Spectral methods in quantum field theory / N. Graham, M. Quandt, H. Weigel.

Format:

Book

Author/Creator:

Graham, N. (Noah)

Contributor:

Quandt, M. (Markus)

Weigel, H. (Herbert)

Series:

Lecture notes in physics ; 777.

Lecture notes in physics, 0075-8450 ; 777

Language:

English

Subjects (All):

Quantum field theory.

Scattering (Physics).

Perturbation (Quantum dynamics).

Physical Description:

xi, 182 pages : illustrations ; 24 cm.

Place of Publication:

Berlin ; New York : Springer, [2009]

Summary:

This concise text introduces techniques from quantum mechanics, especially scattering theory, to computer the effects of an external background on a quantum field in general, and on the properties of the quantum vacuum in particular. This approach can be successfully used in an increasingly large number of situations, ranging from the study of solitons in field theory and cosmology to the determination of Casimir forces in nano-technology.

The method introduced and applied in this book is shown to give an unambiguous connection to perturbation theory, implementing standard renormalization conditions even for non-perturbative backgrounds. It both gives new theoretical insights, for example illuminating longstanding questions regarding Casimir stresses, and also provides an efficient analytic and numerical tool well suited to practical calculations. Last but not least, it elucidates in a concrete context many of the subtleties of quantum field theory, such as divergences, regularization and renormalization, by connecting them to more familiar results in quantum mechanics.

While addressed primarily at young researchers entering the field and nonspecialist researchers with backgrounds in theoretical and mathematical physics, introductory chapters on the theoretical aspects of the method make the book self-contained and thus suitable for advanced graduate students.

Contents:

1 Introduction 1

1.1 Background and Motivation 1

1.2 Invitation: A Sample Calculation 3

1.2.1 Zero-Point Energies: Summing $$ 3

1.2.2 From Discrete to Continuous: Phase Shifts and the Density of States 5

1.2.3 Counting States: Levinson's Theorem 6

1.2.4 Divergences: The Born Approximation 7

1.2.5 Computational Techniques 11

References 12

2 Review of Scattering Theory 15

2.1 Scattering Theory in Arbitrary Dimension 15

2.2 Green's Functions from Scattering Data 18

2.3 Phase Shifts and Density of States 20

2.4 Levinson's Theorem and Finite Energy Sum Rules 23

2.4.1 Overview and Simplified Derivation 24

2.4.2 Proof of the Regular Sum Rules 25

2.4.3 The Symmetric Channel in One Dimension 29

References 32

3 Quantum Field Theory and the Spectral Method 33

3.1 Small-Amplitude Quantum Corrections 33

3.2 The Canonical Formalism 34

3.2.1 The Vacuum Energy Density 35

3.2.2 The Vacuum Energy 40

3.3 The Path Integral Approach 41

3.4 Connecting the Functional and Canonical Formalisms 43

3.5 Feynman Diagrams and the Born Series 44

3.6 Some Remarks on Renormalization 52

3.7 Quantum Energy of Interfaces 55

3.7.1 The Interface Formula 56

References 60

4 Applications in One Space Dimension 63

4.1 Vacuum Polarization Energy in Exactly Solvable Models 63

4.2 Fermions in One Spatial Dimension 67

4.2.1 Parity-Invariant Background Fields 67

4.2.2 Fermions in the sine-Gordon Background 70

4.2.3 Fermions in the Kink Background 72

4.3 Bosons, Fermions, Supersymmetry, Central Charge, and the BPS Bound 76

4.3.1 Fermions 76

4.3.2 Supersymmetry and Central Charge 77

4.3.3 SVV Anomaly 81

4.4 Soliton Stabilization by Fermions in d = 1+1 82

4.4.1 The One-Loop Effective Energy 85

4.4.2 The Fermion Number 86

4.4.3 Results 86

References 88

5 Spectral Analysis of Charges 91

5.1 Basic Idea and Derivation 91

5.2 Electrostatics and the Need for Regularization 93

5.3 Chiral Bag Model in One Space Dimension 96

5.4 Chiral Bag Model in Three Space Dimensions 97

References 101

6 Hedgehog Configurations in d = 3+1 103

6.1 Chiral Fermions 103

6.1.1 The Model 103

6.1.2 The Fermion Loop 104

6.1.3 Numerical Analysis 110

6.2 SUL(2) Gauge Theory 116

6.2.1 Classical Sphalerons 117

6.2.2 Energetically Stabilized Solitons 119

6.2.3 The Search for the Soliton 120

6.2.4 Beyond the Spherical Ansatz 127

References 127

7 Boundary Conditions and Casimir Forces 129

7.1 Dirichlet Conditions from Quantum Field Theory 131

7.2 Rigid Bodies: Dirichlet Points and Parallel Plates 132

7.3 The Casimir Stress on a Dirichlet Ring 136

7.4 Oversubtraction and Diagrammatic Analysis 139

7.5 The Dirichlet Sphere 140

References 142

8 String-Type Configurations 143

8.1 Flux Tubes in Quantum Electrodynamics 143

8.1.1 The Vortex Configuration 144

8.1.2 The Quantum Energy of the Vortex 144

8.1.3 Subtleties of Configurations with Net Flux and Embedding 146

8.1.4 Numerical Results for the Quantum Energy 148

8.1.5 Quantum Energy Density 151

8.2 Flux Tubes and Strings in the Electroweak Standard Model 153

8.2.1 The Bosonic Sector 154

8.2.2 The String Solutions 155

8.2.3 The Sphaleron Square 155

8.2.4 The Fermion Action 157

8.2.5 Fermions on Strings 158

8.2.6 The Vacuum Polarization Energy 159

8.2.7 Numerical Results 163

References 168

9 Quantum Corrections to Q-Balls 171

9.1 The Q-Ball 171

9.2 Quantum Corrections 173

9.3 Applications 176

References 177.

Notes:

Includes bibliographical references and index.

ISBN:

9783642001383

3642001386

OCLC:

310400848