The Principles of Quantum Mechanics
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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 . -
Perturbative Versus Non-Perturbative Quantum Field Theory: Tao’S Method, the Casimir Effect, and Interacting Wightman Theories
universe Review Perturbative versus Non-Perturbative Quantum Field Theory: Tao’s Method, the Casimir Effect, and Interacting Wightman Theories Walter Felipe Wreszinski Instituto de Física, Universidade de São Paulo, São Paulo 05508-090, SP, Brazil; [email protected] Abstract: We dwell upon certain points concerning the meaning of quantum field theory: the problems with the perturbative approach, and the question raised by ’t Hooft of the existence of the theory in a well-defined (rigorous) mathematical sense, as well as some of the few existent mathematically precise results on fully quantized field theories. Emphasis is brought on how the mathematical contributions help to elucidate or illuminate certain conceptual aspects of the theory when applied to real physical phenomena, in particular, the singular nature of quantum fields. In a first part, we present a comprehensive review of divergent versus asymptotic series, with qed as background example, as well as a method due to Terence Tao which conveys mathematical sense to divergent series. In a second part, we apply Tao’s method to the Casimir effect in its simplest form, consisting of perfectly conducting parallel plates, arguing that the usual theory, which makes use of the Euler-MacLaurin formula, still contains a residual infinity, which is eliminated in our approach. In the third part, we revisit the general theory of nonperturbative quantum fields, in the form of newly proposed (with Christian Jaekel) Wightman axioms for interacting field theories, with applications to “dressed” electrons in a theory with massless particles (such as qed), as well as Citation: Wreszinski, W.F. unstable particles. Various problems (mostly open) are finally discussed in connection with concrete Perturbative versus Non-Perturbative models. -
Quantum Theory Cannot Consistently Describe the Use of Itself
ARTICLE DOI: 10.1038/s41467-018-05739-8 OPEN Quantum theory cannot consistently describe the use of itself Daniela Frauchiger1 & Renato Renner1 Quantum theory provides an extremely accurate description of fundamental processes in physics. It thus seems likely that the theory is applicable beyond the, mostly microscopic, domain in which it has been tested experimentally. Here, we propose a Gedankenexperiment 1234567890():,; to investigate the question whether quantum theory can, in principle, have universal validity. The idea is that, if the answer was yes, it must be possible to employ quantum theory to model complex systems that include agents who are themselves using quantum theory. Analysing the experiment under this presumption, we find that one agent, upon observing a particular measurement outcome, must conclude that another agent has predicted the opposite outcome with certainty. The agents’ conclusions, although all derived within quantum theory, are thus inconsistent. This indicates that quantum theory cannot be extrapolated to complex systems, at least not in a straightforward manner. 1 Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland. Correspondence and requests for materials should be addressed to R.R. (email: [email protected]) NATURE COMMUNICATIONS | (2018) 9:3711 | DOI: 10.1038/s41467-018-05739-8 | www.nature.com/naturecommunications 1 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-05739-8 “ 1”〉 “ 1”〉 irect experimental tests of quantum theory are mostly Here, | z ¼À2 D and | z ¼þ2 D denote states of D depending restricted to microscopic domains. Nevertheless, quantum on the measurement outcome z shown by the devices within the D “ψ ”〉 “ψ ”〉 theory is commonly regarded as being (almost) uni- lab. -
Appendix: the Interaction Picture
Appendix: The Interaction Picture The technique of expressing operators and quantum states in the so-called interaction picture is of great importance in treating quantum-mechanical problems in a perturbative fashion. It plays an important role, for example, in the derivation of the Born–Markov master equation for decoherence detailed in Sect. 4.2.2. We shall review the basics of the interaction-picture approach in the following. We begin by considering a Hamiltonian of the form Hˆ = Hˆ0 + V.ˆ (A.1) Here Hˆ0 denotes the part of the Hamiltonian that describes the free (un- perturbed) evolution of a system S, whereas Vˆ is some added external per- turbation. In applications of the interaction-picture formalism, Vˆ is typically assumed to be weak in comparison with Hˆ0, and the idea is then to deter- mine the approximate dynamics of the system given this assumption. In the following, however, we shall proceed with exact calculations only and make no assumption about the relative strengths of the two components Hˆ0 and Vˆ . From standard quantum theory, we know that the expectation value of an operator observable Aˆ(t) is given by the trace rule [see (2.17)], & ' & ' ˆ ˆ Aˆ(t) =Tr Aˆ(t)ˆρ(t) =Tr Aˆ(t)e−iHtρˆ(0)eiHt . (A.2) For reasons that will immediately become obvious, let us rewrite this expres- sion as & ' ˆ ˆ ˆ ˆ ˆ ˆ Aˆ(t) =Tr eiH0tAˆ(t)e−iH0t eiH0te−iHtρˆ(0)eiHte−iH0t . (A.3) ˆ In inserting the time-evolution operators e±iH0t at the beginning and the end of the expression in square brackets, we have made use of the fact that the trace is cyclic under permutations of the arguments, i.e., that Tr AˆBˆCˆ ··· =Tr BˆCˆ ···Aˆ , etc. -
Quantum Violation of Classical Physics in Macroscopic Systems
Quantum VIOLATION OF CLASSICAL PHYSICS IN MACROSCOPIC SYSTEMS Lucas Clemente Ludwig Maximilian University OF Munich Max Planck INSTITUTE OF Quantum Optics Quantum VIOLATION OF CLASSICAL PHYSICS IN MACROSCOPIC SYSTEMS Lucas Clemente Dissertation AN DER Fakultät für Physik DER Ludwig-Maximilians-Universität München VORGELEGT VON Lucas Clemente AUS München München, IM NoVEMBER 2015 TAG DER mündlichen Prüfung: 26. Januar 2016 Erstgutachter: Prof. J. IGNACIO Cirac, PhD Zweitgutachter: Prof. Dr. Jan VON Delft WEITERE Prüfungskommissionsmitglieder: Prof. Dr. HarALD Weinfurter, Prof. Dr. Armin Scrinzi Reality is that which, when you stop believing in it, doesn’t go away. “ — Philip K. Dick How To Build A Universe That Doesn’t Fall Apart Two Days Later, a speech published in the collection I Hope I Shall Arrive Soon Contents Abstract xi Zusammenfassung xiii List of publications xv Acknowledgments xvii 0 Introduction 1 0.1 History and motivation . 3 0.2 Local realism and Bell’s theorem . 5 0.3 Contents of this thesis . 10 1 Conditions for macrorealism 11 1.1 Macroscopic realism . 13 1.2 Macrorealism per se following from strong non-invasive measurability 15 1.3 The Leggett-Garg inequality . 17 1.4 No-signaling in time . 19 1.5 Necessary and sufficient conditions for macrorealism . 21 1.6 No-signaling in time for quantum measurements . 25 1.6.1 Without time evolution . 26 1.6.2 With time evolution . 27 1.7 Conclusion and outlook . 28 Appendix 31 1.A Proof that NSIT0(1)2 is sufficient for NIC0(1)2 . 31 2 Macroscopic classical dynamics from microscopic quantum behavior 33 2.1 Quantifying violations of classicality . -
Particle Or Wave: There Is No Evidence of Single Photon Delayed Choice
Particle or wave: there is no evidence of single photon delayed choice. Michael Devereux* Los Alamos National Laboratory (Retired) Abstract Wheeler supposed that the way in which a single photon is measured in the present could determine how it had behaved in the past. He named such retrocausation delayed choice. Over the last forty years many experimentalists have claimed to have observed single-photon delayed choice. Recently, however, researchers have proven that the quantum wavefunction of a single photon assumes the identical mathematical form of the solution to Maxwell’s equations for that photon. This efficacious understanding allows for a trenchant analysis of delayed-choice experiments and denies their retrocausation conclusions. It is now usual for physicists to employ Bohr’s wave-particle complementarity theory to distinguish wave from particle aspects in delayed- choice observations. Nevertheless, single-photon, delayed-choice experiments, provide no evidence that the photon actually acts like a particle, or, instead, like a wave, as a function of a future measurement. And, a recent, careful, Stern-Gerlach analysis has shown that the supposition of concurrent wave and particle characteristics in the Bohm-DeBroglie theory is not tenable. PACS 03.65.Ta, 03.65.Ud 03.67.-a, 42.50.Ar, 42.50.Xa 1. Introduction. Almost forty years ago Wheeler suggested that there exists a type of retrocausation, which he called delayed choice, in certain physical phenomena [1]. Specifically, he said that the past behavior of some quantum systems could be determined by how they are observed in the present. “The past”, he wrote, “has no existence except as it is recorded in the present” [2]. -
New Journal of Physics the Open–Access Journal for Physics
New Journal of Physics The open–access journal for physics The rise and fall of redundancy in decoherence and quantum Darwinism C Jess Riedel1,2,5, Wojciech H Zurek1,3 and Michael Zwolak1,4 1 Theoretical Division/CNLS, LANL, Los Alamos, NM 87545, USA 2 Department of Physics, University of California, Santa Barbara, CA 93106, USA 3 Santa Fe Institute, Santa Fe, NM 87501, USA 4 Department of Physics, Oregon State University, Corvallis, OR 97331, USA E-mail: [email protected] New Journal of Physics 14 (2012) 083010 (20pp) Received 9 March 2012 Published 10 August 2012 Online at http://www.njp.org/ doi:10.1088/1367-2630/14/8/083010 Abstract. A state selected at random from the Hilbert space of a many-body system is overwhelmingly likely to exhibit highly non-classical correlations. For these typical states, half of the environment must be measured by an observer to determine the state of a given subsystem. The objectivity of classical reality—the fact that multiple observers can agree on the state of a subsystem after measuring just a small fraction of its environment—implies that the correlations found in nature between macroscopic systems and their environments are exceptional. Building on previous studies of quantum Darwinism showing that highly redundant branching states are produced ubiquitously during pure decoherence, we examine the conditions needed for the creation of branching states and study their demise through many-body interactions. We show that even constrained dynamics can suppress redundancy to the values typical of random states on relaxation timescales, and prove that these results hold exactly in the thermodynamic limit. -
Quantum Mechanics Propagator
Quantum Mechanics_propagator This article is about Quantum field theory. For plant propagation, see Plant propagation. In Quantum mechanics and quantum field theory, the propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum. In Feynman diagrams, which calculate the rate of collisions in quantum field theory, virtual particles contribute their propagator to the rate of the scattering event described by the diagram. They also can be viewed as the inverse of the wave operator appropriate to the particle, and are therefore often called Green's functions. Non-relativistic propagators In non-relativistic quantum mechanics the propagator gives the probability amplitude for a particle to travel from one spatial point at one time to another spatial point at a later time. It is the Green's function (fundamental solution) for the Schrödinger equation. This means that, if a system has Hamiltonian H, then the appropriate propagator is a function satisfying where Hx denotes the Hamiltonian written in terms of the x coordinates, δ(x)denotes the Dirac delta-function, Θ(x) is the Heaviside step function and K(x,t;x',t')is the kernel of the differential operator in question, often referred to as the propagator instead of G in this context, and henceforth in this article. This propagator can also be written as where Û(t,t' ) is the unitary time-evolution operator for the system taking states at time t to states at time t'. The quantum mechanical propagator may also be found by using a path integral, where the boundary conditions of the path integral include q(t)=x, q(t')=x' . -
Schrödinger Equation Revisited
Schrödinger equation revisited Wolfgang P. Schleicha,b, Daniel M. Greenbergera,c, Donald H. Kobeb, and Marlan O. Scullyd,e,f,1 aInstitut für Quantenphysik and Center for Integrated Quantum Science and Technology (IQST), Universität Ulm, D-89069 Ulm, Germany; bDepartment of Physics, University of North Texas, Denton, TX 76203-1427; cCity College of the City University of New York, New York, NY 10031; dTexas A&M University, College Station, TX 77843; ePrinceton University, Princeton, NJ 08544; and fBaylor University, Waco, TX 97096 Contributed by Marlan O. Scully, February 8, 2013 (sent for review November 8, 2012) The time-dependent Schrödinger equation is a cornerstone of quan- i) Starting from the nonlinear wave equation (Eq. 2), we as- tum physics and governs all phenomena of the microscopic world. sume that we search for a wave equation for a scalar wave However, despite its importance, its origin is still not widely ap- containing only a first-order derivative in time and a second- preciated and properly understood. We obtain the Schrödinger order derivative in space, and we establish a mathematical equation from a mathematical identity by a slight generalization identity involving derivatives of a complex-valued field. of the formulation of classical statistical mechanics based on the ii) A description (16–20) of classical statistical mechanics in Hamilton–Jacobi equation. This approach brings out most clearly terms of a classical matter wave whose phase is given by the fact that the linearity of quantum mechanics is intimately con- the classical action and is governed by the Hamilton–Jacobi nected to the strong coupling between the amplitude and phase equation (21), and whose amplitude is defined by the square of a quantum wave. -
Winter 2017/18)
Thorsten Ohl 2018-02-07 14:05:03 +0100 subject to change! i Relativistic Quantum Field Theory (Winter 2017/18) Thorsten Ohl Institut f¨urTheoretische Physik und Astrophysik Universit¨atW¨urzburg D-97070 W¨urzburg Personal Manuscript! Use at your own peril! git commit: c8138b2 Thorsten Ohl 2018-02-07 14:05:03 +0100 subject to change! i Abstract 1. Symmetrien 2. Relativistische Einteilchenzust¨ande 3. Langrangeformalismus f¨urFelder 4. Feldquantisierung 5. Streutheorie und S-Matrix 6. Eichprinzip und Wechselwirkung 7. St¨orungstheorie 8. Feynman-Regeln 9. Quantenelektrodynamische Prozesse in Born-N¨aherung 10. Strahlungskorrekturen (optional) 11. Renormierung (optional) Thorsten Ohl 2018-02-07 14:05:03 +0100 subject to change! i Contents 1 Introduction1 Lecture 01: Tue, 17. 10. 2017 1.1 Limitations of Quantum Mechanics (QM).......... 1 1.2 (Special) Relativistic Quantum Field Theory (QFT)..... 2 1.3 Limitations of QFT...................... 3 2 Symmetries4 2.1 Principles ofQM....................... 4 2.2 Symmetries inQM....................... 6 2.3 Groups............................. 6 Lecture 02: Wed, 18. 10. 2017 2.3.1 Lie Groups....................... 7 2.3.2 Lie Algebras...................... 8 2.3.3 Homomorphisms.................... 8 2.3.4 Representations.................... 9 2.4 Infinitesimal Generators.................... 10 2.4.1 Unitary and Conjugate Representations........ 12 Lecture 03: Tue, 24. 10. 2017 2.5 SO(3) and SU(2) ........................ 13 2.5.1 O(3) and SO(3) .................... 13 2.5.2 SU(2) .......................... 15 2.5.3 SU(2) ! SO(3) .................... 16 2.5.4 SO(3) and SU(2) Representations........... 17 Lecture 04: Wed, 25. 10. 2017 2.6 Lorentz- and Poincar´e-Group................ -
INFORMATION– CONSCIOUSNESS– REALITY How a New Understanding of the Universe Can Help Answer Age-Old Questions of Existence the FRONTIERS COLLECTION
THE FRONTIERS COLLECTION James B. Glattfelder INFORMATION– CONSCIOUSNESS– REALITY How a New Understanding of the Universe Can Help Answer Age-Old Questions of Existence THE FRONTIERS COLLECTION Series editors Avshalom C. Elitzur, Iyar, Israel Institute of Advanced Research, Rehovot, Israel Zeeya Merali, Foundational Questions Institute, Decatur, GA, USA Thanu Padmanabhan, Inter-University Centre for Astronomy and Astrophysics (IUCAA), Pune, India Maximilian Schlosshauer, Department of Physics, University of Portland, Portland, OR, USA Mark P. Silverman, Department of Physics, Trinity College, Hartford, CT, USA Jack A. Tuszynski, Department of Physics, University of Alberta, Edmonton, AB, Canada Rüdiger Vaas, Redaktion Astronomie, Physik, bild der wissenschaft, Leinfelden-Echterdingen, Germany THE FRONTIERS COLLECTION The books in this collection are devoted to challenging and open problems at the forefront of modern science and scholarship, including related philosophical debates. In contrast to typical research monographs, however, they strive to present their topics in a manner accessible also to scientifically literate non-specialists wishing to gain insight into the deeper implications and fascinating questions involved. Taken as a whole, the series reflects the need for a fundamental and interdisciplinary approach to modern science and research. Furthermore, it is intended to encourage active academics in all fields to ponder over important and perhaps controversial issues beyond their own speciality. Extending from quantum physics and relativity to entropy, conscious- ness, language and complex systems—the Frontiers Collection will inspire readers to push back the frontiers of their own knowledge. More information about this series at http://www.springer.com/series/5342 For a full list of published titles, please see back of book or springer.com/series/5342 James B. -
Quarks and Gluons in the Phase Diagram of Quantum Chromodynamics
Dissertation Quarks and Gluons in the Phase Diagram of Quantum Chromodynamics Christian Andreas Welzbacher July 2016 Justus-Liebig-Universitat¨ Giessen Fachbereich 07 Institut fur¨ theoretische Physik Dekan: Prof. Dr. Bernhard M¨uhlherr Erstgutachter: Prof. Dr. Christian S. Fischer Zweitgutachter: Prof. Dr. Lorenz von Smekal Vorsitzende der Pr¨ufungskommission: Prof. Dr. Claudia H¨ohne Tag der Einreichung: 13.05.2016 Tag der m¨undlichen Pr¨ufung: 14.07.2016 So much universe, and so little time. (Sir Terry Pratchett) Quarks und Gluonen im Phasendiagramm der Quantenchromodynamik Zusammenfassung In der vorliegenden Dissertation wird das Phasendiagramm von stark wechselwirk- ender Materie untersucht. Dazu wird im Rahmen der Quantenchromodynamik der Quarkpropagator ¨uber seine quantenfeldtheoretischen Bewegungsgleichungen bes- timmt. Diese sind bekannt als Dyson-Schwinger Gleichungen und konstituieren einen funktionalen Zugang, welcher mithilfe des Matsubara-Formalismus bei endlicher Tem- peratur und endlichem chemischen Potential angewendet wird. Theoretische Hin- tergr¨undeder Quantenchromodynamik werden erl¨autert,wobei insbesondere auf die Dyson-Schwinger Gleichungen eingegangen wird. Chirale Symmetrie sowie Confine- ment und zugeh¨origeOrdnungsparameter werden diskutiert, welche eine Unterteilung des Phasendiagrammes in verschiedene Phasen erm¨oglichen. Zudem wird der soge- nannte Columbia Plot erl¨autert,der die Abh¨angigkeit verschiedener Phasen¨uberg¨ange von der Quarkmasse skizziert. Zun¨achst werden Ergebnisse f¨urein System mit zwei entarteten leichten Quarks und einem Strange-Quark mit vorangegangenen Untersuchungen verglichen. Eine Trunkierung, welche notwendig ist um das System aus unendlich vielen gekoppelten Gleichungen auf eine endliche Anzahl an Gleichungen zu reduzieren, wird eingef¨uhrt. Die Ergebnisse f¨urdie Propagatoren und das Phasendiagramm stimmen gut mit vorherigen Arbeiten ¨uberein. Einige zus¨atzliche Ergebnisse werden pr¨asentiert, wobei insbesondere auf die Abh¨angigkeit des Phasendiagrammes von der Quarkmasse einge- gangen wird.