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9783642001383

Spectral Methods in Quantum Field Theory

by ; ;
  • ISBN13:

    9783642001383

  • ISBN10:

    3642001386

  • Format: Hardcover
  • Copyright: 2009-05-01
  • Publisher: Springer Verlag
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Summary

This concise text introduces techniques from quantum mechanics, especially scattering theory, to compute 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.

Table of Contents

Introductionp. 1
Background and Motivationp. 1
Invitation: A Sample Calculationp. 3
Zero-Point Energies: Summing $$p. 3
From Discrete to Continuous: Phase Shifts and the Density of Statesp. 5
Counting States: Levinson's Theoremp. 6
Divergences: The Born Approximationp. 7
Computational Techniquesp. 11
Referencesp. 12
Review of Scattering Theoryp. 15
Scattering Theory in Arbitrary Dimensionp. 15
Green's Functions from Scattering Datap. 18
Phase Shifts and Density of Statesp. 20
Levinson's Theorem and Finite Energy Sum Rulesp. 23
Overview and Simplified Derivationp. 24
Proof of the Regular Sum Rulesp. 25
The Symmetric Channel in One Dimensionp. 29
Referencesp. 32
Quantum Field Theory and the Spectral Methodp. 33
Small-Amplitude Quantum Correctionsp. 33
The Canonical Formalismp. 34
The Vacuum Energy Densityp. 35
The Vacuum Energyp. 40
The Path Integral Approachp. 41
Connecting the Functional and Canonical Formalismsp. 43
Feynman Diagrams and the Born Seriesp. 44
Some Remarks on Renormalizationp. 52
Quantum Energy of Interfacesp. 55
The Interface Formulap. 56
Referencesp. 60
Applications in One Space Dimensionp. 63
Vacuum Polarization Energy in Exactly Solvable Modelsp. 63
Fermions in One Spatial Dimensionp. 67
Parity-Invariant Background Fieldsp. 67
Fermions in the sine-Gordon Backgroundp. 70
Fermions in the Kink Backgroundp. 72
Bosons, Fermions, Supersymmetry, Central Charge, and the BPS Boundp. 76
Fermionsp. 76
Supersymmetry and Central Chargep. 77
SVV Anomalyp. 81
Soliton Stabilization by Fermions in d = 1+1p. 82
The One-Loop Effective Energyp. 85
The Fermion Numberp. 86
Resultsp. 86
Referencesp. 88
Spectral Analysis of Chargesp. 91
Basic Idea and Derivationp. 91
Electrostatics and the Need for Regularizationp. 93
Chiral Bag Model in One Space Dimensionp. 96
Chiral Bag Model in Three Space Dimensionsp. 97
Referencesp. 101
Hedgehog Configurations in d = 3+1p. 103
Chiral Fermionsp. 103
The Modelp. 103
The Fermion Loopp. 104
Numerical Analysisp. 110
SUL(2) Gauge Theoryp. 116
Classical Sphaleronsp. 117
Energetically Stabilized Solitonsp. 119
The Search for the Solitonp. 120
Beyond the Spherical Ansatzp. 127
Referencesp. 127
Boundary Conditions and Casimir Forcesp. 129
Dirichlet Conditions from Quantum Field Theoryp. 131
Rigid Bodies: Dirichlet Points and Parallel Platesp. 132
The Casimir Stress on a Dirichlet Ringp. 136
Oversubtraction and Diagrammatic Analysisp. 139
The Dirichlet Spherep. 140
Referencesp. 142
String-Type Configurationsp. 143
Flux Tubes in Quantum Electrodynamicsp. 143
The Vortex Configurationp. 144
The Quantum Energy of the Vortexp. 144
Subtleties of Configurations with Net Flux and Embeddingp. 146
Numerical Results for the Quantum Energyp. 148
Quantum Energy Densityp. 151
Flux Tubes and Strings in the Electroweak Standard Modelp. 153
The Bosonic Sectorp. 154
The String Solutionsp. 155
The Sphaleron Squarep. 155
The Fermion Actionp. 157
Fermions on Stringsp. 158
The Vacuum Polarization Energyp. 159
Numerical Resultsp. 163
Referencesp. 168
Quantum Corrections to Q-Ballsp. 171
The Q-Ballp. 171
Quantum Correctionsp. 173
Applicationsp. 176
Referencesp. 177
Indexp. 179
Table of Contents provided by Ingram. All Rights Reserved.

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