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Path Integrals in Quantum Mechanics
Path Integrals in Quantum Mechanics Dennis V. Perepelitsa MIT Department of Physics 70 Amherst Ave. Cambridge, MA 02142 Abstract We present the path integral formulation of quantum mechanics and demon- strate its equivalence to the Schr¨odinger picture. We apply the method to the free particle and quantum harmonic oscillator, investigate the Euclidean path integral, and discuss other applications. 1 Introduction A fundamental question in quantum mechanics is how does the state of a particle evolve with time? That is, the determination the time-evolution ψ(t) of some initial | i state ψ(t ) . Quantum mechanics is fully predictive [3] in the sense that initial | 0 i conditions and knowledge of the potential occupied by the particle is enough to fully specify the state of the particle for all future times.1 In the early twentieth century, Erwin Schr¨odinger derived an equation specifies how the instantaneous change in the wavefunction d ψ(t) depends on the system dt | i inhabited by the state in the form of the Hamiltonian. In this formulation, the eigenstates of the Hamiltonian play an important role, since their time-evolution is easy to calculate (i.e. they are stationary). A well-established method of solution, after the entire eigenspectrum of Hˆ is known, is to decompose the initial state into this eigenbasis, apply time evolution to each and then reassemble the eigenstates. That is, 1In the analysis below, we consider only the position of a particle, and not any other quantum property such as spin. 2 D.V. Perepelitsa n=∞ ψ(t) = exp [ iE t/~] n ψ(t ) n (1) | i − n h | 0 i| i n=0 X This (Hamiltonian) formulation works in many cases. -
An Introduction to Quantum Field Theory
AN INTRODUCTION TO QUANTUM FIELD THEORY By Dr M Dasgupta University of Manchester Lecture presented at the School for Experimental High Energy Physics Students Somerville College, Oxford, September 2009 - 1 - - 2 - Contents 0 Prologue....................................................................................................... 5 1 Introduction ................................................................................................ 6 1.1 Lagrangian formalism in classical mechanics......................................... 6 1.2 Quantum mechanics................................................................................... 8 1.3 The Schrödinger picture........................................................................... 10 1.4 The Heisenberg picture............................................................................ 11 1.5 The quantum mechanical harmonic oscillator ..................................... 12 Problems .............................................................................................................. 13 2 Classical Field Theory............................................................................. 14 2.1 From N-point mechanics to field theory ............................................... 14 2.2 Relativistic field theory ............................................................................ 15 2.3 Action for a scalar field ............................................................................ 15 2.4 Plane wave solution to the Klein-Gordon equation ........................... -
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. -
Feynman Diagrams, Momentum Space Feynman Rules, Disconnected Diagrams, Higher Correlation Functions
PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 Dr. Anosh Joseph, IISER Mohali LECTURE 05 Tuesday, January 14, 2020 Topics: Feynman Diagrams, Momentum Space Feynman Rules, Disconnected Diagrams, Higher Correlation Functions. Feynman Diagrams We can use the Wick’s theorem to express the n-point function h0jT fφ(x1)φ(x2) ··· φ(xn)gj0i as a sum of products of Feynman propagators. We will be interested in developing a diagrammatic interpretation of such expressions. Let us look at the expression containing four fields. We have h0jT fφ1φ2φ3φ4g j0i = DF (x1 − x2)DF (x3 − x4) +DF (x1 − x3)DF (x2 − x4) +DF (x1 − x4)DF (x2 − x3): (1) We can interpret Eq. (1) as the sum of three diagrams, if we represent each of the points, x1 through x4 by a dot, and each factor DF (x − y) by a line joining x to y. These diagrams are called Feynman diagrams. In Fig. 1 we provide the diagrammatic interpretation of Eq. (1). The interpretation of the diagrams is the following: particles are created at two spacetime points, each propagates to one of the other points, and then they are annihilated. This process can happen in three different ways, and they correspond to the three ways to connect the points in pairs, as shown in the three diagrams in Fig. 1. Thus the total amplitude for the process is the sum of these three diagrams. PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 ⟨0|T {ϕ1ϕ2ϕ3ϕ4}|0⟩ = DF(x1 − x2)DF(x3 − x4) + DF(x1 − x3)DF(x2 − x4) +DF(x1 − x4)DF(x2 − x3) 1 2 1 2 1 2 + + 3 4 3 4 3 4 Figure 1: Diagrammatic interpretation of Eq. -
Effective Field Theories, Reductionism and Scientific Explanation Stephan
To appear in: Studies in History and Philosophy of Modern Physics Effective Field Theories, Reductionism and Scientific Explanation Stephan Hartmann∗ Abstract Effective field theories have been a very popular tool in quantum physics for almost two decades. And there are good reasons for this. I will argue that effec- tive field theories share many of the advantages of both fundamental theories and phenomenological models, while avoiding their respective shortcomings. They are, for example, flexible enough to cover a wide range of phenomena, and concrete enough to provide a detailed story of the specific mechanisms at work at a given energy scale. So will all of physics eventually converge on effective field theories? This paper argues that good scientific research can be characterised by a fruitful interaction between fundamental theories, phenomenological models and effective field theories. All of them have their appropriate functions in the research process, and all of them are indispens- able. They complement each other and hang together in a coherent way which I shall characterise in some detail. To illustrate all this I will present a case study from nuclear and particle physics. The resulting view about scientific theorising is inherently pluralistic, and has implications for the debates about reductionism and scientific explanation. Keywords: Effective Field Theory; Quantum Field Theory; Renormalisation; Reductionism; Explanation; Pluralism. ∗Center for Philosophy of Science, University of Pittsburgh, 817 Cathedral of Learning, Pitts- burgh, PA 15260, USA (e-mail: [email protected]) (correspondence address); and Sektion Physik, Universit¨at M¨unchen, Theresienstr. 37, 80333 M¨unchen, Germany. 1 1 Introduction There is little doubt that effective field theories are nowadays a very popular tool in quantum physics. -
Coupling Constant Unification in Extensions of Standard Model
Coupling Constant Unification in Extensions of Standard Model Ling-Fong Li, and Feng Wu Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213 May 28, 2018 Abstract Unification of electromagnetic, weak, and strong coupling con- stants is studied in the extension of standard model with additional fermions and scalars. It is remarkable that this unification in the su- persymmetric extension of standard model yields a value of Weinberg angle which agrees very well with experiments. We discuss the other possibilities which can also give same result. One of the attractive features of the Grand Unified Theory is the con- arXiv:hep-ph/0304238v2 3 Jun 2003 vergence of the electromagnetic, weak and strong coupling constants at high energies and the prediction of the Weinberg angle[1],[3]. This lends a strong support to the supersymmetric extension of the Standard Model. This is because the Standard Model without the supersymmetry, the extrapolation of 3 coupling constants from the values measured at low energies to unifi- cation scale do not intercept at a single point while in the supersymmetric extension, the presence of additional particles, produces the convergence of coupling constants elegantly[4], or equivalently the prediction of the Wein- berg angle agrees with the experimental measurement very well[5]. This has become one of the cornerstone for believing the supersymmetric Standard 1 Model and the experimental search for the supersymmetry will be one of the main focus in the next round of new accelerators. In this paper we will explore the general possibilities of getting coupling constants unification by adding extra particles to the Standard Model[2] to see how unique is the Supersymmetric Standard Model in this respect[?]. -
5 the Dirac Equation and Spinors
5 The Dirac Equation and Spinors In this section we develop the appropriate wavefunctions for fundamental fermions and bosons. 5.1 Notation Review The three dimension differential operator is : ∂ ∂ ∂ = , , (5.1) ∂x ∂y ∂z We can generalise this to four dimensions ∂µ: 1 ∂ ∂ ∂ ∂ ∂ = , , , (5.2) µ c ∂t ∂x ∂y ∂z 5.2 The Schr¨odinger Equation First consider a classical non-relativistic particle of mass m in a potential U. The energy-momentum relationship is: p2 E = + U (5.3) 2m we can substitute the differential operators: ∂ Eˆ i pˆ i (5.4) → ∂t →− to obtain the non-relativistic Schr¨odinger Equation (with = 1): ∂ψ 1 i = 2 + U ψ (5.5) ∂t −2m For U = 0, the free particle solutions are: iEt ψ(x, t) e− ψ(x) (5.6) ∝ and the probability density ρ and current j are given by: 2 i ρ = ψ(x) j = ψ∗ ψ ψ ψ∗ (5.7) | | −2m − with conservation of probability giving the continuity equation: ∂ρ + j =0, (5.8) ∂t · Or in Covariant notation: µ µ ∂µj = 0 with j =(ρ,j) (5.9) The Schr¨odinger equation is 1st order in ∂/∂t but second order in ∂/∂x. However, as we are going to be dealing with relativistic particles, space and time should be treated equally. 25 5.3 The Klein-Gordon Equation For a relativistic particle the energy-momentum relationship is: p p = p pµ = E2 p 2 = m2 (5.10) · µ − | | Substituting the equation (5.4), leads to the relativistic Klein-Gordon equation: ∂2 + 2 ψ = m2ψ (5.11) −∂t2 The free particle solutions are plane waves: ip x i(Et p x) ψ e− · = e− − · (5.12) ∝ The Klein-Gordon equation successfully describes spin 0 particles in relativistic quan- tum field theory. -
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]. -
On the Limits of Effective Quantum Field Theory
RUNHETC-2019-15 On the Limits of Effective Quantum Field Theory: Eternal Inflation, Landscapes, and Other Mythical Beasts Tom Banks Department of Physics and NHETC Rutgers University, Piscataway, NJ 08854 E-mail: [email protected] Abstract We recapitulate multiple arguments that Eternal Inflation and the String Landscape are actually part of the Swampland: ideas in Effective Quantum Field Theory that do not have a counterpart in genuine models of Quantum Gravity. 1 Introduction Most of the arguments and results in this paper are old, dating back a decade, and very little of what is written here has not been published previously, or presented in talks. I was motivated to write this note after spending two weeks at the Vacuum Energy and Electroweak Scale workshop at KITP in Santa Barbara. There I found a whole new generation of effective field theorists recycling tired ideas from the 1980s about the use of effective field theory in gravitational contexts. These were ideas that I once believed in, but since the beginning of the 21st century my work in string theory and the dynamics of black holes, convinced me that they arXiv:1910.12817v2 [hep-th] 6 Nov 2019 were wrong. I wrote and lectured about this extensively in the first decade of the century, but apparently those arguments have not been accepted, and effective field theorists have concluded that the main lesson from string theory is that there is a vast landscape of meta-stable states in the theory of quantum gravity, connected by tunneling transitions in the manner envisioned by effective field theorists in the 1980s. -
TASI 2008 Lectures: Introduction to Supersymmetry And
TASI 2008 Lectures: Introduction to Supersymmetry and Supersymmetry Breaking Yuri Shirman Department of Physics and Astronomy University of California, Irvine, CA 92697. [email protected] Abstract These lectures, presented at TASI 08 school, provide an introduction to supersymmetry and supersymmetry breaking. We present basic formalism of supersymmetry, super- symmetric non-renormalization theorems, and summarize non-perturbative dynamics of supersymmetric QCD. We then turn to discussion of tree level, non-perturbative, and metastable supersymmetry breaking. We introduce Minimal Supersymmetric Standard Model and discuss soft parameters in the Lagrangian. Finally we discuss several mech- anisms for communicating the supersymmetry breaking between the hidden and visible sectors. arXiv:0907.0039v1 [hep-ph] 1 Jul 2009 Contents 1 Introduction 2 1.1 Motivation..................................... 2 1.2 Weylfermions................................... 4 1.3 Afirstlookatsupersymmetry . .. 5 2 Constructing supersymmetric Lagrangians 6 2.1 Wess-ZuminoModel ............................... 6 2.2 Superfieldformalism .............................. 8 2.3 VectorSuperfield ................................. 12 2.4 Supersymmetric U(1)gaugetheory ....................... 13 2.5 Non-abeliangaugetheory . .. 15 3 Non-renormalization theorems 16 3.1 R-symmetry.................................... 17 3.2 Superpotentialterms . .. .. .. 17 3.3 Gaugecouplingrenormalization . ..... 19 3.4 D-termrenormalization. ... 20 4 Non-perturbative dynamics in SUSY QCD 20 4.1 Affleck-Dine-Seiberg -
PHY 2404S Lecture Notes
PHY 2404S Lecture Notes Michael Luke Winter, 2003 These notes are perpetually under construction. Please let me know of any typos or errors. Once again, large portions of these notes have been plagiarized from Sidney Coleman’s field theory lectures from Harvard, written up by Brian Hill. 1 Contents 1 Preliminaries 3 1.1 Counterterms and Divergences: A Simple Example . 3 1.2 CountertermsinScalarFieldTheory . 11 2 Reformulating Scattering Theory 18 2.1 Feynman Diagrams with External Lines off the Mass Shell . .. 18 2.1.1 AnswerOne: PartofaLargerDiagram . 19 2.1.2 Answer Two: The Fourier Transform of a Green Function 20 2.1.3 Answer Three: The VEV of a String of Heisenberg Fields 22 2.2 GreenFunctionsandFeynmanDiagrams. 23 2.3 TheLSZReductionFormula . 27 2.3.1 ProofoftheLSZReductionFormula . 29 3 Renormalizing Scalar Field Theory 36 3.1 The Two-Point Function: Wavefunction Renormalization .... 37 3.2 The Analytic Structure of G(2), and1PIGreenFunctions . 40 3.3 Calculation of Π(k2) to order g2 .................. 44 3.4 The definition of g .......................... 50 3.5 Unstable Particles and the Optical Theorem . 51 3.6 Renormalizability. 57 4 Renormalizability 60 4.1 DegreesofDivergence . 60 4.2 RenormalizationofQED. 65 4.2.1 TroubleswithVectorFields . 65 4.2.2 Counterterms......................... 66 2 1 Preliminaries 1.1 Counterterms and Divergences: A Simple Example In the last semester we considered a variety of field theories, and learned to calculate scattering amplitudes at leading order in perturbation theory. Unfor- tunately, although things were fine as far as we went, the whole basis of our scattering theory had a gaping hole in it. -
Critical Coupling for Dynamical Chiral-Symmetry Breaking with an Infrared Finite Gluon Propagator *
BR9838528 Instituto de Fisica Teorica IFT Universidade Estadual Paulista November/96 IFT-P.050/96 Critical coupling for dynamical chiral-symmetry breaking with an infrared finite gluon propagator * A. A. Natale and P. S. Rodrigues da Silva Instituto de Fisica Teorica Universidade Estadual Paulista Rua Pamplona 145 01405-900 - Sao Paulo, S.P. Brazil *To appear in Phys. Lett. B t 2 9-04 Critical Coupling for Dynamical Chiral-Symmetry Breaking with an Infrared Finite Gluon Propagator A. A. Natale l and P. S. Rodrigues da Silva 2 •r Instituto de Fisica Teorica, Universidade Estadual Paulista Rua Pamplona, 145, 01405-900, Sao Paulo, SP Brazil Abstract We compute the critical coupling constant for the dynamical chiral- symmetry breaking in a model of quantum chromodynamics, solving numer- ically the quark self-energy using infrared finite gluon propagators found as solutions of the Schwinger-Dyson equation for the gluon, and one gluon prop- agator determined in numerical lattice simulations. The gluon mass scale screens the force responsible for the chiral breaking, and the transition occurs only for a larger critical coupling constant than the one obtained with the perturbative propagator. The critical coupling shows a great sensibility to the gluon mass scale variation, as well as to the functional form of the gluon propagator. 'e-mail: [email protected] 2e-mail: [email protected] 1 Introduction The idea that quarks obtain effective masses as a result of a dynamical breakdown of chiral symmetry (DBCS) has received a great deal of attention in the last years [1, 2]. One of the most common methods used to study the quark mass generation is to look for solutions of the Schwinger-Dyson equation for the fermionic propagator.