# Quantum Mechanics Propagator

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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.
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• 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 ...........................
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• Lectures on Quantum Field Theory
QUANTUM FIELD THEORY Notes taken from a course of R. E. Borcherds, Fall 2001, Berkeley Richard E. Borcherds, Mathematics department, Evans Hall, UC Berkeley, CA 94720, U.S.A. e–mail: [email protected] home page: www.math.berkeley.edu/~reb Alex Barnard, Mathematics department, Evans Hall, UC Berkeley, CA 94720, U.S.A. e–mail: [email protected] home page: www.math.berkeley.edu/~barnard Contents 1 Introduction 4 1.1 Life Cycle of a Theoretical Physicist . ....... 4 1.2 Historical Survey of the Standard Model . ....... 5 1.3 SomeProblemswithNeutrinos . ... 9 1.4 Elementary Particles in the Standard Model . ........ 10 2 Lagrangians 11 arXiv:math-ph/0204014v1 8 Apr 2002 2.1 WhatisaLagrangian?.............................. 11 2.2 Examples ...................................... 11 2.3 The General Euler–Lagrange Equation . ...... 13 3 Symmetries and Currents 14 3.1 ObviousSymmetries ............................... 14 3.2 Not–So–ObviousSymmetries . 16 3.3 TheElectromagneticField. 17 1 3.4 Converting Classical Field Theory to Homological Algebra........... 19 4 Feynman Path Integrals 21 4.1 Finite Dimensional Integrals . ...... 21 4.2 TheFreeFieldCase ................................ 22 4.3 Free Field Green’s Functions . 23 4.4 TheNon-FreeCase................................. 24 5 0-Dimensional QFT 25 5.1 BorelSummation.................................. 28 5.2 OtherGraphSums................................. 28 5.3 TheClassicalField............................... 29 5.4 TheEﬀectiveAction ............................... 31 6 Distributions and Propagators 35 6.1 EuclideanPropagators . 36 6.2 LorentzianPropagators . 38 6.3 Wavefronts and Distribution Products . ....... 40 7 Higher Dimensional QFT 44 7.1 AnExample..................................... 44 7.2 Renormalisation Prescriptions . ....... 45 7.3 FiniteRenormalisations . 47 7.4 A Group Structure on Finite Renormalisations . ........ 50 7.5 More Conditions on Renormalisation Prescriptions .
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• 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.
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• 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.
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• 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.
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• 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.
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• 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.
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• 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.
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• On the Definition of the Renormalization Constants in Quantum Electrodynamics
On the Definition of the Renormalization Constants in Quantum Electrodynamics Autor(en): Källén, Gunnar Objekttyp: Article Zeitschrift: Helvetica Physica Acta Band(Jahr): 25(1952) Heft IV Erstellt am: Mar 18, 2014 Persistenter Link: http://dx.doi.org/10.5169/seals-112316 Nutzungsbedingungen Mit dem Zugriff auf den vorliegenden Inhalt gelten die Nutzungsbedingungen als akzeptiert. Die angebotenen Dokumente stehen für nicht-kommerzielle Zwecke in Lehre, Forschung und für die private Nutzung frei zur Verfügung. Einzelne Dateien oder Ausdrucke aus diesem Angebot können zusammen mit diesen Nutzungsbedingungen und unter deren Einhaltung weitergegeben werden. Die Speicherung von Teilen des elektronischen Angebots auf anderen Servern ist nur mit vorheriger schriftlicher Genehmigung möglich. Die Rechte für diese und andere Nutzungsarten der Inhalte liegen beim Herausgeber bzw. beim Verlag. Ein Dienst der ETH-Bibliothek Rämistrasse 101, 8092 Zürich, Schweiz [email protected] http://retro.seals.ch On the Definition of the Renormalization Constants in Quantum Electrodynamics by Gunnar Källen.*) Swiss Federal Institute of Technology, Zürich. (14.11.1952.) Summary. A formulation of quantum electrodynamics in terms of the renor- malized Heisenberg operators and the experimental mass and charge of the electron is given. The renormalization constants are implicitly defined and ex¬ pressed as integrals over finite functions in momentum space. No discussion of the convergence of these integrals or of the existence of rigorous solutions is given. Introduction. The renormalization method in quantum electrodynamics has been investigated by many authors, and it has been proved by Dyson1) that every term in a formal expansion in powers of the coupling constant of various expressions is a finite quantity.
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• THE DEVELOPMENT of the SPACE-TIME VIEW of QUANTUM ELECTRODYNAMICS∗ by Richard P
THE DEVELOPMENT OF THE SPACE-TIME VIEW OF QUANTUM ELECTRODYNAMICS∗ by Richard P. Feynman California Institute of Technology, Pasadena, California Nobel Lecture, December 11, 1965. We have a habit in writing articles published in scientiﬁc journals to make the work as ﬁnished as possible, to cover all the tracks, to not worry about the blind alleys or to describe how you had the wrong idea ﬁrst, and so on. So there isn’t any place to publish, in a digniﬁed manner, what you actually did in order to get to do the work, although, there has been in these days, some interest in this kind of thing. Since winning the prize is a personal thing, I thought I could be excused in this particular situation, if I were to talk personally about my relationship to quantum electrodynamics, rather than to discuss the subject itself in a reﬁned and ﬁnished fashion. Furthermore, since there are three people who have won the prize in physics, if they are all going to be talking about quantum electrodynamics itself, one might become bored with the subject. So, what I would like to tell you about today are the sequence of events, really the sequence of ideas, which occurred, and by which I ﬁnally came out the other end with an unsolved problem for which I ultimately received a prize. I realize that a truly scientiﬁc paper would be of greater value, but such a paper I could publish in regular journals. So, I shall use this Nobel Lecture as an opportunity to do something of less value, but which I cannot do elsewhere.
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• Finite Quantum Field Theory and Renormalization Group
Finite Quantum Field Theory and Renormalization Group M. A. Greena and J. W. Moﬀata;b aPerimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada bDepartment of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada September 15, 2021 Abstract Renormalization group methods are applied to a scalar ﬁeld within a ﬁnite, nonlocal quantum ﬁeld theory formulated perturbatively in Euclidean momentum space. It is demonstrated that the triviality problem in scalar ﬁeld theory, the Higgs boson mass hierarchy problem and the stability of the vacuum do not arise as issues in the theory. The scalar Higgs ﬁeld has no Landau pole. 1 Introduction An alternative version of the Standard Model (SM), constructed using an ultraviolet ﬁnite quantum ﬁeld theory with nonlocal ﬁeld operators, was investigated in previous work [1]. In place of Dirac delta-functions, δ(x), the theory uses distributions (x) based on ﬁnite width Gaussians. The Poincar´eand gauge invariant model adapts perturbative quantumE ﬁeld theory (QFT), with a ﬁnite renormalization, to yield ﬁnite quantum loops. For the weak interactions, SU(2) U(1) is treated as an ab initio broken symmetry group with non- zero masses for the W and Z intermediate× vector bosons and for left and right quarks and leptons. The model guarantees the stability of the vacuum. Two energy scales, ΛM and ΛH , were introduced; the rate of asymptotic vanishing of all coupling strengths at vertices not involving the Higgs boson is controlled by ΛM , while ΛH controls the vanishing of couplings to the Higgs. Experimental tests of the model, using future linear or circular colliders, were proposed.
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