This thesis is devoted to the systematic study of non-local theories that respect Lorentz invariance and are devoid of new, unphysical degrees of freedom. Such theories are attractive for phenomenological applications since they are mostly unconstrained by current experiments. Non-locality has played an increasingly important role in the physics of the last decades, appearing in effective actions in quantum field theory, and arising naturally in string theory and non-commutative geometry. It may even be a necessary ingredient for quantum theories of gravity. It is a feature of quantum entanglement, and may even solve the long-standing black hole information loss problem. “Non-locality” is a broad concept with many promising and fruitful applications in theoretical and mathematical physics. After a historical and pedagogical introduction into the concept of non-locality the author develops the notion of non-local Green functions to study various non-local weak-field problems in quantum mechanics, quantum field theory, gravity, and quantum field theory in curved spacetime. This thesis fills a gap in the literature by providing a self-contained exploration of weak-field effects in non-local theories, thereby establishing a “non-local intuition” which may serve as a stepping stone for studies of the full, non-linear problem of non-locality.
Author(s): Jens Boos
Publisher: Springer Nature
Year: 2021
Language: English
Pages: 210
Supervisor's Foreword
Acknowledgments
Acknowledgment of Traditional Territory
Contents
Parts of this Thesis Have Been Published in the Following Journal Articles
List of Symbols
1 Introduction
1.1 Why Non-locality?
1.1.1 Regularization and Singularity Resolution
1.1.2 Black Hole Information Loss Problem
1.1.3 Non-locality in Quantum Effective Actions
1.2 Historical Aspects
1.3 Non-local Form Factors and Kernel Representations
1.4 On-shell vs. Off-shell Properties
1.5 Remarks on Non-locality and the Variational Principle
1.6 Initial Value Problem
1.7 Recent Work
1.7.1 Quantum Field Theory
1.7.2 Cosmology
1.7.3 Black Hole Physics and Gravitational Physics
1.7.4 Related Approaches
1.8 Overview of Thesis
References
2 Green Functions in Non-local Theories
2.1 Introduction
2.2 Green Functions in Classical Field Theory
2.3 Green Functions in Quantum Field Theory
2.4 Causality from Analyticity: Local Case
2.4.1 Retarded Green Function
2.4.2 Advanced Green Function
2.4.3 Feynman Green Function
2.4.4 Physical Interpretations
2.4.5 Homogeneous Green Functions
2.5 Asymptotic Causality Condition on Green Functions
2.6 Causality from Analyticity: Non-local Case
2.6.1 Explicit Expressions
2.6.2 General Asymptotics
2.7 Non-local Green Function Contributions: Some Results
2.7.1 Four-Dimensional Case
2.7.2 Two-Dimensional Case
2.8 Static Green Functions
2.8.1 Higher and Lower Dimensions
2.8.2 Local Limit
2.8.3 Coincidence Limit
2.8.4 Explicit Expressions
2.8.5 Heat Kernel Representation of Static Green Functions
2.9 Concluding Remarks
References
3 Static and Stationary Solutions in Weak-Field Gravity
3.1 Weak-Field Limit of Non-Local Gravity
3.1.1 Gauge Freedom
3.2 Staticity and Stationarity
3.2.1 Overview of the Rest of the Chapter
3.3 Gravitational Sources
3.4 Point Particles
3.4.1 Regularity
3.4.2 Asymptotics
3.5 Friedel Oscillations Around Point Particles
3.5.1 Oscillations in Higher-Derivative Gravity
3.5.2 Physical Interpretation: An Attempt
3.6 Extended Objects: p-Branes
3.6.1 Regularity
3.6.2 Curvature Expressions
3.6.3 Concrete Examples
3.6.3.1 The Point Particle, Revisited
3.6.3.2 A Cosmic String
3.6.3.3 A Domain Wall
3.6.3.4 Angle Deficit Configurations
3.7 Geometry of a Cosmic String in Non-Local Gravity
3.7.1 No Distributional Curvature
3.7.2 Angle Deficit
3.7.3 A Tale of Two Cones
3.8 Stationary Rotating Objects (General Solution)
3.9 Angular Momentum in Higher Dimensions
3.10 Spinning Point Particles
3.10.1 Spinning Point Particle in Four Spacetime Dimensions
3.10.2 Spinning Point Particle in Higher Dimensions
3.11 Spinning Strings and p-Branes
3.11.1 Cosmic String in Four Dimensions
3.11.2 Angle Deficit Configurations
3.11.3 Curvature Expressions
3.12 Concluding Remarks
References
4 Ultrarelativistic Objects
4.1 The Aichelburg–Sexl Metric and the Penrose Limit
4.2 Gyratons
4.2.1 Geometrical Setup
4.2.2 Boost and Penrose Limit
4.2.3 Gyraton Solutions in d=3
4.2.3.1 General Relativity
4.2.3.2 Ghost-Free Infinite-Derivative Gravity
4.2.4 Curvature
4.2.5 Gyraton Solutions in d≥4
4.3 Gyratonic p-Branes
4.3.1 p-Brane Metric
4.3.2 Boost and Penrose Limit
4.3.3 Examples in d=3?
4.3.4 Example in d=4
4.3.5 Higher Dimensions
4.4 Concluding Remarks
References
5 Quantum-Mechanical Scattering
5.1 Introduction
5.2 A Non-Local Scalar Field in Quantum Mechanics
5.3 Lippmann–Schwinger Method
5.4 Transmission and Reflection Coefficients for a δ-Potential
5.4.1 Local Case
5.4.2 Non-Local Case
5.4.3 Properties of the Scattering Coefficients
5.5 Quasinormal Modes
5.5.1 Local Case
5.5.2 Non-Local Case
5.6 Multiple δ-Potentials
5.6.1 Quasinormal Modes
5.7 Concluding Remarks
References
6 Vacuum Polarization and the Fluctuation-Dissipation Theorem
6.1 Introduction
6.2 A Model of a Non-local Scalar Quantum Field
6.3 Vacuum Fluctuations Around a δ-Potential
6.3.1 Free Green Functions
6.3.2 Interacting Green Functions
6.3.2.1 Hadamard Prescription
6.3.2.2 Lippmann–Schwinger Method for Green Functions
6.3.2.3 Equivalence of Both Methods
6.3.3 Vacuum Polarization
6.3.3.1 Local Theory
6.3.3.2 Non-local GF1 Theory
6.3.3.3 Non-local GF2 Theory
6.4 Stability Properties of Non-local QFT
6.5 Thermal Fluctuations Around a δ-Potential
6.5.1 Local Theory
6.5.2 Non-local GF2 Theory
6.6 Fluctuation-Dissipation Theorem
6.7 Concluding Remarks
References
7 Black Holes, Generalized Polyakov Action, and HawkingRadiation
7.1 Introduction
7.2 2D Conformal Anomaly and the Polyakov Action
7.3 Ghost-Free Modification of the Polyakov Action
7.3.1 Effective Energy-Momentum Tensor
7.3.1.1 Trace
7.3.1.2 Tensorial Components
7.3.2 State Dependence
7.4 Black Hole Entropy
7.5 Hawking Flux
7.6 Example: Two-Dimensional Dilaton Black Hole
7.6.1 Black Hole Metric
7.6.2 Analytical Considerations
7.6.3 Quasilocal Approximation
7.6.4 Non-local Corrections to the Trace Anomaly
7.6.5 Non-local Corrections to Black Hole Entropy
7.7 Concluding Remarks
References
8 Conclusions
8.1 Summary of Key Results
8.2 Open Problems
A Calculational Details
A.1 Retarded Green Function Two Dimensions
A.2 Two-Dimensional Massive Green Functions
A.2.1 Inhomogeneous Green Functions
A.2.1.1 Retarded Green Function
A.2.1.2 Advanced Green Function
A.2.1.3 Feynman Green Function
A.2.1.4 Anti-Feynman Green Function
A.2.2 Homogeneous Green Functions
A.3 Proof of Eq.(6.46)
A.4 Gω in GF2 Theory
A.4.1 Analytical Evaluation of Gω(0)
A.4.2 Numerical Evaluation of Gω(x)
A.4.3 Asymptotics of Gω(x)
A.4.4 Remarks on "426830A φ2(x)"526930B ren in GF2n Theories for Larger n
References
B Two-Dimensional Ghost-Free Modification of the Polyakov Action
B.1 General Relations for Static Geometries
B.2 Energy-Momentum Tensor
B.2.1 Boulware Vacuum
B.2.2 Hartle–Hawking Vacuum
B.2.3 Unruh Vacuum
B.3 Spectral Representation of F(s,R)
B.3.1 Orthogonality and Normalization of Eigenfunctions
B.3.2 Spectral Measure
References
