Quantum Field Theory in Flat Robertson-Walker Space-Time: Functional Schrõdinger Picture*
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1 Notation for States
The S-Matrix1 D. E. Soper2 University of Oregon Physics 634, Advanced Quantum Mechanics November 2000 1 Notation for states In these notes we discuss scattering nonrelativistic quantum mechanics. We will use states with the nonrelativistic normalization h~p |~ki = (2π)3δ(~p − ~k). (1) Recall that in a relativistic theory there is an extra factor of 2E on the right hand side of this relation, where E = [~k2 + m2]1/2. We will use states in the “Heisenberg picture,” in which states |ψ(t)i do not depend on time. Often in quantum mechanics one uses the Schr¨odinger picture, with time dependent states |ψ(t)iS. The relation between these is −iHt |ψ(t)iS = e |ψi. (2) Thus these are the same at time zero, and the Schrdinger states obey d i |ψ(t)i = H |ψ(t)i (3) dt S S In the Heisenberg picture, the states do not depend on time but the op- erators do depend on time. A Heisenberg operator O(t) is related to the corresponding Schr¨odinger operator OS by iHt −iHt O(t) = e OS e (4) Thus hψ|O(t)|ψi = Shψ(t)|OS|ψ(t)iS. (5) The Heisenberg picture is favored over the Schr¨odinger picture in the case of relativistic quantum mechanics: we don’t have to say which reference 1Copyright, 2000, D. E. Soper [email protected] 1 frame we use to define t in |ψ(t)iS. For operators, we can deal with local operators like, for instance, the electric field F µν(~x, t). -
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 ........................... -
Physical Masses and the Vacuum Expectation Value
View metadata, citation and similar papers at core.ac.uk brought to you by CORE PHYSICAL MASSES AND THE VACUUM EXPECTATION VALUE provided by CERN Document Server OF THE HIGGS FIELD Hung Cheng Department of Mathematics, Massachusetts Institute of Technology Cambridge, MA 02139, U.S.A. and S.P.Li Institute of Physics, Academia Sinica Nankang, Taip ei, Taiwan, Republic of China Abstract By using the Ward-Takahashi identities in the Landau gauge, we derive exact relations between particle masses and the vacuum exp ectation value of the Higgs eld in the Ab elian gauge eld theory with a Higgs meson. PACS numb ers : 03.70.+k; 11.15-q 1 Despite the pioneering work of 't Ho oft and Veltman on the renormalizability of gauge eld theories with a sp ontaneously broken vacuum symmetry,a numb er of questions remain 2 op en . In particular, the numb er of renormalized parameters in these theories exceeds that of bare parameters. For such theories to b e renormalizable, there must exist relationships among the renormalized parameters involving no in nities. As an example, the bare mass M of the gauge eld in the Ab elian-Higgs theory is related 0 to the bare vacuum exp ectation value v of the Higgs eld by 0 M = g v ; (1) 0 0 0 where g is the bare gauge coupling constant in the theory.We shall show that 0 v u u D (0) T t g (0)v: (2) M = 2 D (M ) T 2 In (2), g (0) is equal to g (k )atk = 0, with the running renormalized gauge coupling constant de ned in (27), and v is the renormalized vacuum exp ectation value of the Higgs eld de ned in (29) b elow. -
On the Time Evolution of Wavefunctions in Quantum Mechanics C
On the Time Evolution of Wavefunctions in Quantum Mechanics C. David Sherrill School of Chemistry and Biochemistry Georgia Institute of Technology October 1999 1 Introduction The purpose of these notes is to help you appreciate the connection between eigenfunctions of the Hamiltonian and classical normal modes, and to help you understand the time propagator. 2 The Classical Coupled Mass Problem Here we will review the results of the coupled mass problem, Example 1.8.6 from Shankar. This is an example from classical physics which nevertheless demonstrates some of the essential features of coupled degrees of freedom in quantum mechanical problems and a general approach for removing such coupling. The problem involves two objects of equal mass, connected to two different walls andalsotoeachotherbysprings.UsingF = ma and Hooke’s Law( F = −kx) for the springs, and denoting the displacements of the two masses as x1 and x2, it is straightforward to deduce equations for the acceleration (second derivative in time,x ¨1 andx ¨2): k k x −2 x x ¨1 = m 1 + m 2 (1) k k x x − 2 x . ¨2 = m 1 m 2 (2) The goal of the problem is to solve these second-order differential equations to obtain the functions x1(t)andx2(t) describing the motion of the two masses at any given time. Since they are second-order differential equations, we need two initial conditions for each variable, i.e., x1(0), x˙ 1(0),x2(0), andx ˙ 2(0). Our two differential equations are clearly coupled,since¨x1 depends not only on x1, but also on x2 (and likewise forx ¨2). -
Discrete-Continuous and Classical-Quantum
Under consideration for publication in Math. Struct. in Comp. Science Discrete-continuous and classical-quantum T. Paul C.N.R.S., D.M.A., Ecole Normale Sup´erieure Received 21 November 2006 Abstract A discussion concerning the opposition between discretness and continuum in quantum mechanics is presented. In particular this duality is shown to be present not only in the early days of the theory, but remains actual, featuring different aspects of discretization. In particular discreteness of quantum mechanics is a key-stone for quantum information and quantum computation. A conclusion involving a concept of completeness linking dis- creteness and continuum is proposed. 1. Discrete-continuous in the old quantum theory Discretness is obviously a fundamental aspect of Quantum Mechanics. When you send white light, that is a continuous spectrum of colors, on a mono-atomic gas, you get back precise line spectra and only Quantum Mechanics can explain this phenomenon. In fact the birth of quantum physics involves more discretization than discretness : in the famous Max Planck’s paper of 1900, and even more explicitly in the 1905 paper by Einstein about the photo-electric effect, what is really done is what a computer does: an integral is replaced by a discrete sum. And the discretization length is given by the Planck’s constant. This idea will be later on applied by Niels Bohr during the years 1910’s to the atomic model. Already one has to notice that during that time another form of discretization had appeared : the atomic model. It is astonishing to notice how long it took to accept the atomic hypothesis (Perrin 1905). -
An S-Matrix for Massless Particles Arxiv:1911.06821V2 [Hep-Th]
An S-Matrix for Massless Particles Holmfridur Hannesdottir and Matthew D. Schwartz Department of Physics, Harvard University, Cambridge, MA 02138, USA Abstract The traditional S-matrix does not exist for theories with massless particles, such as quantum electrodynamics. The difficulty in isolating asymptotic states manifests itself as infrared divergences at each order in perturbation theory. Building on insights from the literature on coherent states and factorization, we construct an S-matrix that is free of singularities order-by-order in perturbation theory. Factorization guarantees that the asymptotic evolution in gauge theories is universal, i.e. independent of the hard process. Although the hard S-matrix element is computed between well-defined few particle Fock states, dressed/coherent states can be seen to form as intermediate states in the calculation of hard S-matrix elements. We present a framework for the perturbative calculation of hard S-matrix elements combining Lorentz-covariant Feyn- man rules for the dressed-state scattering with time-ordered perturbation theory for the asymptotic evolution. With hard cutoffs on the asymptotic Hamiltonian, the cancella- tion of divergences can be seen explicitly. In dimensional regularization, where the hard cutoffs are replaced by a renormalization scale, the contribution from the asymptotic evolution produces scaleless integrals that vanish. A number of illustrative examples are given in QED, QCD, and N = 4 super Yang-Mills theory. arXiv:1911.06821v2 [hep-th] 24 Aug 2020 Contents 1 Introduction1 2 The hard S-matrix6 2.1 SH and dressed states . .9 2.2 Computing observables using SH ........................... 11 2.3 Soft Wilson lines . 14 3 Computing the hard S-matrix 17 3.1 Asymptotic region Feynman rules . -
Exact Solutions to the Interacting Spinor and Scalar Field Equations in the Godel Universe
XJ9700082 E2-96-367 A.Herrera 1, G.N.Shikin2 EXACT SOLUTIONS TO fHE INTERACTING SPINOR AND SCALAR FIELD EQUATIONS IN THE GODEL UNIVERSE 1 E-mail: [email protected] ^Department of Theoretical Physics, Russian Peoples ’ Friendship University, 117198, 6 Mikluho-Maklaya str., Moscow, Russia ^8 as ^; 1996 © (XrbCAHHeHHuft HHCTtrryr McpHMX HCCJicAOB&Htift, fly 6na, 1996 1. INTRODUCTION Recently an increasing interest was expressed to the search of soliton-like solu tions because of the necessity to describe the elementary particles as extended objects [1]. In this work, the interacting spinor and scalar field system is considered in the external gravitational field of the Godel universe in order to study the influence of the global properties of space-time on the interaction of one dimensional fields, in other words, to observe what is the role of gravitation in the interaction of elementary particles. The Godel universe exhibits a number of unusual properties associated with the rotation of the universe [2]. It is ho mogeneous in space and time and is filled with a perfect fluid. The main role of rotation in this universe consists in the avoidance of the cosmological singu larity in the early universe, when the centrifugate forces of rotation dominate over gravitation and the collapse does not occur [3]. The paper is organized as follows: in Sec. 2 the interacting spinor and scalar field system with £jnt = | <ptp<p'PF(Is) in the Godel universe is considered and exact solutions to the corresponding field equations are obtained. In Sec. 3 the properties of the energy density are investigated. -
Advanced Quantum Mechanics
Advanced Quantum Mechanics Rajdeep Sensarma [email protected] Quantum Dynamics Lecture #9 Schrodinger and Heisenberg Picture Time Independent Hamiltonian Schrodinger Picture: A time evolving state in the Hilbert space with time independent operators iHtˆ i@t (t) = Hˆ (t) (t) = e− (0) | i | i | i | i i✏nt Eigenbasis of Hamiltonian: Hˆ n = ✏ n (t) = cn(t) n c (t)=c (0)e− n | i | i n n | i | i n X iHtˆ iHtˆ Operators and Expectation: A(t)= (t) Aˆ (t) = (0) e Aeˆ − (0) = (0) Aˆ(t) (0) h | | i h | | i h | | i Heisenberg Picture: A static initial state and time dependent operators iHtˆ iHtˆ Aˆ(t)=e Aeˆ − Schrodinger Picture Heisenberg Picture (0) (t) (0) (0) | i!| i | i!| i Equivalent description iHtˆ iHtˆ Aˆ Aˆ Aˆ Aˆ(t)=e Aeˆ − ! ! of a quantum system i@ (t) = Hˆ (t) i@ Aˆ(t)=[Aˆ(t), Hˆ ] t| i | i t Time Evolution and Propagator iHtˆ (t) = e− (0) Time Evolution Operator | i | i (x, t)= x (t) = x Uˆ(t) (0) = dx0 x Uˆ(t) x0 x0 (0) h | i h | | i h | | ih | i Z Propagator: Example : Free Particle The propagator satisfies and hence is often called the Green’s function Retarded and Advanced Propagator The following propagators are useful in different contexts Retarded or Causal Propagator: This propagates states forward in time for t > t’ Advanced or Anti-Causal Propagator: This propagates states backward in time for t < t’ Both the retarded and the advanced propagator satisfies the same diff. eqn., but with different boundary conditions Propagators in Frequency Space Energy Eigenbasis |n> The integral is ill defined due to oscillatory nature -
B1. the Generating Functional Z[J]
B1. The generating functional Z[J] The generating functional Z[J] is a key object in quantum field theory - as we shall see it reduces in ordinary quantum mechanics to a limiting form of the 1-particle propagator G(x; x0jJ) when the particle is under thje influence of an external field J(t). However, before beginning, it is useful to look at another closely related object, viz., the generating functional Z¯(J) in ordinary probability theory. Recall that for a simple random variable φ, we can assign a probability distribution P (φ), so that the expectation value of any variable A(φ) that depends on φ is given by Z hAi = dφP (φ)A(φ); (1) where we have assumed the normalization condition Z dφP (φ) = 1: (2) Now let us consider the generating functional Z¯(J) defined by Z Z¯(J) = dφP (φ)eφ: (3) From this definition, it immediately follows that the "n-th moment" of the probability dis- tribution P (φ) is Z @nZ¯(J) g = hφni = dφP (φ)φn = j ; (4) n @J n J=0 and that we can expand Z¯(J) as 1 X J n Z Z¯(J) = dφP (φ)φn n! n=0 1 = g J n; (5) n! n ¯ so that Z(J) acts as a "generator" for the infinite sequence of moments gn of the probability distribution P (φ). For future reference it is also useful to recall how one may also expand the logarithm of Z¯(J); we write, Z¯(J) = exp[W¯ (J)] Z W¯ (J) = ln Z¯(J) = ln dφP (φ)eφ: (6) 1 Now suppose we make a power series expansion of W¯ (J), i.e., 1 X 1 W¯ (J) = C J n; (7) n! n n=0 where the Cn, known as "cumulants", are just @nW¯ (J) C = j : (8) n @J n J=0 The relationship between the cumulants Cn and moments gn is easily found. -
The Liouville Equation in Atmospheric Predictability
The Liouville Equation in Atmospheric Predictability Martin Ehrendorfer Institut fur¨ Meteorologie und Geophysik, Universitat¨ Innsbruck Innrain 52, A–6020 Innsbruck, Austria [email protected] 1 Introduction and Motivation It is widely recognized that weather forecasts made with dynamical models of the atmosphere are in- herently uncertain. Such uncertainty of forecasts produced with numerical weather prediction (NWP) models arises primarily from two sources: namely, from imperfect knowledge of the initial model condi- tions and from imperfections in the model formulation itself. The recognition of the potential importance of accurate initial model conditions and an accurate model formulation dates back to times even prior to operational NWP (Bjerknes 1904; Thompson 1957). In the context of NWP, the importance of these error sources in degrading the quality of forecasts was demonstrated to arise because errors introduced in atmospheric models, are, in general, growing (Lorenz 1982; Lorenz 1963; Lorenz 1993), which at the same time implies that the predictability of the atmosphere is subject to limitations (see, Errico et al. 2002). An example of the amplification of small errors in the initial conditions, or, equivalently, the di- vergence of initially nearby trajectories is given in Fig. 1, for the system discussed by Lorenz (1984). The uncertainty introduced into forecasts through uncertain initial model conditions, and uncertainties in model formulations, has been the subject of numerous studies carried out in parallel to the continuous development of NWP models (e.g., Leith 1974; Epstein 1969; Palmer 2000). In addition to studying the intrinsic predictability of the atmosphere (e.g., Lorenz 1969a; Lorenz 1969b; Thompson 1985a; Thompson 1985b), efforts have been directed at the quantification or predic- tion of forecast uncertainty that arises due to the sources of uncertainty mentioned above (see the review papers by Ehrendorfer 1997 and Palmer 2000, and Ehrendorfer 1999). -
An Introduction to Supersymmetry
An Introduction to Supersymmetry Ulrich Theis Institute for Theoretical Physics, Friedrich-Schiller-University Jena, Max-Wien-Platz 1, D–07743 Jena, Germany [email protected] This is a write-up of a series of five introductory lectures on global supersymmetry in four dimensions given at the 13th “Saalburg” Summer School 2007 in Wolfersdorf, Germany. Contents 1 Why supersymmetry? 1 2 Weyl spinors in D=4 4 3 The supersymmetry algebra 6 4 Supersymmetry multiplets 6 5 Superspace and superfields 9 6 Superspace integration 11 7 Chiral superfields 13 8 Supersymmetric gauge theories 17 9 Supersymmetry breaking 22 10 Perturbative non-renormalization theorems 26 A Sigma matrices 29 1 Why supersymmetry? When the Large Hadron Collider at CERN takes up operations soon, its main objective, besides confirming the existence of the Higgs boson, will be to discover new physics beyond the standard model of the strong and electroweak interactions. It is widely believed that what will be found is a (at energies accessible to the LHC softly broken) supersymmetric extension of the standard model. What makes supersymmetry such an attractive feature that the majority of the theoretical physics community is convinced of its existence? 1 First of all, under plausible assumptions on the properties of relativistic quantum field theories, supersymmetry is the unique extension of the algebra of Poincar´eand internal symmtries of the S-matrix. If new physics is based on such an extension, it must be supersymmetric. Furthermore, the quantum properties of supersymmetric theories are much better under control than in non-supersymmetric ones, thanks to powerful non- renormalization theorems. -
Finally Making Sense of the Double-Slit Experiment
Finally making sense of the double-slit experiment Yakir Aharonova,b,c,1, Eliahu Cohend,1,2, Fabrizio Colomboe, Tomer Landsbergerc,2, Irene Sabadinie, Daniele C. Struppaa,b, and Jeff Tollaksena,b aInstitute for Quantum Studies, Chapman University, Orange, CA 92866; bSchmid College of Science and Technology, Chapman University, Orange, CA 92866; cSchool of Physics and Astronomy, Tel Aviv University, Tel Aviv 6997801, Israel; dH. H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, United Kingdom; and eDipartimento di Matematica, Politecnico di Milano, 9 20133 Milan, Italy Contributed by Yakir Aharonov, March 20, 2017 (sent for review September 26, 2016; reviewed by Pawel Mazur and Neil Turok) Feynman stated that the double-slit experiment “. has in it the modular momentum operator will arise as particularly signifi- heart of quantum mechanics. In reality, it contains the only mys- cant in explaining interference phenomena. This approach along tery” and that “nobody can give you a deeper explanation of with the use of Heisenberg’s unitary equations of motion intro- this phenomenon than I have given; that is, a description of it” duce a notion of dynamical nonlocality. Dynamical nonlocality [Feynman R, Leighton R, Sands M (1965) The Feynman Lectures should be distinguished from the more familiar kinematical non- on Physics]. We rise to the challenge with an alternative to the locality [implicit in entangled states (10) and previously ana- wave function-centered interpretations: instead of a quantum lyzed in the Heisenberg picture by Deutsch and Hayden (11)], wave passing through both slits, we have a localized particle with because dynamical nonlocality has observable effects on prob- nonlocal interactions with the other slit.