 Title Pages
 Preface
 Acknowledgments
 General References

1 Algebraic Preliminaries 
2 Euclidean Path Integrals In Quantum Mechanics 
3 Path Integrals In Quantum Mechanics: Generalizations 
4 Stochastic Differential Equations: Langevin, Fokker –Planck Equations 
5 Path And Functional Integrals In Quantum Statistical Physics 
6 Quantum Evolution: From Particles To Fields 
7 Quantum Field Theory: Functional Methods and Perturbation Theory 
8 Relativistic Fermions 
9 Quantum Field Theory: Divergences and Regularization 
10 Introduction to Renormalization Theory. Renormalization Group Equations 
11 Dimensional Regularization, Minimal Subtraction: RG Functions 
12 Renormalization Of Composite Operators. Short Distance Expansion 
13 Symmetries And Renormalization 
14 The NonLinear σModel: An Example Of a NonLinear Symmetry 
15 General NonLinear Models In Two Dimensions 
16 St And Brs Symmetries, Stochastic Field Equations 
17 From Langevin Equation To Supersymmetry 
18 Abelian Gauge Theories 
19 NonAbelian Gauge Theories: Introduction 
20 The Standard Model. Anomalies 
21 Gauge Theories: Master Equation And Renormalization 
22 Classical And Quantum Gravity. Riemannian Manifolds And Tensors 
23 Critical Phenomena: General Considerations 
24 Mean Field Theory For Ferromagnetic Systems 
25 General Renormalization Group. The Critical Theory Near Dimension Four 
26 Scaling Behaviour In The Critical Domain 
27 Corrections to Scaling Behaviour 
28 NonMagnetic Systems and The (φ^{2})^{2}Field Theory 
29 Calculation Of Universal Quantities 
30 The O(N) Vector Model For N Large 
31 Phase Transitions Near Two Dimensions 
32 TwoDimensional Models and Bosonization Method 
33 The O(2) Classical Spin Model In Two Dimensions 
34 Critical Properties Of Gauge Theories 
35 Uv Fixed Points In Quantum Field Theory 
36 Critical Dynamics 
37 Field Theory in a Finite Geometry: Finite Size Scaling 
38 Quantum Field Theory At Finite Temperature: Equilibrium Properties 
39 Instantons In Quantum Mechanics 
40 Unstable Vacua In Quantum Field Theory 
41 Degenerate Classical Minima And Instantons 
42 Perturbation Series At Large Orders. Summation Methods 
43 MultiInstantons In Quantum Mechanics 
Index
Path And Functional Integrals In Quantum Statistical Physics
Path And Functional Integrals In Quantum Statistical Physics
 Chapter:
 (p.83) 5 PATH AND FUNCTIONAL INTEGRALS IN QUANTUM STATISTICAL PHYSICS
 Source:
 Quantum Field Theory and Critical Phenomena
 Author(s):
JEAN ZINNJUSTIN
 Publisher:
 Oxford University Press
Hamiltonians in quantum mechanics can be expressed in terms of creation and annihilation operators, instead of the more usual position and momentum operators, a method well adapted to the study of perturbed harmonic oscillators. In the holomorphic formalism these operators act by multiplication and differentiation on a vector space of analytic functions. Alternatively, they can also be represented by kernels, functions of complex variables which in the classical limit correspond to a complex parametrization of phase space. To this formalism corresponds a path integral representation of the statistical operator (the density matrix at thermal equilibrium) where paths belong to complex spaces instead of the more usual positionmomentum phase space. Its construction provides a useful introduction to the construction of the path integral for fermion degrees of freedom. Both formalisms can then be generalized to a quantum gas of Bose or Fermi particles in the grand canonical formulation. A functional integral representation of quantum partition functions can be derived, a topic discussed in the second part of the chapter. Finally, these formalisms allow the construction of the functional integral representation of the scattering Smatrix in quantum field theory shown in Chapters 6 and 8.
Keywords: quantum mechanics, holomorphic formalism, functional integrals, Bose particles, Fermi particles
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 Title Pages
 Preface
 Acknowledgments
 General References

1 Algebraic Preliminaries 
2 Euclidean Path Integrals In Quantum Mechanics 
3 Path Integrals In Quantum Mechanics: Generalizations 
4 Stochastic Differential Equations: Langevin, Fokker –Planck Equations 
5 Path And Functional Integrals In Quantum Statistical Physics 
6 Quantum Evolution: From Particles To Fields 
7 Quantum Field Theory: Functional Methods and Perturbation Theory 
8 Relativistic Fermions 
9 Quantum Field Theory: Divergences and Regularization 
10 Introduction to Renormalization Theory. Renormalization Group Equations 
11 Dimensional Regularization, Minimal Subtraction: RG Functions 
12 Renormalization Of Composite Operators. Short Distance Expansion 
13 Symmetries And Renormalization 
14 The NonLinear σModel: An Example Of a NonLinear Symmetry 
15 General NonLinear Models In Two Dimensions 
16 St And Brs Symmetries, Stochastic Field Equations 
17 From Langevin Equation To Supersymmetry 
18 Abelian Gauge Theories 
19 NonAbelian Gauge Theories: Introduction 
20 The Standard Model. Anomalies 
21 Gauge Theories: Master Equation And Renormalization 
22 Classical And Quantum Gravity. Riemannian Manifolds And Tensors 
23 Critical Phenomena: General Considerations 
24 Mean Field Theory For Ferromagnetic Systems 
25 General Renormalization Group. The Critical Theory Near Dimension Four 
26 Scaling Behaviour In The Critical Domain 
27 Corrections to Scaling Behaviour 
28 NonMagnetic Systems and The (φ^{2})^{2}Field Theory 
29 Calculation Of Universal Quantities 
30 The O(N) Vector Model For N Large 
31 Phase Transitions Near Two Dimensions 
32 TwoDimensional Models and Bosonization Method 
33 The O(2) Classical Spin Model In Two Dimensions 
34 Critical Properties Of Gauge Theories 
35 Uv Fixed Points In Quantum Field Theory 
36 Critical Dynamics 
37 Field Theory in a Finite Geometry: Finite Size Scaling 
38 Quantum Field Theory At Finite Temperature: Equilibrium Properties 
39 Instantons In Quantum Mechanics 
40 Unstable Vacua In Quantum Field Theory 
41 Degenerate Classical Minima And Instantons 
42 Perturbation Series At Large Orders. Summation Methods 
43 MultiInstantons In Quantum Mechanics 
Index