- 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 Non-Linear σ-Model: An Example Of a Non-Linear Symmetry -
15 General Non-Linear Models In Two Dimensions -
16 St And Brs Symmetries, Stochastic Field Equations -
17 From Langevin Equation To Supersymmetry -
18 Abelian Gauge Theories -
19 Non-Abelian 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 Non-Magnetic Systems and The (φ2)2Field Theory -
29 Calculation Of Universal Quantities -
30 The O(N) Vector Model For N Large -
31 Phase Transitions Near Two Dimensions -
32 Two-Dimensional 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 Multi-Instantons 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 ZINN-JUSTIN
- 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 position-momentum 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 S-matrix 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 Non-Linear σ-Model: An Example Of a Non-Linear Symmetry -
15 General Non-Linear Models In Two Dimensions -
16 St And Brs Symmetries, Stochastic Field Equations -
17 From Langevin Equation To Supersymmetry -
18 Abelian Gauge Theories -
19 Non-Abelian 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 Non-Magnetic Systems and The (φ2)2Field Theory -
29 Calculation Of Universal Quantities -
30 The O(N) Vector Model For N Large -
31 Phase Transitions Near Two Dimensions -
32 Two-Dimensional 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 Multi-Instantons In Quantum Mechanics -
Index
