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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. -
2007-2008 Physics at Brown Newsletter
Physics at Brown NEWS FOR ALUM N I an D FRIE N DS 2007 ISSUE GREETINGS FROM THE CHAIR - SP RING 2008 elcome to another issue of the Brown Physics newsletter. the rank of Associate Professor with tenure. We also report on WI wrote three years ago, during my first term as the some notable faculty achievements for the past year. department chair--with a committed faculty, dedicated staff, enthusiastic students, supportive administration, and engaged e continue the tradition of highlighting the research of alumni and friends--that the future of physics at Brown looked Wour 2007 Galkin Foundation Fellow on page 2. Also bright. Many things have taken place since then. Here we the effort in enriching our physics instruction continues. Three highlight some of the activities of the past year. new courses are offered this year and proposals for three new physics concentrations are under way. Other noteworthy 007 marked the 50th anniversary of the BCS Theory activities include WiSE, Poster Session, UTRA Awards, 2of Superconductivity. We honored Prof. Leon Resource Center, etc. In addition, community outreach Cooper with a two-day symposium on April remains a priority for the Department with a weekly 12-13. A brief description of this event is open house at Ladd and a greatly expanded five- provided on page 3. year NSF supported GK-12 program. e also report on the establishment hanks to a generous gift from his family, an Wof the Institute for Molecular and TAnthony Houghton Prize will be awarded Nanoscale Innovation, which represents an annually for the best theoretical thesis. -
Ettore Majorana: Genius and Mystery
«ETTORE MAJORANA» FOUNDATION AND CENTRE FOR SCIENTIFIC CULTURE TO PAY A PERMANENT TRIBUTE TO GALILEO GALILEI, FOUNDER OF MODERN SCIENCE AND TO ENRICO FERMI, THE "ITALIAN NAVIGATOR", FATHER OF THE WEAK FORCES ETTORE MAJORANA CENTENARY ETTORE MAJORANA: GENIUS AND MYSTERY Antonino Zichichi ETTORE MAJORANA: GENIUS AND MYSTERY Antonino Zichichi ABSTRACT The geniality of Ettore Majorana is discussed in the framework of the crucial problems being investigated at the time of his activity. These problems are projected to our present days, where the number of space-time dimensions is no longer four and where the unification of the fundamental forces needs the Majorana particle: neutral, with spin ½ and identical to its antiparticle. The mystery of the way Majorana disappeared is restricted to few testimonies, while his geniality is open to all eminent physicists of the XXth century, who had the privilege of knowing him, directly or indirectly. 3 44444444444444444444444444444444444 ETTORE MAJORANA: GENIUS AND MYSTERY Antonino Zichichi CONTENTS 1 LEONARDO SCIASCIA’S IDEA 5 2 ENRICO FERMI: FEW OTHERS IN THE WORLD COULD MATCH MAJORANA’S DEEP UNDERSTANDING OF THE PHYSICS OF THE TIME 7 3 RECOLLECTIONS BY ROBERT OPPENHEIMER 19 4 THE DISCOVERY OF THE NEUTRON – RECOLLECTIONS BY EMILIO SEGRÉ AND GIANCARLO WICK 21 5 THE MAJORANA ‘NEUTRINOS’ – RECOLLECTIONS BY BRUNO PONTECORVO – THE MAJORANA DISCOVERY ON THE DIRAC γ- MATRICES 23 6 THE FIRST COURSE OF THE SUBNUCLEAR PHYSICS SCHOOL (1963): JOHN BELL ON THE DIRAC AND MAJORANA NEUTRINOS 45 7 THE FIRST STEP TO RELATIVISTICALLY DESCRIBE PARTICLES WITH ARBITRARY SPIN 47 8 THE CENTENNIAL OF THE BIRTH OF A GENIUS – A HOMAGE BY THE INTERNATIONAL SCIENTIFIC COMMUNITY 53 REFERENCES 61 4 44444444444444444444444444444444444 Ettore Majorana’s photograph taken from his university card dated 3rd November 1923. -
J. Robert Schrieffer Strange Quantum Numbers in Condensed Matter
Wednesday, May 1, 2002 3:00 pm APS Auditorium, Building 402, Argonne National Laboratory APS Colloquium home J. Robert Schrieffer Nobel Laureate National High Magnetic Field Laboratory Florida State University, Tallahassee [email protected] http://www.physics.fsu.edu/research/NHMFL.htm Strange Quantum Numbers in Condensed Matter Physics The origin of peculiar quantum numbers in condensed matter physics will be reviewed. The source of spin-charge separation and fractional charge in conducting polymers has to do with solitons in broken symmetry states. For superconductors with an energy gap, which is odd under time reversal, reverse spin-orbital angular momentum pairing occurs. In the fractional quantum Hall effect, quasi particles of fractional charge occur. In superfluid helium 3, a one-way branch of excitations exists if a domain wall occurs in the system. Many of these phenomena occur due to vacuum flow of particles without crossing the excitation of the energy gap. John Robert Schrieffer received his bachelor's degree from Massachusetts Institute of Technology in 1953 and his Ph.D. from the University of Illinois in 1957. In addition, he holds honorary Doctor of Science degrees from universities in Germany, Switzerland, and Israel, and from the University of Pennsylvania, the University of Cincinnati, and the University of Alabama. Since 1992, Dr. Schrieffer has been a professor of Physics at Florida State University and the University of Florida and the Chief Scientist of the National High Magnetic Field Laboratory. He also holds the FSU Eminent Scholar Chair in Physics. Before moving to Florida in 1991, he served as director for the Institute for Theoretical Physics from 1984-1989 and was the Chancellor's Professor at the University of California in Santa Barbara from 1984-1991. -
Basics of Thermal Field Theory
September 2021 Basics of Thermal Field Theory A Tutorial on Perturbative Computations 1 Mikko Lainea and Aleksi Vuorinenb aAEC, Institute for Theoretical Physics, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland bDepartment of Physics, University of Helsinki, P.O. Box 64, FI-00014 University of Helsinki, Finland Abstract These lecture notes, suitable for a two-semester introductory course or self-study, offer an elemen- tary and self-contained exposition of the basic tools and concepts that are encountered in practical computations in perturbative thermal field theory. Selected applications to heavy ion collision physics and cosmology are outlined in the last chapter. 1An earlier version of these notes is available as an ebook (Springer Lecture Notes in Physics 925) at dx.doi.org/10.1007/978-3-319-31933-9; a citable eprint can be found at arxiv.org/abs/1701.01554; the very latest version is kept up to date at www.laine.itp.unibe.ch/basics.pdf. Contents Foreword ........................................... i Notation............................................ ii Generaloutline....................................... iii 1 Quantummechanics ................................... 1 1.1 Path integral representation of the partition function . ....... 1 1.2 Evaluation of the path integral for the harmonic oscillator . ..... 6 2 Freescalarfields ..................................... 13 2.1 Path integral for the partition function . ... 13 2.2 Evaluation of thermal sums and their low-temperature limit . .... 16 2.3 High-temperatureexpansion. 23 3 Interactingscalarfields............................... ... 30 3.1 Principles of the weak-coupling expansion . 30 3.2 Problems of the naive weak-coupling expansion . .. 38 3.3 Proper free energy density to (λ): ultraviolet renormalization . 40 O 3 3.4 Proper free energy density to (λ 2 ): infraredresummation . -
SHELDON LEE GLASHOW Lyman Laboratory of Physics Harvard University Cambridge, Mass., USA
TOWARDS A UNIFIED THEORY - THREADS IN A TAPESTRY Nobel Lecture, 8 December, 1979 by SHELDON LEE GLASHOW Lyman Laboratory of Physics Harvard University Cambridge, Mass., USA INTRODUCTION In 1956, when I began doing theoretical physics, the study of elementary particles was like a patchwork quilt. Electrodynamics, weak interactions, and strong interactions were clearly separate disciplines, separately taught and separately studied. There was no coherent theory that described them all. Developments such as the observation of parity violation, the successes of quantum electrodynamics, the discovery of hadron resonances and the appearance of strangeness were well-defined parts of the picture, but they could not be easily fitted together. Things have changed. Today we have what has been called a “standard theory” of elementary particle physics in which strong, weak, and electro- magnetic interactions all arise from a local symmetry principle. It is, in a sense, a complete and apparently correct theory, offering a qualitative description of all particle phenomena and precise quantitative predictions in many instances. There is no experimental data that contradicts the theory. In principle, if not yet in practice, all experimental data can be expressed in terms of a small number of “fundamental” masses and cou- pling constants. The theory we now have is an integral work of art: the patchwork quilt has become a tapestry. Tapestries are made by many artisans working together. The contribu- tions of separate workers cannot be discerned in the completed work, and the loose and false threads have been covered over. So it is in our picture of particle physics. Part of the picture is the unification of weak and electromagnetic interactions and the prediction of neutral currents, now being celebrated by the award of the Nobel Prize. -
Some Comments on Physical Mathematics
Preprint typeset in JHEP style - HYPER VERSION Some Comments on Physical Mathematics Gregory W. Moore Abstract: These are some thoughts that accompany a talk delivered at the APS Savannah meeting, April 5, 2014. I have serious doubts about whether I deserve to be awarded the 2014 Heineman Prize. Nevertheless, I thank the APS and the selection committee for their recognition of the work I have been involved in, as well as the Heineman Foundation for its continued support of Mathematical Physics. Above all, I thank my many excellent collaborators and teachers for making possible my participation in some very rewarding scientific research. 1 I have been asked to give a talk in this prize session, and so I will use the occasion to say a few words about Mathematical Physics, and its relation to the sub-discipline of Physical Mathematics. I will also comment on how some of the work mentioned in the citation illuminates this emergent field. I will begin by framing the remarks in a much broader historical and philosophical context. I hasten to add that I am neither a historian nor a philosopher of science, as will become immediately obvious to any expert, but my impression is that if we look back to the modern era of science then major figures such as Galileo, Kepler, Leibniz, and New- ton were neither physicists nor mathematicans. Rather they were Natural Philosophers. Even around the turn of the 19th century the same could still be said of Bernoulli, Euler, Lagrange, and Hamilton. But a real divide between Mathematics and Physics began to open up in the 19th century. -
Real-Time Dynamics of Chern-Simons Fluctuations Near a Critical Point
PHYSICAL REVIEW D 103, L071502 (2021) Letter Real-time dynamics of Chern-Simons fluctuations near a critical point † ‡ Kazuki Ikeda ,1,* Dmitri E. Kharzeev ,2,3,4, and Yuta Kikuchi 3, 1Research Institute for Advanced Materials and Devices, Kyocera Corporation, Soraku, Kyoto 619-0237, Japan 2Center for Nuclear Theory, Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794-3800, USA 3Department of Physics, Brookhaven National Laboratory, Upton, New York 11973-5000 4RIKEN-BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973-5000 (Received 16 December 2020; accepted 22 March 2021; published 21 April 2021) The real-time topological susceptibility is studied in ð1 þ 1Þ-dimensional massive Schwinger model with a θ-term. We evaluate the real-time correlation function of electric field that represents the topological Chern-Pontryagin number density in (1 þ 1) dimensions. Near the parity-breaking critical point located at θ ¼ π and fermion mass m to coupling g ratio of m=g ≈ 0.33, we observe a sharp maximum in the topological susceptibility. We interpret this maximum in terms of the growth of critical fluctuations near the critical point, and draw analogies between the massive Schwinger model, QCD near the critical point, and ferroelectrics near the Curie point. DOI: 10.1103/PhysRevD.103.L071502 I. INTRODUCTION Coleman [59] that the line of the first order phase transition terminates at some critical value mÃ, where the phase The Schwinger model [1] is quantum electrodynamics in transition is second order. The position of this critical point ð1 þ 1Þ space-time dimensions. -
An Introduction to Effective Field Theory
An Introduction to Effective Field Theory Thinking Effectively About Hierarchies of Scale c C.P. BURGESS i Preface It is an everyday fact of life that Nature comes to us with a variety of scales: from quarks, nuclei and atoms through planets, stars and galaxies up to the overall Universal large-scale structure. Science progresses because we can understand each of these on its own terms, and need not understand all scales at once. This is possible because of a basic fact of Nature: most of the details of small distance physics are irrelevant for the description of longer-distance phenomena. Our description of Nature’s laws use quantum field theories, which share this property that short distances mostly decouple from larger ones. E↵ective Field Theories (EFTs) are the tools developed over the years to show why it does. These tools have immense practical value: knowing which scales are important and why the rest decouple allows hierarchies of scale to be used to simplify the description of many systems. This book provides an introduction to these tools, and to emphasize their great generality illustrates them using applications from many parts of physics: relativistic and nonrelativistic; few- body and many-body. The book is broadly appropriate for an introductory graduate course, though some topics could be done in an upper-level course for advanced undergraduates. It should interest physicists interested in learning these techniques for practical purposes as well as those who enjoy the beauty of the unified picture of physics that emerges. It is to emphasize this unity that a broad selection of applications is examined, although this also means no one topic is explored in as much depth as it deserves. -
Screening in Two-Dimensional Qcd
IC/96/51 hep-th /9604063 INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS lllllfilllllllllllll XA9743416 SCREENING IN TWO-DIMENSIONAL QCD E. Abdalla R. Mohayaee and A. Zadra MIRAMARE-TRIESTE IC/96/51 hep-th/9604063 United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS SCREENING IN TWO-DIMENSIONAL QCD E. Abdalla 1 , R. Mohayaee2 and A. Zadra 3 International Centre for Theoretical Physics, Trieste, Italy. ABSTRACT We discuss the issue of screening and confinement of external colour charges in bosonised two-dimensional quantum chromodynamics. Our computation relies on the static solu tions of the semi-classical equations of motion. The significance of the different repre sentations of the matter field is explicitly studied. We arrive at the conclusion that the screening phase prevails, even in the presence of a small mass term for the fermions. To confirm this result further, we outline the construction of operators corresponding to screened quarks. MIRAMARE - TRIESTE April 1996 iPermanent address: Institute de Ffsica-USP, C.P. 20516, Sao Paulo, Brazil. E-mail address: [email protected] 2E-mail address: [email protected]; ^Permanent address: Institute de Fisica, Universidade de Sao Paulo, Caixa Postal 66 318, Cidade Universitaria, Sao Paulo, Brazil. E-mail address: [email protected] 1 Introduction Theissue of confinement of fundamental constituents of matter is a long-standing prob lem of theoretical physics whose solution has evaded full comprehension up to now. How ever, significant progress has been made towards making a clear distinction between the apparently related phenomena of screening and confinement. -
Arxiv:1711.04534V4 [Hep-Ph]
UCI-HEP-TR-2017-15 Millicharged Scalar Fields, Massive Photons and the Breaking of SU(3)C U(1)EM × Jennifer Rittenhouse West∗ Department of Physics and Astronomy, University of California, Irvine, CA 92697, USA and SLAC National Accelerator Laboratory, Stanford University, Stanford, California 94309, USA (Dated: April 10, 2019) Under the assumption that the current epoch of the Universe is not special, i.e. is not the final state of a long history of processes in particle physics, the cosmological fate of SU(3)C × U(1)EM is investigated. Spontaneous symmetry breaking of U(1)EM at the temperature of the Universe today is carried out. The charged scalar field φEM which breaks the symmetry is found to be ruled out for the charge of the electron, q = e. − Scalar fields with millicharges are viable and limits on their masses and charges are found to be q . 10 3e and −5 mφEM . 10 eV. Furthermore, it is possible that U(1)EM has already been broken at temperatures higher −18 than T = 2.7K given the nonzero limits on the mass of the photon. A photon mass of mγ = 10 eV, the ∼ −13 current upper limit, is found to require a spontaneous symmetry breaking scalar mass of mφEM 10 eV with charge q = 10−6e, well within the allowed parameter space of the model. Finally, the cosmological fate of the strong interaction is studied. SU(3)C is tested for complementarity in which the confinement phase of QCD + colored scalars is equivalent to a spontaneously broken SU(3) gauge theory. -
Complete Normal Ordering 1: Foundations
KCL-PH-TH/2015-55, LCTS/2015-43, CERN-PH-TH/2015-295 Complete Normal Ordering 1: Foundations a,b a,b a,c John Ellisp q, Nick E. Mavromatosp q and Dimitri P. Sklirosp q (a) Theoretical Particle Physics and Cosmology Group Department of Physics, King’s College London London WC2R 2LS, United Kingdom (b) Theory Division, CERN, CH-1211 Geneva 23, Switzerland (c) School of Physics and Astronomy, University of Nottingham Nottingham, NG7 2RD, UK [email protected]; [email protected]; [email protected] Abstract We introduce a new prescription for quantising scalar field theories (in generic spacetime dimension and background) perturbatively around a true minimum of the full quantum ef- fective action, which is to ‘complete normal order’ the bare action of interest. When the true vacuum of the theory is located at zero field value, the key property of this prescription is the automatic cancellation, to any finite order in perturbation theory, of all tadpole and, more generally, all ‘cephalopod’ Feynman diagrams. The latter are connected diagrams that can be disconnected into two pieces by cutting one internal vertex, with either one or both pieces free from external lines. In addition, this procedure of ‘complete normal ordering’ (which is an extension of the standard field theory definition of normal ordering) reduces by a substantial factor the number of Feynman diagrams to be calculated at any given loop order. We illustrate explicitly the complete normal ordering procedure and the cancellation of cephalopod diagrams in scalar field theories with non-derivative interactions, and by using a point splitting ‘trick’ we extend this result to theories with derivative interactions, such as those appearing as non-linear σ-models in the world-sheet formulation of string theory.