DOCSLIB.ORG

Infrared Divergences

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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. The general principle by which infrared divergences cancel is the same as the principle by which ultraviolet divergences cancel: only physical, observable quantities are guaranteed to be finite. For ultraviolet divergences, it turns out that a simple proxy for the set of observables is the set of Green’s functions of renormalized fields φ1 ( x1 ) φ2 ( x2 ) . These Green’s functions are not observable, and often not gauge invariant, buth are still UV finite.i For infrared divergences, Green’s functions are not good enough. In fact, S-matrix elements or even differences of S- matrix elements at different scales are not good enough. As we will see, infrared divergences only generally cancel after cross sections for processes involving different initial or final states are combined. In this lecture, we will perform one of the most important calculations in QED. We will show that although the cross section for the 2 2 process e+ e − µ+ µ− is infrared divergent at → → order e4 , as is the cross section for the related 2 3 process e+ e − µ+ µ− γ, their sum is R → → infrared finite. More precisely, we will find from calculating 2 2Re + + (2) × ¡ ¡ ¡ ¡ that 3e2 σ( e+ e − µ+ µ− (+ γ)) σ( e+ e − µ+ µ− ) + σ( e+ e − µ+ µ− γ) = σ 1 + R (3) → ≡ → → 0 16π2 4 e R + − + − where σ = 2 is the tree-level cross section for e e µ µ at E = Q. Other diagrams 0 12 πQ → CM contributing to the total e+ e − µ+ µ− cross section at 1-loop are also discussed, but this combi- nation is the most important.→ While this QED cross section is very difficult to measure, its analog in QCD: e+ e − q¯q(+ g) , to be discussed in Lecture IV-2, is an important precision cal- culation which has been→ well confirmed by data and provides strong constraints on beyond-the- standard model physics. 1 2 Section 2 We will see how having to sum over final states (and sometimes initial states) with different particle multiplicities is related to a muon not being physically separable from its surrounding cloud of soft photons. Trying to make this photon cloud more precise leads naturally to the notion of jets. Similarly, trying to understand the initial state radiation contribution leads natu- rally to the notion of parton distribution functions. The total cross section calculation is so important that we will calculate it two ways, with a Pauli-Villars UV regulator and a photon mass IR regulator, and with dimensional regularization for both the UV and the IR, showing that the total cross section is regulator independent. 2 e + e − → µ+ µ − (+ γ) At leading order, the cross section for e+ e − µ+ µ− involves a single Feynman diagram → p1 p3 2 eR µ µ i 0 = = i 2 v¯( p2 ) γ u( p1 ) u¯( p3) γ v( p4) (4) M p2 p4 Q ¡ 2 2 2 where Q = ( p1 + p2 ) = ECM = s is the square of the center-of-mass energy. We already studied this process at tree-level in Lecture II-6 and found that, in the high energy limit Q me , m µ, the differential cross section is ≫ dσ e4 = R (1 + cos2 θ) (5) dΩ 64π2 Q 2 The total tree-level cross section is then a simple integral 2 π 1 dσ e4 σ ( Q 2 ) = dφ d cosθ = R (6) 0 dΩ 12πQ 2 Z0 Z− 1 What we would like to calculate is the next-to-leading order correction to σ0, which begins at ( α3) . O For an S-matrix calculation only amputated graphs are necessary (see Lecture III-4). In this case, there are five relevant 1-loop graphs in QED + + + + (7) ¡ ¡ ¡ ¡ ¡ The next-to-leading order ( α3 ) result is the interference between these graphs (of order α2 ) and the original graph (of orderO α). In addition to loop corrections to the 4-point function, we will also need to calculate real emission graphs to cancel the infrared divergences. Real emission graphs correspond to pro- cesses which are the same order in perturbation theory as the loops but involve more final state particles. We will do the loops first, then the real emission graphs, and then show that we can + − take m γ 0 after all the contributions are combined into the full cross section σ = σ( e e → tot → µ+ µ− ( + γ)) . An important simplifying observation is that since, as far as QED is concerned, the electron and muon charges Q e and Q µ can be anything, the infrared divergence must cancel order by 2 2 order in Q e and Q µ separately. The tree-level cross section scales like σ0 Q e Q µ. The loops in 3 3 2 2 2 2 2 ∼ Eq. (7) scale like Q e Q µ, Q eQ µ, Q e Q µ, Q e Q µ and Q e Q µQ X respectively, where Q X is the charge of the particles going around the vacuum polarization loop, which can be anything. In 3 particular, we will focus on the cancellation of divergences proportional to σ0 Q eQ µ. This cancel- lation gives the critical demonstration of infrared finiteness, and is phenomenologically relevant. Other loop contributions will be discussed afterwards. e+ e− → µ+ µ− (+ γ) 3 2.1 Vertex Correction The vertex correction is p3 p1 2 eR µ µ i Γ = p + =i 2 v¯( p2 ) γ u( p1 ) u¯( p3)Γ2 ( p) v( p4) (8) M p2 Q ¡ p4 ¡ µ µ µ 2 2 where p = p1 + p2 is the photon momentum entering the vertex with p = Q . In this equation, µ 2 Γ2 ( p) refers to the ( e ) contribution to 1PI vertex function, for which we do not introduce any O µ new subscripts for readability. Conveniently, we have already computed Γ2 ( p) for a general off- shell photon in Lectures III-3 and III-5. Thus we can just copy over those results. Recall the general vertex function Γ µ( p) was parameterized in terms of two form factors iσ µν Γ µ( p) = F ( p2 ) γµ + p F ( p2 ) (9) 1 2 m ν 2 Here m can represent either the electron or muon mass. We also do not write mR, since mass renormalization will not be relevant to the calculations in this lecture. We found that the second 2 form factor at order eRwas 2 1 2 2 eR z(1 z) m 4 F2 ( p ) = dxdydzδ( x + y + z 1) − + ( eR) (10) 4π2 − (1 z) 2 m2 xyp2 O Z0 − − m 2 In the high-energy limit, 2 0, this form factor vanishes, F ( p ) 0. This makes sense, since p → 2 → F2 couples right- and left-handed spinors which are uncoupled in massless QED. The first form factor was both ultraviolet and infrared divergent. Regulating the ultraviolet 2 divergence in F1 ( p ) with Pauli-Villars and the infrared divergence with a photon mass, we found that F ( p2 )=1+ f( p2 ) + δ + ( e4 ) (11) 1 1 O R where e2 1 zΛ2 p2 (1 x)(1 y) + m2 (1 4z + z 2 ) f( p2 ) = R dxdydzδ( x + y + z 1) ln + − − − 8π2 − ∆ ∆ 0 with Z ∆ = (1 z) 2 m2 xyp2 + zm2 (12) − − γ For e+ e − µ+ µ− we need p2 = Q 2 and we can take m = 0 for the high-energy limit ( Q m), → ≫ The counterterm is set by F1 (0) = 1 which normalizes the electric charge to what is measured at large distances. In the previous lecture we calculated δ1 for finite m. Now with m = 0 we find 2 2 eR 1 Λ δ1 = f(0) = 2 ln 2 (13) − − 8π 2 m γ ! Evaluating f( Q 2 ) is more challenging. It has the form e2 1 1 − x (1 x y)Λ2 Q 2 (1 x)(1 y) f( Q 2 ) = R dx dy ln − − + − − (14) 8π2 xyQ 2 + (1 x y) m2 xyQ 2 + (1 x y) m2 Z0 Z0 " − − − γ − − − γ # The first term is IR finite and gives 1 1 − x (1 x y)Λ2 3 1 Λ2 dx dy ln − − = + ln + ( m γ) (15) xyQ 2 + (1 x y) m2 4 2 Q 2 O Z0 Z0 " − − − γ # − 4 Section 2 2 Note that the ln Λ has the right coefficient to be canceled by δ1 .
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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 .
    [Show full text]
  • Notas De Física CBPF-NF-008/10 February 2010
    ISSN 0029-3865 CBPF - CENTRO BRASILEIRO DE PESQUISAS FíSICAS Rio de Janeiro Notas de Física CBPF-NF-008/10 February 2010 Pascual Jordan's legacy and the ongoing research in quantum field theory Bert Schroer Ministét"io da Ciência e Tecnologia Pascual Jordan’s legacy and the ongoing research in quantum field theory to be published in EPJH - Historical Perspectives on Contemporary Physics Bert Schroer present address: CBPF, Rua Dr. Xavier Sigaud 150, 22290-180 Rio de Janeiro, Brazil email [email protected] permanent address: Institut f¨urTheoretische Physik FU-Berlin, Arnimallee 14, 14195 Berlin, Germany December 2009 Abstract After recalling Pascual Jordan’s pathbreaking work in shaping quantum mechan- ics I explain his role as the protagonist of quantum field theory (QFT). Particular emphasis is given to the 1929 Kharkov conference where Jordan not only presents a quite modern looking panorama about the state of art, but were some of his ideas already preempt an intrinsic point of view about a future QFT liberated from the classical parallelism and quantum field theory, a new approach for which the conceptional basis began to emerge ony 30 years later. Two quite profound subjects in which Jordan was far ahead of his contempo- raries will be presented in separate sections: ”Bosonization and Re-fermionization instead of Neutrino theory of Light” and ”Nonlocal gauge invariants and an alge- braic monopole quantization”. The last section contains scientific episodes mixed with biographical details. It includes remarks about his much criticized conduct during the NS regime. Without knowing about his entanglement with the Nazis it is not possible to understand that such a giant of particle physics dies without having received a Nobel prize.
    [Show full text]
  • Springer Series on Atomic, Optical, and Plasma Physics
    Springer Series on Atomic, Optical, and Plasma Physics Volume 63 Series Editors W.E. Baylis Uwe Becker Philip George Burke Robert N. Compton M.R. Flannery Charles J. Joachain Brian R. Judd Kate Kirby Peter Lambropoulos Gerd Leuchs Pierre Meystre For further volumes: http://www.springer.com/series/411 Ingvar Lindgren Relativistic Many-Body Theory A New Field-Theoretical Approach With 97 Figures ABC Ingvar Lindgren Department of Physics University of Gothenburg SE-412 96 Go¨teborg Sweden [email protected] ISSN 1615-5653 ISBN 978-1-4419-8308-4 e-ISBN 978-1-4419-8309-1 DOI 10.1007/978-1-4419-8309-1 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011921721 c Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) To Eva Preface It is now almost 30 years since the first edition of my book together with John Morrison, Atomic Many-Body Theory [6], appeared, and the second edition ap- peared some years later.
    [Show full text]
  • Notas De Física CBPF-NF-008J10 February 2010
    ISSN 0029-3865 CBPF - CENTRO BRASILEIRO DE PESQUISAS FíSICAS Rio de Janeiro Notas de Física CBPF-NF-008j10 February 2010 Pascual Jordan's legacy and the ongoing research in quantum field theory Bert Schroer Ministét"io da Ciência e Tecnologia Pascual Jordan’s legacy and the ongoing research in quantum field theory to be published in EPJH - Historical Perspectives on Contemporary Physics Bert Schroer present address: CBPF, Rua Dr. Xavier Sigaud 150, 22290-180 Rio de Janeiro, Brazil email [email protected] permanent address: Institut f¨urTheoretische Physik FU-Berlin, Arnimallee 14, 14195 Berlin, Germany December 2009 Abstract After recalling Pascual Jordan’s pathbreaking work in shaping quantum mechan- ics I explain his role as the protagonist of quantum field theory (QFT). Particular emphasis is given to the 1929 Kharkov conference where Jordan not only presents a quite modern looking panorama about the state of art, but were some of his ideas already preempt an intrinsic point of view about a future QFT liberated from the classical parallelism and quantum field theory, a new approach for which the conceptional basis began to emerge ony 30 years later. Two quite profound subjects in which Jordan was far ahead of his contempo- raries will be presented in separate sections: ”Bosonization and Re-fermionization instead of Neutrino theory of Light” and ”Nonlocal gauge invariants and an alge- braic monopole quantization”. The last section contains scientific episodes mixed with biographical details. It includes remarks about his much criticized conduct during the NS regime. Without knowing about his entanglement with the Nazis it is not possible to understand that such a giant of particle physics dies without having received a Nobel prize.
    [Show full text]
  • Planck Scale, Dirac Delta Function and Ultraviolet Divergence
    Planck scale, Dirac delta function and ultraviolet divergence Hua Zhang,a;1 Mingshun Yuanb aShenzhen Forms Syntron Information Co.,Ltd., Shenzhen 518000, P.R. China bFujian University of Technology,Mathematics and Physics Institute Fuzhou 350118, P.R. China E-mail: [email protected] Abstract: The Planck length is the minimum length which physical law do not fail. The Dirac delta function was created to deal with continuous range issue, and it is zero except for one point. Thus contradict the Planck length. Renormalization method is the usual way to deal with divergence difficulties. The authors proposed a new way to solve the problem of ultraviolet divergence and this method is self-consistent with the Planck length. For this purpose a redefine function δP in position representation was introduced to han- dle with the canonical quantization. The function δP tends to the Dirac delta function if the Planck length goes to zero. By logical deduction the authors obtains new commuta- tion/anticommutation relations and new Feynman propagators which are convergent. This article will deduce the new Feynman propagators for the Klein-Gordon field, the Dirac field and the Maxwell field. Through the new Feynman propagators, we can eliminate ultraviolet divergence. arXiv:1911.01789v5 [physics.gen-ph] 2 Jan 2020 1Corresponding author. Contents 1 Introduction2 1.1 History of renormalization2 1.2 The Planck scale & the Dirac delta function2 2 The renormalization function3 2.1 Rectangular coordinate system3 2.2 Cylindrical coordinate system4 2.3 Spherical coordinate
    [Show full text]
  • Ground States and Spectral Properties in Quantum Field Theories
    Ground states and spectral properties in quantum field theories Dissertation zur Erlangung des akademischen Grades doctor rerum naturalium vorgelegt dem Rat der Fakultät für Mathematik und Informatik der Friedrich–Schiller–Universität Jena von M.Sc. Markus Lange geboren am 10.02.1987 in Münster 1. Gutachter: Prof. Dr. David Hasler (Friedrich-Schiller-University Jena, Ernst-Abbe- Platz 2, 07743 Jena, Germany) 2. Gutachter: Prof. Dr. Marcel Griesemer (University Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany) 3. Gutachter: Prof. Dr. Jacob Schach Møller (Aarhus University, Ny Munkegade 118, 8000 Aarhus C, Denmark) Tag der öffentlichen Verteidigung: 29. Juni 2018 Abstract In this thesis we consider non-relativistic quantum electrodynamics in dipole approximation and study low-energy phenomenons of quantum mechanical systems. We investigate the analytic dependence of the lowest-energy eigenvalue and eigenvector on spectral parameters of the system. In particular we study situations where the ground-state eigenvalue is assumed to be degenerate. In the first situation the eigenspace of a degenerate ground-state eigenvalue is assumed to split up in a specific way in second order formal perturbation theory. We show, using a mild infrared assumption, that the emerging unique ground state and the corresponding ground-state eigenvalue are analytic functions of the coupling constant in a cone with apex at the origin. Secondly we analyse the situation that the degeneracy is protected by a set of symmetries for the considered quantum mechanical system. We prove, in accordance with known results for the non-degenerate situation, that the ground-state eigenvalue and eigenvectors depend analytically on the coupling constant. In order to show these results we extend operator-theoretic renormalization to such degenerate situations.
    [Show full text]
  • [Hep-Th] 11 May 2017 Infrared Divergences in QED, Revisited
    Infrared Divergences in QED, Revisited Daniel Kapec†, Malcolm Perry∗, Ana-Maria Raclariu† and Andrew Strominger† Abstract Recently it has been shown that the vacuum state in QED is infinitely degenerate. Moreover a transition among the degenerate vacua is induced in any nontrivial scatter- ing process and determined from the associated soft factor. Conventional computations of scattering amplitudes in QED do not account for this vacuum degeneracy and there- fore always give zero. This vanishing of all conventional QED amplitudes is usually attributed to infrared divergences. Here we show that if these vacuum transitions are properly accounted for, the resulting amplitudes are nonzero and infrared finite. Our construction of finite amplitudes is mathematically equivalent to, and amounts to a physical reinterpretation of, the 1970 construction of Faddeev and Kulish. arXiv:1705.04311v1 [hep-th] 11 May 2017 † Center for the Fundamental Laws of Nature, Harvard University, Cambridge, MA, USA ∗ Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, UK Contents 1 Introduction 1 2 Vacuum selection rules 2 3 Dressed quantum states 7 4 FK states 10 5 Massive Particles 12 6 Charged states 15 1 Introduction Recently it has been shown [1–5] (see [6] for a review) that the infrared (IR) sector of all abelian gauge theories, including QED, is governed by an infinite-dimensional symmetry group. The symmetry group is generated by large gauge transformations that approach angle-dependent constants at null infinity. The soft photon theorem is the matrix element of the associated conservation laws. This large gauge symmetry is spontaneously broken, resulting in an infinite vacuum degeneracy.
    [Show full text]
  • A Function-Analytic Development of Field Theory
    1 A Function-Analytic Development of Field Theory David K. Houseman Thesis submitted for the degree of Doctor of Philosophy of UCL Department of Physics and Astronomy University College London 25 June 2012 2 I confirm that the work presented in this thesis is my own. Where information has been derived from other sources, I confirm that this has been indicated in the thesis. David Houseman University College London 25 June 2012 Abstract 3 Abstract This thesis presents a system of coupled differential equations as a simple model of quantum electrodynamics (QED). A key feature of the model is the Riemann- Silberstein (RS) representation of the photon. The RS representation leads to a natural configuration-space description for a system of multiple, non-interacting electrons and photons. Relativistic covariance is shown by extending the dynamics to a representation of the Poincare´ group on the space of configuration-space amplitudes. Because the dif- ferential system forms a well-posed initial-value problem, this model features a natural concept of time evolution, and concretely parametrises the system even at intermediate times during scattering processes. If QED could be formulated in a framework of this type, both the analysis and rigorous formulation of quantum field theory may bene- fit from a useful new perspective. Towards this aim, I consider deformation of the free theory, preserving the initial-value nature while incorporating interactions between par- ticles. The deformation takes the form of a coupling between states of different particle content. I present some simple criteria to show whether the deformation is compatible with relativity.
    [Show full text]