A Function-Analytic Development of Field Theory
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Perturbative Versus Non-Perturbative Quantum Field Theory: Tao’S Method, the Casimir Effect, and Interacting Wightman Theories
universe Review Perturbative versus Non-Perturbative Quantum Field Theory: Tao’s Method, the Casimir Effect, and Interacting Wightman Theories Walter Felipe Wreszinski Instituto de Física, Universidade de São Paulo, São Paulo 05508-090, SP, Brazil; [email protected] Abstract: We dwell upon certain points concerning the meaning of quantum field theory: the problems with the perturbative approach, and the question raised by ’t Hooft of the existence of the theory in a well-defined (rigorous) mathematical sense, as well as some of the few existent mathematically precise results on fully quantized field theories. Emphasis is brought on how the mathematical contributions help to elucidate or illuminate certain conceptual aspects of the theory when applied to real physical phenomena, in particular, the singular nature of quantum fields. In a first part, we present a comprehensive review of divergent versus asymptotic series, with qed as background example, as well as a method due to Terence Tao which conveys mathematical sense to divergent series. In a second part, we apply Tao’s method to the Casimir effect in its simplest form, consisting of perfectly conducting parallel plates, arguing that the usual theory, which makes use of the Euler-MacLaurin formula, still contains a residual infinity, which is eliminated in our approach. In the third part, we revisit the general theory of nonperturbative quantum fields, in the form of newly proposed (with Christian Jaekel) Wightman axioms for interacting field theories, with applications to “dressed” electrons in a theory with massless particles (such as qed), as well as Citation: Wreszinski, W.F. unstable particles. Various problems (mostly open) are finally discussed in connection with concrete Perturbative versus Non-Perturbative models. -
Numerical Evaluation of Feynman Loop Integrals by Reduction to Tree Graphs
Numerical Evaluation of Feynman Loop Integrals by Reduction to Tree Graphs Dissertation zur Erlangung des Doktorgrades des Departments Physik der Universit¨atHamburg vorgelegt von Tobias Kleinschmidt aus Duisburg Hamburg 2007 Gutachter des Dissertation: Prof. Dr. W. Kilian Prof. Dr. J. Bartels Gutachter der Disputation: Prof. Dr. W. Kilian Prof. Dr. G. Sigl Datum der Disputation: 18. 12. 2007 Vorsitzender des Pr¨ufungsausschusses: Dr. H. D. R¨uter Vorsitzender des Promotionsausschusses: Prof. Dr. G. Huber Dekan der Fakult¨atMIN: Prof. Dr. A. Fr¨uhwald Abstract We present a method for the numerical evaluation of loop integrals, based on the Feynman Tree Theorem. This states that loop graphs can be expressed as a sum of tree graphs with additional external on-shell particles. The original loop integral is replaced by a phase space integration over the additional particles. In cross section calculations and for event generation, this phase space can be sampled simultaneously with the phase space of the original external particles. Since very sophisticated matrix element generators for tree graph amplitudes exist and phase space integrations are generically well understood, this method is suited for a future implementation in a fully automated Monte Carlo event generator. A scheme for renormalization and regularization is presented. We show the construction of subtraction graphs which cancel ultraviolet divergences and present a method to cancel internal on-shell singularities. Real emission graphs can be naturally included in the phase space integral of the additional on-shell particles to cancel infrared divergences. As a proof of concept, we apply this method to NLO Bhabha scattering in QED. -
Lectures on Conformal Field Theory Arxiv:1511.04074V2 [Hep-Th] 19
Prepared for submission to JHEP Lectures on Conformal Field Theory Joshua D. Quallsa aDepartment of Physics, National Taiwan University, Taipei, Taiwan E-mail: [email protected] Abstract: These lectures notes are based on courses given at National Taiwan University, National Chiao-Tung University, and National Tsing Hua University in the spring term of 2015. Although the course was offered primarily for graduate students, these lecture notes have been prepared for a more general audience. They are intended as an introduction to conformal field theories in various dimensions working toward current research topics in conformal field theory. We assume the reader to be familiar with quantum field theory. Familiarity with string theory is not a prerequisite for this lectures, although it can only help. These notes include over 80 homework problems and over 45 longer exercises for students. arXiv:1511.04074v2 [hep-th] 19 May 2016 Contents 1 Lecture 1: Introduction and Motivation2 1.1 Introduction and outline2 1.2 Conformal invariance: What?5 1.3 Examples of classical conformal invariance7 1.4 Conformal invariance: Why?8 1.4.1 CFTs in critical phenomena8 1.4.2 Renormalization group 12 1.5 A preview for future courses 16 1.6 Conformal quantum mechanics 17 2 Lecture 2: CFT in d ≥ 3 22 2.1 Conformal transformations for d ≥ 3 22 2.2 Infinitesimal conformal transformations for d ≥ 3 24 2.3 Special conformal transformations and conformal algebra 26 2.4 Conformal group 28 2.5 Representations of the conformal group 29 2.6 Constraints of Conformal -
Infrared Divergences
2012 Matthew Schwartz III-6: Infrared divergences 1 Introduction We have shown that the 1, 2 and 3 point functions in QED are UV finite at one loop. We were able to introduce 4 counterterms ( δm , δ1 , δ2 , δ3) which canceled all the infinities. Now let us move on to four point functions, such as Ω T ψ( x ) ψ¯ ( x ) ψ( x ) ψ¯ ( x ) Ω (1) h | { 1 2 3 4 }| i This could represent, for example, Møller scattering ( e − e − e − e − ) or Bhabha scattering → ( e+ e − e+ e − ). We will take it to be e+ e − µ+ µ− for simplicity, since at tree-level this process only has→ an s-channel diagram. Looking→ at these 4-point functions at one-loop will help us understand how to combine previous loop calculations and counterterms into new observables, and will also illustrate a new feature: cancellation of infrared divergences. Although important results and calculational techniques are introduced in this lecture, it can be skipped without much loss of continuity with the rest of the text. Recall that in the on-shell subtraction scheme we found δ1 and δ2 depended on a fictitious photon mass m γ. This mass was introduced to make the loops finite and is an example of an infrared regulator. As we will see, the dependence on IR regulators, like m γ, drops out not in differences between the Green’s functions at different scales, as with UV regulators, but in the sum of different types of Green’s functions contributing to the same observable at the same scale. -
Introduction to Renormalization
Introduction to Renormalization A lecture given by Prof. Wayne Repko Notes: Michael Flossdorf April 4th 2007 Contents 1 Introduction 1 2 RenormalizationoftheQEDLagrangian 2 3 One Loop Correction to the Fermion Propagator 3 4 Dimensional Regularization 6 4.1 SuperficialDegreeofDivergence. 6 4.2 TheProcedureofDimensionalRegularization . .. 8 4.3 AUsefulGeneralIntegral. 9 4.4 Dirac Matrices in n Dimensions .................. 11 4.5 Dimensional Regularization of Σ2 ................. 11 5 Renormalization 12 5.1 Summary ofLastSection and OutlineoftheNextSteps . .. 12 5.2 MomentumSubtractionSchemes. 13 5.3 BacktoRenormalizationinQED. 14 5.4 Leading Order Expression of Σ2 for Small ǫ ............ 17 6 Outline of the Following Procedure 19 7 The Vertex Correction 19 8 Vacuum Polarization 22 CONTENTS 2 9 The Beta Function of QED 25 Appendix 27 References 30 1 INTRODUCTION 1 First Part of the Lecture 1 Introduction Starting with the Lagrangian of any Quantum Field Theory, we have seen in some of the previous lectures how to obtain the Feynman rules, which sufficiently de- scribe how to do pertubation theory in that particular theory. But beyond tree level, the naive calculation of diagrams involving loops will often yield infinity, since the integrals have to be performed over the whole momentum space. Renormal- ization Theory deals with the systematic isolation and removing of these infinities from physical observables. The first important insight is, that it is not the fields or the coupling constants which represent measurable quantities. Measured are cross sections, decay width, etc.. As long as we make sure that this observables are finite in the end and can be unambiguously derived from the Lagrangian, we are free to introduce new quantities, called renormalized quantities for every, so called, bare quantity. -
Scattering Theory of Quantum Systems with Infinitely Many Degrees of Freedom
Technische Universit¨at Munchen¨ Zentrum Mathematik Scattering theory of quantum systems with infinitely many degrees of freedom Wojciech Dybalski Habilitation Thesis Januar 2019 Fachmentorat: Prof. Dr. Chris Fewster (University of York) Prof. Dr. Herbert Spohn (TU Munchen)¨ Prof. Dr. Simone Warzel (TU Munchen)¨ Preface Since the early days of Quantum Mechanics, scattering theory has been a central tool for comparison of theory with experiment. Mathematical foundations of scattering theory were laid by giants of Mathematical Physics such as L.D. Faddeev or T. Kato, as discussed, e.g., in [Si18, De18]. The problem of asymptotic completeness, i.e., the question of particle interpretation of all states in the physical Hilbert space of the theory, emerged as one of the main problems of the mathematical scat- tering theory. Its solution in N-body Quantum Mechanics for particles with quadratic dispersion relations, interacting with possibly long-range forces, is an impressive chapter of the 20th century Mathematical Physics whose milestones are [En78, SiSo87, Gr90, De93]. However, if the assumption of quadratic dispersion relations is dropped, even in Quantum Mechanics the problem of complete particle interpretation is largely open beyond the two-body scattering. It is therefore not a surprise that in quantum field theory (QFT) or for quantum spin systems, where basic excitations typically have non-quadratic dispersion relations, asymptotic completeness is rather poorly understood. An additional problem for quantum systems with infinitely many degrees of freedom is a possible break- down of the Stone-von Neumann uniqueness theorem and the resulting multitude of ‘charged sectors’. This aspect undermines the conventional property of asymptotic completeness, inherited from Quan- tum Mechanics, and calls for more suitable concepts. -
Equations of Motion As Constraints: Superselection Rules, Ward Identities
Published for SISSA by Springer Received: January 3, 2017 Revised: February 23, 2017 Accepted: March 13, 2017 Published: March 27, 2017 Equations of motion as constraints: superselection rules, Ward identities JHEP03(2017)136 M. Asorey,a A.P. Balachandran,b;c F. Lizzid;e;f and G. Marmod;e aDepartamento de F´ısica Te´orica, Universidad de Zaragoza, C/Pedro Cerbuna 12, E-50009 Zaragoza, Spain bPhysics Department, Syracuse University, Physics Building Syracuse, NY 13244, U.S.A. cInstitute of Mathematical Sciences, C.I.T Campus, Taramani Chennai 600113, India dDipartimento di Fisica \E. Pancini" Universit`adi Napoli Federico II, Via Cintia, 80126 Napoli, Italy eINFN | Sezione di Napoli, Via Cintia, 80126 Napoli, Italy f Departament de Estructura i Constituents de la Mat`eria,Institut de Ci´enciesdel Cosmos, Universitat de Barcelona, Diagonal 647, 08028 Barcelona, Catalonia, Spain E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: The meaning of local observables is poorly understood in gauge theories, not to speak of quantum gravity. As a step towards a better understanding we study asymptotic (infrared) transformations in local quantum physics. Our observables are smeared by test functions, at first vanishing at infinity. In this context we show that the equations of motion can be seen as constraints, which generate a group, the group of space and time dependent gauge transformations. This is one of the main points of the paper. Infrared nontrivial effects are captured allowing test functions which do not vanish at infinity. These extended operators generate a larger group. -
A Criterion to Characterize Interacting Theories in the Wightman Framework
Quantum Stud.: Math. Found. (2021) 8:51–68 CHAPMAN INSTITUTE FOR https://doi.org/10.1007/s40509-020-00227-5 UNIVERSITY QUANTUM STUDIES REGULAR PAPER A criterion to characterize interacting theories in the Wightman framework Christian D. Jäkel · Walter F. Wreszinski Received: 26 August 2019 / Accepted: 1 April 2020 / Published online: 21 April 2020 © Chapman University 2020 Abstract We propose that a criterion to characterize interacting theories in a suitable Wightman framework of relativistic quantum field theories which incorporates a “singularity hypothesis”, which has been conjectured for a long time, is supported by renormalization group theory, but has never been formulated mathematically. The (nonperturbative) wave function renormalization Z occurring in these theories is shown not to be necessarily equal to zero, except if the equal time commutation relations (ETCR) are assumed. Since the ETCR are not justified in general (because the interacting fields cannot in general be restricted to sharp times, as is known from model studies), the condition Z = 0 is not of general validity in interacting theories. We conjecture that it characterizes either unstable (composite) particles or the charge-carrying particles, which become infraparticles in the presence of massless particles. In the case of QED, such “dressed” electrons are not expected to be confined, but in QCD, we propose a quark confinement criterion, which follows naturally from lines suggested by the works of Casher, Kogut and Susskind, and Lowenstein and Swieca. Keywords Interacting quantum field theories · Wightman framework · Singularity hypothesis · Wave function renormalization constant · Infraparticles · Unstable particles 1 Introduction In his recent recollections, ’tHooft ([30], Sect. 5) emphasizes that an asymptotic (divergent) series, such as the α = 1 power series for the scattering (S) matrix in the coupling constant 137 in quantum electrodynamics (qed1+3) does not define a theory rigorously (or mathematically). -
QCD Breaks Lorentz Invariance and Colour
QCD Breaks Lorentz Invariance and Colour A. P. Balachandran ∗ Physics Department, Syracuse University, Syracuse, New York 13244-1130, U.S.A. Abstract In a previous work [1], we have argued that the algebra of non-abelian superselection rules is spontaneously broken to its maximal abelian subalgebra, that is, the algebra generated by its completing commuting set (the two Casimirs and a basis of its Cartan subalgebra). In this paper, alternative arguments confirming these results are presented. In addition, Lorentz invariance is shown to be broken in QCD, just as it is in QED. The experimental consequences of these results include fuzzy mass and spin shells of coloured particles like quarks, and decay life times which depend on the frame of observation [2– 4]. In a paper under preparation, these results are extended to the ADM Poincar´e group and the local Lorentz group of frames. The renormalisation of the ADM energy by infrared gravitons is also studied and estimated. 1 Introduction Quantum field theory (QFT) is defined by the algebra of local observables and an ir- A arXiv:1509.05235v2 [hep-th] 27 Jan 2016 reducible representation (IRR) π of on a Hilbert space . In general, there are many A H inequivalent IRR’s π ,π ,... of defining its superselection sectors. 0 1 A For example, in QED, π0 can be the sector with total charge q0 = 0, while πn can be the sector with total charge qn. No observation can mix these sectors. In QED, the charge operator Q generates the abelian U(1) group. In QCD, the U(1) is replaced that the non-abelian SU(3) of colour. -
Chapter 1 Quantum Information in Fundamental Physics
Aspects of quantum information in quantum field theory and quantum gravity by Dominik Neuenfeld a thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the faculty of graduate and postdoctoral studies (Physics) The University of British Columbia (Vancouver) July 2019 © Dominik Neuenfeld, 2019 The following individuals certify that they have read, and recommend to the Faculty of Graduate and Postdoctoral Studies for acceptance, the thesis entitled: Aspects of quantum information in quantum field theory and quantum gravity submitted by Dominik Neuenfeld in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics. Examining Committee: Gordon Semenoff, Physics Co-supervisor Ian Affleck, Physics University Examiner Joel Feldman, Math University Examiner Alison Lister, Physics Supervisory Committee Member Robert Raussendorf, Physics Supervisory Committee Member Additional Supervisory Committee Members: Mark Van Raamsdonk, Physics Co-supervisor ii Abstract In this thesis we discuss applications of quantum information theoretic concepts to quantum gravity and the low-energy regime of quantum field theories. The first part of this thesis is concerned with how quantum information spreads in four-dimensional scattering experiments for theories coupled to quantum electro- dynamics or perturbative quantum gravity. In these cases, every scattering process is accompanied by the emission of an infinite number of soft photons or gravi- tons, which cause infrared divergences in the calculation of scattering probabilities. There are two methods to deal with IR divergences: the inclusive and dressed formalisms. We demonstrate that in the late-time limit, independent of the method, the hard outgoing particles are entangled with soft particles in such a way that the reduced density matrix of the hard particles is essentially completely decohered. -
Infrared Divergence Phenomena, High-Energy Processes, and Regge Poles
INFRARED DIVERGENCE PHENOMENA, HIGH-ENERGY PROCESSES, AND REGGE POLES S. FRAUTSCHI NEWMAN LABORATORY OF NUCIEAR STUDIES, CORNELL UNIVERSITY, ITHACA, N.Y., UNITED STATES OP AMERICA 1. INTRODUCTION Infrared divergence phenomena are already well known from semi- classical arguments. For example, suppose an electron in motion is de flected due to its interaction with a potential, the Lorentz-contracted proper field of the electron will be altered by the collision, and the change in the proper field will be emitted as electromagnetic radiation. For sufficiently long wavelengths (kR < < 1, where k is the wave number and R is a dimension of the scattering region), the radiation can be calculated without knowledge of the details of the trajectory in the scattering region. It depends only on the initial and final momenta of the electron and the direction in which the radiation is observed (assuming the electron suffers no time delay in the scattering region). As is well known, the energy emitted per unit frequency is constant in this limit. Making the transcription to the photon description, it is clear that the number of photons emitted per unit frequency range is inversely proportional to the frequency; i. e., the photon spectrum is of the form dk/k, which diverges as k — 0. This is the infrared divergence for real photons. The angular distribution can also be understood by the semiclassical argument. In the extreme relativistic limit, the proper fields will be Lorentz - contracted in a small region near the plane perpendicular to the direction of motion of the charge and moving along with the charge. -
Cancellation of Infrared Divergences in QCD
Cancellation of infrared divergences in QCD Daniel Wennberg Physics specalization project Submission date: January 2014 Supervisor: Michael Kachelrieß Norwegian University of Science and Technology Department of Physics Abstract The scattering process hadrons is considered at next-to- + − leading order, . It is demonstrated푒 푒 → that the cross sections for both the real gluon emission풪(훼푠) and virtual gluon exchange have infrared singu- larities. Using dimensional regularization it is shown that the divergent terms cancel exactly when adding the contributions, such that the total cross section for hadrons is finite at this order. This result, and + − the general implications푒 푒 → of infrared divergences and their cancellation, is discussed in light of the Kinoshita-Lee-Nauenberg theorem and the concept of infrared safety. 2 Contents List of figures 5 List of tables 6 List of symbols and notation 7 Preface 9 1 Introduction 10 1.1 Perturbative quantum chromodynamics . 10 1.2 Infrared divergences . 13 2 Cancellation of infrared divergences in hadrons 14 + − 2.1 Leading order amplitude for 푒 푒 →. 15 + − 2.2 Cross sections for hadrons푒 푒 → . 푞 .푞 . 19 + − 2.3 Real gluon emission:푒 푒 → at leading order . 21 + − 2.4 Virtual gluon exchange:푒 푒 → 푞푞푔 at next-to-leading order . 26 + − 2.5 Cancellation: hadrons푒 푒 → at 푞 next-to-leading푞 order . 34 + − 2.6 The ultraviolet푒 divergence푒 → . 35 3 Discussion 38 3.1 Infrared divergences in general quantum field theories . 38 3.2 The optical theorem and cutting rules . 41 3.3 Infrared safety in QCD . 44 4 Summary 46 A Feynman rules for QED and QCD 47 A.1 General remarks .