3. Nonlinear u-model 178
4. Lattice QCD 179
5. Micro-canonical method 181
Chapter XII Minkowski Stochastic Quantization and Complex Langevin Equation 183
1. Langevin equation with a complexdrift 183
2. Minkowski stochastic quantization 186
2.1 Naive Minkowski stochastic quantization 186
2.2 Use ofkerneled Langevin equations 191
3. Numerical applicationofthe complex Langevin equation 194
3.1 Positivity ofthe Fokker-Planckoperator 194
3.2 Blow-up solution 195
3.3 A kernel and unphysical solutions 197
3.4 Violation ofergodicity 200
Appendix A Differential and Integral CalculusofGrassmann Variables 204
1. Differentiation 204
2. Integration 205
x
Appendix B Stochastic Differential Calculus - Ito and Stratonovich Calculus 206
1. Wiener process and stochastic convergence 206
2. Itocalculus 207
3. Stratonovich calculus 209
References 211
