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On the Backreaction of Scalar and Spinor Quantum Fields in Curved Spacetimes

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On the Backreaction of Scalar and Spinor Quantum Fields in Curved Spacetimes Dissertation zur Erlangung des Doktorgrades des Department Physik der Universität Hamburg vorgelegt von Thomas-Paul Hack aus Timisoara Hamburg 2010 Gutachter der Dissertation: Prof. Dr. K. Fredenhagen Prof. Dr. V. Moretti Prof. Dr. R. M. Wald Gutachter der Disputation: Prof. Dr. K. Fredenhagen Prof. Dr. W. Buchmüller Datum der Disputation: Mittwoch, 19. Mai 2010 Vorsitzender des Prüfungsausschusses: Prof. Dr. J. Bartels Vorsitzender des Promotionsausschusses: Prof. Dr. J. Bartels Dekan der Fakultät für Mathematik, Informatik und Naturwissenschaften: Prof. Dr. H. Graener Zusammenfassung In der vorliegenden Arbeit werden zunächst einige Konstruktionen und Resultate in Quantenfeldtheorie auf gekrümmten Raumzeiten, die bisher nur für das Klein-Gordon Feld behandelt und erlangt worden sind, für Dirac Felder verallgemeinert. Es wird im Rahmen des algebraischen Zugangs die erweiterte Algebra der Observablen konstruiert, die insbesondere normalgeordnete Wickpolynome des Diracfeldes enthält. Anschließend wird ein ausgezeichnetes Element dieser erweiterten Algebra, der Energie-Impuls Tensor, analysiert. Unter Zuhilfenahme ausführlicher Berechnungen der Hadamardkoe ?zienten des Diracfeldes wird gezeigt, dass eine lokale, kovariante und kovariant erhaltene Konstruktion des Energie- Impuls Tensors möglich ist. Anschließend wird das Verhältnis der mathematisch fundierten Hadamardreg- ularisierung des Energie-Impuls Tensors mit der mathematisch weniger rigorosen DeWitt-Schwinger Reg- ularisierung verglichen. Man findet, dass die beiden Regularisierungen im wesentlichen äquivalent sind, insbesondere lässt sich die DeWitt-Schwinger Regularisierung mathematisch exakt formulieren. Während die bisher angeführten Resultate auf allgemeinen gekrümmten Raumzeiten gültig sind, werden zusätzliche Untersuchungen auf einer Klasse von flachen Robertson-Walker Raumzeiten, die eine lichtartige Urk- nallhyperfläche besitzen, angestellt. Mit hilfe holographischer Methoden werden auf solchen Raumzeiten Hadamardzustände für das Klein-Gordon- und Diracfeld konstruiert, die dadurch ausgezeichnet sind, dass sie im Sinne eines Limes zur Urknallhyperfläche asymptotische Gleichgewichtszustände darstellen. Ab- schließend werden Lösungen der semiklassischen Einsteingleichungen für Quantenfelder mit beliebigem Spin im flachen Robertson-Walker Fall untersucht. Es stellt sich heraus, dass die gefundene Lösungen Supernova Typ Ia Messdaten ebenso gut wie das ΛCDM Modell erklären. Damit ist eine natürliche Erklärung für dunkle Energie und ein einfaches Modell für kosmologische dunkle Materie gefunden. Abstract In the first instance, the present work is concerned with generalising constructions and results in quantum field theory on curved spacetimes from the well-known case of the Klein-Gordon field to Dirac fields. To this end, the enlarged algebra of observables of the Dirac field is constructed in the algebraic framework. This algebra contains normal-ordered Wick polynomials in particular, and an extended analysis of one of its elements, the stress-energy tensor, is performed. Based on detailed calculations of the Hadamard coe ?cients of the Dirac field, it is found that a local, covariant, and covariantly conserved construction of the stress-energy tensor is possible. Additionally, the mathematically sound Hadamard regularisation prescription of the stress-energy tensor is compared to the mathematically less rigorous DeWitt-Schwinger regularisation. It is found that both prescriptions are essentially equivalent, particularly, it turns out to be possible to formulate the DeWitt-Schwinger prescription in a well-defined way. While the aforemen- tioned results hold in generic curved spacetimes, particular attention is also devoted to a specific class of Robertson-Walker spacetimes with a lightlike Big Bang hypersurface. Employing holographic methods, Hadamard states for the Klein-Gordon and the Dirac field are constructed. These states are preferred in the sense that they constitute asymptotic equilibrium states in the limit to the Big Bang hypersurface. Finally, solutions of the semiclassical Einstein equation for quantum fields of arbitrary spin are analysed in the flat Robertson-Walker case. One finds that these solutions explain the measured supernova Ia data as good as the ΛCDM model. Hence, one arrives at a natural explanation of dark energy and a simple quantum model of cosmological dark matter. It seems plain and self-evident, yet it needs to be said: the isolated knowledge obtained by a group of specialists in a narrow field has in itself no value whatsoever, but only in its synthesis with all the rest of knowledge and only inasmuch as it really contributes in this synthesis toward answering the demand τ´ινǫζ δ ǫ` ηµǫ´ ¯ιζ ; (‘who are we?’) Erwin Schrödinger, [Sch96, p. 109] Contents Introduction . 1 I Spacetimes and Spacetime Structures 11 I.1 Spacetimes . 13 I.2 Spinors on Curved Spacetimes . 17 I.2.1 Spin Structures . 17 I.2.2 The Spin Connection . 21 I.3 Null Big Bang Spacetimes . 28 I.3.1 Cosmological Spacetimes and Power-Law Inflation . 28 I.3.2 The Conformal Boundary of Null Big Bang Spacetimes . 32 I.3.3 Spinors in Flat Friedmann-Lemaître-Robertson-Walker Spacetimes . 39 I.4 Spacetime Categories . 44 II Classical Fields in Curved Spacetimes 47 II.1 The Free Scalar Field in General Curved Spacetimes . 49 II.1.1 The Klein-Gordon Equation and its Fundamental Solutions . 49 II.1.2 The Symplectic Space of Solutions . 52 II.1.3 The Algebra of Classical Observables . 55 II.2 The Free Scalar Field in Null Big Bang Spacetimes . 57 II.2.1 The Mode Expansion of Solutions and the Causal Propagator . 58 II.2.2 The Bulk and Boundary Symplectic Spaces of Solutions . 66 II.3 The Free Dirac Field in General Curved Spacetimes . 72 II.3.1 The Dirac Equation and its Fundamental Solutions . 72 II.3.2 The Inner Product Space of Solutions . 77 II.3.3 The Algebra of Classical Observables . 80 II.4 The Free Dirac Field in Null Big Bang Spacetimes . 82 II.4.1 The Mode Expansion of Solutions and the Causal Propagator . 83 II.4.2 The Bulk and Boundary Inner Product Spaces of Solutions . 90 7 III Quantized Fields in Curved Spacetimes 97 III.1 The Free Scalar Field in General Curved Spacetimes . 99 III.1.1 The Algebras of Observables . 99 III.1.2 Hadamard States . 105 III.1.3 The Enlarged Algebra of Observables . 118 III.1.4 Locality and General Covariance . 126 III.2 The Free Scalar Field in Null Big Bang Spacetimes . 129 III.2.1 The Bulk and Boundary Algebras . 129 III.2.2 Preferred Asymptotic Ground States and Thermal States of Hadamard Type .............................................. 131 III.3 The Free Dirac Field in General Curved Spacetimes . 148 III.3.1 The Algebras of Observables . 148 III.3.2 Hadamard States . 153 III.3.3 The Enlarged Algebra of Observables . 165 III.3.4 Locality and General Covariance . 171 III.4 The Free Dirac Field in Null Big Bang Spacetimes . 173 III.4.1 The Bulk and Boundary Algebras . 173 III.4.2 Preferred Asymptotic Ground States and Thermal States of Hadamard Type .............................................. 175 IV The Backreaction of Quantized Fields in Curved Spacetimes 187 IV.1 The Semiclassical Einstein Equation and Wald’s Axioms . 189 IV.2 The Quantum Stress-Energy Tensor of a Scalar Field . 197 IV.3 The Quantum Stress-Energy Tensor of a Dirac Field . 200 IV.4 The Relation between DeWitt-Schwinger and Hadamard Point-Splitting . 203 IV.5 Cosmological Solutions of the Semiclassical Einstein Equation . 212 IV.5.1 The Energy Density of Quantum Fields . 213 IV.5.2 Computation of the State-Dependent Contributions . 215 IV.5.3 Classification of the Cosmological Solutions . 223 IV.6 Comparison with Supernova Ia Measurements . 229 Conclusions . 237 Bibliography . 242 Introduction Quantum field theory in curved spacetimes is a framework in which the spacetime is treated as a curved Lorentzian manifold in accord with General Relativity, whereas matter is treated as a quantum field. This framework is expected to provide a semiclassical approximation to a full quantum theory of both gravity and matter, and, hence, finds its applications in cases where the spacetime curvature is low enough for quantum gravity e :ects to be negligible, but too large for Minkowskian quantum field theory to make sense. Well, these are the words one often finds as introductory phrases in works on quantum field theory in curved spacetimes. Unfortunately, they somehow suggest that quantum field theory on curved spacetimes ultimately needs to be legitimated by a full-fledged quantum gravity theory. Although this is certainly as correct as it is already for flat spacetime quantum field theory, this point of view seems to disregard a maybe obvious fact. Namely, we inevitably live in a curved spacetime, although our local curvature is quite low. Hence, if one wants to treat matter as quantum fields in our whole universe at all, then one is forced to consider quantum field theory in curved spacetime. Therefore, quantum field theory on curved spacetimes is more fundamental than quantum field theory on Minkowski spacetime, the latter being undoubtedly a good local approximation in cases where the relevant energy scales are much larger than the curvature scales. Thus, whichever new lessons quantum field theory on curved space teaches us, we should take them serious and should not raise an eyebrow if they seem to contradict our Minkowskian intuition. Two situations in which the curvature of the spacetime can certainly not be neglected are
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