The Casimir Effect: Some Aspects
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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 ........................... -
Pauli Crystals–Interplay of Symmetries
S S symmetry Article Pauli Crystals–Interplay of Symmetries Mariusz Gajda , Jan Mostowski , Maciej Pylak , Tomasz Sowi ´nski ∗ and Magdalena Załuska-Kotur Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02668 Warsaw, Poland; [email protected] (M.G.); [email protected] (J.M.); [email protected] (M.P.); [email protected] (M.Z.-K.) * Correspondence: [email protected] Received: 20 October 2020; Accepted: 13 November 2020; Published: 16 November 2020 Abstract: Recently observed Pauli crystals are structures formed by trapped ultracold atoms with the Fermi statistics. Interactions between these atoms are switched off, so their relative positions are determined by joined action of the trapping potential and the Pauli exclusion principle. Numerical modeling is used in this paper to find the Pauli crystals in a two-dimensional isotropic harmonic trap, three-dimensional harmonic trap, and a two-dimensional square well trap. The Pauli crystals do not have the symmetry of the trap—the symmetry is broken by the measurement of positions and, in many cases, by the quantum state of atoms in the trap. Furthermore, the Pauli crystals are compared with the Coulomb crystals formed by electrically charged trapped particles. The structure of the Pauli crystals differs from that of the Coulomb crystals, this provides evidence that the exclusion principle cannot be replaced by a two-body repulsive interaction but rather has to be considered to be a specifically quantum mechanism leading to many-particle correlations. Keywords: pauli exclusion; ultracold fermions; quantum correlations 1. Introduction Recent advances of experimental capabilities reached such precision that simultaneous detection of many ultracold atoms in a trap is possible [1,2]. -
ABSTRACT Investigation Into Compactified Dimensions: Casimir
ABSTRACT Investigation into Compactified Dimensions: Casimir Energies and Phenomenological Aspects Richard K. Obousy, Ph.D. Chairperson: Gerald B. Cleaver, Ph.D. The primary focus of this dissertation is the study of the Casimir effect and the possibility that this phenomenon may serve as a mechanism to mediate higher dimensional stability, and also as a possible mechanism for creating a small but non- zero vacuum energy density. In chapter one we review the nature of the quantum vacuum and discuss the different contributions to the vacuum energy density arising from different sectors of the standard model. Next, in chapter two, we discuss cosmology and the introduction of the cosmological constant into Einstein's field equations. In chapter three we explore the Casimir effect and study a number of mathematical techniques used to obtain a finite physical result for the Casimir energy. We also review the experiments that have verified the Casimir force. In chapter four we discuss the introduction of extra dimensions into physics. We begin by reviewing Kaluza Klein theory, and then discuss three popular higher dimensional models: bosonic string theory, large extra dimensions and warped extra dimensions. Chapter five is devoted to an original derivation of the Casimir energy we derived for the scenario of a higher dimensional vector field coupled to a scalar field in the fifth dimension. In chapter six we explore a range of vacuum scenarios and discuss research we have performed regarding moduli stability. Chapter seven explores a novel approach to spacecraft propulsion we have proposed based on the idea of manipulating the extra dimensions of string/M theory. -
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. -
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. -
Coherent State Path Integral for the Harmonic Oscillator 11
COHERENT STATE PATH INTEGRAL FOR THE HARMONIC OSCILLATOR AND A SPIN PARTICLE IN A CONSTANT MAGNETIC FLELD By MARIO BERGERON B. Sc. (Physique) Universite Laval, 1987 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 1989 © MARIO BERGERON, 1989 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis . for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada DE-6 (2/88) Abstract The definition and formulas for the harmonic oscillator coherent states and spin coherent states are reviewed in detail. The path integral formalism is also reviewed with its relation and the partition function of a sytem is also reviewed. The harmonic oscillator coherent state path integral is evaluated exactly at the discrete level, and its relation with various regularizations is established. The use of harmonic oscillator coherent states and spin coherent states for the computation of the path integral for a particle of spin s put in a magnetic field is caried out in several ways, and a careful analysis of infinitesimal terms (in 1/N where TV is the number of time slices) is done explicitly. -
The Mechanics of the Fermionic and Bosonic Fields: an Introduction to the Standard Model and Particle Physics
The Mechanics of the Fermionic and Bosonic Fields: An Introduction to the Standard Model and Particle Physics Evan McCarthy Phys. 460: Seminar in Physics, Spring 2014 Aug. 27,! 2014 1.Introduction 2.The Standard Model of Particle Physics 2.1.The Standard Model Lagrangian 2.2.Gauge Invariance 3.Mechanics of the Fermionic Field 3.1.Fermi-Dirac Statistics 3.2.Fermion Spinor Field 4.Mechanics of the Bosonic Field 4.1.Spin-Statistics Theorem 4.2.Bose Einstein Statistics !5.Conclusion ! 1. Introduction While Quantum Field Theory (QFT) is a remarkably successful tool of quantum particle physics, it is not used as a strictly predictive model. Rather, it is used as a framework within which predictive models - such as the Standard Model of particle physics (SM) - may operate. The overarching success of QFT lends it the ability to mathematically unify three of the four forces of nature, namely, the strong and weak nuclear forces, and electromagnetism. Recently substantiated further by the prediction and discovery of the Higgs boson, the SM has proven to be an extraordinarily proficient predictive model for all the subatomic particles and forces. The question remains, what is to be done with gravity - the fourth force of nature? Within the framework of QFT theoreticians have predicted the existence of yet another boson called the graviton. For this reason QFT has a very attractive allure, despite its limitations. According to !1 QFT the gravitational force is attributed to the interaction between two gravitons, however when applying the equations of General Relativity (GR) the force between two gravitons becomes infinite! Results like this are nonsensical and must be resolved for the theory to stand. -
Charged Quantum Fields in Ads2
Charged Quantum Fields in AdS2 Dionysios Anninos,1 Diego M. Hofman,2 Jorrit Kruthoff,2 1 Department of Mathematics, King’s College London, the Strand, London WC2R 2LS, UK 2 Institute for Theoretical Physics and ∆ Institute for Theoretical Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands Abstract We consider quantum field theory near the horizon of an extreme Kerr black hole. In this limit, the dynamics is well approximated by a tower of electrically charged fields propagating in an SL(2, R) invariant AdS2 geometry endowed with a constant, symmetry preserving back- ground electric field. At large charge the fields oscillate near the AdS2 boundary and no longer admit a standard Dirichlet treatment. From the Kerr black hole perspective, this phenomenon is related to the presence of an ergosphere. We discuss a definition for the quantum field theory whereby we ‘UV’ complete AdS2 by appending an asymptotically two dimensional Minkowski region. This allows the construction of a novel observable for the flux-carrying modes that resembles the standard flat space S-matrix. We relate various features displayed by the highly charged particles to the principal series representations of SL(2, R). These representations are unitary and also appear for massive quantum fields in dS2. Both fermionic and bosonic fields are studied. We find that the free charged massless fermion is exactly solvable for general background, providing an interesting arena for the problem at hand. arXiv:1906.00924v2 [hep-th] 7 Oct 2019 Contents 1 Introduction 2 2 Geometry near the extreme Kerr horizon 4 2.1 Unitary representations of SL(2, R).......................... -
Coherent and Squeezed States on Physical Basis
PRAMANA © Printed in India Vol. 48, No. 3, __ journal of March 1997 physics pp. 787-797 Coherent and squeezed states on physical basis R R PURI Theoretical Physics Division, Central Complex, Bhabha Atomic Research Centre, Bombay 400 085, India MS received 6 April 1996; revised 24 December 1996 Abstract. A definition of coherent states is proposed as the minimum uncertainty states with equal variance in two hermitian non-commuting generators of the Lie algebra of the hamil- tonian. That approach classifies the coherent states into distinct classes. The coherent states of a harmonic oscillator, according to the proposed approach, are shown to fall in two classes. One is the familiar class of Glauber states whereas the other is a new class. The coherent states of spin constitute only one class. The squeezed states are similarly defined on the physical basis as the states that give better precision than the coherent states in a process of measurement of a force coupled to the given system. The condition of squeezing based on that criterion is derived for a system of spins. Keywords. Coherent state; squeezed state; Lie algebra; SU(m, n). PACS Nos 03.65; 02-90 1. Introduction The hamiltonian of a quantum system is generally a linear combination of a set of operators closed under the operation of commutation i.e. they constitute a Lie algebra. Since their introduction for a hamiltonian linear in harmonic oscillator (HO) oper- ators, the notion of coherent states [1] has been extended to other hamiltonians (see [2-9] and references therein). Physically, the HO coherent states are the minimum uncertainty states of the two quadratures of the bosonic operator with equal variance in the two. -
What Is the Dirac Equation?
What is the Dirac equation? M. Burak Erdo˘gan ∗ William R. Green y Ebru Toprak z July 15, 2021 at all times, and hence the model needs to be first or- der in time, [Tha92]. In addition, it should conserve In the early part of the 20th century huge advances the L2 norm of solutions. Dirac combined the quan- were made in theoretical physics that have led to tum mechanical notions of energy and momentum vast mathematical developments and exciting open operators E = i~@t, p = −i~rx with the relativis- 2 2 2 problems. Einstein's development of relativistic the- tic Pythagorean energy relation E = (cp) + (E0) 2 ory in the first decade was followed by Schr¨odinger's where E0 = mc is the rest energy. quantum mechanical theory in 1925. Einstein's the- Inserting the energy and momentum operators into ory could be used to describe bodies moving at great the energy relation leads to a Klein{Gordon equation speeds, while Schr¨odinger'stheory described the evo- 2 2 2 4 lution of very small particles. Both models break −~ tt = (−~ ∆x + m c ) : down when attempting to describe the evolution of The Klein{Gordon equation is second order, and does small particles moving at great speeds. In 1927, Paul not have an L2-conservation law. To remedy these Dirac sought to reconcile these theories and intro- shortcomings, Dirac sought to develop an operator1 duced the Dirac equation to describe relativitistic quantum mechanics. 2 Dm = −ic~α1@x1 − ic~α2@x2 − ic~α3@x3 + mc β Dirac's formulation of a hyperbolic system of par- tial differential equations has provided fundamental which could formally act as a square root of the Klein- 2 2 2 models and insights in a variety of fields from parti- Gordon operator, that is, satisfy Dm = −c ~ ∆ + 2 4 cle physics and quantum field theory to more recent m c . -
1 Coherent States and Path Integral Quantiza- Tion. 1.1 Coherent States Let Q and P Be the Coordinate and Momentum Operators
1 Coherent states and path integral quantiza- tion. 1.1 Coherent States Let q and p be the coordinate and momentum operators. They satisfy the Heisenberg algebra, [q,p] = i~. Let us introduce the creation and annihilation operators a† and a, by their standard relations ~ q = a† + a (1) r2mω m~ω p = i a† a (2) r 2 − Let us consider a Hilbert space spanned by a complete set of harmonic os- cillator states n , with n =0,..., . Leta ˆ† anda ˆ be a pair of creation and annihilation operators{| i} acting on that∞ Hilbert space, and satisfying the commu- tation relations a,ˆ aˆ† =1 , aˆ†, aˆ† =0 , [ˆa, aˆ] = 0 (3) These operators generate the harmonic oscillators states n in the usual way, {| i} 1 n n = aˆ† 0 (4) | i √n! | i aˆ 0 = 0 (5) | i where 0 is the vacuum state of the oscillator. Let| usi denote by z the coherent state | i † z = ezaˆ 0 (6) | i | i z = 0 ez¯aˆ (7) h | h | where z is an arbitrary complex number andz ¯ is the complex conjugate. The coherent state z has the defining property of being a wave packet with opti- mal spread, i.e.,| ithe Heisenberg uncertainty inequality is an equality for these coherent states. How doesa ˆ act on the coherent state z ? | i ∞ n z n aˆ z = aˆ aˆ† 0 (8) | i n! | i n=0 X Since n n−1 a,ˆ aˆ† = n aˆ† (9) we get h i ∞ n z n n aˆ z = a,ˆ aˆ† + aˆ† aˆ 0 (10) | i n! | i n=0 X h i 1 Thus, we find ∞ n z n−1 aˆ z = n aˆ† 0 z z (11) | i n! | i≡ | i n=0 X Therefore z is a right eigenvector ofa ˆ and z is the (right) eigenvalue.