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Quantum Scalar Field Theory in Ads and the Ads/CFT Correspondence

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Quantum Scalar Field Theory in Ads and the Ads/CFT Correspondence Quantum Scalar Field Theory in AdS and the AdS/CFT Correspondence Igor Bertan Munich, September 2019 Quantum Scalar Field Theory in AdS and the AdS/CFT Correspondence Igor Bertan Dissertation an der Fakultät für Physik der Ludwig–Maximilians–Universität München vorgelegt von Igor Bertan München, den 2. September 2019 “It is reminiscent of what distinguishes the good theorists from the bad ones. The good ones always make an even number of sign errors, and the bad ones always make an odd number.” – Anthony Zee Erstgutachter: Prof. Dr. Ivo Sachs Zweitgutachter: Prof. Dr. Gerhard Buchalla Tag der mündlichen Prüfung: 11. November 2019 i Contents Zusammenfassung iii Abstract v I Introduction1 1.1 Motivation . .1 1.2 Research statement and results . .7 1.3 Content of the thesis . 10 1.4 List of published papers . 11 1.5 Acknowledgments . 12 II QFT in flat space-time 13 2.1 Axiomatic quantum field theory . 13 2.2 The Wightman functions . 18 2.3 Analytic continuation of correlation functions . 22 2.4 Free scalar quantum field theory . 27 III QFT in curved space-times 41 3.1 Free scalar quantum field theory . 41 3.2 Generalized Wightman axioms . 52 IV The anti–de Sitter space-time 55 4.1 Geometry of AdS . 56 4.2 Symmetries . 61 4.3 Conformal boundary . 62 V Free scalar QFT in (E)AdS 65 5.1 Free scalar quantum field theory . 65 5.2 Correlation functions . 71 5.3 The holographic correlators and the CFT dual . 75 VI Interacting scalar QFT in EAdS 79 6.1 Correlation functions . 80 6.2 Two-point function . 82 6.2.1 The mass shift diagram . 82 6.2.2 The tadpole diagram . 84 6.2.3 The double tadpole diagram . 85 6.2.4 The sunset diagram . 88 ii 6.3 Four-point function . 92 6.3.1 The cross diagram . 93 6.3.2 The one loop diagram . 95 6.4 The holographic correlators . 101 6.5 The CFT dual . 103 VII Conclusions 109 A Ambient space approach 113 B Spinor helicity formalism in AdS 119 C Collection of identities 125 D Bosonic higher spin propagators 129 D.1 Propagation of a scalar particle . 129 D.2 Propagation of a vector particle . 130 D.3 Propagation of a higher spin particle . 133 E Expansions in the conformal invariants 137 F OPE coefficients 141 References 145 iii Zusammenfassung In dieser Arbeit berechnen wir Quantenkorrekturen zu den Zwei- und Vierpunktsfunktio- nen bis zur zweiten Ordnung in der Kopplungskonstante für eine konform gekoppelte Skalarfeldtheorie mit quartischer Selbstwechselwirkung in der vierdimensionalen Anti–de Sitter-Raumzeit (AdS). Unsere Berechnungen werden durchgeführt, indem die übliche Feynman-Störungstheorie in flacher Raumzeit auf den Poincaré-Patch des Euklidischen AdS verallgemeinert wird. Insbesondere wenden wir keine Kenntnisse in konformer Feldtheorie (CFT) an. Die erhaltenen Ergebnisse für die Zwei- und Vierpunktsfunktionen sind miteinander konsistent. Darüber hinaus argumentieren wir, dass die kritischen Exponenten von Korrelationsfunktionen nahe des dreidimensionalen konformen Randes von AdS die erforderlichen Daten für die Renormierungsbedingungen liefern und somit die üblichen on-shell Bedingungen ersetzen. Die holographische Vierpunktsfunktion kann systematisch in den konformen Invari- anten entwickelt und mit der konformen Block-Entwicklung auf dem Rand von AdS verglichen werden. Dies wird hier in niedriger Ordnung in den konformen Invarianten durchgeführt, wobei gezeigt wird, dass die entsprechenden Expansionskoeffizienten die Daten der konformen Block-Entwicklung eindeutig festlegen. Trotz Feinheiten bei UV- und (manchmal) IR-Divergenzen tritt kein Widerspruch auf. Wir zeigen ferner, dass die resultierende duale Randtheorie, stark eingeschränkt aufgrund der konformen Symmetrie und daher einer Reihe nichttrivialer Bedingungen unterliegend, tatsächlich eine mathe- matisch und physikalisch konsistente CFT ist. Unsere Theorie liefert daher eine erste explizite Bestätigung einer Quanten-AdS/CFT-Korrespondenz. Schließlich wird die Struktur der Operatorproduktentwicklung (OPE) der dualen CFT, einer deformierten verallgemeinerten freien Feldtheorie, zusammen mit den Korrek- turen sowohl der OPE-Koeffizienten als auch der konformen Dimensionen der primären Operatoren dargelegt. Insbesondere wird das Fehlen des Energie-Impuls-Tensors und jeglicher erhaltener Ströme deutlich. Analytische Ausdrücke für die anomalen Dimen- sionen werden bei einer Loop-Ordnung gefunden, sowohl für Neumann- als auch für Dirichlet-Randbedingungen. v Abstract In this thesis we compute quantum corrections to the two- and four-point correlation functions up to second order in the coupling constant for a conformally coupled scalar field theory with quartic selfinteraction in four-dimensional anti–de Sitter space-time (AdS). Our calculations are performed by generalizing the usual flat space-time Feynman perturbation theory to the Poincaré patch of Euclidean AdS. In particular, we do not exert any conformal field theory (CFT) knowledge. The obtained results for the two- and four-point functions are mutually consistent. In addition, we argue that the critical exponents of correlation functions near the three-dimensional conformal boundary of AdS provide the necessary data for the renormalization conditions, thus replacing the usual on-shell condition. The holographic four-point function can systematically be expanded in the conformal invariants and compared with the conformal block expansion on the boundary of AdS. This is carried out here at low order in the conformal invariants, where the corresponding expansion coefficients are shown to uniquely fix the data for the conformal block expansion. No contradiction arises despite subtleties with UV and (sometimes) IR divergences. We also show that the disclosed boundary dual, subject to a set of nontrivial conditions dictated by the strong constraint of conformal symmetry, is indeed a mathematically and physically consistent CFT. Hence, our theory provides a first explicit confirmation of a quantum AdS/CFT correspondence. Finally, the operator product expansion (OPE) structure of the dual CFT, a deformed generalized free field theory, is revealed, along with the corrections to both the OPE coefficients and conformal dimensions of primary operators. In particular, the absence of the stress tensor and of any conserved current becomes explicit. Analytic expressions for the anomalous dimensions are found at one loop, both for Neumann and Dirichlet boundary conditions. 1.1 Motivation 1 I Introduction 1.1 Motivation Without any doubts, Quantum Field Theory (QFT) is one of the most successful theoret- ical frameworks in physics at present. It combines field theory with quantum mechanics in a consistent manner and, in contrast to the conventional quantum formulation à la Heisenberg and Schrödinger, it allows for a sensible (special) relativistic formulation. From its inception almost a century ago it fueled theoretical and experimental physicists with deep insights into particle physics. Just to mention a few, the first developed QFT, the theory of Quantum Electrodynamics describing the electromagnetic interaction, was able to explain the phenomenon of spontaneous emission of photons from atoms and to infer the existence of anti–matter. But perhaps, the most impressive achievement of Quantum Electrodynamics is the extraordinary precision in the prediction of the electron magnetic moment [1]. Another significant accomplishment was the implementation of the weak interactions within the framework of QFT. The emergent theory, Quantum Flavordynamics, is however better understood in terms of the Electroweak Theory, a theory unifying the quantum theories of Electrodynamics and Flavordynamics. This unified theory agrees with experiments showing that, at very high energies, the electro- magnetic and weak interaction merge into a single electroweak interaction. Also the theory of strong interactions was realized in terms of a QFT and is known today as Quantum Chromodynamics. Although Quantum Chromodynamics was less prolific than the Electroweak Theory, it predicted the asymptotic freedom and has been the missing piece of the puzzle in the formulation of the Standard Model, a theory describing three of the four fundamental interactions between elementary particles. Many predictions of the Standard Model have been met with remarkable experimental precision [2], conveying the impression that we are close to a complete characterization of particle physics. Moreover, particle physics is not the only subject QFT has been successfully applied to. For instance, QFT has proven to supply a precious formulation of emergent phenomena at macroscopic scales of many-particle systems. Indeed, it found applications in a plethora of fields describing condensed matter, ranging from the simplest crystal lattices to the hardly manageable theories of superconductivity and the fractional quantum hall effect [3]. Notably, inalienable concepts of particle physics like the Higgs mechanism for spontaneous symmetry breaking and the renormalization procedure, were developed respectively improved by the study of Condensed Matter Physics within the framework of QFT. This certifies QFT as a universal concept, and not as a mere descriptive framework. On the other hand, a slightly older theory than QFT amazed generations of physi- cists with its puzzling implications. The theory is Einstein’s General Relativity (GR), which describes the fourth fundamental interaction, gravitation, the Standard Model is not capable of. It provided us with a flawless theoretical clarification of gravitational 2 I Introduction phenomena as well as with data of striking precision [4]. Long
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