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An Alternate Constructive Approach to the 43 Quantum Field Theory, and A
ANNALES DE L’I. H. P., SECTION A ALAN D. SOKAL 4 An alternate constructive approach to the j3 quantum 4 field theory, and a possible destructive approach to j4 Annales de l’I. H. P., section A, tome 37, no 4 (1982), p. 317-398 <http://www.numdam.org/item?id=AIHPA_1982__37_4_317_0> © Gauthier-Villars, 1982, tous droits réservés. L’accès aux archives de la revue « Annales de l’I. H. P., section A » implique l’accord avec les conditions générales d’utilisation (http://www.numdam. org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. Inst. Henri Poincaré, Vol. XXXVII, n° 4, 1982 317 An alternate constructive approach to the 03C643 quantum field theory, and a possible destructive approach to 03C644 (*) Alan D. SOKAL Courant Institute of Mathematical Sciences New York University, 251 Mercer Street, New York, New York 10012, USA ABSTRACT. - I study the construction of cp4 quantum field theories by means of lattice approximations. It is easy to prove the existence of the continuum limit (by subsequences) ; the key question is whether this limit is something other than a (generalized) free field. I use correlation inequalities, infrared bounds and field equations to investigate this question. For space-time dimension d less than four, I give a simple proof that the continuum-limit theory is indeed nontrivial ; it relies, however, on a conjec- tured correlation inequality closely related to the T6 conjecture of Glimm and Jaffe. -
The State of the Multiverse: the String Landscape, the Cosmological Constant, and the Arrow of Time
The State of the Multiverse: The String Landscape, the Cosmological Constant, and the Arrow of Time Raphael Bousso Center for Theoretical Physics University of California, Berkeley Stephen Hawking: 70th Birthday Conference Cambridge, 6 January 2011 RB & Polchinski, hep-th/0004134; RB, arXiv:1112.3341 The Cosmological Constant Problem The Landscape of String Theory Cosmology: Eternal inflation and the Multiverse The Observed Arrow of Time The Arrow of Time in Monovacuous Theories A Landscape with Two Vacua A Landscape with Four Vacua The String Landscape Magnitude of contributions to the vacuum energy graviton (a) (b) I Vacuum fluctuations: SUSY cutoff: ! 10−64; Planck scale cutoff: ! 1 I Effective potentials for scalars: Electroweak symmetry breaking lowers Λ by approximately (200 GeV)4 ≈ 10−67. The cosmological constant problem −121 I Each known contribution is much larger than 10 (the observational upper bound on jΛj known for decades) I Different contributions can cancel against each other or against ΛEinstein. I But why would they do so to a precision better than 10−121? Why is the vacuum energy so small? 6= 0 Why is the energy of the vacuum so small, and why is it comparable to the matter density in the present era? Recent observations Supernovae/CMB/ Large Scale Structure: Λ ≈ 0:4 × 10−121 Recent observations Supernovae/CMB/ Large Scale Structure: Λ ≈ 0:4 × 10−121 6= 0 Why is the energy of the vacuum so small, and why is it comparable to the matter density in the present era? The Cosmological Constant Problem The Landscape of String Theory Cosmology: Eternal inflation and the Multiverse The Observed Arrow of Time The Arrow of Time in Monovacuous Theories A Landscape with Two Vacua A Landscape with Four Vacua The String Landscape Many ways to make empty space Topology and combinatorics RB & Polchinski (2000) I A six-dimensional manifold contains hundreds of topological cycles. -
On Asymptotically De Sitter Space: Entropy & Moduli Dynamics
On Asymptotically de Sitter Space: Entropy & Moduli Dynamics To Appear... [1206.nnnn] Mahdi Torabian International Centre for Theoretical Physics, Trieste String Phenomenology Workshop 2012 Isaac Newton Institute , CAMBRIDGE 1 Tuesday, June 26, 12 WMAP 7-year Cosmological Interpretation 3 TABLE 1 The state of Summarythe ofart the cosmological of Observational parameters of ΛCDM modela Cosmology b c Class Parameter WMAP 7-year ML WMAP+BAO+H0 ML WMAP 7-year Mean WMAP+BAO+H0 Mean 2 +0.056 Primary 100Ωbh 2.227 2.253 2.249−0.057 2.255 ± 0.054 2 Ωch 0.1116 0.1122 0.1120 ± 0.0056 0.1126 ± 0.0036 +0.030 ΩΛ 0.729 0.728 0.727−0.029 0.725 ± 0.016 ns 0.966 0.967 0.967 ± 0.014 0.968 ± 0.012 τ 0.085 0.085 0.088 ± 0.015 0.088 ± 0.014 2 d −9 −9 −9 −9 ∆R(k0) 2.42 × 10 2.42 × 10 (2.43 ± 0.11) × 10 (2.430 ± 0.091) × 10 +0.030 Derived σ8 0.809 0.810 0.811−0.031 0.816 ± 0.024 H0 70.3km/s/Mpc 70.4km/s/Mpc 70.4 ± 2.5km/s/Mpc 70.2 ± 1.4km/s/Mpc Ωb 0.0451 0.0455 0.0455 ± 0.0028 0.0458 ± 0.0016 Ωc 0.226 0.226 0.228 ± 0.027 0.229 ± 0.015 2 +0.0056 Ωmh 0.1338 0.1347 0.1345−0.0055 0.1352 ± 0.0036 e zreion 10.4 10.3 10.6 ± 1.210.6 ± 1.2 f t0 13.79 Gyr 13.76 Gyr 13.77 ± 0.13 Gyr 13.76 ± 0.11 Gyr a The parameters listed here are derived using the RECFAST 1.5 and version 4.1 of the WMAP[WMAP-7likelihood 1001.4538] code. -
Black Hole Singularity Resolution Via the Modified Raychaudhuri
Black hole singularity resolution via the modified Raychaudhuri equation in loop quantum gravity Keagan Blanchette,a Saurya Das,b Samantha Hergott,a Saeed Rastgoo,a aDepartment of Physics and Astronomy, York University 4700 Keele Street, Toronto, Ontario M3J 1P3 Canada bTheoretical Physics Group and Quantum Alberta, Department of Physics and Astronomy, University of Lethbridge, 4401 University Drive, Lethhbridge, Alberta T1K 3M4, Canada E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: We derive loop quantum gravity corrections to the Raychaudhuri equation in the interior of a Schwarzschild black hole and near the classical singularity. We show that the resulting effective equation implies defocusing of geodesics due to the appearance of repulsive terms. This prevents the formation of conjugate points, renders the singularity theorems inapplicable, and leads to the resolution of the singularity for this spacetime. arXiv:2011.11815v3 [gr-qc] 21 Apr 2021 Contents 1 Introduction1 2 Interior of the Schwarzschild black hole3 3 Classical dynamics6 3.1 Classical Hamiltonian and equations of motion6 3.2 Classical Raychaudhuri equation 10 4 Effective dynamics and Raychaudhuri equation 11 4.1 ˚µ scheme 14 4.2 µ¯ scheme 17 4.3 µ¯0 scheme 19 5 Discussion and outlook 22 1 Introduction It is well known that General Relativity (GR) predicts that all reasonable spacetimes are singular, and therefore its own demise. While a similar situation in electrodynamics was resolved in quantum electrodynamics, quantum gravity has not been completely formulated yet. One of the primary challenges of candidate theories such as string theory and loop quantum gravity (LQG) is to find a way of resolving the singularities. -
8.962 General Relativity, Spring 2017 Massachusetts Institute of Technology Department of Physics
8.962 General Relativity, Spring 2017 Massachusetts Institute of Technology Department of Physics Lectures by: Alan Guth Notes by: Andrew P. Turner May 26, 2017 1 Lecture 1 (Feb. 8, 2017) 1.1 Why general relativity? Why should we be interested in general relativity? (a) General relativity is the uniquely greatest triumph of analytic reasoning in all of science. Simultaneity is not well-defined in special relativity, and so Newton's laws of gravity become Ill-defined. Using only special relativity and the fact that Newton's theory of gravity works terrestrially, Einstein was able to produce what we now know as general relativity. (b) Understanding gravity has now become an important part of most considerations in funda- mental physics. Historically, it was easy to leave gravity out phenomenologically, because it is a factor of 1038 weaker than the other forces. If one tries to build a quantum field theory from general relativity, it fails to be renormalizable, unlike the quantum field theories for the other fundamental forces. Nowadays, gravity has become an integral part of attempts to extend the standard model. Gravity is also important in the field of cosmology, which became more prominent after the discovery of the cosmic microwave background, progress on calculations of big bang nucleosynthesis, and the introduction of inflationary cosmology. 1.2 Review of Special Relativity The basic assumption of special relativity is as follows: All laws of physics, including the statement that light travels at speed c, hold in any inertial coordinate system. Fur- thermore, any coordinate system that is moving at fixed velocity with respect to an inertial coordinate system is also inertial. -
Raychaudhuri and Optical Equations for Null Geodesic Congruences With
Raychaudhuri and optical equations for null geodesic congruences with torsion Simone Speziale1 1 Aix Marseille Univ., Univ. de Toulon, CNRS, CPT, UMR 7332, 13288 Marseille, France October 17, 2018 Abstract We study null geodesic congruences (NGCs) in the presence of spacetime torsion, recovering and extending results in the literature. Only the highest spin irreducible component of torsion gives a proper acceleration with respect to metric NGCs, but at the same time obstructs abreastness of the geodesics. This means that it is necessary to follow the evolution of the drift term in the optical equations, and not just shear, twist and expansion. We show how the optical equations depend on the non-Riemannian components of the curvature, and how they reduce to the metric ones when the highest spin component of torsion vanishes. Contents 1 Introduction 1 2 Metric null geodesic congruences and optical equations 3 2.1 Null geodesic congruences and kinematical quantities . ................. 3 2.2 Dynamics: Raychaudhuri and optical equations . ............. 6 3 Curvature, torsion and their irreducible components 8 4 Torsion-full null geodesic congruences 9 4.1 Kinematical quantities and the congruence’s geometry . ................ 11 arXiv:1808.00952v3 [gr-qc] 16 Oct 2018 5 Raychaudhuri equation with torsion 13 6 Optical equations with torsion 16 7 Comments and conclusions 18 A Newman-Penrose notation 19 1 Introduction Torsion plays an intriguing role in approaches to gravity where the connection is given an indepen- dent status with respect to the metric. This happens for instance in the first-order Palatini and in the Einstein-Cartan versions of general relativity (see [1, 2, 3] for reviews and references therein), and in more elaborated theories with extra gravitational degrees of freedom like the Poincar´egauge 1 theory of gravity, see e.g. -
Analog Raychaudhuri Equation in Mechanics [14]
Analog Raychaudhuri equation in mechanics Rajendra Prasad Bhatt∗, Anushree Roy and Sayan Kar† Department of Physics, Indian Institute of Technology Kharagpur, 721 302, India Abstract Usually, in mechanics, we obtain the trajectory of a particle in a given force field by solving Newton’s second law with chosen initial conditions. In contrast, through our work here, we first demonstrate how one may analyse the behaviour of a suitably defined family of trajectories of a given mechanical system. Such an approach leads us to develop a mechanics analog following the well-known Raychaudhuri equation largely studied in Riemannian geometry and general relativity. The idea of geodesic focusing, which is more familiar to a relativist, appears to be analogous to the meeting of trajectories of a mechanical system within a finite time. Applying our general results to the case of simple pendula, we obtain relevant quantitative consequences. Thereafter, we set up and perform a straightforward experiment based on a system with two pendula. The experimental results on this system are found to tally well with our proposed theoretical model. In summary, the simple theory, as well as the related experiment, provides us with a way to understand the essence of a fairly involved concept in advanced physics from an elementary standpoint. arXiv:2105.04887v1 [gr-qc] 11 May 2021 ∗ Present Address: Inter-University Centre for Astronomy and Astrophysics, Post Bag 4, Ganeshkhind, Pune 411 007, India †Electronic address: [email protected], [email protected], [email protected] 1 I. INTRODUCTION Imagine two pendula of the same length hung from a common support. -
On the Limits of Effective Quantum Field Theory
RUNHETC-2019-15 On the Limits of Effective Quantum Field Theory: Eternal Inflation, Landscapes, and Other Mythical Beasts Tom Banks Department of Physics and NHETC Rutgers University, Piscataway, NJ 08854 E-mail: [email protected] Abstract We recapitulate multiple arguments that Eternal Inflation and the String Landscape are actually part of the Swampland: ideas in Effective Quantum Field Theory that do not have a counterpart in genuine models of Quantum Gravity. 1 Introduction Most of the arguments and results in this paper are old, dating back a decade, and very little of what is written here has not been published previously, or presented in talks. I was motivated to write this note after spending two weeks at the Vacuum Energy and Electroweak Scale workshop at KITP in Santa Barbara. There I found a whole new generation of effective field theorists recycling tired ideas from the 1980s about the use of effective field theory in gravitational contexts. These were ideas that I once believed in, but since the beginning of the 21st century my work in string theory and the dynamics of black holes, convinced me that they arXiv:1910.12817v2 [hep-th] 6 Nov 2019 were wrong. I wrote and lectured about this extensively in the first decade of the century, but apparently those arguments have not been accepted, and effective field theorists have concluded that the main lesson from string theory is that there is a vast landscape of meta-stable states in the theory of quantum gravity, connected by tunneling transitions in the manner envisioned by effective field theorists in the 1980s. -
Geometric Origin of Coincidences and Hierarchies in the Landscape
Geometric origin of coincidences and hierarchies in the landscape The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Bousso, Raphael et al. “Geometric origin of coincidences and hierarchies in the landscape.” Physical Review D 84.8 (2011): n. pag. Web. 26 Jan. 2012. © 2011 American Physical Society As Published http://dx.doi.org/10.1103/PhysRevD.84.083517 Publisher American Physical Society (APS) Version Final published version Citable link http://hdl.handle.net/1721.1/68666 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. PHYSICAL REVIEW D 84, 083517 (2011) Geometric origin of coincidences and hierarchies in the landscape Raphael Bousso,1,2 Ben Freivogel,3 Stefan Leichenauer,1,2 and Vladimir Rosenhaus1,2 1Center for Theoretical Physics and Department of Physics, University of California, Berkeley, California 94720-7300, USA 2Lawrence Berkeley National Laboratory, Berkeley, California 94720-8162, USA 3Center for Theoretical Physics and Laboratory for Nuclear Science, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Received 2 August 2011; published 14 October 2011) We show that the geometry of cutoffs on eternal inflation strongly constrains predictions for the time scales of vacuum domination, curvature domination, and observation. We consider three measure proposals: the causal patch, the fat geodesic, and the apparent horizon cutoff, which is introduced here for the first time. We impose neither anthropic requirements nor restrictions on landscape vacua. For vacua with positive cosmological constant, all three measures predict the double coincidence that most observers live at the onset of vacuum domination and just before the onset of curvature domination. -
Arxiv:Hep-Th/0209231 V1 26 Sep 2002 Tbs,Daai N Uigi Eurdfrsae Te Hntetemlvcu Olead to Vacuum Thermal the Anisotropy
SLAC-PUB-9533 BRX TH-505 October 2002 SU-ITP-02/02 Initial conditions for inflation Nemanja Kaloper1;2, Matthew Kleban1, Albion Lawrence1;3;4, Stephen Shenker1 and Leonard Susskind1 1Department of Physics, Stanford University, Stanford, CA 94305 2Department of Physics, University of California, Davis, CA 95616 3Brandeis University Physics Department, MS 057, POB 549110, Waltham, MA 02454y 4SLAC Theory Group, MS 81, 2575 Sand Hill Road, Menlo Park, CA 94025 Free scalar fields in de Sitter space have a one-parameter family of states invariant under the de Sitter group, including the standard thermal vacuum. We show that, except for the thermal vacuum, these states are unphysical when gravitational interactions are arXiv:hep-th/0209231 v1 26 Sep 2002 included. We apply these observations to the quantum state of the inflaton, and find that at best, dramatic fine tuning is required for states other than the thermal vacuum to lead to observable features in the CMBR anisotropy. y Present and permanent address. *Work supported in part by Department of Energy Contract DE-AC03-76SF00515. 1. Introduction In inflationary cosmology, cosmic microwave background (CMB) data place a tanta- lizing upper bound on the vacuum energy density during the inflationary epoch: 4 V M 4 1016 GeV : (1:1) ∼ GUT ∼ Here MGUT is the \unification scale" in supersymmetric grand unified models, as predicted by the running of the observed strong, weak and electromagnetic couplings above 1 T eV in the minimal supersymmetric standard model. If this upper bound is close to the truth, the vacuum energy can be measured directly with detectors sensitive to the polarization of the CMBR. -
Light Rays, Singularities, and All That
Light Rays, Singularities, and All That Edward Witten School of Natural Sciences, Institute for Advanced Study Einstein Drive, Princeton, NJ 08540 USA Abstract This article is an introduction to causal properties of General Relativity. Topics include the Raychaudhuri equation, singularity theorems of Penrose and Hawking, the black hole area theorem, topological censorship, and the Gao-Wald theorem. The article is based on lectures at the 2018 summer program Prospects in Theoretical Physics at the Institute for Advanced Study, and also at the 2020 New Zealand Mathematical Research Institute summer school in Nelson, New Zealand. Contents 1 Introduction 3 2 Causal Paths 4 3 Globally Hyperbolic Spacetimes 11 3.1 Definition . 11 3.2 Some Properties of Globally Hyperbolic Spacetimes . 15 3.3 More On Compactness . 18 3.4 Cauchy Horizons . 21 3.5 Causality Conditions . 23 3.6 Maximal Extensions . 24 4 Geodesics and Focal Points 25 4.1 The Riemannian Case . 25 4.2 Lorentz Signature Analog . 28 4.3 Raychaudhuri’s Equation . 31 4.4 Hawking’s Big Bang Singularity Theorem . 35 5 Null Geodesics and Penrose’s Theorem 37 5.1 Promptness . 37 5.2 Promptness And Focal Points . 40 5.3 More On The Boundary Of The Future . 46 1 5.4 The Null Raychaudhuri Equation . 47 5.5 Trapped Surfaces . 52 5.6 Penrose’s Theorem . 54 6 Black Holes 58 6.1 Cosmic Censorship . 58 6.2 The Black Hole Region . 60 6.3 The Horizon And Its Generators . 63 7 Some Additional Topics 66 7.1 Topological Censorship . 67 7.2 The Averaged Null Energy Condition . -
Geometric Approaches to Quantum Field Theory
GEOMETRIC APPROACHES TO QUANTUM FIELD THEORY A thesis submitted to The University of Manchester for the degree of Doctor of Philosophy in the Faculty of Science and Engineering 2020 Kieran T. O. Finn School of Physics and Astronomy Supervised by Professor Apostolos Pilaftsis BLANK PAGE 2 Contents Abstract 7 Declaration 9 Copyright 11 Acknowledgements 13 Publications by the Author 15 1 Introduction 19 1.1 Unit Independence . 20 1.2 Reparametrisation Invariance in Quantum Field Theories . 24 1.3 Example: Complex Scalar Field . 25 1.4 Outline . 31 1.5 Conventions . 34 2 Field Space Covariance 35 2.1 Riemannian Geometry . 35 2.1.1 Manifolds . 35 2.1.2 Tensors . 36 2.1.3 Connections and the Covariant Derivative . 37 2.1.4 Distances on the Manifold . 38 2.1.5 Curvature of a Manifold . 39 2.1.6 Local Normal Coordinates and the Vielbein Formalism 41 2.1.7 Submanifolds and Induced Metrics . 42 2.1.8 The Geodesic Equation . 42 2.1.9 Isometries . 43 2.2 The Field Space . 44 2.2.1 Interpretation of the Field Space . 48 3 2.3 The Configuration Space . 50 2.4 Parametrisation Dependence of Standard Approaches to Quan- tum Field Theory . 52 2.4.1 Feynman Diagrams . 53 2.4.2 The Effective Action . 56 2.5 Covariant Approaches to Quantum Field Theory . 59 2.5.1 Covariant Feynman Diagrams . 59 2.5.2 The Vilkovisky–DeWitt Effective Action . 62 2.6 Example: Complex Scalar Field . 66 3 Frame Covariance in Quantum Gravity 69 3.1 The Cosmological Frame Problem .