DOCSLIB.ORG

Ground States and Spectral Properties in Quantum Field Theories

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read 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. To complement the analyticity results we additionally show that an asymptotic expansion of the ground state and the ground-state eigenvalue exists up to arbitrary order. The infrared assumption needed for the asymptotic expansion is weaker than the usual assumptions required for other methods such as operator theoretic renormalization to be applicable. Zusammenfassung in deutscher Sprache In der vorliegenden Arbeit betrachten wir Modelle der nicht-relativistischen Quantenelektrodynamik in Dipol-Approximation und studieren Phänomene, die bei kleinen Energien in diesen quanten-mechanischen Systemen auftreten. Wir untersuchen die analytische Abhängigkeit des kleinsten Energie-Eigenwertes und der zugehörigen Eigenvektoren im Bezug auf spektrale Parameter des Systems. Insbesondere un- tersuchen wir Situationen, in denen der Grundzustandseigenwert entartet ist. In der ersten Situation wird angenommen, dass sich der Eigenraum des Grundzustandseigenwertes in zweiter Ordnung in for- maler Störungstheorie in bestimmter Weise aufspaltet. Wir zeigen unter Annahme einer schwachen Infrarot-Bedingung, dass der durch die Aufspaltung entstandene nicht-entartete Grundzustand und die zugehörige Grundzustandsenergie analytische Funktionen der Kopplungskonstante in einem Kegel mit Spitze im Ursprung sind. Im zweiten Fall untersuchen wir die Situation, dass die Entartung durch eine Menge von Symmetrien im Systems erzeugt wird. Unter bestimmten Voraussetzungen zeigen wir, dass der Grundzustand und die Grundzustandsenergie analytische Funktionen der Kopplungskonstante sind. Dieses Ergebnis stimmt mit entsprechenden Ergebnissen für nicht-entartete Situationen überein. Um diese Resultate zu zeigen erweitern wir die Methode ’operator-theoretic renormalization’ auf diese entarteten Situationen. Ergänzend zu den obrigen Analytizitätsergebnisse zeigen wir, dass eine asympto- tische Entwicklung, zu beliebiger Ordnung, des Grundzustandes und der Grundzustandsenergie existiert. Die dafür benötigte Infrarot-Bedingung ist schwächer als die üblichen Bedingungen die gebraucht werden damit andere Methoden, wie zum Beispiel ’operator-theoretic renormalization’, anwendbar sind. Acknowledgements I genuinely thank my adviser David Hasler for his guidance and support during my doctorate. We had a lot of fruitful and enlightening discussions and he always had an open door for my questions and very good advises. Moreover I thank Gerhard Bräunlich for interesting discussions and a fruitful collaboration that resulted in a joint work with him and David Hasler. I am thankful to Professor Israel Michael Sigal for his hospitality in September 2016 at the University of Toronto. I learned a lot about Born-Oppenheimer approximation during that time. Additionally I want to thank him for connecting me with other researchers in the community on multiple occasions. I am very grateful to Siegfried Beckus, Johanna Nabrotzki and Andreas Vollmer for reading parts of the thesis in advance. Their useful remarks and suggestions were very helpful in the formation process of this thesis. In addition, I want to thank my colleagues Alexandra, Andreas, Benjamin, Daniel, Jannis, Marcel, Melchior, Nina, Oliver, Siegfried and Therese for the nice and pleasant working atmosphere and the many relaxing breaks we had together. I thank the Research Training Group (1523/2) “Quantum and Gravitational Fields” which is located at the Friedrich-Schiller-University Jena, Germany, for funding my doctoral position for three years and for their financial support on most of my business trips during my nearly four years inJena. Last but not least I want to thank my family and friends for their mental and emotional support. In particular I am deeply thankful to my parents Mechthild and Dieter Lange for their faith in me and for laying a very reliable foundation for my ongoing life journey. In addition I am thankful to Johanna Nabrotzki for being there and diverting my thoughts at the right moments. Moreover I want to thank my dearest friends Simon Frerking, Thomas Goldbeck, Florian Peschka and Hendrik Sommer for all the joyful times we had and will have together. Contents Abstract iii Acknowledgements iv 1 Introduction 1 2 The Spin-Boson model 4 2.1 The atomic space . .4 2.2 The Fock space . .6 2.2.1 Definition of the Fock space . .6 2.2.2 The bosonic Fock space . .9 2.3 Properties of the Spin-Boson Hamiltonian . 11 2.3.1 Examples of generalized Spin-Boson Models . 13 3 Operator-theoretic renormalization 15 3.1 Perturbation theory . 15 3.2 The operator-theoretic renormalization group method . 17 3.2.1 The Feshbach map . 17 3.2.2 The operator-theoretic renormalization group method . 19 3.3 Operator-valued integral kernels . 20 3.3.1 Banach spaces of integral kernels . 21 3.3.2 Field operators, Pull-Through Formula and Wick’s theorem . 23 4 Degenerate perturbation theory 27 4.1 Second order split-up for the Spin-Boson model . 27 4.1.1 Definition of the model and statement of result . 28 4.1.2 Initial Feshbach step . 30 4.1.3 Banach space estimate for the first step . 32 4.1.4 Second Feshbach step . 37 4.1.5 Banach Space estimate for the second step . 41 4.1.6 Proof of main result for the split-up Spin-Boson model . 47 4.2 Symmetries for the Spin-Boson Hamiltonian . 50 4.2.1 Preliminaries . 50 4.2.2 Description of the model and statement of result . 53 4.2.3 Construction of an effective Hamiltonian . 56 4.2.4 The renormalization transformation . 60 4.2.5 Iterating the renormalization transformation Rρ ................... 61 4.2.6 Proof of main result for the symmetry-protected Spin-Boson model . 69 4.3 The hydrogen atom in dipole approximation . 71 5 Asymptotic expansions 75 5.1 Model and statement of results . 75 5.2 Asymptotic perturbation theory . 79 5.2.1 Expansion method . 79 5.2.2 Resolvent method . 81 5.3 Existence of an asymptotic expansion . 84 1 Introduction The general theory of quantum mechanics, which includes the quantum theory of fields is a fundamental theory of nature. This theory gives us a way to describe what happens at small scales and energy levels of atoms and subatomic particles. For example the process of absorption and emission of photons by electrons that are bounded to an atomic nucleus can be described using tools from quantum mechanics. Physicists already worked a lot on important and interesting questions involving quantum mechanics. Moreover a part of the physics community has shifted its research focus to questions concerning relativistic field theories and string theory. From a mathematical point of view even the seemingly simple question: “Does the lowest-energy eigenvalue of a system of non-relativistic matter that interacts with a quantized field of massless particles depend analytically on the spectral parameters of the overall system?”is conceptually very complicated and there are still open problems concerning this question. Likewise similar questions regarding excited energy eigenvalues of such interacting matter-radiation systems do only have partial answers. In this thesis we extend pre-existing answers concerning the above mentioned question to degenerate situations. In particular, we generalize the operator-theoretic renormalization group method so that we can handle Hamiltonians with degenerate eigenvalues. Such renormalization group methods play an essential role in the spectral analysis of quantum mechanical systems. About 100 years ago Erwin Schrödinger, Werner Heisenberg, Max Born and others developed an early version of quantum theory that is based on work of Max Planck on black body radiation [113] and Albert Einstein on the photoelectric effect [44]. Later Paul Dirac, David Hilbert, John von Neumann and Hermann Weyl developed a mathematically
Load more

Recommended publications
  • 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.
    [Show full text]
  • Configuration Interaction Study of the Ground State of the Carbon Atom
    Configuration Interaction Study of the Ground State of the Carbon Atom María Belén Ruiz* and Robert Tröger Department of Theoretical Chemistry Friedrich-Alexander-University Erlangen-Nürnberg Egerlandstraße 3, 91054 Erlangen, Germany In print in Advances Quantum Chemistry Vol. 76: Novel Electronic Structure Theory: General Innovations and Strongly Correlated Systems 30th July 2017 Abstract Configuration Interaction (CI) calculations on the ground state of the C atom are carried out using a small basis set of Slater orbitals [7s6p5d4f3g]. The configurations are selected according to their contribution to the total energy. One set of exponents is optimized for the whole expansion. Using some computational techniques to increase efficiency, our computer program is able to perform partially-parallelized runs of 1000 configuration term functions within a few minutes. With the optimized computer programme we were able to test a large number of configuration types and chose the most important ones. The energy of the 3P ground state of carbon atom with a wave function of angular momentum L=1 and ML=0 and spin eigenfunction with S=1 and MS=0 leads to -37.83526523 h, which is millihartree accurate. We discuss the state of the art in the determination of the ground state of the carbon atom and give an outlook about the complex spectra of this atom and its low-lying states. Keywords: Carbon atom; Configuration Interaction; Slater orbitals; Ground state *Corresponding author: e-mail address: [email protected] 1 1. Introduction The spectrum of the isolated carbon atom is the most complex one among the light atoms. The ground state of carbon atom is a triplet 3P state and its low-lying excited states are singlet 1D, 1S and 1P states, more stable than the corresponding triplet excited ones 3D and 3S, against the Hund’s rule of maximal multiplicity.
    [Show full text]
  • 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.
    [Show full text]
  • 1 the LOCALIZED QUANTUM VACUUM FIELD D. Dragoman
    1 THE LOCALIZED QUANTUM VACUUM FIELD D. Dragoman – Univ. Bucharest, Physics Dept., P.O. Box MG-11, 077125 Bucharest, Romania, e-mail: [email protected] ABSTRACT A model for the localized quantum vacuum is proposed in which the zero-point energy of the quantum electromagnetic field originates in energy- and momentum-conserving transitions of material systems from their ground state to an unstable state with negative energy. These transitions are accompanied by emissions and re-absorptions of real photons, which generate a localized quantum vacuum in the neighborhood of material systems. The model could help resolve the cosmological paradox associated to the zero-point energy of electromagnetic fields, while reclaiming quantum effects associated with quantum vacuum such as the Casimir effect and the Lamb shift; it also offers a new insight into the Zitterbewegung of material particles. 2 INTRODUCTION The zero-point energy (ZPE) of the quantum electromagnetic field is at the same time an indispensable concept of quantum field theory and a controversial issue (see [1] for an excellent review of the subject). The need of the ZPE has been recognized from the beginning of quantum theory of radiation, since only the inclusion of this term assures no first-order temperature-independent correction to the average energy of an oscillator in thermal equilibrium with blackbody radiation in the classical limit of high temperatures. A more rigorous introduction of the ZPE stems from the treatment of the electromagnetic radiation as an ensemble of harmonic quantum oscillators. Then, the total energy of the quantum electromagnetic field is given by E = åk,s hwk (nks +1/ 2) , where nks is the number of quantum oscillators (photons) in the (k,s) mode that propagate with wavevector k and frequency wk =| k | c = kc , and are characterized by the polarization index s.
    [Show full text]
  • The Heisenberg Uncertainty Principle*
    OpenStax-CNX module: m58578 1 The Heisenberg Uncertainty Principle* OpenStax This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 Abstract By the end of this section, you will be able to: • Describe the physical meaning of the position-momentum uncertainty relation • Explain the origins of the uncertainty principle in quantum theory • Describe the physical meaning of the energy-time uncertainty relation Heisenberg's uncertainty principle is a key principle in quantum mechanics. Very roughly, it states that if we know everything about where a particle is located (the uncertainty of position is small), we know nothing about its momentum (the uncertainty of momentum is large), and vice versa. Versions of the uncertainty principle also exist for other quantities as well, such as energy and time. We discuss the momentum-position and energy-time uncertainty principles separately. 1 Momentum and Position To illustrate the momentum-position uncertainty principle, consider a free particle that moves along the x- direction. The particle moves with a constant velocity u and momentum p = mu. According to de Broglie's relations, p = }k and E = }!. As discussed in the previous section, the wave function for this particle is given by −i(! t−k x) −i ! t i k x k (x; t) = A [cos (! t − k x) − i sin (! t − k x)] = Ae = Ae e (1) 2 2 and the probability density j k (x; t) j = A is uniform and independent of time. The particle is equally likely to be found anywhere along the x-axis but has denite values of wavelength and wave number, and therefore momentum.
    [Show full text]
  • 8 the Variational Principle
    8 The Variational Principle 8.1 Approximate solution of the Schroedinger equation If we can’t find an analytic solution to the Schroedinger equation, a trick known as the varia- tional principle allows us to estimate the energy of the ground state of a system. We choose an unnormalized trial function Φ(an) which depends on some variational parameters, an and minimise hΦ|Hˆ |Φi E[a ] = n hΦ|Φi with respect to those parameters. This gives an approximation to the wavefunction whose accuracy depends on the number of parameters and the clever choice of Φ(an). For more rigorous treatments, a set of basis functions with expansion coefficients an may be used. The proof is as follows, if we expand the normalised wavefunction 1/2 |φ(an)i = Φ(an)/hΦ(an)|Φ(an)i in terms of the true (unknown) eigenbasis |ii of the Hamiltonian, then its energy is X X X ˆ 2 2 E[an] = hφ|iihi|H|jihj|φi = |hφ|ii| Ei = E0 + |hφ|ii| (Ei − E0) ≥ E0 ij i i ˆ where the true (unknown) ground state of the system is defined by H|i0i = E0|i0i. The inequality 2 arises because both |hφ|ii| and (Ei − E0) must be positive. Thus the lower we can make the energy E[ai], the closer it will be to the actual ground state energy, and the closer |φi will be to |i0i. If the trial wavefunction consists of a complete basis set of orthonormal functions |χ i, each P i multiplied by ai: |φi = i ai|χii then the solution is exact and we just have the usual trick of expanding a wavefunction in a basis set.
    [Show full text]
  • Recent Experimental Progress of Fractional Quantum Hall Effect: 5/2 Filling State and Graphene
    Recent Experimental Progress of Fractional Quantum Hall Effect: 5/2 Filling State and Graphene X. Lin, R. R. Du and X. C. Xie International Center for Quantum Materials, Peking University, Beijing, People’s Republic of China 100871 ABSTRACT The phenomenon of fractional quantum Hall effect (FQHE) was first experimentally observed 33 years ago. FQHE involves strong Coulomb interactions and correlations among the electrons, which leads to quasiparticles with fractional elementary charge. Three decades later, the field of FQHE is still active with new discoveries and new technical developments. A significant portion of attention in FQHE has been dedicated to filling factor 5/2 state, for its unusual even denominator and possible application in topological quantum computation. Traditionally FQHE has been observed in high mobility GaAs heterostructure, but new materials such as graphene also open up a new area for FQHE. This review focuses on recent progress of FQHE at 5/2 state and FQHE in graphene. Keywords: Fractional Quantum Hall Effect, Experimental Progress, 5/2 State, Graphene measured through the temperature dependence of I. INTRODUCTION longitudinal resistance. Due to the confinement potential of a realistic 2DEG sample, the gapped QHE A. Quantum Hall Effect (QHE) state has chiral edge current at boundaries. Hall effect was discovered in 1879, in which a Hall voltage perpendicular to the current is produced across a conductor under a magnetic field. Although Hall effect was discovered in a sheet of gold leaf by Edwin Hall, Hall effect does not require two-dimensional condition. In 1980, quantum Hall effect was observed in two-dimensional electron gas (2DEG) system [1,2].
    [Show full text]
  • Quantum Mechanics by Numerical Simulation of Path Integral
    Quantum Mechanics by Numerical Simulation of Path Integral Author: Bruno Gim´enezUmbert Facultat de F´ısica, Universitat de Barcelona, Diagonal 645, 08028 Barcelona, Spain.∗ Abstract: The Quantum Mechanics formulation of Feynman is based on the concept of path integrals, allowing to express the quantum transition between two space-time points without using the bra and ket formalism in the Hilbert space. A particular advantage of this approach is the ability to provide an intuitive representation of the classical limit of Quantum Mechanics. The practical importance of path integral formalism is being a powerful tool to solve quantum problems where the analytic solution of the Schr¨odingerequation is unknown. For this last type of physical systems, the path integrals can be calculated with the help of numerical integration methods suitable for implementation on a computer. Thus, they provide the development of arbitrarily accurate solutions. This is particularly important for the numerical simulation of strong interactions (QCD) which cannot be solved by a perturbative treatment. This thesis will focus on numerical techniques to calculate path integral on some physical systems of interest. I. INTRODUCTION [1]. In the first section of our writeup, we introduce the basic concepts of path integral and numerical simulation. Feynman's space-time approach based on path inte- Next, we discuss some specific examples such as the har- grals is not too convenient for attacking practical prob- monic oscillator. lems in non-relativistic Quantum Mechanics. Even for the simple harmonic oscillator it is rather cumbersome to evaluate explicitly the relevant path integral. However, II. QUANTUM MECHANICS BY PATH INTEGRAL his approach is extremely gratifying from a conceptual point of view.
    [Show full text]
  • 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]
  • 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.
    [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]