Stochastic quanitization (namiki)( 1992)

Mikio Namiki STOCHASTIC QUANTIZATION In Collaboration with Ichiro Ohba, Keisuke Okano, Yoshiya Yamanaka, Ashok K Kapoor, Hiromichi Nakazato, and Satoshi Tanaka Springer-Verlag Berlin 1992 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 of quantization; 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 of quantum mechanics and of quantum field 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 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 Contents Preface II Chapter I Background Ideas Chapter II Elements of the Theory of Stochastic Processes Brownian motion Langevin equation and Fokker-Planck equation Eigenvalue problem of the Fokker-Planck operator 12 Path-integral representation and randomization condition 15 Operator formalism 20 Perturbation theory 25 Generating functional and Green's function 27 Chapter III General Prescription of Stochastic Quantization 31 Basic ideas of SQM 31 Simple examples 35 2.1 Harmonic oscillator 35 2.2 Free neutral scalar field 37 2.3 Anharmonic oscillator and interacting field 39 Fermion field 41 Abelian gauge field 45 Finite temperature problem 47 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 Generalized path-integral formulation Chapter IV Perturbative Approach to Scalar Field Theory Stochastic diagrams from Langevin equation Stochastic diagrams from operator formalism Reduction supersymmetry Chapter V Perturbative Approach to Gauge Fields Stochastic quantization without gauge fixing 1.1 Vacuum polarization tensor of QED 1.2 Gluon self-energy in non-Abelian gauge theory Stochastic quantization with gauge fixing 2.1 Stochastic gauge fixing 2.2 Perturbation theory of non-Abelian gauge field with stochastic gauge fixing 2.3 Discussion on the Gribov problem Chapter VI Stochastic Quantization of Constrained Systems Stochastic quantization of constrained systems Constrained Hamiltonian systems 2.1 Stochastic quantization in phase space 2.2 Systems with first class constraints Stochastic quantization of compact gauge field Chapter VII Superfield Formulation Superfield formulation of stochastic quantization Supersymmetry and Ward-Takahashi identities Dimensional reduction Connection with operator formalism Chapter VIII Renormalization Scheme in Stochastic Quantization General discussion Power counting approach to renormalization Superspace approach to renormalization 3.1 Superspace formulation of stochastic quantization 3.2 Renormaliz ability of the stochastic dynamics 3.3 Renormalization scheme and Ward identities — Scalar theory in 4dimension 3.4 Problem of the boundary condition — twisted boundary condition 3.4.1 Superspace Feynman rules and boundary conditions 3.4.2 Determinant matching and boundary conditions 3.5 Higher order calculations 3.5.1 First order results 3.5.2 Second order contributions Gauge theory 4.1 Generating functional and stochastic Ward identity 4.2 Gauge Ward identity and restricted gauge invariance 4.3 The background field method 4.3.1 The background gauge invariant stochastic generating functional 57 62 62 68 74 78 78 78 81 87 87 89 91 95 95 100 100 102 106 108 108 110 111 113 117 117 118 127 127 129 131 133 134 136 138 139 140 144 144 146 147 148 Chapter IX New Regularizations in Stochastic Quantization General approach to regularization and fictitious- time-smearing regularization Fictitious-time-smearing regularization II Continuum regularization Chapter X Generalized Langevin Equation and Anomaly Generalized Langevin equation 1.1 Basic ideas of generalized Langevin equation 1.2 SU(N) lattice gauge theory 1.3 Fermion field theory Anomaly 2.1 Chiral anomaly 2.2 Conformal anomaly Chapter XI Application to Numerical Simulations Basic procedure of Langevin simulation Langevin source method Nonlinear σ -model Lattice QCD Micro-canonical method Chapter XII Minkowski Stochastic Quantization and Complex Langevin Equation Langevin equation with a complex drift Minkowski stochastic quantization 2.1 Naive Minkowski stochastic quantization 2.2 Use of kerneled Langevin equations Numerical application of the complex Langevin equation 3.1 Positivity of the Fokker- Planck operator 3.2 Blow-up solution 3.3 A kernel and unphysical solutions 3.4 Violation of ergodicity Appendix A Differential and Integral Calculus of Grassmann Variables Differentiation Integration Appendix B Stochastic Differential Calculus — Ito and Stratonovich Calculus Wiener process and stochastic convergence Ito calculus Stratonovich calculus References 154 154 158 160 164 164 164 166 167 169 169 173 176 176 177 178 179 181 183 183 186 186 191 194 194 195 197 200 204 204 205 206 206 207 209 211 ... Scalar Field Theory Stochastic diagrams from Langevin equation Stochastic diagrams from operator formalism Reduction supersymmetry Chapter V Perturbative Approach to Gauge Fields Stochastic quantization... self-energy in non-Abelian gauge theory Stochastic quantization with gauge fixing 2.1 Stochastic gauge fixing 2.2 Perturbation theory of non-Abelian gauge field with stochastic gauge fixing 2.3 Discussion... the Gribov problem Chapter VI Stochastic Quantization of Constrained Systems Stochastic quantization of constrained systems Constrained Hamiltonian systems 2.1 Stochastic quantization in phase