Introduction to Quantum Field Theory,9789056992378

Introduction to Quantum Field Theory

by ;
Edition: 1st
ISBN13: 9789056992378
ISBN10: 9056992376
Format: Hardcover
Pub. Date: 2000-11-17
Publisher(s): CRC Press
List Price: $194.25

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Summary

This text explains the features of quantum and statistical field systems that result from their field-theoretic nature and are common to different physical contexts. It supplies the practical tools for carrying out calculations and discusses the meaning of the results. The central concept is that of effective action (or free energy), and the main technical tool is the path integral, although other formalisms are also mentioned. The author emphasizes the simplest models first, then progresses to discussions of real systems before addressing more general and rigorous conclusions. The book is structured around carefully selected problems, which are solved in detail.

Author Biography

Valerij G. Kiselev is a scientist at the Section of Medical Physics, Department of Diagnostic Radiology, University of Freiburg (Germany). Yakov Shnir is a leading scientist at the Institute for Theoretical Physics, University of Cologne (Germany) Arthur Ya. Tregubovich is a senior scientist at the Institute of Physics, National Academy of Sciences of Belarus

Table of Contents

Preface xiii
I THE PATH INTEGRAL IN QUANTUM MECHANICS 1(70)
Action in Classical Mechanics
3(14)
The Variational Principle and Equations of Motion
3(3)
A Mathematical Note: The Notion of the Functional
6(3)
The Action as a Function of the Boundary Conditions
9(4)
Symmetries of the Action and Conservation Laws
13(4)
The Path Integral in Quantum Mechanics
17(22)
The Green Function of the Schrodinger Equation
17(4)
The Path Integral
21(4)
The Path Integral for Free Motion
25(6)
Free Motion: Straightforward Calculation of the Path Integral
26(1)
Free Motion: Path Integral Calculation by the Stationary Phase Method
27(4)
The Path Integral for the Harmonic Oscillator
31(2)
Imaginary Time and the Ground State Energy
33(6)
The Euclidean Path Integral
39(32)
The Symmetric Double Well
39(22)
Quantum Mechanical Instantons
42(7)
The Contribution from the Vicinity of the Instanton Trajectory
49(4)
Calculation of the Functional Determinant
53(8)
A Particle in a Periodic Potential. Band Structure
61(4)
A Particle on a Circle
65(2)
Conclusions
67(4)
II INTRODUCTION TO QUANTUM FIELD THEORY 71(154)
Classical and Quantum Fields
73(26)
From Large Number of Degrees of Freedom to Particles
73(4)
Energy-Momentum Tensor
77(2)
Field Quantization
79(4)
Canonical Quantization
79(2)
Quantization via Path Integrals
81(2)
The Equivalence of QFT & Statistical Physics
83(3)
Free Field Quantization: From Fields to Particles
86(13)
Momentum Space
86(2)
Normal Modes
88(1)
Zero-Point Energy
89(2)
Elementary Excitations of the Field
91(8)
Vacuum Energy in &phis;4 Theory
99(32)
Casimir Effect
100(7)
Simple Calculation of Casimir Energy
100(3)
Casimir Energy: Calculation Via Path Integral
103(4)
Effective Potential of &phis;4 Theory
107(24)
Calculation of Ueff(&phis;)
109(2)
The Explicit Form of Ueff
111(2)
Renormalization of Mass and Coupling Constant
113(4)
Running Coupling Constant, Dimensional Transmutation and Anomalous Dimensions
117(7)
Effective Potential of the Massive Theory
124(7)
The Effective Action in &phis;4 Theory
131(28)
Correlation Functions and the Generating Functional
132(3)
Z[J], W[J] and Correlation Functions of the Free Field
135(6)
The Classical Green Function
136(2)
Correlation Functions
138(3)
Generating Functionals in &phis;4 Theory
141(7)
&phis;4 Theory
141(1)
Generating Functionals: Expansion in λ
141(5)
Generating Functionals: the Loop Expansion
146(2)
Effective Action
148(11)
Expansion of the Functional Determinant
151(8)
Renormalization of the Effective Action
159(30)
Momentum Space
159(6)
Explicit Form of the Diagrams
162(3)
The Structure of Ultraviolet Divergencies
165(3)
Pauli-Villars Regularization
168(6)
Calculation of Integrals
171(2)
About Dimensional Regularization
173(1)
The Regularized Inverse Propagator
174(5)
Analytic Continuation to Minkowski Space
176(3)
Renormalization
179(6)
Renormalization of Mass
179(2)
Renormalization of the Coupling Constant
181(1)
Renormalization of the Wave Function
182(3)
Conclusion
185(4)
Renormalization Group
189(18)
Renormalization Group
189(8)
Renormalization Group Equation
189(3)
General Solution of RG Equation
192(3)
Explicit Example
195(2)
Scale Transformations
197(3)
Scale Transformations at the Tree Level
197(2)
Gell-Mann - Low Equation
199(1)
Asymptotic Regimes
200(7)
Concluding Remarks
207(18)
Correlators in Terms of Γ[&phis;]
207(3)
On the Properties of Perturbation Series
210(9)
On the Loop Expansion Parameter
210(4)
On the Asymptotic Nature of Perturbation Series
214(5)
On &phis;4 Theory with Large Coupling Constant
219(2)
The Cases d = 2 and d = 3: Second-Order Phase transitions
219(1)
The Cases d = 4: Possible Triviality of &phis;4 Theory
220(1)
Conclusion
221(4)
III MORE COMPLEX FIELDS AND OBJECTS 225(180)
Second Quantisation: From Particles to Fields
227(20)
Identical Particles and Symmetry of Wave Functions
227(3)
Bosons
230(10)
One-Particle Hamiltonian
231(2)
Creation and Annihilation Operators
233(2)
Total Hamiltonian
235(1)
The Field Operator
236(2)
Result: Recipe for Quantisation
238(2)
Fermions
240(7)
One-Particle Hamiltonian
240(1)
Creation and Annihilation Operators
241(1)
Many-Particle Hamiltonian
242(1)
Field Operator
242(5)
Path Integral for Fermions
247(42)
On the Formal Classical Limit for Fermions
247(2)
Grassmann Algebras: A Short Introduction
249(9)
Path Integral For Non-Relativistic Fermions
258(9)
Classical Pseudomechanics
259(4)
Path Integral Quantisation
263(4)
Generating Functional For Fermionic Fields
267(5)
Coupling of the Dirac Spinor and the &phis;4 Scalar Fields
272(9)
Loop Expansion and Diagram Techniques
273(5)
Analysis of Divergences
278(3)
Fermion Contribution to the Effective Potential
281(8)
Gauge Fields
289(54)
Gauge Invariance
289(9)
The Basic Idea
289(1)
Example of a Globally Invariant Lagrangian
290(2)
Example of a Locally Invariant Lagrangian
292(1)
Lagrangian of Gauge Fields
293(5)
Dynamics of Gauge Invariant Fields
298(6)
Equations of Motion
298(1)
The Yang-Mills Equations
299(1)
The Total Energy
300(1)
Gauge Freedom and Gauge Conditions
301(3)
Spontaneously Broken Symmetry
304(6)
Vacuum and its Structure
304(1)
Goldstone Modes and Higgs Mechanism
305(2)
Elimination of Goldstone Modes. Goldstone Theorem
307(1)
Examples
308(2)
Quantization of Systems With Constraints
310(12)
Primary Constraints
310(2)
On Constrained Mechanical Systems
312(1)
Secondary Constraints
312(1)
The Matrix of Poisson Brackets
313(1)
First and Second Order Constraints
314(3)
Quantization
317(2)
Examples
319(3)
Hamiltonian Quantization of Yang - Mills Fields
322(8)
Quantization of Gauge Fields: Faddeev-Popov Method
330(3)
Coleman-Weinberg Effect
333(10)
Topological Objects in Field Theory
343(62)
Kink in 1 + 1 Dimensions
344(3)
A Few Words about Solitons
347(3)
Abrikosov Vortex
350(14)
Ginzburg-Landau Model of Superconductivity
351(1)
Nontrivial Solution
352(4)
Aharonov-Bohm Effect
356(1)
A Few Words about Topology and an Exotic String
357(6)
Vortex Solution in Other Contexts
363(1)
The 't Hooft-Polyakov Monopole
364(11)
Magnetic Properties of the Solution
366(2)
Lower Boundary on the Monopole Mass
368(2)
Dyons
370(1)
A Few Words About the Topology
371(2)
Do Monopoles Exist?
373(2)
SU (2) Instanton
375(8)
Nontrivial Solution
375(4)
On the Vacuum Structure of Yang-Mills Theory
379(4)
Quantum Kink
383(22)
Quantum Correction to the Mass of the Kink
385(4)
Physical Contents of Fluctuations around the Kink
389(2)
Elimination of Zero Mode
391(4)
Generating Functional
395(10)
A Some Integrals and Products 405(8)
Gaussian Integrals
405(1)
Calculation of Πn (1 - x2/n2-&pi2)
406(2)
Calculation of 1ƒ1 dx/x In (1 - x)
408(1)
Calculation of ∞ƒ∞ dx/x2 In (1 + x2)
409(2)
Feynman Parametrization
411(2)
B Splitting of Energy Levels in Double-Well Potential 413(4)
C Lie Algebras 417(15)
Elementary Definitions
417(2)
Examples of Lie Algebras
419(1)
The Idea of Classification. Levi-Maltsev Decomposition
420(4)
The Adjoint Representation
420(1)
Solvable and Nilpotent Algebras
421(1)
Reductive and Semisimple Algebras
422(2)
Classification of Complex Semisimple Lie Algebras
424(8)
The Cartan Subalgebra. Roots
424(1)
Properties of Roots. Cartan-Weyl Basis
425(2)
Cartan Matrix. Dynkin Schemes
427(2)
Compact Algebras
429(3)
Index 432

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