An Introduction to Effective Field Theory
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Electro Magnetic Fields Lecture Notes B.Tech
ELECTRO MAGNETIC FIELDS LECTURE NOTES B.TECH (II YEAR – I SEM) (2019-20) Prepared by: M.KUMARA SWAMY., Asst.Prof Department of Electrical & Electronics Engineering MALLA REDDY COLLEGE OF ENGINEERING & TECHNOLOGY (Autonomous Institution – UGC, Govt. of India) Recognized under 2(f) and 12 (B) of UGC ACT 1956 (Affiliated to JNTUH, Hyderabad, Approved by AICTE - Accredited by NBA & NAAC – ‘A’ Grade - ISO 9001:2015 Certified) Maisammaguda, Dhulapally (Post Via. Kompally), Secunderabad – 500100, Telangana State, India ELECTRO MAGNETIC FIELDS Objectives: • To introduce the concepts of electric field, magnetic field. • Applications of electric and magnetic fields in the development of the theory for power transmission lines and electrical machines. UNIT – I Electrostatics: Electrostatic Fields – Coulomb’s Law – Electric Field Intensity (EFI) – EFI due to a line and a surface charge – Work done in moving a point charge in an electrostatic field – Electric Potential – Properties of potential function – Potential gradient – Gauss’s law – Application of Gauss’s Law – Maxwell’s first law, div ( D )=ρv – Laplace’s and Poison’s equations . Electric dipole – Dipole moment – potential and EFI due to an electric dipole. UNIT – II Dielectrics & Capacitance: Behavior of conductors in an electric field – Conductors and Insulators – Electric field inside a dielectric material – polarization – Dielectric – Conductor and Dielectric – Dielectric boundary conditions – Capacitance – Capacitance of parallel plates – spherical co‐axial capacitors. Current density – conduction and Convection current densities – Ohm’s law in point form – Equation of continuity UNIT – III Magneto Statics: Static magnetic fields – Biot‐Savart’s law – Magnetic field intensity (MFI) – MFI due to a straight current carrying filament – MFI due to circular, square and solenoid current Carrying wire – Relation between magnetic flux and magnetic flux density – Maxwell’s second Equation, div(B)=0, Ampere’s Law & Applications: Ampere’s circuital law and its applications viz. -
Probing the Minimal Length Scale by Precision Tests of the Muon G − 2
Physics Letters B 584 (2004) 109–113 www.elsevier.com/locate/physletb Probing the minimal length scale by precision tests of the muon g − 2 U. Harbach, S. Hossenfelder, M. Bleicher, H. Stöcker Institut für Theoretische Physik, J.W. Goethe-Universität, Robert-Mayer-Str. 8-10, 60054 Frankfurt am Main, Germany Received 25 November 2003; received in revised form 20 January 2004; accepted 21 January 2004 Editor: P.V. Landshoff Abstract Modifications of the gyromagnetic moment of electrons and muons due to a minimal length scale combined with a modified fundamental scale Mf are explored. First-order deviations from the theoretical SM value for g − 2 due to these string theory- motivated effects are derived. Constraints for the fundamental scale Mf are given. 2004 Elsevier B.V. All rights reserved. String theory suggests the existence of a minimum • the need for a higher-dimensional space–time and length scale. An exciting quantum mechanical impli- • the existence of a minimal length scale. cation of this feature is a modification of the uncer- tainty principle. In perturbative string theory [1,2], Naturally, this minimum length uncertainty is re- the feature of a fundamental minimal length scale lated to a modification of the standard commutation arises from the fact that strings cannot probe dis- relations between position and momentum [6,7]. Ap- tances smaller than the string scale. If the energy of plication of this is of high interest for quantum fluc- a string reaches the Planck mass mp, excitations of the tuations in the early universe and inflation [8–16]. string can occur and cause a non-zero extension [3]. -
Gravitational Potential Energy
Briefly review the concepts of potential energy and work. •Potential Energy = U = stored work in a system •Work = energy put into or taken out of system by forces •Work done by a (constant) force F : v v v v F W = F ⋅∆r =| F || ∆r | cosθ θ ∆r Gravitational Potential Energy Lift a book by hand (Fext) at constant velocity. F = mg final ext Wext = Fext h = mgh h Wgrav = -mgh Fext Define ∆U = +Wext = -Wgrav= mgh initial Note that get to define U=0, mg typically at the ground. U is for potential energy, do not confuse with “internal energy” in Thermo. Gravitational Potential Energy (cont) For conservative forces Mechanical Energy is conserved. EMech = EKin +U Gravity is a conservative force. Coulomb force is also a conservative force. Friction is not a conservative force. If only conservative forces are acting, then ∆EMech=0. ∆EKin + ∆U = 0 Electric Potential Energy Charge in a constant field ∆Uelec = change in U when moving +q from initial to final position. ∆U = U f −Ui = +Wext = −W field FExt=-qE + Final position v v ∆U = −W = −F ⋅∆r fieldv field FField=qE v ∆r → ∆U = −qE ⋅∆r E + Initial position -------------- General case What if the E-field is not constant? v v ∆U = −qE ⋅∆r f v v Integral over the path from initial (i) position to final (f) ∆U = −q∫ E ⋅dr position. i Electric Potential Energy Since Coulomb forces are conservative, it means that the change in potential energy is path independent. f v v ∆U = −q∫ E ⋅dr i Electric Potential Energy Positive charge in a constant field Electric Potential Energy Negative charge in a constant field Observations • If we need to exert a force to “push” or “pull” against the field to move the particle to the new position, then U increases. -
Astro 282: Problem Set 1 1 Problem 1: Natural Units
Astro 282: Problem Set 1 Due April 7 1 Problem 1: Natural Units Cosmologists and particle physicists like to suppress units by setting the fundamental constants c,h ¯, kB to unity so that there is one fundamental unit. Structure formation cosmologists generally prefer Mpc as a unit of length, time, inverse energy, inverse temperature. Early universe people use GeV. Familiarize yourself with the elimination and restoration of units in the cosmological context. Here are some fundamental constants of nature Planck’s constant ¯h = 1.0546 × 10−27 cm2 g s−1 Speed of light c = 2.9979 × 1010 cm s−1 −16 −1 Boltzmann’s constant kB = 1.3807 × 10 erg K Fine structure constant α = 1/137.036 Gravitational constant G = 6.6720 × 10−8 cm3 g−1 s−2 Stefan-Boltzmann constant σ = a/4 = π2/60 a = 7.5646 × 10−15 erg cm−3K−4 2 2 −25 2 Thomson cross section σT = 8πα /3me = 6.6524 × 10 cm Electron mass me = 0.5110 MeV Neutron mass mn = 939.566 MeV Proton mass mp = 938.272 MeV −1/2 19 Planck mass mpl = G = 1.221 × 10 GeV and here are some unit conversions: 1 s = 9.7157 × 10−15 Mpc 1 yr = 3.1558 × 107 s 1 Mpc = 3.0856 × 1024 cm 1 AU = 1.4960 × 1013 cm 1 K = 8.6170 × 10−5 eV 33 1 M = 1.989 × 10 g 1 GeV = 1.6022 × 10−3 erg = 1.7827 × 10−24 g = (1.9733 × 10−14 cm)−1 = (6.5821 × 10−25 s)−1 −1 −1 • Define the Hubble constant as H0 = 100hkm s Mpc where h is a dimensionless number observed to be −1 −1 h ≈ 0.7. -
Soft Gravitons and the Flat Space Limit of Anti-Desitter Space
RUNHETC-2016-15, UTTG-18-16 Soft Gravitons and the Flat Space Limit of Anti-deSitter Space Tom Banks Department of Physics and NHETC Rutgers University, Piscataway, NJ 08854 E-mail: [email protected] Willy Fischler Department of Physics and Texas Cosmology Center University of Texas, Austin, TX 78712 E-mail: fi[email protected] Abstract We argue that flat space amplitudes for the process 2 n gravitons at center of mass → energies √s much less than the Planck scale, will coincide approximately with amplitudes calculated from correlators of a boundary CFT for AdS space of radius R LP , only ≫ when n < R/LP . For larger values of n in AdS space, insisting that all the incoming energy enters “the arena”[20] , implies the production of black holes, whereas there is no black hole production in the flat space amplitudes. We also argue, from unitarity, that flat space amplitudes for all n are necessary to reconstruct the complicated singularity arXiv:1611.05906v3 [hep-th] 8 Apr 2017 at zero momentum in the 2 2 amplitude, which can therefore not be reproduced as → the limit of an AdS calculation. Applying similar reasoning to amplitudes for real black hole production in flat space, we argue that unitarity of the flat space S-matrix cannot be assessed or inferred from properties of CFT correlators. 1 Introduction The study of the flat space limit of correlation functions in AdS/CFT was initiated by the work of Polchinski and Susskind[20] and continued in a host of other papers. Most of those papers agree with the contention, that the limit of Witten diagrams for CFT correlators smeared with appropriate test functions, for operators carrying vanishing angular momentum on the compact 1 space,1 converge to flat space S-matrix elements between states in Fock space. -
Small Angle Scattering in Neutron Imaging—A Review
Journal of Imaging Review Small Angle Scattering in Neutron Imaging—A Review Markus Strobl 1,2,*,†, Ralph P. Harti 1,†, Christian Grünzweig 1,†, Robin Woracek 3,† and Jeroen Plomp 4,† 1 Paul Scherrer Institut, PSI Aarebrücke, 5232 Villigen, Switzerland; [email protected] (R.P.H.); [email protected] (C.G.) 2 Niels Bohr Institute, University of Copenhagen, Copenhagen 1165, Denmark 3 European Spallation Source ERIC, 225 92 Lund, Sweden; [email protected] 4 Department of Radiation Science and Technology, Technical University Delft, 2628 Delft, The Netherlands; [email protected] * Correspondence: [email protected]; Tel.: +41-56-310-5941 † These authors contributed equally to this work. Received: 6 November 2017; Accepted: 8 December 2017; Published: 13 December 2017 Abstract: Conventional neutron imaging utilizes the beam attenuation caused by scattering and absorption through the materials constituting an object in order to investigate its macroscopic inner structure. Small angle scattering has basically no impact on such images under the geometrical conditions applied. Nevertheless, in recent years different experimental methods have been developed in neutron imaging, which enable to not only generate contrast based on neutrons scattered to very small angles, but to map and quantify small angle scattering with the spatial resolution of neutron imaging. This enables neutron imaging to access length scales which are not directly resolved in real space and to investigate bulk structures and processes spanning multiple length scales from centimeters to tens of nanometers. Keywords: neutron imaging; neutron scattering; small angle scattering; dark-field imaging 1. Introduction The largest and maybe also broadest length scales that are probed with neutrons are the domains of small angle neutron scattering (SANS) and imaging. -
Complete Normal Ordering 1: Foundations
KCL-PH-TH/2015-55, LCTS/2015-43, CERN-PH-TH/2015-295 Complete Normal Ordering 1: Foundations a,b a,b a,c John Ellisp q, Nick E. Mavromatosp q and Dimitri P. Sklirosp q (a) Theoretical Particle Physics and Cosmology Group Department of Physics, King’s College London London WC2R 2LS, United Kingdom (b) Theory Division, CERN, CH-1211 Geneva 23, Switzerland (c) School of Physics and Astronomy, University of Nottingham Nottingham, NG7 2RD, UK [email protected]; [email protected]; [email protected] Abstract We introduce a new prescription for quantising scalar field theories (in generic spacetime dimension and background) perturbatively around a true minimum of the full quantum ef- fective action, which is to ‘complete normal order’ the bare action of interest. When the true vacuum of the theory is located at zero field value, the key property of this prescription is the automatic cancellation, to any finite order in perturbation theory, of all tadpole and, more generally, all ‘cephalopod’ Feynman diagrams. The latter are connected diagrams that can be disconnected into two pieces by cutting one internal vertex, with either one or both pieces free from external lines. In addition, this procedure of ‘complete normal ordering’ (which is an extension of the standard field theory definition of normal ordering) reduces by a substantial factor the number of Feynman diagrams to be calculated at any given loop order. We illustrate explicitly the complete normal ordering procedure and the cancellation of cephalopod diagrams in scalar field theories with non-derivative interactions, and by using a point splitting ‘trick’ we extend this result to theories with derivative interactions, such as those appearing as non-linear σ-models in the world-sheet formulation of string theory. -
Collective Coordinates for D-Branes 1 Introduction
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server UTTG-01-96 Collective Co ordinates for D-branes Willy Fischler, Sonia Paban and Moshe Rozali Theory Group, Department of Physics University of Texas, Austin, Texas 78712 Abstract We develop a formalism for the scattering o D-branes that includes collective co- ordinates. This allows a systematic expansion in the string coupling constant for such pro cesses, including a worldsheet calculation for the D-brane's mass. 1 Intro duction In the recent developments of string theory, solitons play a central role. For example, in the case of duality among various string theories, solitons of a weakly coupled string theory b ecome elementary excitations of the dual theory [1]. Another example, among many others, is the role of solitons in resolving singularities in compacti ed geometries, as they b ecome light[2]. In order to gain more insight in the various asp ects of these recent developments, it is imp ortant to understand the dynamics of solitons in the context of string theory.In this quest one of the to ols at our disp osal is the use of scattering involving these solitons. This includes the scattering of elementary string states o these solitons as well as the scattering among solitons. An imp ortant class of solitons that have emerged are D-branes [4]. In an in uential pap er [3]itwas shown that these D-branes carry R R charges, and are therefore a central ingredient in the aforementioned dualities. -
King's Research Portal
King’s Research Portal DOI: 10.1088/1742-6596/631/1/012089 Document Version Publisher's PDF, also known as Version of record Link to publication record in King's Research Portal Citation for published version (APA): Sarkar, S. (2015). String theory backgrounds with torsion in the presence of fermions and implications for leptogenesis. Journal of Physics: Conference Series, 631(1), [012089]. 10.1088/1742-6596/631/1/012089 Citing this paper Please note that where the full-text provided on King's Research Portal is the Author Accepted Manuscript or Post-Print version this may differ from the final Published version. If citing, it is advised that you check and use the publisher's definitive version for pagination, volume/issue, and date of publication details. And where the final published version is provided on the Research Portal, if citing you are again advised to check the publisher's website for any subsequent corrections. General rights Copyright and moral rights for the publications made accessible in the Research Portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognize and abide by the legal requirements associated with these rights. •Users may download and print one copy of any publication from the Research Portal for the purpose of private study or research. •You may not further distribute the material or use it for any profit-making activity or commercial gain •You may freely distribute the URL identifying the publication in the Research Portal Take down policy If you believe that this document breaches copyright please contact [email protected] providing details, and we will remove access to the work immediately and investigate your claim. -
Arxiv:1901.04741V2 [Hep-Th] 15 Feb 2019 2
Quantum scale symmetry C. Wetterich [email protected] Universität Heidelberg, Institut für Theoretische Physik, Philosophenweg 16, D-69120 Heidelberg Quantum scale symmetry is the realization of scale invariance in a quantum field theory. No parameters with dimension of length or mass are present in the quantum effective action. Quantum scale symmetry is generated by quantum fluctuations via the presence of fixed points for running couplings. As for any global symmetry, the ground state or cosmological state may be scale invariant or not. Spontaneous breaking of scale symmetry leads to massive particles and predicts a massless Goldstone boson. A massless particle spectrum follows from scale symmetry of the effective action only if the ground state is scale symmetric. Approximate scale symmetry close to a fixed point leads to important predictions for observations in various areas of fundamental physics. We review consequences of scale symmetry for particle physics, quantum gravity and cosmology. For particle physics, scale symmetry is closely linked to the tiny ratio between the Fermi scale of weak interactions and the Planck scale for gravity. For quantum gravity, scale symmetry is associated to the ultraviolet fixed point which allows for a non-perturbatively renormalizable quantum field theory for all known interactions. The interplay between gravity and particle physics at this fixed point permits to predict couplings of the standard model or other “effective low energy models” for momenta below the Planck mass. In particular, quantum gravity determines the ratio of Higgs boson mass and top quark mass. In cosmology, approximate scale symmetry explains the almost scale-invariant primordial fluctuation spectrum which is at the origin of all structures in the universe. -
Minimal Length Scale Scenarios for Quantum Gravity
Living Rev. Relativity, 16, (2013), 2 LIVINGREVIEWS http://www.livingreviews.org/lrr-2013-2 doi:10.12942/lrr-2013-2 in relativity Minimal Length Scale Scenarios for Quantum Gravity Sabine Hossenfelder Nordita Roslagstullsbacken 23 106 91 Stockholm Sweden email: [email protected] Accepted: 11 October 2012 Published: 29 January 2013 Abstract We review the question of whether the fundamental laws of nature limit our ability to probe arbitrarily short distances. First, we examine what insights can be gained from thought experiments for probes of shortest distances, and summarize what can be learned from different approaches to a theory of quantum gravity. Then we discuss some models that have been developed to implement a minimal length scale in quantum mechanics and quantum field theory. These models have entered the literature as the generalized uncertainty principle or the modified dispersion relation, and have allowed the study of the effects of a minimal length scale in quantum mechanics, quantum electrodynamics, thermodynamics, black-hole physics and cosmology. Finally, we touch upon the question of ways to circumvent the manifestation of a minimal length scale in short-distance physics. Keywords: minimal length, quantum gravity, generalized uncertainty principle This review is licensed under a Creative Commons Attribution-Non-Commercial 3.0 Germany License. http://creativecommons.org/licenses/by-nc/3.0/de/ Imprint / Terms of Use Living Reviews in Relativity is a peer reviewed open access journal published by the Max Planck Institute for Gravitational Physics, Am M¨uhlenberg 1, 14476 Potsdam, Germany. ISSN 1433-8351. This review is licensed under a Creative Commons Attribution-Non-Commercial 3.0 Germany License: http://creativecommons.org/licenses/by-nc/3.0/de/. -
Collective Coordinates for D-Branes
UTTG-01-96 Collective Coordinates for D-branes Willy Fischler, Sonia Paban and Moshe Rozali∗ Theory Group, Department of Physics University of Texas, Austin, Texas 78712 Abstract We develop a formalism for the scattering off D-branes that includes collective co- ordinates. This allows a systematic expansion in the string coupling constant for such processes, including a worldsheet calculation for the D-brane’s mass. 1 Introduction In the recent developments of string theory, solitons play a central role. For example, in the case of duality among various string theories, solitons of a weakly coupled string theory become elementary excitations of the dual theory [1]. Another example, among many others, is the role of solitons in resolving singularities in compactified geometries, as they become light [2]. In order to gain more insight in the various aspects of these recent developments, it is important to understand the dynamics of solitons in the context of string theory. In this quest one of the tools at our disposal is the use of scattering involving these solitons. This includes the scattering of elementary string states off these solitons as well as the arXiv:hep-th/9604014v2 11 Apr 1996 scattering among solitons. An important class of solitons that have emerged are D-branes [4]. In an influential paper [3] it was shown that these D-branes carry R R charges, − and are therefore a central ingredient in the aforementioned dualities. These objects are described by simple conformal field theories, which makes them particularly suited for explicit calculations. In this paper we use some of the preliminary ideas about collective coordinates de- veloped in a previous paper [5], to describe the scattering of elementary string states off D-branes.