The Chiral Anomaly and the Dirac Sea
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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. -
Thermal Evolution of the Axial Anomaly
Thermal evolution of the axial anomaly Gergely Fej}os Research Center for Nuclear Physics Osaka University The 10th APCTP-BLTP/JINR-RCNP-RIKEN Joint Workshop on Nuclear and Hadronic Physics 18th August, 2016 G. Fejos & A. Hosaka, arXiv: 1604.05982 Gergely Fej}os Thermal evolution of the axial anomaly Outline aaa Motivation Functional renormalization group Chiral (linear) sigma model and axial anomaly Extension with nucleons Summary Gergely Fej}os Thermal evolution of the axial anomaly Motivation Gergely Fej}os Thermal evolution of the axial anomaly Chiral symmetry is spontaneuously broken in the ground state: < ¯ > = < ¯R L > + < ¯L R > 6= 0 SSB pattern: SUL(Nf ) × SUR (Nf ) −! SUV (Nf ) Anomaly: UA(1) is broken by instantons Details of chiral symmetry restoration? Critical temperature? Axial anomaly? Is it recovered at the critical point? Motivation QCD Lagrangian with quarks and gluons: 1 L = − G a G µνa + ¯ (iγ Dµ − m) 4 µν i µ ij j Approximate chiral symmetry for Nf = 2; 3 flavors: iT aθa iT aθa L ! e L L; R ! e R R [vector: θL + θR , axialvector: θL − θR ] Gergely Fej}os Thermal evolution of the axial anomaly Details of chiral symmetry restoration? Critical temperature? Axial anomaly? Is it recovered at the critical point? Motivation QCD Lagrangian with quarks and gluons: 1 L = − G a G µνa + ¯ (iγ Dµ − m) 4 µν i µ ij j Approximate chiral symmetry for Nf = 2; 3 flavors: iT aθa iT aθa L ! e L L; R ! e R R [vector: θL + θR , axialvector: θL − θR ] Chiral symmetry is spontaneuously broken in the ground state: < ¯ > = < ¯R L > + < ¯L R -
Dirac Equation - Wikipedia
Dirac equation - Wikipedia https://en.wikipedia.org/wiki/Dirac_equation Dirac equation From Wikipedia, the free encyclopedia In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it 1 describes all spin-2 massive particles such as electrons and quarks for which parity is a symmetry. It is consistent with both the principles of quantum mechanics and the theory of special relativity,[1] and was the first theory to account fully for special relativity in the context of quantum mechanics. It was validated by accounting for the fine details of the hydrogen spectrum in a completely rigorous way. The equation also implied the existence of a new form of matter, antimatter, previously unsuspected and unobserved and which was experimentally confirmed several years later. It also provided a theoretical justification for the introduction of several component wave functions in Pauli's phenomenological theory of spin; the wave functions in the Dirac theory are vectors of four complex numbers (known as bispinors), two of which resemble the Pauli wavefunction in the non-relativistic limit, in contrast to the Schrödinger equation which described wave functions of only one complex value. Moreover, in the limit of zero mass, the Dirac equation reduces to the Weyl equation. Although Dirac did not at first fully appreciate the importance of his results, the entailed explanation of spin as a consequence of the union of quantum mechanics and relativity—and the eventual discovery of the positron—represents one of the great triumphs of theoretical physics. -
Chiral Anomaly Without Relativity Arxiv:1511.03621V1 [Physics.Pop-Ph]
Chiral anomaly without relativity A.A. Burkov Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada, and ITMO University, Saint Petersburg 197101, Russia Perspective on J. Xiong et al., Science 350, 413 (2015). The Dirac equation, which describes relativistic fermions, has a mathematically inevitable, but puzzling feature: negative energy solutions. The physical reality of these solutions is unques- tionable, as one of their direct consequences, the existence of antimatter, is confirmed by ex- periment. It is their interpretation that has always been somewhat controversial. Dirac’s own idea was to view the vacuum as a state in which all the negative energy levels are physically filled by fermions, which is now known as the Dirac sea. This idea seems to directly contradict a common-sense view of the vacuum as a state in which matter is absent and is thus generally disliked among high-energy physicists, who prefer to regard the Dirac sea as not much more than a useful metaphor. On the other hand, the Dirac sea is a very natural concept from the point of view of a condensed matter physicist, since there is a direct and simple analogy: filled valence bands of an insulating crystal. There exists, however, a phenomenon within the con- arXiv:1511.03621v1 [physics.pop-ph] 26 Oct 2015 text of the relativistic quantum field theory itself, whose satisfactory understanding seems to be hard to achieve without assigning physical reality to the Dirac sea. This phenomenon is the chiral anomaly, a quantum-mechanical violation of chiral symmetry, which was first observed experimentally in the particle physics setting as a decay of a neutral pion into two photons. -
Effective Lagrangian for Axial Anomaly and Its Applications in Dirac And
www.nature.com/scientificreports OPEN Efective lagrangian for axial anomaly and its applications in Dirac and Weyl semimetals Received: 25 April 2018 Chih-Yu Chen, C. D. Hu & Yeu-Chung Lin Accepted: 9 July 2018 A gauge invariant efective lagrangian for the fermion axial anomaly is constructed. The dynamical Published: xx xx xxxx degree of freedom for fermion feld is preserved. Using the anomaly lagrangian, the scattering cross section of pair production γγ → e−e+ in Dirac or Weyl semimetal is computed. The result is compared with the corresponding result from Dirac lagrangian. It is found that anomaly lagrangain and Dirac lagrangian exhibit the same E B pattern, therefore the E B signature may not serve a good indicator of the existence of axial anomaly. Because anomaly generates excessive right-handed electrons and positrons, pair production can give rise to spin current by applying gate voltage and charge current with depositing spin flters. These experiments are able to discern genuine anomaly phenomena. Axial anomaly1,2, sometimes referred to as chiral anomaly in proper context, arises from the non-invariance of fermion measure under axial γ5 transformation3, it is a generic property of quantum fermion feld theory, massive or massless. Te realization of axial anomaly in condensed matter is frst explored by Nielsen and Ninomiya4. It is argued that in the presence of external parallel electric and magnetic felds, there is net production of chiral charge and electrons move from lef-handed (LH) Weyl cone to right-handed (RH) Weyl cone, the induced anomaly current causes magneto-conductivity prominent. Aji5 investigated pyrochlore iridates under magnetic feld. -
A Relativistic Symmetrical Interpretation of the Dirac Equation in (1+1) Dimensions
A RELATIVISTIC SYMMETRICAL INTERPRETATION OF THE DIRAC EQUATION IN (1+1) DIMENSIONS MICHAEL B. HEANEY 3182 STELLING DRIVE PALO ALTO, CA 94303 [email protected] Abstract. This paper presents a new Relativistic Symmetrical Interpretation (RSI) of the Dirac equation in (1+1)D which postulates: quantum mechanics is intrinsically time-symmetric, with no arrow of time; the fundamental objects of quantum mechanics are transitions; a transition is fully described by a complex transition amplitude density with specified initial and final boundary condi- tions; and transition amplitude densities never collapse. This RSI is compared to the Copenhagen 1 Interpretation (CI) for the analysis of Einstein's bubble experiment with a spin- 2 particle. This RSI can predict the future and retrodict the past, has no zitterbewegung, resolves some inconsistencies of the CI, and eliminates some of the conceptual problems of the CI. 1. Introduction At the 1927 Solvay Congress, Einstein presented a thought-experiment to illustrate what he saw as a flaw in the Copenhagen Interpretation (CI) of quantum mechanics [1]. In his thought-experiment (later known as Einstein's bubble experiment) a single particle is released from a source, evolves freely for a time, then is captured by a detector. Let us assume the single particle is a massive 1 spin- 2 particle. The particle is localized at the source immediately before release, as shown in Figure 1(a). The CI says that, after release from the source, the particle's probability density evolves continuously and deterministically into a progressively delocalized distribution, up until immediately before capture at the detector, as shown in Figures 1(b) and 1(c). -
Π0 Decay Precision-Tests the Chiral Anomaly Harvey B
NUCLEAR PHYSICS p0 decay precision-tests the chiral anomaly More precise neutral pion lifetime measurements probe quantum symmetry breaking By Harvey B. Meyer point, it was assumed that a symmetry of the errors in quadrature, this amounts to a classical Lagrangian would protect p0 from tension of 1.8 standard deviations versus ost subatomic particles are decaying in the limit of massless up and the combined result of the PrimEx-I and strongly interacting composites down quarks and lead to a longer lifetime. -II experiments, 8.34 (60.13) × 10−17 s. Al- called hadrons. Most hadrons are However, it turns out to be impossible to reg- though this difference could be a statisti- unstable and decay on extremely ularize quantum chromodynamics without cal fluctuation, it provides motivation to short time scales (10−22 s) to lighter breaking that symmetry. Therefore, the latter revisit the theory prediction. The gp pp hadrons. The electrically neutral is not respected by the quantum fluctuations reaction also has a sharp low-energy pre- Mpion, p0, is the lightest hadron and decays of the quantum chromodynamics fields and diction based on the chiral anomaly (6, 7) on a time scale of 10−16 s in 98.8% of cases does not protect p0 from decaying. and is being investigated by the COMPASS into two photons, gg, through the electro- The quantum origin of the symmetry experiment (8). magnetic interaction. Historically, under- breaking leads to an exact prediction for the The gg decay width of p0 has recently standing this time scale presented a major strength of p0 coupling to gg and hence the been evaluated from first principles using Downloaded from challenge to theoreticians. -
Definition of the Dirac Sea in the Presence of External Fields
c 1998 International Press Adv. Theor. Math. Phys. 2 (1998) 963 – 985 Definition of the Dirac Sea in the Presence of External Fields Felix Finster 1 Mathematics Department Harvard University Abstract It is shown that the Dirac sea can be uniquely defined for the Dirac equation with general interaction, if we impose a causality condition on the Dirac sea. We derive an explicit formula for the Dirac sea in terms of a power series in the bosonic potentials. The construction is extended to systems of Dirac seas. If the system contains chiral fermions, the causality condition yields a restriction for the bosonic potentials. 1 Introduction arXiv:hep-th/9705006v5 6 Jan 2009 The Dirac equation has solutions of negative energy, which have no mean- ingful physical interpretation. This popular problem of relativistic quantum mechanics was originally solved by Dirac’s concept that all negative-energy states are occupied in the vacuum forming the so-called Dirac sea. Fermions and anti-fermions are then described by positive-energy states and “holes” in the Dirac sea, respectively. Although this vivid picture of a sea of interacting particles is nowadays often considered not to be taken too literally, the con- struction of the Dirac sea also plays a crucial role in quantum field theory. There it corresponds to the formal exchanging of creation and annihilation operators for the negative-energy states of the free field theory. e-print archive: http://xxx.lanl.gov/abs/hep-th/9705006 1Supported by the Deutsche Forschungsgemeinschaft, Bonn. 964 DEFINITION OF THE DIRAC SEA ... Usually, the Dirac sea is only constructed in the vacuum. -
Scenario for the Origin of Matter (According to the Theory of Relation)
Journal of Modern Physics, 2019, 10, 163-175 http://www.scirp.org/journal/jmp ISSN Online: 2153-120X ISSN Print: 2153-1196 Scenario for the Origin of Matter (According to the Theory of Relation) Russell Bagdoo Saint-Bruno-de-Montarville, Quebec, Canada How to cite this paper: Bagdoo, R. (2019) Abstract Scenario for the Origin of Matter (According to the Theory of Relation). Journal of Modern Where did matter in the universe come from? Where does the mass of matter Physics, 10, 163-175. come from? Particle physicists have used the knowledge acquired in matter https://doi.org/10.4236/jmp.2019.102013 and space to imagine a standard scenario to provide satisfactory answers to Received: February 20, 2019 these major questions. The dominant thought to explain the absence of anti- Accepted: February 24, 2019 matter in nature is that we had an initially symmetrical universe made of Published: February 27, 2019 matter and antimatter and that a dissymmetry would have sufficed for more matter having constituted our world than antimatter. This dissymmetry Copyright © 2019 by author(s) and Scientific Research Publishing Inc. would arise from an anomaly in the number of neutrinos resulting from nu- This work is licensed under the Creative clear reactions which suggest the existence of a new type of titanic neutrino Commons Attribution International who would exceed the possibilities of the standard model and would justify License (CC BY 4.0). the absence of antimatter in the macrocosm. We believe that another scenario http://creativecommons.org/licenses/by/4.0/ could better explain why we observe only matter. -
Anomaly Constraint on Massless QCD and the Role of Skyrmions in Chiral Symmetry Breaking
Prepared for submission to JHEP Anomaly constraint on massless QCD and the role of Skyrmions in chiral symmetry breaking Yuya Tanizaki RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973 USA E-mail: [email protected] Abstract: We discuss consequences of the ’t Hooft anomaly matching condition for Quantum Chromodynamics (QCD) with massless fundamental quarks. We derive the new discrete ’t Hooft anomaly of massless QCD for generic numbers of color Nc and flavor Nf , and an exotic chiral-symmetry broken phase without quark-bilinear condensate is ruled out from possible QCD vacua. We show that the U(1)B baryon number symmetry is anomalously broken when the (Z2Nf )A discrete axial symmetry and the flavor symmetry are gauged. In the ordinary chiral symmetry breaking, the Skyrmion current turns out to reproduce this ’t Hooft anomaly of massless QCD. In the exotic chiral symmetry break- ing, however, the anomalous breaking of U(1)B does not take the correct form, and it is inconsistent with anomaly matching. This no-go theorem is based only on symmetries and anomalies, and thus has a wider range of applicability to the QCD phase diagram than the previous one obtained by QCD inequalities. Lastly, as another application, we check that duality of N = 1 supersymmetric QCD with Nf ≥ Nc + 1 satisfies the new anomaly matching. arXiv:1807.07666v3 [hep-th] 23 Aug 2018 Contents 1 Introduction1 2 Symmetry of massless QCD4 2.1 Symmetry group of massless Nf -flavor QCD5 2.2 Other equivalent expressions of the flavor symmetry6 2.3 Background -
Lecture Notes Wave Equations of Relativistic Quantum Mechanics
Lecture Notes Wave Equations of Relativistic Quantum Mechanics Dr. Matthias Lienert Eberhard-Karls Universität Tübingen Winter Semester 2018/19 2 Contents 1 Background 5 1.1 Basics of non-relativistic quantum mechanics . .6 1.2 Basics of special relativity . .9 2 The Poincaré group 13 2.1 Lorentz transformations. 13 2.2 Poincaré transformations . 14 2.3 Invariance of wave equations . 17 2.4 Wish list for a relativistic wave equation . 18 3 The Klein-Gordon equation 19 3.1 Derivation . 19 3.2 Physical properties . 20 3.3 Solution theory . 23 4 The Dirac equation 33 4.1 Derivation . 33 4.2 Relativistic invariance of the Dirac equation . 37 4.3 Discrete Transformations. 44 4.4 Physical properties . 45 4.5 Solution theory . 50 4.5.1 Classical solutions . 50 4.5.2 Hilbert space valued solutions . 51 5 The multi-time formalism 57 5.1 Motivation . 57 5.2 Evolution equations . 59 5.3 Probability conservation and generalized Born rule . 63 5.4 An interacting multi-time model in 1+1 dimensions . 67 3 4 CONTENTS 1. Background In this course, we will discuss relativistic quantum mechanical wave equations such as the Klein-Gordon equation 1 @ m2c2 − ∆ + (t; x) = 0 (1.1) c2 @t2 ~2 and the Dirac equation 3 X @ i γµ − mc = 0; (1.2) ~ @xµ µ=0 where γµ; µ = 0; 1; 2; 3 are certain 4 × 4 matrices. The basic motivation is that the Schrödinger equation 2 i @ (t; q) = − ~ ∆ + V (t; q) (t; q) (1.3) ~ t 2m is not invariant under the basic symmetry transformations of (special) relativity, Lorentz transformations. -
Symmetric Point Flavour Singlet Axial Vector Current Renormalization At
LTH 1228 Symmetric point flavour singlet axial vector current renormalization at two loops J.A. Gracey, Theoretical Physics Division, Department of Mathematical Sciences, University of Liverpool, P.O. Box 147, Liverpool, L69 3BX, United Kingdom. Abstract. We calculate the two loop correction to the quark 2-point function with the non-zero momentum insertion of the flavour singlet axial vector current at the fully symmetric subtraction point for massless quarks in the modified minimal subtraction (MS) scheme. The Larin method is used to handle γ5 within dimensional regularization at this loop order ensuring that the effect of the chiral anomaly is properly included within the construction. arXiv:2001.11282v2 [hep-ph] 21 Jul 2020 1 1 Introduction. One of the more curious experimental results over a generation ago was that of the EMC collabo- ration, [1]. They measured the origin of the proton spin and discovered that against expectations it was not due in a major part to the valence quarks. As the proton is a bound state of three quarks it was widely assumed that the combination of their quark spins would be the source 1 of the overall spin- 2 of the proton. Instead the experiment observed that the gluons binding the quarks together give a sizeable contribution. This was surprising due to the fact that in some sense the gluons are sea partons. While the original experiment was subsequently refined and improved to confirm the original observation, [2, 3, 4, 5], a clear theoretical understand- ing was sought to explain the phenomenon. As such a venture requires the use of the strong sector of the Standard Model described by Quantum Chromodynamics (QCD), tools had to be developed and refined to tackle the problem.