Cosmological Applications of Algebraic Quantum Field Theory in Curved Spacetimes

129 Pages·2016·1.713 MB·English

by  Thomas-Paul Hack (auth.)

SPRINGER BRIEFS IN MATHEMATICAL PHYSICS 6
Thomas-Paul Hack
Cosmological
Applications
of Algebraic
Quantum Field
Theory in Curved
Spacetimes
123
SpringerBriefs in Mathematical Physics
Volume 6
Series editors
Nathanaël Berestycki, Cambridge, UK
Mihalis Dafermos, Cambridge, UK
Tohru Eguchi, Tokyo, Japan
Atsuo Kuniba, Tokyo, Japan
Matilde Marcolli, Pasadena, USA
Bruno Nachtergaele, Davis, USA
More information about this series at http://www.springer.com/series/11953
Thomas-Paul Hack
Cosmological Applications
of Algebraic Quantum Field
Theory in Curved Spacetimes
123
Thomas-Paul Hack
Institute for Theoretical Physics
University of Leipzig
Leipzig
Germany
ISSN 2197-1757 ISSN 2197-1765 (electronic)
SpringerBriefs inMathematical Physics
ISBN978-3-319-21893-9 ISBN978-3-319-21894-6 (eBook)
DOI 10.1007/978-3-319-21894-6
LibraryofCongressControlNumber:2015946991
SpringerChamHeidelbergNewYorkDordrechtLondon
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Preface
Thismonographprovidesalargelyself-containedandbroadlyaccessibleexposition
oftwocosmologicalapplicationsofalgebraicquantumfieldtheory(QFT)incurved
spacetime: a fundamental analysis of the cosmological evolution according to the
Standard Model of Cosmology and a fundamental study of the perturbations in
Inflation. The two central sections of the book dealing with these applications are
precededbysectionscontainingapedagogicalintroductiontothesubjectaswellas
introductory material on the construction of linear QFTs on general curved
spacetimeswithandwithoutgaugesymmetryinthealgebraicapproach,physically
meaningful quantum states on general curved spacetimes, and the backreaction of
quantum fields in curved spacetimes via the semiclassical Einstein equation. The
target reader should have a basic understanding of General Relativity and QFT on
Minkowski spacetime, but does not need to have a background in QFT on curved
spacetimes or the algebraic approach to QFT. In particular, I took a great deal of
care to provide a thorough motivation for all concepts of algebraic QFT touched
upon in this monograph, as they partly may seem rather abstract at first glance.
Thus,itismyhopethatthisworkcanhelpnon-expertstomake‘firstcontact’with
the algebraic approach to QFT.
I would like to thank my colleagues and friends Claudio Dappiaggi, Klaus
Fredenhagen, Hanno Gottschalk, Valter Moretti, Nicola Pinamonti and Alexander
Schenkel, among others, for their past and ongoing support and the fruitful
collaborationsonsomeofthetopicscoveredinthismonograph.Specialthanksare
duetoJanMöllerforthepersistentencouragementtoapplyalgebraicquantumfield
theory to cosmology. I would also like to thank Aldo Rampioni and Kirsten
Theunissen at Springer for their patient collaboration on the realisation of this
monograph.
Leipzig, Germany Thomas-Paul Hack
v
Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 A Pedagogical Introduction to Algebraic Quantum Field
Theory on Curved Spacetimes. . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Outline of the Cosmological Applications. . . . . . . . . . . . . . . . . 7
1.2.1 The Cosmological Expansion in QFT
on Curved Spacetimes . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.2 A Birds-Eye View of Perturbations in Inflation . . . . . . . . 9
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Algebraic Quantum Field Theory on Curved Spacetimes. . . . . . . . 13
2.1 Globally Hyperbolic Spacetimes and Related
Geometric Notions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Linear Classical Fields on Curved Spacetimes . . . . . . . . . . . . . . 18
2.2.1 Models Without Gauge Symmetry. . . . . . . . . . . . . . . . . 18
2.2.2 Models with Gauge Symmetry. . . . . . . . . . . . . . . . . . . . 27
2.3 Linear Quantum Fields on Curved Spacetimes. . . . . . . . . . . . . . 33
2.4 Hadamard States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.5 Locality and General Covariance . . . . . . . . . . . . . . . . . . . . . . . 50
2.6 The Quantum Stress-Energy Tensor and the Semiclassical
Einstein Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.6.1 Local and Covariant Wick Polynomials . . . . . . . . . . . . . 55
2.6.2 The Semiclassical Einstein Equation
and Wald’s Axioms. . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.6.3 A Conserved Quantum Stress-Energy Tensor. . . . . . . . . . 66
2.7 Further Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
vii
viii Contents
3 Cosmological Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.1 A Brief Introduction to Cosmology . . . . . . . . . . . . . . . . . . . . . 77
3.2 The Cosmological Expansion in QFT on Curved Spacetimes. . . . 81
3.2.1 The Renormalisation Freedom of the Quantum
Stress-Energy Tensor in the Context of Cosmology . . . . . 82
3.2.2 States of Interest in Cosmological Spacetimes . . . . . . . . . 84
3.2.3 Setup and (Computational) Strategy for Approximately
Solving the Semiclassical Einstein Equation . . . . . . . . . . 88
3.2.4 Computation of the Energy Density . . . . . . . . . . . . . . . . 92
3.2.5 Deviations from the Standard Model and Their
Phenomenological Consequences: εJ
00
and Dark Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.3 A Birds-Eye View of Perturbations in Inflation . . . . . . . . . . . . . 105
3.3.1 The Linearised Einstein-Klein-Gordon System
on General Curved Spacetimes . . . . . . . . . . . . . . . . . . . 106
3.3.2 The Standard Approach to Perturbations in Inflation
and Comparison of Approaches. . . . . . . . . . . . . . . . . . . 113
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Chapter 1
Introduction
Abstract Inthischapterwegiveapedagogicalintroductiontoalgebraicquantum
fieldtheoryandexplaintheconceptsandtherelevanceofthisframeworkinquantum
fieldtheoryoncurvedspacetimes.Thisintroductionshouldserveasaguideforthe
nextchapter,wheremanyconceptsofandconstructionsinalgebraicquantumfield
theory on curved spacetimes are reviewed in detail. Afterwards, we give a non-
technicaloverviewofthecosmologicalapplicationsdiscussedinthefinalchapterof
thismonograph.
1.1 APedagogicalIntroductiontoAlgebraicQuantumField
TheoryonCurvedSpacetimes
Algebraic Quantum Field Theory (AQFT) [12] is a framework which focusses on
the local and algebraic properties of QFT and thus aims for understanding struc-
tural properties of relativistic quantum field theory from first principles and in a
model-independentfashion.InstandardtextbooktreatmentsofQFTinMinkowski
spacetime,theformalismofQFTisdevelopedbyconstructingoperatorsandderiv-
ingrelationsbasedonthevacuumstateandtheassociatedHilbertspace.However,
thisapproachisnotdirectlygeneralisabletocurvedspacetimesasweshallexplain
now.
To this avail, we consider a quantized Hermitean scalar field φ(x) and assume
thatitcanbedecomposedintwodifferentwaysas
(cid:2) (cid:2)
φ(x)= A (x)a + A (x)a† = B (x)b +B (x)b† (1.1)
i i i i i i i i
i i
where a†,a and b†,b are two sets of creation and annihilation operators with
i i i i
correspondingmodes A (x), B (x)andvacua|Ω (cid:2),Ω (cid:2)
i i a b
a |Ω (cid:2)=0, b |Ω (cid:2)=0.
i a i b
©TheAuthor(s)2016 1
T.-P.Hack,CosmologicalApplicationsofAlgebraicQuantumFieldTheory
inCurvedSpacetimes,SpringerBriefsinMathematicalPhysics6,
DOI10.1007/978-3-319-21894-6_1
2 1 Introduction
The two sets of creating and annihilation operators are related by a Bogoliubov
transformation
b =α a +β a†.
i i i i i
Mathematically the two possible decompositions of φ(x) seem to be equivalent,
whereas physically we would ask the question: Which of the two decompositions
ofφ(x)isbetter,or,alternatively,isthereapreferredwaytodecompose φ(x)?In
Minkowski spacetime, we have Poincaré symmetry at our disposal, in particular
Minkowskispacetimeistime-translationinvariantandwecanconstructaHamilton
operator H andobtainarelatednotionof‘energy’.If H|Ω (cid:2)=0,but H|Ω (cid:2)(cid:3)=0,
a b
wewouldcall|Ω (cid:2)thegroundstateorvacuumstateandchoosetoworkwiththe
a
decomposition of φ(x) in terms of a†,a . In other words, we would consider—
i i
i.e.represent—φ(x)asanoperatorintheFockspaceofthevacuumstate.Incurved
spacetimes,theseideasfailingeneralbecausegenericcurvedspacetimesarenottime-
translation invariant; prominent examples of such backgrounds are cosmological
spacetimeswithametriclineelement
ds2 =−dt2+a(t)2dx(cid:4)2, (1.2)
where thefunctiona(t)isnon-constant intime.Inthe absence of time-translation
invariance,nomeaningfulnotionofaHamiltonoperatoror‘energy’exists,andthus
wehavenomeanstoselectordefineavacuumstate.Evenunderthesecircumstances,
onemightstillthinkthatvariouspossibledecompositionsoftheform(1.1)areina
senseequivalent,sothatitdoesnotreallymatterwhichoftheseonechoosestowork
with.However,ingeneraloneisfacingtheadditionalproblemthat
(cid:2)
.
N = (cid:5)Ω |b†b |Ω (cid:2)=∞,
ba a i i a
i
i.e. the ‘a-vacuum’ contains infinitely many ‘b-particles’, which in mathematical
termsimpliesthat|Ω (cid:2)and|Ω (cid:2)cannotlieinthesameHilbertspace.Anexample
a b
ofthissituationcanbeconstructedbyconsideringacosmologicalspacetimeofthe
form (1.2) with a(t) = tanh(ct) where c is a constant with dimension of inverse
time. The asymptotic regions of this spacetime for t → ±∞ are time-translation
invariantandonecandefinecorrespondingasymptoticvacua|Ω±(cid:2).Onemaythen
computethate.g.the‘+’-vacuumcontainsinfinitelymany‘−’-particleswhichcanbe
physicallyinterpretedbysayingthattheexpansionofspaceencodedinthefunctional
formofa(t)createsinfinitelymanyparticles.Thisoccursbecausethequantumfield
φ(x)hasinfinitelymanydegreesoffreedom,whichareallexcitedbytheexpansion.
Consequently,thesecondequalitysignin(1.1)isingeneralpurelyheuristicbecause
thetwodecompositionsofφ(x)listedthereareingeneralnotrelatedinaphysically
andmathematicallymeaningfulway.
Fromtheabovediscussionwecaninferthatinthecontextofcurvedspacetimes
the very notion of ‘particle’ is strictly speaking meaningless, because it relies on
apreferredchoiceofvacuumstate,whichingeneraldoesnotexist.Consequently,