Whether an Enormously Large Energy Density of the Quantum Vacuum Is Catastrophic
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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 ........................... -
Kaluza-Klein Gravity, Concentrating on the General Rel- Ativity, Rather Than Particle Physics Side of the Subject
Kaluza-Klein Gravity J. M. Overduin Department of Physics and Astronomy, University of Victoria, P.O. Box 3055, Victoria, British Columbia, Canada, V8W 3P6 and P. S. Wesson Department of Physics, University of Waterloo, Ontario, Canada N2L 3G1 and Gravity Probe-B, Hansen Physics Laboratories, Stanford University, Stanford, California, U.S.A. 94305 Abstract We review higher-dimensional unified theories from the general relativity, rather than the particle physics side. Three distinct approaches to the subject are identi- fied and contrasted: compactified, projective and noncompactified. We discuss the cosmological and astrophysical implications of extra dimensions, and conclude that none of the three approaches can be ruled out on observational grounds at the present time. arXiv:gr-qc/9805018v1 7 May 1998 Preprint submitted to Elsevier Preprint 3 February 2008 1 Introduction Kaluza’s [1] achievement was to show that five-dimensional general relativity contains both Einstein’s four-dimensional theory of gravity and Maxwell’s the- ory of electromagnetism. He however imposed a somewhat artificial restriction (the cylinder condition) on the coordinates, essentially barring the fifth one a priori from making a direct appearance in the laws of physics. Klein’s [2] con- tribution was to make this restriction less artificial by suggesting a plausible physical basis for it in compactification of the fifth dimension. This idea was enthusiastically received by unified-field theorists, and when the time came to include the strong and weak forces by extending Kaluza’s mechanism to higher dimensions, it was assumed that these too would be compact. This line of thinking has led through eleven-dimensional supergravity theories in the 1980s to the current favorite contenders for a possible “theory of everything,” ten-dimensional superstrings. -
Conformal Symmetry in Field Theory and in Quantum Gravity
universe Review Conformal Symmetry in Field Theory and in Quantum Gravity Lesław Rachwał Instituto de Física, Universidade de Brasília, Brasília DF 70910-900, Brazil; [email protected] Received: 29 August 2018; Accepted: 9 November 2018; Published: 15 November 2018 Abstract: Conformal symmetry always played an important role in field theory (both quantum and classical) and in gravity. We present construction of quantum conformal gravity and discuss its features regarding scattering amplitudes and quantum effective action. First, the long and complicated story of UV-divergences is recalled. With the development of UV-finite higher derivative (or non-local) gravitational theory, all problems with infinities and spacetime singularities might be completely solved. Moreover, the non-local quantum conformal theory reveals itself to be ghost-free, so the unitarity of the theory should be safe. After the construction of UV-finite theory, we focused on making it manifestly conformally invariant using the dilaton trick. We also argue that in this class of theories conformal anomaly can be taken to vanish by fine-tuning the couplings. As applications of this theory, the constraints of the conformal symmetry on the form of the effective action and on the scattering amplitudes are shown. We also remark about the preservation of the unitarity bound for scattering. Finally, the old model of conformal supergravity by Fradkin and Tseytlin is briefly presented. Keywords: quantum gravity; conformal gravity; quantum field theory; non-local gravity; super- renormalizable gravity; UV-finite gravity; conformal anomaly; scattering amplitudes; conformal symmetry; conformal supergravity 1. Introduction From the beginning of research on theories enjoying invariance under local spacetime-dependent transformations, conformal symmetry played a pivotal role—first introduced by Weyl related changes of meters to measure distances (and also due to relativity changes of periods of clocks to measure time intervals). -
Carbon – Science and Technology
© Applied Science Innovations Pvt. Ltd., India Carbon – Sci. Tech. 1 (2010) 139 - 143 Carbon – Science and Technology ASI ISSN 0974 – 0546 http://www.applied-science-innovations.com ARTICLE Received :29/03/2010, Accepted :02/09/2010 ----------------------------------------------------------------------------------------------------------------------------- Ablation morphologies of different types of carbon in carbon/carbon composites Jian Yin, Hongbo Zhang, Xiang Xiong, Jinlv Zuo State Key Laboratory of Powder Metallurgy, Central South University, Lushan South Road, Changsha, China. --------------------------------------------------------------------------------------------------------------------------------------- 1. Introduction : Carbon/carbon (C/C) composites the ablation morphology and formation mechanism of combine good mechanical properties and designable each types of carbon. capabilities of composites and excellent ultrahigh temperature properties of carbon materials. They have In this study, ablation morphologies of resin-based low densities, high specific strength, good thermal carbon, carbon fibers, pyrolytic carbon with smooth stability, high thermal conductivity, low thermal laminar structure and rough laminar structure has been expansion coefficient and excellent ablation properties investigated in detail and their formation mechanisms [1]. As C/C composites show excellent characteristics were discussed. in both structural design and functional application, 2 Experimental : they have become one of the most competitive high temperature materials widely used in aviation and 2.1 Preparation of C/C composites : Bulk needled polyacrylonitrile (PAN) carbon fiber felts spacecraft industry [2 - 4]. In particular, they are were used as reinforcements. Three kinds of C/C considered to be the most suitable materials for solid composites, labeled as sample A, B and C, were rocket motor nozzles. prepared. Sample A is a C/C composite mainly with C/C composites are composed of carbon fibers and smooth laminar pyrolytic carbon, sample B is a C/C carbon matrices. -
Equilibrium Thermodynamics
Equilibrium Thermodynamics Instructor: - Clare Yu (e-mail [email protected], phone: 824-6216) - office hours: Wed from 2:30 pm to 3:30 pm in Rowland Hall 210E Textbooks: - primary: Herbert Callen “Thermodynamics and an Introduction to Thermostatistics” - secondary: Frederick Reif “Statistical and Thermal Physics” - secondary: Kittel and Kroemer “Thermal Physics” - secondary: Enrico Fermi “Thermodynamics” Grading: - weekly homework: 25% - discussion problems: 10% - midterm exam: 30% - final exam: 35% Equilibrium Thermodynamics Material Covered: Equilibrium thermodynamics, phase transitions, critical phenomena (~ 10 first chapters of Callen’s textbook) Homework: - Homework assignments posted on course website Exams: - One midterm, 80 minutes, Tuesday, May 8 - Final, 2 hours, Tuesday, June 12, 10:30 am - 12:20 pm - All exams are in this room 210M RH Course website is at http://eiffel.ps.uci.edu/cyu/p115B/class.html The Subject of Thermodynamics Thermodynamics describes average properties of macroscopic matter in equilibrium. - Macroscopic matter: large objects that consist of many atoms and molecules. - Average properties: properties (such as volume, pressure, temperature etc) that do not depend on the detailed positions and velocities of atoms and molecules of macroscopic matter. Such quantities are called thermodynamic coordinates, variables or parameters. - Equilibrium: state of a macroscopic system in which all average properties do not change with time. (System is not driven by external driving force.) Why Study Thermodynamics ? - Thermodynamics predicts that the average macroscopic properties of a system in equilibrium are not independent from each other. Therefore, if we measure a subset of these properties, we can calculate the rest of them using thermodynamic relations. - Thermodynamics not only gives the exact description of the state of equilibrium but also provides an approximate description (to a very high degree of precision!) of relatively slow processes. -
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. -
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 Stability of Classical Orbits of the Hydrogen Ground State in Stochastic Electrodynamics
entropy Article On the Stability of Classical Orbits of the Hydrogen Ground State in Stochastic Electrodynamics Theodorus M. Nieuwenhuizen 1,2 1 Institute for Theoretical Physics, P.O. Box 94485, 1098 XH Amsterdam, The Netherlands; [email protected]; Tel.: +31-20-525-6332 2 International Institute of Physics, UFRG, Av. O. Gomes de Lima, 1722, 59078-400 Natal-RN, Brazil Academic Editors: Gregg Jaeger and Andrei Khrennikov Received: 19 February 2016; Accepted: 31 March 2016; Published: 13 April 2016 Abstract: De la Peña 1980 and Puthoff 1987 show that circular orbits in the hydrogen problem of Stochastic Electrodynamics connect to a stable situation, where the electron neither collapses onto the nucleus nor gets expelled from the atom. Although the Cole-Zou 2003 simulations support the stability, our recent numerics always lead to self-ionisation. Here the de la Peña-Puthoff argument is extended to elliptic orbits. For very eccentric orbits with energy close to zero and angular momentum below some not-small value, there is on the average a net gain in energy for each revolution, which explains the self-ionisation. Next, an 1/r2 potential is added, which could stem from a dipolar deformation of the nuclear charge by the electron at its moving position. This shape retains the analytical solvability. When it is enough repulsive, the ground state of this modified hydrogen problem is predicted to be stable. The same conclusions hold for positronium. Keywords: Stochastic Electrodynamics; hydrogen ground state; stability criterion PACS: 11.10; 05.20; 05.30; 03.65 1. Introduction Stochastic Electrodynamics (SED) is a subquantum theory that considers the quantum vacuum as a true physical vacuum with its zero-point modes being physical electromagnetic modes (see [1,2]). -
Identifying the Elastic Isotropy of Architectured Materials Based On
Identifying the elastic isotropy of architectured materials based on deep learning method Anran Wei a, Jie Xiong b, Weidong Yang c, Fenglin Guo a, d, * a Department of Engineering Mechanics, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China b Department of Mechanical Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China c School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China d State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, China * Corresponding author, E-mail: [email protected] Abstract: With the achievement on the additive manufacturing, the mechanical properties of architectured materials can be precisely designed by tailoring microstructures. As one of the primary design objectives, the elastic isotropy is of great significance for many engineering applications. However, the prevailing experimental and numerical methods are normally too costly and time-consuming to determine the elastic isotropy of architectured materials with tens of thousands of possible microstructures in design space. The quick mechanical characterization is thus desired for the advanced design of architectured materials. Here, a deep learning-based approach is developed as a portable and efficient tool to identify the elastic isotropy of architectured materials directly from the images of their representative microstructures with arbitrary component distributions. The measure of elastic isotropy for architectured materials is derived firstly in this paper to construct a database with associated images of microstructures. Then a convolutional neural network is trained with the database. It is found that the convolutional neural network shows good performance on the isotropy identification. -
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. -
Gravity, Inertia, and Quantum Vacuum Zero Point Fields
Foundations of Physics, Vol.31,No. 5, 2001 Gravity, Inertia, and Quantum Vacuum Zero Point Fields James F. Woodward1 Received July 27, 2000 Over the past several years Haisch, Rueda, and others have made the claim that the origin of inertial reaction forces can be explained as the interaction of electri- cally charged elementary particles with the vacuum electromagnetic zero-point field expected on the basis of quantum field theory. After pointing out that this claim, in light of the fact that the inertial masses of the hadrons reside in the electrically chargeless, photon-like gluons that bind their constituent quarks, is untenable, the question of the role of quantum zero-point fields generally in the origin of inertia is explored. It is shown that, although non-gravitational zero-point fields might be the cause of the gravitational properties of normal matter, the action of non-gravitational zero-point fields cannot be the cause of inertial reac- tion forces. The gravitational origin of inertial reaction forces is then briefly revisited. Recent claims critical of the gravitational origin of inertial reaction forces by Haisch and his collaborators are then shown to be without merit. 1. INTRODUCTION Several years ago Haisch, Rueda, and Puthoff (1) (hereafter, HRP) pub- lished a lengthy paper in which they claimed that a substantial part, indeed perhaps all of normal inertial reaction forces could be understood as the action of the electromagnetic zero-point field (EZPF), expected on the basis of quantum field theory, on electric charges of normal matter. In several subsequent papers Haisch and Rueda particularly have pressed this claim, making various modifications to the fundamental argument to try to deflect several criticisms.