The de Sitter (dS) Group and its Representations: An Introduction to Elementary Systems and Modeling the Dark Energy Universe

223 Pages·2022·3.13 MB·English

by  Mohammad Enayati, Jean-Pierre Gazeau, Hamed Pejhan, Anzhong Wang

Synthesis Lectures on
Mathematics & Statistics
Mohammad Enayati · Jean-Pierre Gazeau ·
Hamed Pejhan · Anzhong Wang
The de Sitter (dS)
Group and its
Representations
An Introduction to Elementary Systems
and Modeling the Dark Energy Universe
Synthesis Lectures on Mathematics &
Statistics
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USA
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Mohammad Enayati · Jean-Pierre Gazeau ·
Hamed Pejhan · Anzhong Wang
The de Sitter (dS) Group
and its Representations
An Introduction to Elementary Systems
and Modeling the Dark Energy Universe
MohammadEnayati Jean-PierreGazeau
DepartmentofPhysics AstroparticuleetCosmologie,CNRS
RaziUniversity UniversitéParisCité
Kermanshah,Iran Paris,France
HamedPejhan AnzhongWang
InstituteforTheoreticalPhysics GCAP-CASPER,PhysicsDepartment
andCosmology BaylorUniversity
ZhejiangUniversityofTechnology Waco,TX,USA
Hangzhou,China
ISSN1938-1743 ISSN1938-1751 (electronic)
SynthesisLecturesonMathematics&Statistics
ISBN978-3-031-16044-8 ISBN978-3-031-16045-5 (eBook)
https://doi.org/10.1007/978-3-031-16045-5
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Preface
After a quick search in the literature, a researcher can find hundreds of books, articles,
etc., through which readers are provided with pedagogical or advanced materials needed
to understand various aspects of a consistent quantum field theory (QFT) reading of ele-
mentary systems in curved spacetimes, including the de Sitter (dS) spacetime which is
of interest in the present study. [We point out in passing that dS spacetime is one of the
components of the (cid:2)CDM standard model, where (cid:2) stands for the currently observed
acceleration of the Universe expansion (positive “cosmological constant”) and CDM for
“Cold Dark Matter”. It is also the prototype of curved spacetimes for which QFT can
be,toacertainextent,constructedonfirmmathematicalbackgrounds.]But,almostallof
these references totally ignore a critical point in the formulation of elementary systems,
thatis,consistencywithregardtoWigner’sapproach(!),aswefurtherexplainnow.Both
fieldtheoreticalformulationandphenomenologicaltreatmentofanelementarysystem,at
the level of interpretation in particular, rest on the concepts of energy, momentum, mass,
andspin,whoseexistenceliterallystemsfrominvarianceprinciples,strictlyspeaking,the
principle of invariance under the Poincaré group (the relativity ~ kinematical group of
flat Minkowski spacetime); the rest mass m and the spin s of an (Einsteinian) elemen-
tary system living in flat Minkowski spacetime are the two invariants that specify the
respective unitary irreducible representation (UIR) of the Poincaré group. In the pres-
ence of gravity, however, any interpretation with reference to the relativity group of flat
Minkowski spacetime is physically irrelevant. On the other hand, in curved spacetimes
generally (with the exception of dS and anti-dS (AdS) spacetimes), no nontrivial groups
of motion, and consequently, no literal or unique extension of the aforementioned phys-
ical concepts exists (?!). Therefore, although, one may simply generalize the important
differential equations describing elementary systems (such as Klein-Gordon and Dirac
equations) to forms that enjoy general covariance in curved spacetimes, in the end such
mathematicalconstructionscannot be associatedwith physical elementarysystems in the
sense given above.
This missing point is exactly the main motive for writing the present manuscript,
through which we review the construction of (“free”) elementary systems in dS space-
time, in the Wigner sense, as associated with UIR’s of the dS (relativity) group. This
v
vi Preface
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 van-
ishing 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.
We review three decompositions of the dS group physically relevant for the descrip-
tion of dS spacetime and classical phase spaces of elementary systems living on it. We
reviewtheconstructionof(projective)dSUIR’sissuedfromthesegroupdecompositions.
(Projective)HilbertspacescarryingtheUIR’s(insomerestrictedsense)identifyquantum
(“one-particle”)statespacesofdSelementarysystems.Adoptingawell-establishedFock
procedure, based on the Wightman-Gärding axioms and on analyticity requirements in
the complexified Riemannian manifold, we proceed with a consistent QFT formulation
of elementary systems in dS spacetime. This dS QFT formulation closely parallels the
corresponding Minkowskian one, while the usual spectral condition is replaced by a cer-
tain geometric Kubo-Martin-Schwinger (KMS) condition equivalent to a precise thermal
manifestation of the associated “vacuum” states.
We end our study by reviewing a consistent and univocal definition of mass in dS
relativity. This definition, presented in terms of invariant parameters characterizing the
dS UIR’s, accurately gives sense to terms like “massive” and “massless” fields in dS
relativity according to their Minkowskian counterparts, yielded by the group contraction
procedures.
Going through the above process, this manuscript provides a wide readership in
theoretical and mathematical physics, in QFT in curved spacetime, in cosmology, in
quantization, with comprehensive and pedagogical materials needed to understand the
various mathematical aspects of the dS group, its Lie manifold, its Lie algebra, its
(co-)adjoint orbits (as possible classical elementary systems in dS), their quantum
counterparts, namely, the dS UIR’s (as possible quantum elementary systems in dS
spacetime).
Since success is never an orphan, we would like to thank here M. R. Abolhasani,
K. Bamba, M. V. Takook, M. R. Tanhayi, S. Parsamehr, A. Rabeie, S. Rahbardehghan,
M. Sharma, Q. Wu, T. Zhu, for their support and encouragement over the years leading
to this work. During writing this book, Hamed Pejhan has had the privilege to work at
the Zhejiang University of Technology (ZJUT), which provided him with the flexibility
to follow scientific ideas wherever they go, on land or sea. He is extremely grateful for
this freedom. He also gratefully acknowledges the financial support from ZJUT.
Kermanshah, Iran Mohammad Enayati
Paris, France Jean-Pierre Gazeau
Hangzhou, China Hamed Pejhan
Waco, USA Anzhong Wang
Contents
PartI DeSitterianElementarySystems
1 Introduction ......................................................... 3
1.1 Brief Description ................................................. 3
1.2 Motivations ...................................................... 4
1.3 Content at a Glance .............................................. 5
1.4 Reading Guide and Conventions ................................... 7
References ........................................................... 8
PartII AsaPreliminary:1+1-DimensionaldS(dS )Geometryand
2
Relativity
2 TheClassicalContext ................................................. 13
2.1 DS Manifold and Its Symmetry Group ............................. 13
2
2.1.1 Precision on SO (1,2): A Geometric Viewpoint ............... 14
0
2.1.2 Homomorphism Between SO (1,2) and SU(1,1) .............. 14
0
2.2 Relativistic Meaning of the dS Group: Group Decomposition ......... 16
2
2.2.1 Space-Time-Lorentz Decomposition .......................... 16
2.2.2 Cartan Decomposition ...................................... 19
2.2.3 Iwasawa Decomposition .................................... 21
2.3 DS Lie Algebra and Classical Phase Spaces ........................ 23
2
2.3.1 Co-adjoint Orbits: A Brief Introduction ....................... 24
2.3.2 DS (co-)Adjoint Orbits as Possible Phase Spaces
2
for Motions on dS Spacetime ............................... 26
2
References ........................................................... 30
3 TheQuantum-MechanicalContext .................................... 31
3.1 UIR’s of the dS Group and Quantum Version of dS Motions ......... 31
2 2
3.2 UIR’s of the dS Group: Global Realization ......................... 33
2
3.2.1 Group Representations: A Brief Introduction .................. 34
3.2.2 Principal Series ............................................ 36
vii
viii Contents
3.2.3 Complementary Series ...................................... 42
3.2.4 Discrete Series ............................................ 43
References ........................................................... 44
PartIII 1+3-DimensionaldS(dS )GeometryandRelativity
4
4 TheClassicalContext ................................................. 47
4.1 DS Manifold and Its Causal Structure .............................. 47
4
4.2 DS Relativity Group SO (1,4) and Its Covering Sp(2,2) ............ 48
4 0
4.2.1 Homomorphism Between SO (1,4) and Sp(2,2) and Some
0
Discrete Symmetries ....................................... 50
4.3 Relativistic Meaning of the dS Group: Group Decomposition ......... 52
4
4.3.1 Space-Time-Lorentz Decomposition .......................... 52
4.3.2 Cartan Decomposition ...................................... 56
4.3.3 Iwasawa Decomposition .................................... 58
4.3.4 Four Integration Formulas on Sp(2,2) ........................ 60
4.4 Relativistic Meaning of the dS Group: Group (Algebra) Contraction ... 61
4
4.4.1 Group (Algebra) Contraction: A Brief Introduction ............. 61
4.4.2 DS Group (Algebra) Contractions ........................... 62
4
4.5 DS Lie Algebra and Classical Phase Spaces ........................ 65
4
4.5.1 Phase Space for Scalar “Massive”/“Massless” Elementary
Systems in dS Spacetime .................................. 67
4
4.5.2 Phase Space for “Spin” “Massive” Elementary Systems
in dS Spacetime .......................................... 71
4
4.5.3 Phase Space for “Spin” (or Helicity) “Massless” Elementary
Systems in dS Spacetime .................................. 73
4
References ........................................................... 77
5 TheQuantum-MechanicalContext .................................... 79
5.1 UIR’s of the dS Group and Quantum Version of dS Motions ......... 79
4 4
5.1.1 Discrete Series ............................................ 84
5.1.2 Principal Series ............................................ 84
5.1.3 Complementary Series ...................................... 85
5.1.4 Discussion: A Natural Fuzzyness of dS Spacetime ............ 85
4
5.2 UIR’s of the dS Group: Global Realization ......................... 86
4
5.2.1 Principal Series: Scalar Case ................................ 86
5.2.2 Principal Series: General Case ............................... 92
5.2.3 Principal Series: Restriction to the Maximal Compact
Subgroup SU(2)×SU(2) ................................... 93
5.2.4 Complementary Series ...................................... 94
5.2.5 Discrete Series ............................................ 95
Contents ix
5.3 “Massive”/“Massless” dS UIR’s and the Poincaré Contraction ........ 98
4
5.3.1 Group Contraction (The Representation Level): A Brief
Introduction ............................................... 98
5.3.2 DS Massive UIR’s ........................................ 99
4
5.3.3 DS Massless UIR’s ....................................... 100
4
5.3.4 Discussion: Rehabilitating the dS Physics from the Point
4
of View of a Local Minkowskian Observer .................... 101
5.3.5 For Comparison: AdS UIR’s and the Poincaré Contraction ..... 102
4
References ........................................................... 125
6 TheQFTContext .................................................... 127
6.1 An Overview .................................................... 127
6.2 DS Wave Equations ............................................. 128
4
6.2.1 Computation in Ambient Notations .......................... 129
6.2.2 Link to Intrinsic Coordinates ................................ 133
6.3 Plane-Wave Type Solutions ........................................ 134
6.3.1 A Global Definition: dS Plane Waves in Their Tube
4
Domains .................................................. 135
6.3.2 Precision on Orbital Basis of the Cone ....................... 137
6.3.3 DS Plane Waves as Generating Functions
4
for Square-Integrable Eigenfunctions ......................... 138
6.3.4 DS Plane Waves and the Zero-Curvature Limit ............... 149
4
6.4 QFT in dS Spacetime ............................................ 151
4
6.4.1 Local dS Scalar Fields: Generalized Free Fields .............. 151
4
6.4.2 (Analytic) Wightman Two-Point Functions for the dS
4
(Principal and Complementary) Klein–Gordon Scalar Fields ..... 158
6.4.3 Minimally Coupled Scalar Field as an Illustration of a Krein
Structure .................................................. 170
References ........................................................... 184
7 NotionofMassin(A)dS Relativity .................................... 187
4
7.1 An Overview .................................................... 187
7.2 Discussion/Reminder: Mass and Symmetries ......................... 188
7.3 Garidi Mass: Definition ........................................... 189
7.4 Garidi Mass: A More Elaborate Discussion .......................... 192
7.4.1 DS Case ................................................. 192
4
7.4.2 AdS Case ................................................ 194
4
7.4.3 Discussion: Dark Matter as a Relic AdS Curvature
4
Energy (?) ................................................ 195