This book reviews the construction of elementary systems living in de Sitter (dS) spacetime, in both the classical and quantum senses. Field theories on dS spacetime are among the most studied mathematical models of the Universe, whether for its earlier period (inflationary phase) or for its current phase of expansion acceleration (dark energy or cosmological constant). Classical elementary systems are Hamiltonian phase spaces, which are associated with co-adjoint orbits of the relativity group. On the other hand, quantum elementary systems are associated with (projective) unitary irreducible representations of the (possibly extended) relativity group (or one of its covering). This study emphasizes the conceptual issues arising in the formulation of such systems and discusses known results in a mathematically rigorous way. Particular attention is paid to: “smooth” transition from classical to quantum theory; physical content under vanishing curvature, from the point of view of a local (“tangent”) Minkowskian observer; and thermal interpretation (on the quantum level), in the sense of the Gibbons-Hawking temperature. Such a mathematical construction is of paramount importance to the understanding of the early Universe (due to the critical role that the dS metric plays in the inflationary cosmological scenarii) as well as to the construction of possible models for late-time cosmology (since a small positive cosmological constant or dark energy seems to be required by recent data). In this sense, this book uniquely blends mathematical physics (spacetime symmetry on classical and quantum levels) and theoretical physics (quantization, quantum field theory, and cosmology). Moreover, the level of exposition varies in different parts of the book so that both experts and beginners alike can utilize the book.
Author(s): Mohammad Enayati, Jean-Pierre Gazeau, Hamed Pejhan, Anzhong Wang
Series: Synthesis Lectures on Mathematics & Statistics
Publisher: Springer
Year: 2022
Language: English
Pages: 222
City: Cham
Preface
Contents
Acronyms
Part I De Sitterian Elementary Systems
1 Introduction
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1.1 Brief Description
1.2 Motivations
1.3 Content at a Glance
1.4 Reading Guide and Conventions
Part II As a Preliminary: 1 + 1-Dimensional dS (dS2) Geometry and Relativity
2 The Classical Context
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2.1 DS2 Manifold and Its Symmetry Group
2.1.1 Precision on SO0(1,2): A Geometric Viewpoint
2.1.2 Homomorphism Between SO0(1,2) and SU(1,1)
2.2 Relativistic Meaning of the dS2 Group: Group Decomposition
2.2.1 Space-Time-Lorentz Decomposition
2.2.2 Cartan Decomposition
2.2.3 Iwasawa Decomposition
2.3 DS2 Lie Algebra and Classical Phase Spaces
2.3.1 Co-adjoint Orbits: A Brief Introduction
2.3.2 DS2 (co-)Adjoint Orbits as Possible Phase Spaces for Motions on dS2 Spacetime
3 The Quantum-Mechanical Context
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3.1 UIR's of the dS2 Group and Quantum Version of dS2 Motions
3.2 UIR's of the dS2 Group: Global Realization
3.2.1 Group Representations: A Brief Introduction
3.2.2 Principal Series
3.2.3 Complementary Series
3.2.4 Discrete Series
Part III 1 + 3-Dimensional dS (dS4) Geometry and Relativity
4 The Classical Context
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4.1 DS4 Manifold and Its Causal Structure
4.2 DS4 Relativity Group SO0(1,4) and Its Covering Sp(2,2)
4.2.1 Homomorphism Between SO0(1,4) and Sp(2,2) and Some Discrete Symmetries
4.3 Relativistic Meaning of the dS4 Group: Group Decomposition
4.3.1 Space-Time-Lorentz Decomposition
4.3.2 Cartan Decomposition
4.3.3 Iwasawa Decomposition
4.3.4 Four Integration Formulas on Sp(2,2)
4.4 Relativistic Meaning of the dS4 Group: Group (Algebra) Contraction
4.4.1 Group (Algebra) Contraction: A Brief Introduction
4.4.2 DS4 Group (Algebra) Contractions
4.5 DS4 Lie Algebra and Classical Phase Spaces
4.5.1 Phase Space for Scalar ``Massive''/``Massless'' Elementary Systems in dS4 Spacetime
4.5.2 Phase Space for ``Spin''``Massive'' Elementary Systems in dS4 Spacetime
4.5.3 Phase Space for ``Spin'' (or Helicity) ``Massless'' Elementary Systems in dS4 Spacetime
5 The Quantum-Mechanical Context
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5.1 UIR's of the dS4 Group and Quantum Version of dS4 Motions
5.1.1 Discrete Series
5.1.2 Principal Series
5.1.3 Complementary Series
5.1.4 Discussion: A Natural Fuzzyness of dS4 Spacetime
5.2 UIR's of the dS4 Group: Global Realization
5.2.1 Principal Series: Scalar Case
5.2.2 Principal Series: General Case
5.2.3 Principal Series: Restriction to the Maximal Compact Subgroup SU(2) timesSU(2)
5.2.4 Complementary Series
5.2.5 Discrete Series
5.3 ``Massive''/``Massless'' dS4 UIR's and the Poincaré Contraction
5.3.1 Group Contraction (The Representation Level): A Brief Introduction
5.3.2 DS4 Massive UIR's
5.3.3 DS4 Massless UIR's
5.3.4 Discussion: Rehabilitating the dS4 Physics from the Point of View of a Local Minkowskian Observer
5.3.5 For Comparison: AdS4 UIR's and the Poincaré Contraction
6 The QFT Context
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6.1 An Overview
6.2 DS4 Wave Equations
6.2.1 Computation in Ambient Notations
6.2.2 Link to Intrinsic Coordinates
6.3 Plane-Wave Type Solutions
6.3.1 A Global Definition: dS4 Plane Waves in Their Tube Domains
6.3.2 Precision on Orbital Basis of the Cone
6.3.3 DS4 Plane Waves as Generating Functions for Square-Integrable Eigenfunctions
6.3.4 DS4 Plane Waves and the Zero-Curvature Limit
6.4 QFT in dS4 Spacetime
6.4.1 Local dS4 Scalar Fields: Generalized Free Fields
6.4.2 (Analytic) Wightman Two-Point Functions for the dS4 (Principal and Complementary) Klein–Gordon Scalar Fields
6.4.3 Minimally Coupled Scalar Field as an Illustration of a Krein Structure
7 Notion of Mass in (A)dS4 Relativity
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7.1 An Overview
7.2 Discussion/Reminder: Mass and Symmetries
7.3 Garidi Mass: Definition
7.4 Garidi Mass: A More Elaborate Discussion
7.4.1 DS4 Case
7.4.2 AdS4 Case
7.4.3 Discussion: Dark Matter as a Relic AdS4 Curvature Energy (?)
7.5 Garidi Mass: Examples and Applications
7.5.1 ``Partially Massless'' Fields
7.5.2 The Question of Graviton ``Mass''
Index
Index
