DOCSLIB.ORG

Classical Theory of Gauge Fields This Page Intentionally Left Blank Classical Theory of Gauge Fields

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Classical Theory of Gauge Fields This Page Intentionally Left Blank Classical Theory of Gauge Fields Valery Rubakov Translated by Stephen S Wilson PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD Copyright c 2002 by Princeton University Press Original title: Klassicheskie Kalibrovochnye Polia c V.A. Rubakov, 1999 c Editorial´ URSS, 1999 Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 3 Market Place, Woodstock, Oxfordshire OX20 1SY All Rights Reserved Library of Congress Control Number 2002102049 ISBN 0-691-05927-6 (hardcover : alk. paper) The publisher would like to acknowledge the translator of this volume for providing the camera-ready copy from which this book was printed Printed on acid-free paper. ∞ www.pupress.princeton.edu Printed in the United States of America 10987654321 Contents Preface ix Part I 1 1 Gauge Principle in Electrodynamics 3 1.1Electromagnetic-fieldactioninvacuum............ 3 1.2Gaugeinvariance........................ 5 1.3 General solution of Maxwell’s equations in vacuum ..... 6 1.4Choiceofgauge......................... 8 2 Scalar and Vector Fields 11 2.1 System of units = c =1................... 11 2.2Scalarfieldaction........................ 12 2.3Massivevectorfield....................... 15 2.4Complexscalarfield...................... 17 2.5Degreesoffreedom....................... 18 2.6Interactionoffieldswithexternalsources.......... 19 2.7 Interacting fields. Gauge-invariant interaction in scalar electrodynamics......................... 21 2.8Noether’stheorem....................... 26 3 Elements of the Theory of Lie Groups and Algebras 33 3.1Groups.............................. 33 3.2Liegroupsandalgebras.................... 41 3.3RepresentationsofLiegroupsandLiealgebras....... 48 3.4CompactLiegroupsandalgebras............... 53 4 Non-Abelian Gauge Fields 57 4.1Non-Abelianglobalsymmetries................ 57 4.2 Non-Abelian gauge invariance and gauge fields: the group SU(2).............................. 63 4.3Generalizationtoothergroups................ 69 vi 4.4Fieldequations......................... 75 4.5Cauchyproblemandgaugeconditions............ 81 5 Spontaneous Breaking of Global Symmetry 85 5.1Spontaneousbreakingofdiscretesymmetry......... 86 5.2 Spontaneous breaking of global U(1) symmetry. Nambu– Goldstonebosons........................ 91 5.3 Partial symmetry breaking: the SO(3)model........ 94 5.4Generalcase.Goldstone’stheorem.............. 99 6 Higgs Mechanism 105 6.1ExampleofanAbelianmodel................. 105 6.2 Non-Abelian case: model with complete breaking of SU(2) symmetry............................ 112 6.3 Example of partial breaking of gauge symmetry: bosonic sectorofstandardelectroweaktheory............. 116 Supplementary Problems for Part I 127 Part II 135 7 The Simplest Topological Solitons 137 7.1Kink............................... 138 7.2 Scale transformations and theorems on the absence of solitons............................. 149 7.3Thevortex........................... 155 7.4 Soliton in a model of n-field in (2 + 1)-dimensional space–time........................... 165 8 Elements of Homotopy Theory 173 8.1Homotopyofmappings..................... 173 8.2 The fundamental group . .................. 176 8.3Homotopygroups........................ 179 8.4 Fiber bundles and homotopy groups ............. 184 8.5Summaryoftheresults.................... 189 9 Magnetic Monopoles 193 9.1 The soliton in a model with gauge group SU(2)....... 193 9.2Magneticcharge........................ 200 9.3Generalizationtoothermodels................ 207 9.4 The limit mH /mV → 0 .................... 208 9.5Dyons.............................. 212 vii 10 Non-Topological Solitons 215 11 Tunneling and Euclidean Classical Solutions in Quantum Mechanics 225 11.1Decayofametastablestateinquantummechanicsofone variable............................. 226 11.2Generalizationtothecaseofmanyvariables......... 232 11.3 Tunneling in potentials with classical degeneracy . ..... 240 12 Decay of a False Vacuum in Scalar Field Theory 249 12.1Preliminaryconsiderations................... 249 12.2 Decay probability: Euclidean bubble (bounce) ........ 253 12.3Thin-wallapproximation.................... 259 13 Instantons and Sphalerons in Gauge Theories 263 13.1Euclideangaugetheories.................... 263 13.2 Instantons in Yang–Mills theory ................ 265 13.3 Classical vacua and θ-vacua.................. 272 13.4 Sphalerons in four-dimensional models with the Higgs mechanism........................... 280 Supplementary Problems for Part II 287 Part III 293 14 Fermions in Background Fields 295 14.1FreeDiracequation...................... 295 14.2SolutionsofthefreeDiracequation.Diracsea....... 302 14.3Fermionsinbackgroundbosonicfields............ 308 14.4FermionicsectoroftheStandardModel........... 318 15 Fermions and Topological External Fields in Two-dimensional Models 329 15.1Chargefractionalization.................... 329 15.2 Level crossing and non-conservation of fermion quantum numbers............................. 336 16 Fermions in Background Fields of Solitons and Strings in Four-Dimensional Space–Time 351 16.1 Fermions in a monopole background field: integer angular momentumandfermionnumberfractionalization...... 352 viii 16.2 Scattering of fermions off a monopole: non-conservation of fermionnumbers........................ 359 16.3 Zero modes in a background field of a vortex: superconducting strings . ................... 364 17 Non-Conservation of Fermion Quantum Numbers in Four-dimensional Non-Abelian Theories 373 17.1LevelcrossingandEuclideanfermionzeromodes...... 374 17.2Fermionzeromodeinaninstantonfield........... 378 17.3Selectionrules.......................... 385 17.4 Electroweak non-conservation of baryon and lepton numbers athightemperatures...................... 392 Supplementary Problems for Part III 397 Appendix. Classical Solutions and the Functional Integral 403 A.1 Decay of the false vacuum in the functional integral formalism............................ 404 A.2 Instanton contributions to the fermion Green’s functions . 411 A.3 Instantons in theories with the Higgs mechanism. Integrationalongvalleys.................... 418 A.4Growinginstantoncrosssections............... 423 Bibliography 429 Index 441 Preface This book is based on a lecture course taught over a number of years in the Department of Quantum Statistics and Field Theory of the Moscow State University, Faculty of Physics, to students in the third and fourth years, specializing in theoretical physics. Traditionally, the theory of gauge fields is covered in courses on quantum field theory. However, many concepts and results of gauge theories are already in evidence at the level of classical field theory, which makes it possible and useful to study them in parallel with a study of quantum mechanics. Accordingly, the reading of the first ten chapters of this book does not require a knowledge of quantum mechanics, Chapters 11–13 employ the concepts and methods conventionally presented at the start of a course on quantum mechanics, and a thorough knowledge of quantum mechanics, including the Dirac equation, is only required in order to read the subsequent chapters. A detailed acquaintance with quantum field theory is not mandatory in order to read this text. At the same time, at the very start, it is assumed that the reader has a knowledge of classical mechanics, special relativity and classical electrodynamics. Part I of the book contains a study of the basic ideas of the theory of gauge fields, the construction of gauge-invariant Lagrangians and an analysis of spectra of linear perturbations, including perturbations about a non-trivial ground state. Part II is devoted to the construction and interpretation of solutions, whose existence is entirely due to the nonlinearity of the field equations, namely, solitons, bounces and instantons. In Part III, we consider certain interesting effects, arising in the interactions of fermions with topological scalar and gauge fields. The book has an Appendix, which briefly discusses the role of instantons as saddle points of the Euclidean functional integral in quantum field theory and several related topics. The purpose of the Appendix is to give an initial idea about this rather complex aspect of quantum field theory; the presentation there is schematic and makes no claims to completeness (for example, we do not touch at all upon the important questions relating to supersymmetric gauge theories). To read the Appendix, one needs to be x Preface acquainted with the quantum theory of gauge fields. Of course, the majority of questions touched upon in this book are considered in one way or another in other existing books and reviews on quantum field theory, a far from complete list of which is given at the end of the book. In a sense, this book may serve as an introduction to the subject. The book contains two mathematical digressions, where elements of the theory of Lie groups and algebras and of homotopy theory are studied, without any claim to completeness or mathematical rigor. This should make it possible to read the book without constantly resorting to the specialized literature in these areas. We should like to thank our colleagues F.L. Bezrukov, D.Yu. Grigoriev, M.V. Libanov, D.V. Semikoz, D.T. Son, P.G. Tinyakov and S.V. Troitsky for their great help in the preparation and teaching of the lecture course, attentive reading of the manuscript and preparation for its publication. Classical Theory of Gauge Fields This Page Intentionally
Recommended publications
  • An Introduction to Quantum Field Theory

    An Introduction to Quantum Field Theory

    AN INTRODUCTION TO QUANTUM FIELD THEORY By Dr M Dasgupta University of Manchester Lecture presented at the School for Experimental High Energy Physics Students Somerville College, Oxford, September 2009 - 1 - - 2 - Contents 0 Prologue....................................................................................................... 5 1 Introduction ................................................................................................ 6 1.1 Lagrangian formalism in classical mechanics......................................... 6 1.2 Quantum mechanics................................................................................... 8 1.3 The Schrödinger picture........................................................................... 10 1.4 The Heisenberg picture............................................................................ 11 1.5 The quantum mechanical harmonic oscillator ..................................... 12 Problems .............................................................................................................. 13 2 Classical Field Theory............................................................................. 14 2.1 From N-point mechanics to field theory ............................................... 14 2.2 Relativistic field theory ............................................................................ 15 2.3 Action for a scalar field ............................................................................ 15 2.4 Plane wave solution to the Klein-Gordon equation ...........................
  • Chapter 2 Introduction to Electrostatics

    Chapter 2 Introduction to Electrostatics

    Chapter 2 Introduction to electrostatics 2.1 Coulomb and Gauss’ Laws We will restrict our discussion to the case of static electric and magnetic fields in a homogeneous, isotropic medium. In this case the electric field satisfies the two equations, Eq. 1.59a with a time independent charge density and Eq. 1.77 with a time independent magnetic flux density, D (r)= ρ (r) , (1.59a) ∇ · 0 E (r)=0. (1.77) ∇ × Because we are working with static fields in a homogeneous, isotropic medium the constituent equation is D (r)=εE (r) . (1.78) Note : D is sometimes written : (1.78b) D = ²oE + P .... SI units D = E +4πP in Gaussian units in these cases ε = [1+4πP/E] Gaussian The solution of Eq. 1.59 is 1 ρ0 (r0)(r r0) 3 D (r)= − d r0 + D0 (r) , SI units (1.79) 4π r r 3 ZZZ | − 0| with D0 (r)=0 ∇ · If we are seeking the contribution of the charge density, ρ0 (r) , to the electric displacement vector then D0 (r)=0. The given charge density generates the electric field 1 ρ0 (r0)(r r0) 3 E (r)= − d r0 SI units (1.80) 4πε r r 3 ZZZ | − 0| 18 Section 2.2 The electric or scalar potential 2.2 TheelectricorscalarpotentialFaraday’s law with static fields, Eq. 1.77, is automatically satisfied by any electric field E(r) which is given by E (r)= φ (r) (1.81) −∇ The function φ (r) is the scalar potential for the electric field. It is also possible to obtain the difference in the values of the scalar potential at two points by integrating the tangent component of the electric field along any path connecting the two points E (r) d` = φ (r) d` (1.82) − path · path ∇ · ra rb ra rb Z → Z → ∂φ(r) ∂φ(r) ∂φ(r) = dx + dy + dz path ∂x ∂y ∂z ra rb Z → · ¸ = dφ (r)=φ (rb) φ (ra) path − ra rb Z → The result obtained in Eq.
  • Quantum Field Theory*

    Quantum Field Theory*

    Quantum Field Theory y Frank Wilczek Institute for Advanced Study, School of Natural Science, Olden Lane, Princeton, NJ 08540 I discuss the general principles underlying quantum eld theory, and attempt to identify its most profound consequences. The deep est of these consequences result from the in nite number of degrees of freedom invoked to implement lo cality.Imention a few of its most striking successes, b oth achieved and prosp ective. Possible limitation s of quantum eld theory are viewed in the light of its history. I. SURVEY Quantum eld theory is the framework in which the regnant theories of the electroweak and strong interactions, which together form the Standard Mo del, are formulated. Quantum electro dynamics (QED), b esides providing a com- plete foundation for atomic physics and chemistry, has supp orted calculations of physical quantities with unparalleled precision. The exp erimentally measured value of the magnetic dip ole moment of the muon, 11 (g 2) = 233 184 600 (1680) 10 ; (1) exp: for example, should b e compared with the theoretical prediction 11 (g 2) = 233 183 478 (308) 10 : (2) theor: In quantum chromo dynamics (QCD) we cannot, for the forseeable future, aspire to to comparable accuracy.Yet QCD provides di erent, and at least equally impressive, evidence for the validity of the basic principles of quantum eld theory. Indeed, b ecause in QCD the interactions are stronger, QCD manifests a wider variety of phenomena characteristic of quantum eld theory. These include esp ecially running of the e ective coupling with distance or energy scale and the phenomenon of con nement.
  • Introductory Lectures on Quantum Field Theory

    Introductory Lectures on Quantum Field Theory

    Introductory Lectures on Quantum Field Theory a b L. Álvarez-Gaumé ∗ and M.A. Vázquez-Mozo † a CERN, Geneva, Switzerland b Universidad de Salamanca, Salamanca, Spain Abstract In these lectures we present a few topics in quantum field theory in detail. Some of them are conceptual and some more practical. They have been se- lected because they appear frequently in current applications to particle physics and string theory. 1 Introduction These notes are based on lectures delivered by L.A.-G. at the 3rd CERN–Latin-American School of High- Energy Physics, Malargüe, Argentina, 27 February–12 March 2005, at the 5th CERN–Latin-American School of High-Energy Physics, Medellín, Colombia, 15–28 March 2009, and at the 6th CERN–Latin- American School of High-Energy Physics, Natal, Brazil, 23 March–5 April 2011. The audience on all three occasions was composed to a large extent of students in experimental high-energy physics with an important minority of theorists. In nearly ten hours it is quite difficult to give a reasonable introduction to a subject as vast as quantum field theory. For this reason the lectures were intended to provide a review of those parts of the subject to be used later by other lecturers. Although a cursory acquaintance with the subject of quantum field theory is helpful, the only requirement to follow the lectures is a working knowledge of quantum mechanics and special relativity. The guiding principle in choosing the topics presented (apart from serving as introductions to later courses) was to present some basic aspects of the theory that present conceptual subtleties.
  • Effective Field Theories, Reductionism and Scientific Explanation Stephan

    Effective Field Theories, Reductionism and Scientific Explanation Stephan

    To appear in: Studies in History and Philosophy of Modern Physics Effective Field Theories, Reductionism and Scientific Explanation Stephan Hartmann∗ Abstract Effective field theories have been a very popular tool in quantum physics for almost two decades. And there are good reasons for this. I will argue that effec- tive field theories share many of the advantages of both fundamental theories and phenomenological models, while avoiding their respective shortcomings. They are, for example, flexible enough to cover a wide range of phenomena, and concrete enough to provide a detailed story of the specific mechanisms at work at a given energy scale. So will all of physics eventually converge on effective field theories? This paper argues that good scientific research can be characterised by a fruitful interaction between fundamental theories, phenomenological models and effective field theories. All of them have their appropriate functions in the research process, and all of them are indispens- able. They complement each other and hang together in a coherent way which I shall characterise in some detail. To illustrate all this I will present a case study from nuclear and particle physics. The resulting view about scientific theorising is inherently pluralistic, and has implications for the debates about reductionism and scientific explanation. Keywords: Effective Field Theory; Quantum Field Theory; Renormalisation; Reductionism; Explanation; Pluralism. ∗Center for Philosophy of Science, University of Pittsburgh, 817 Cathedral of Learning, Pitts- burgh, PA 15260, USA (e-mail: [email protected]) (correspondence address); and Sektion Physik, Universit¨at M¨unchen, Theresienstr. 37, 80333 M¨unchen, Germany. 1 1 Introduction There is little doubt that effective field theories are nowadays a very popular tool in quantum physics.
  • Electromagnetic Field Theory

    Electromagnetic Field Theory

    Electromagnetic Field Theory BO THIDÉ Υ UPSILON BOOKS ELECTROMAGNETIC FIELD THEORY Electromagnetic Field Theory BO THIDÉ Swedish Institute of Space Physics and Department of Astronomy and Space Physics Uppsala University, Sweden and School of Mathematics and Systems Engineering Växjö University, Sweden Υ UPSILON BOOKS COMMUNA AB UPPSALA SWEDEN · · · Also available ELECTROMAGNETIC FIELD THEORY EXERCISES by Tobia Carozzi, Anders Eriksson, Bengt Lundborg, Bo Thidé and Mattias Waldenvik Freely downloadable from www.plasma.uu.se/CED This book was typeset in LATEX 2" (based on TEX 3.14159 and Web2C 7.4.2) on an HP Visualize 9000⁄360 workstation running HP-UX 11.11. Copyright c 1997, 1998, 1999, 2000, 2001, 2002, 2003 and 2004 by Bo Thidé Uppsala, Sweden All rights reserved. Electromagnetic Field Theory ISBN X-XXX-XXXXX-X Downloaded from http://www.plasma.uu.se/CED/Book Version released 19th June 2004 at 21:47. Preface The current book is an outgrowth of the lecture notes that I prepared for the four-credit course Electrodynamics that was introduced in the Uppsala University curriculum in 1992, to become the five-credit course Classical Electrodynamics in 1997. To some extent, parts of these notes were based on lecture notes prepared, in Swedish, by BENGT LUNDBORG who created, developed and taught the earlier, two-credit course Electromagnetic Radiation at our faculty. Intended primarily as a textbook for physics students at the advanced undergradu- ate or beginning graduate level, it is hoped that the present book may be useful for research workers
  • An Introduction to Supersymmetry

    An Introduction to Supersymmetry

    An Introduction to Supersymmetry Ulrich Theis Institute for Theoretical Physics, Friedrich-Schiller-University Jena, Max-Wien-Platz 1, D–07743 Jena, Germany [email protected] This is a write-up of a series of five introductory lectures on global supersymmetry in four dimensions given at the 13th “Saalburg” Summer School 2007 in Wolfersdorf, Germany. Contents 1 Why supersymmetry? 1 2 Weyl spinors in D=4 4 3 The supersymmetry algebra 6 4 Supersymmetry multiplets 6 5 Superspace and superfields 9 6 Superspace integration 11 7 Chiral superfields 13 8 Supersymmetric gauge theories 17 9 Supersymmetry breaking 22 10 Perturbative non-renormalization theorems 26 A Sigma matrices 29 1 Why supersymmetry? When the Large Hadron Collider at CERN takes up operations soon, its main objective, besides confirming the existence of the Higgs boson, will be to discover new physics beyond the standard model of the strong and electroweak interactions. It is widely believed that what will be found is a (at energies accessible to the LHC softly broken) supersymmetric extension of the standard model. What makes supersymmetry such an attractive feature that the majority of the theoretical physics community is convinced of its existence? 1 First of all, under plausible assumptions on the properties of relativistic quantum field theories, supersymmetry is the unique extension of the algebra of Poincar´eand internal symmtries of the S-matrix. If new physics is based on such an extension, it must be supersymmetric. Furthermore, the quantum properties of supersymmetric theories are much better under control than in non-supersymmetric ones, thanks to powerful non- renormalization theorems.
  • A Guiding Vector Field Algorithm for Path Following Control Of

    A Guiding Vector Field Algorithm for Path Following Control Of

    1 A guiding vector field algorithm for path following control of nonholonomic mobile robots Yuri A. Kapitanyuk, Anton V. Proskurnikov and Ming Cao Abstract—In this paper we propose an algorithm for path- the integral tracking error is uniformly positive independent following control of the nonholonomic mobile robot based on the of the controller design. Furthermore, in practice the robot’s idea of the guiding vector field (GVF). The desired path may be motion along the trajectory can be quite “irregular” due to an arbitrary smooth curve in its implicit form, that is, a level set of a predefined smooth function. Using this function and the its oscillations around the reference point. For instance, it is robot’s kinematic model, we design a GVF, whose integral curves impossible to guarantee either path following at a precisely converge to the trajectory. A nonlinear motion controller is then constant speed, or even the motion with a perfectly fixed proposed which steers the robot along such an integral curve, direction. Attempting to keep close to the reference point, the bringing it to the desired path. We establish global convergence robot may “overtake” it due to unpredictable disturbances. conditions for our algorithm and demonstrate its applicability and performance by experiments with real wheeled robots. As has been clearly illustrated in the influential paper [11], these performance limitations of trajectory tracking can be Index Terms—Path following, vector field guidance, mobile removed by carefully designed path following algorithms. robot, motion control, nonlinear systems Unlike the tracking approach, path following control treats the path as a geometric curve rather than a function of I.
  • Lo Algebras for Extended Geometry from Borcherds Superalgebras

    Lo Algebras for Extended Geometry from Borcherds Superalgebras

    Commun. Math. Phys. 369, 721–760 (2019) Communications in Digital Object Identifier (DOI) https://doi.org/10.1007/s00220-019-03451-2 Mathematical Physics L∞ Algebras for Extended Geometry from Borcherds Superalgebras Martin Cederwall, Jakob Palmkvist Division for Theoretical Physics, Department of Physics, Chalmers University of Technology, 412 96 Gothenburg, Sweden. E-mail: [email protected]; [email protected] Received: 24 May 2018 / Accepted: 15 March 2019 Published online: 26 April 2019 – © The Author(s) 2019 Abstract: We examine the structure of gauge transformations in extended geometry, the framework unifying double geometry, exceptional geometry, etc. This is done by giving the variations of the ghosts in a Batalin–Vilkovisky framework, or equivalently, an L∞ algebra. The L∞ brackets are given as derived brackets constructed using an underlying Borcherds superalgebra B(gr+1), which is a double extension of the structure algebra gr . The construction includes a set of “ancillary” ghosts. All brackets involving the infinite sequence of ghosts are given explicitly. All even brackets above the 2-brackets vanish, and the coefficients appearing in the brackets are given by Bernoulli numbers. The results are valid in the absence of ancillary transformations at ghost number 1. We present evidence that in order to go further, the underlying algebra should be the corresponding tensor hierarchy algebra. Contents 1. Introduction ................................. 722 2. The Borcherds Superalgebra ......................... 723 3. Section Constraint and Generalised Lie Derivatives ............. 728 4. Derivatives, Generalised Lie Derivatives and Other Operators ....... 730 4.1 The derivative .............................. 730 4.2 Generalised Lie derivative from “almost derivation” .......... 731 4.3 “Almost covariance” and related operators ..............
  • (Aka Second Quantization) 1 Quantum Field Theory

    (Aka Second Quantization) 1 Quantum Field Theory

    221B Lecture Notes Quantum Field Theory (a.k.a. Second Quantization) 1 Quantum Field Theory Why quantum field theory? We know quantum mechanics works perfectly well for many systems we had looked at already. Then why go to a new formalism? The following few sections describe motivation for the quantum field theory, which I introduce as a re-formulation of multi-body quantum mechanics with identical physics content. 1.1 Limitations of Multi-body Schr¨odinger Wave Func- tion We used totally anti-symmetrized Slater determinants for the study of atoms, molecules, nuclei. Already with the number of particles in these systems, say, about 100, the use of multi-body wave function is quite cumbersome. Mention a wave function of an Avogardro number of particles! Not only it is completely impractical to talk about a wave function with 6 × 1023 coordinates for each particle, we even do not know if it is supposed to have 6 × 1023 or 6 × 1023 + 1 coordinates, and the property of the system of our interest shouldn’t be concerned with such a tiny (?) difference. Another limitation of the multi-body wave functions is that it is incapable of describing processes where the number of particles changes. For instance, think about the emission of a photon from the excited state of an atom. The wave function would contain coordinates for the electrons in the atom and the nucleus in the initial state. The final state contains yet another particle, photon in this case. But the Schr¨odinger equation is a differential equation acting on the arguments of the Schr¨odingerwave function, and can never change the number of arguments.
  • Finite Quantum Field Theory and Renormalization Group

    Finite Quantum Field Theory and Renormalization Group

    Finite Quantum Field Theory and Renormalization Group M. A. Greena and J. W. Moffata;b aPerimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada bDepartment of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada September 15, 2021 Abstract Renormalization group methods are applied to a scalar field within a finite, nonlocal quantum field theory formulated perturbatively in Euclidean momentum space. It is demonstrated that the triviality problem in scalar field theory, the Higgs boson mass hierarchy problem and the stability of the vacuum do not arise as issues in the theory. The scalar Higgs field has no Landau pole. 1 Introduction An alternative version of the Standard Model (SM), constructed using an ultraviolet finite quantum field theory with nonlocal field operators, was investigated in previous work [1]. In place of Dirac delta-functions, δ(x), the theory uses distributions (x) based on finite width Gaussians. The Poincar´eand gauge invariant model adapts perturbative quantumE field theory (QFT), with a finite renormalization, to yield finite quantum loops. For the weak interactions, SU(2) U(1) is treated as an ab initio broken symmetry group with non- zero masses for the W and Z intermediate× vector bosons and for left and right quarks and leptons. The model guarantees the stability of the vacuum. Two energy scales, ΛM and ΛH , were introduced; the rate of asymptotic vanishing of all coupling strengths at vertices not involving the Higgs boson is controlled by ΛM , while ΛH controls the vanishing of couplings to the Higgs. Experimental tests of the model, using future linear or circular colliders, were proposed.
  • Arxiv:1809.08749V3 [Quant-Ph] 5 Dec 2018 a General Guiding Principle in Building the Theory of Fun- Damental Interactions (See, E.G., [1,3])

    Arxiv:1809.08749V3 [Quant-Ph] 5 Dec 2018 a General Guiding Principle in Building the Theory of Fun- Damental Interactions (See, E.G., [1,3])

    Resolution of Gauge Ambiguities in Ultrastrong-Coupling Cavity QED Omar Di Stefano,1 Alessio Settineri,2 Vincenzo Macr`ı,1 Luigi Garziano,1 Roberto Stassi,1 Salvatore Savasta,1, 2, ∗ and Franco Nori1, 3 1Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan 2Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze della Terra, Universit`adi Messina, I-98166 Messina, Italy 3Physics Department, The University of Michigan, Ann Arbor, Michigan 48109-1040, USA Gauge invariance is the cornerstone of modern quantum field theory [1{4]. Recently, it has been shown that the quantum Rabi model, describing the dipolar coupling between a two-level atom and a quantized electromagnetic field, violates this principle [5{7]. This widely used model describes a plethora of quantum systems and physical processes under different interaction regimes [8,9]. In the ultrastrong coupling regime, it provides predictions which drastically depend on the chosen gauge. This failure is attributed to the finite-level truncation of the matter system. We show that a careful application of the gauge principle is able to restore gauge invariance even for extreme light- matter interaction regimes. The resulting quantum Rabi Hamiltonian in the Coulomb gauge differs significantly from the standard model and provides the same physical results obtained by using the dipole gauge. It contains field operators to all orders that cannot be neglected when the coupling strength is high. These results shed light on subtleties of gauge invariance in the nonperturbative and extreme interaction regimes, which are now experimentally accessible, and solve all the long-lasting controversies arising from gauge ambiguities in the quantum Rabi and Dicke models [5, 10{18].