DOCSLIB.ORG

Hamiltonian Perturbation Theory (And Transition to Chaos) H 4515 H

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Hamiltonian Perturbation Theory (And Transition to Chaos) H 4515 H Hamiltonian Perturbation Theory (and Transition to Chaos) H 4515 H Hamiltonian Perturbation Theory dle third from a closed interval) designates topological spaces that locally have the structure Rn Cantor dust (and Transition to Chaos) for some n 2 N. Cantor families are parametrized by HENK W. BROER1,HEINZ HANSSMANN2 such Cantor sets. On the real line R one can define 1 Instituut voor Wiskunde en Informatica, Cantor dust of positive measure by excluding around Rijksuniversiteit Groningen, Groningen, each rational number p/q an interval of size The Netherlands 2 2 Mathematisch Instituut, Universiteit Utrecht, ;>0 ;>2 : q Utrecht, The Netherlands Similar Diophantine conditions define Cantor sets Article Outline in Rn. Since these Cantor sets have positive measure their Hausdorff dimension is n.Wheretheunper- Glossary turbed system is stratified according to the co-dimen- Definition of the Subject sion of occurring (bifurcating) tori, this leads to a Can- Introduction tor stratification. One Degree of Freedom Chaos An evolution of a dynamical system is chaotic if its Perturbations of Periodic Orbits future is badly predictable from its past. Examples of Invariant Curves of Planar Diffeomorphisms non-chaotic evolutions are periodic or multi-periodic. KAM Theory: An Overview A system is called chaotic when many of its evolutions Splitting of Separatrices are. One criterion for chaoticity is the fact that one of Transition to Chaos and Turbulence the Lyapunov exponents is positive. Future Directions Diophantine condition, Diophantine frequency vector Bibliography A frequency vector ! 2 Rn is called Diophantine if there are constants >0and>n 1with Glossary jhk;!ij for all k 2 Znnf0g : Bifurcation In parametrized dynamical systems a bifur- jkj cation occurs when a qualitative change is invoked by a change of parameters. In models such a qualitative The Diophantine frequency vectors satisfying this con- change corresponds to transition between dynamical dition for fixed and form a Cantor set of half lines. regimes. In the generic theory a finite list of cases is ob- As the Diophantine parameter tends to zero (while tained, containing elements like ‘saddle-node’, ‘period remains fixed), these half lines extend to the origin. doubling’, ‘Hopf bifurcation’ and many others. The complement in any compact set of frequency vec- Cantor set, Cantor dust, Cantor family, Cantor strati- tors satisfying a Diophantine condition with fixed fication Cantor dust is a separable locally compact space has a measure of order O()as # 0. that is perfect, i. e. every point is in the closure of its Integrable system A Hamiltonian system with n degrees complement, and totally disconnected. This deter- of freedom is (Liouville)-integrable if it has n func- mines Cantor dust up to homeomorphisms. The term tionally independent commuting integrals of motion. Cantor set (originally reserved for the specific form of Locally this implies the existence of a torus action, Cantor dust obtained by repeatedly deleting the mid- a feature that can be generalized to dissipative sys- 4516 H Hamiltonian Perturbation Theory (and Transition to Chaos) tems. In particular a mapping is integrable if it can be planes in !-space and, by means of the frequency interpolated to become the stroboscopic mapping of mapping, also in phase space. The smallest number aflow. jhjDjh1jCCjhk j is the order of the resonance. KAM theory Kolmogorov–Arnold–Moser theory is the Diophantine conditions describe a measure-theoret- perturbation theory of (Diophantine) quasi-periodic ically large complement of a neighborhood of the tori for nearly integrable Hamiltonian systems. In the (dense!) set of all resonances. format of quasi-periodic stability, the unperturbed and Separatrices Consider a hyperbolic equilibrium, fixed or perturbed system, restricted to a Diophantine Cantor periodic point or invariant torus. If the stable and un- set, are smoothly conjugated in the sense of Whitney. stable manifolds of such hyperbolic elements are codi- This theory extends to the world of reversible, volume- mension one immersed manifolds, then they are called preserving or general dissipative systems. In the latter separatrices, since they separate domains of phase KAM theory gives rise to families of quasi-periodic at- space, for instance, basins of attraction. tractors. KAM theory also applies to torus bundles, in Singularity theory AfunctionH : Rn ! R has a crit- which case a global Whitney smooth conjugation can ical point z 2 Rn where DH(z)vanishes.Inlo- be proven to exist, that keeps track of the geometry. In cal coordinates we may arrange z D 0 (and simi- an appropriate sense invariants like monodromy and larly that it is mapped to zero as well). Two germs Chern classes thus also can be defined in the nearly K :(Rn; 0) ! (R; 0) and N :(Rn ; 0) ! (R; 0) rep- integrable case. Also compare with Kolmogorov– resent the same function H locally around z if and only Arnold–Moser (KAM) Theory. if there is a diffeomorphism on Rn satisfying Nearly integrable system In the setting of perturbation theory, a nearly integrable system is a perturbation of N D K ı : an integrable one. The latter then is an integrable ap- proximation of the former. See an above item. The corresponding equivalence class is called a singu- Normal form truncation Consider a dynamical system larity. in the neighborhood of an equilibrium point, a fixed or Structurally stable A system is structurally stable if it is periodic point, or a quasi-periodic torus, reducible to topologically equivalent to all nearby systems, where Floquet form. Then Taylor expansions (and their ana- ‘nearby’ is measured in an appropriate topology on the logues) can be changed gradually into normal forms, space of systems, like the Whitney Ck-topology [72]. that usually reflect the dynamics better. Often these A family is structurally stable if for every nearby family display a (formal) torus symmetry, such that the nor- there is a re-parametrization such that all correspond- mal form truncation becomes an integrable approxi- ing systems are topologically equivalent. mation, thus yielding a perturbation theory setting. See above items. Also compare with Normal Forms in Definition of the Subject Perturbation Theory. Persistent property In the setting of perturbation theory, The fundamental problem of mechanics is to study Hamil- a property is persistent whenever it is inherited from tonian systems that are small perturbations of integrable the unperturbed to the perturbed system. Often the systems. Also, perturbations that destroy the Hamiltonian perturbation is taken in an appropriate topology on the character are important, be it to study the effect of a small space of systems, like the Whitney Ck-topology [72]. amount of friction, or to further the theory of dissipa- Perturbation problem In perturbation theory the unper- tive systems themselves which surprisingly often revolves turbed systems usually are transparent regarding their around certain well-chosen Hamiltonian systems. Fur- dynamics. Examples are integrable systems or normal thermore there are approaches like KAM theory that his- form truncations. In a perturbation problem things are torically were first applied to Hamiltonian systems. Typi- arranged in such a way that the original system is well- cally perturbation theory explains only part of the dynam- approximated by such an unperturbed one. This ar- ics, and in the resulting gaps the orderly unperturbed mo- rangement usually involves both changes of variables tion is replaced by random or chaotic motion. and scalings. Resonance If the frequencies of an invariant torus with Introduction multi- or conditionally periodic flow are rationally dependent, this torus divides into invariant sub-tori. We outline perturbation theory from a general point of Such resonances hh;!iD0, h 2 Zk ,definehyper- view, illustrated by a few examples. Hamiltonian Perturbation Theory (and Transition to Chaos) H 4517 The Perturbation Problem a vector field, we obtain The aim of perturbation theory is to approximate a given t˙ D 1 dynamical system by a more familiar one, regarding the former as a perturbation of the latter. The problem then x˙ D y is to deduce certain dynamical properties from the unper- dV y˙ D (x) C " f (x; y; t) ; turbed to the perturbed case. dx What is familiar may or may not be a matter of taste, at R3 Df; ; g least it depends a lot on the dynamical properties of one’s now on the generalized phase space t x y .In interest. Still the most frequently used unperturbed sys- the case where the t-dependence is periodic, we can take S1 R2 tems are: for (generalized) phase space. Linear systems Remark Integrable Hamiltonian systems, compare with Dy- A small variation of the above driven system concerns namics of Hamiltonian Systems and references therein a parametrically forced oscillator like Normal form truncations, compare with Normal Forms in Perturbation Theory and references therein x¨ C (!2 C " cos t)sinx D 0 ; Etc. which happens to be entirely in the world of Hamilto- To some extent the second category can be seen as a special nian systems. case of the third. To avoid technicalities in this section we It may be useful to study the Poincaré or period map- assume all systems to be sufficiently smooth, say of class ping of such time periodic systems, which happens to C1 or real analytic. Moreover in our considerations " will be a mapping of the plane. We recall that in the Hamil- be a real parameter. The unperturbed case always corre- tonian cases this mapping preserves area. For general sponds to " D 0 and the perturbed one to " ¤ 0or">0. reference in this direction see, e. g., [6,7,27,66]. Examples of Perturbation Problems To begin with There are lots of variations and generalizations. One ex- consider the autonomous differential equation ampleisthesolarsystem,wheretheunperturbedcasecon- dV sists of a number of uncoupled two-body problems con- x¨ C "x˙ C (x) D 0 ; dx cerning the Sun and each of the planets, and where the in- teraction between the planets is considered as small [6,9, modeling an oscillator with small damping.
Load more

Recommended publications
  • Notes on Statistical Field Theory
    Lecture Notes on Statistical Field Theory Kevin Zhou [email protected] These notes cover statistical field theory and the renormalization group. The primary sources were: • Kardar, Statistical Physics of Fields. A concise and logically tight presentation of the subject, with good problems. Possibly a bit too terse unless paired with the 8.334 video lectures. • David Tong's Statistical Field Theory lecture notes. A readable, easygoing introduction covering the core material of Kardar's book, written to seamlessly pair with a standard course in quantum field theory. • Goldenfeld, Lectures on Phase Transitions and the Renormalization Group. Covers similar material to Kardar's book with a conversational tone, focusing on the conceptual basis for phase transitions and motivation for the renormalization group. The notes are structured around the MIT course based on Kardar's textbook, and were revised to include material from Part III Statistical Field Theory as lectured in 2017. Sections containing this additional material are marked with stars. The most recent version is here; please report any errors found to [email protected]. 2 Contents Contents 1 Introduction 3 1.1 Phonons...........................................3 1.2 Phase Transitions......................................6 1.3 Critical Behavior......................................8 2 Landau Theory 12 2.1 Landau{Ginzburg Hamiltonian.............................. 12 2.2 Mean Field Theory..................................... 13 2.3 Symmetry Breaking.................................... 16 3 Fluctuations 19 3.1 Scattering and Fluctuations................................ 19 3.2 Position Space Fluctuations................................ 20 3.3 Saddle Point Fluctuations................................. 23 3.4 ∗ Path Integral Methods.................................. 24 4 The Scaling Hypothesis 29 4.1 The Homogeneity Assumption............................... 29 4.2 Correlation Lengths.................................... 30 4.3 Renormalization Group (Conceptual)..........................
    [Show full text]
  • Effective Field Theories, Reductionism and Scientific Explanation Stephan
    To appear in: Studies in History and Philosophy of Modern Physics Effective Field Theories, Reductionism and Scientific Explanation Stephan Hartmann∗ Abstract Effective field theories have been a very popular tool in quantum physics for almost two decades. And there are good reasons for this. I will argue that effec- tive field theories share many of the advantages of both fundamental theories and phenomenological models, while avoiding their respective shortcomings. They are, for example, flexible enough to cover a wide range of phenomena, and concrete enough to provide a detailed story of the specific mechanisms at work at a given energy scale. So will all of physics eventually converge on effective field theories? This paper argues that good scientific research can be characterised by a fruitful interaction between fundamental theories, phenomenological models and effective field theories. All of them have their appropriate functions in the research process, and all of them are indispens- able. They complement each other and hang together in a coherent way which I shall characterise in some detail. To illustrate all this I will present a case study from nuclear and particle physics. The resulting view about scientific theorising is inherently pluralistic, and has implications for the debates about reductionism and scientific explanation. Keywords: Effective Field Theory; Quantum Field Theory; Renormalisation; Reductionism; Explanation; Pluralism. ∗Center for Philosophy of Science, University of Pittsburgh, 817 Cathedral of Learning, Pitts- burgh, PA 15260, USA (e-mail: [email protected]) (correspondence address); and Sektion Physik, Universit¨at M¨unchen, Theresienstr. 37, 80333 M¨unchen, Germany. 1 1 Introduction There is little doubt that effective field theories are nowadays a very popular tool in quantum physics.
    [Show full text]
  • Hamiltonian Perturbation Theory on a Lie Algebra. Application to a Non-Autonomous Symmetric Top
    Hamiltonian Perturbation Theory on a Lie Algebra. Application to a non-autonomous Symmetric Top. Lorenzo Valvo∗ Michel Vittot Dipartimeno di Matematica Centre de Physique Théorique Università degli Studi di Roma “Tor Vergata” Aix-Marseille Université & CNRS Via della Ricerca Scientifica 1 163 Avenue de Luminy 00133 Roma, Italy 13288 Marseille Cedex 9, France January 6, 2021 Abstract We propose a perturbation algorithm for Hamiltonian systems on a Lie algebra V, so that it can be applied to non-canonical Hamiltonian systems. Given a Hamil- tonian system that preserves a subalgebra B of V, when we add a perturbation the subalgebra B will no longer be preserved. We show how to transform the perturbed dynamical system to preserve B up to terms quadratic in the perturbation. We apply this method to study the dynamics of a non-autonomous symmetric Rigid Body. In this example our algebraic transform plays the role of Iterative Lemma in the proof of a KAM-like statement. A dynamical system on some set V is a flow: a one-parameter group of mappings associating to a given element F 2 V (the initial condition) another element F (t) 2 V, for any value of the parameter t. A flow on V is determined by a linear mapping H from V to itself. However, the flow can be rarely computed explicitly. In perturbation theory we aim at computing the flow of H + V, where the flow of H is known, and V is another linear mapping from V to itself. In physics, the set V is often a Lie algebra: for instance in classical mechanics [4], fluid dynamics and plasma physics [20], quantum mechanics [24], kinetic theory [18], special and general relativity [17].
    [Show full text]
  • 1 Introduction 2 Classical Perturbation Theory
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server Classical and Quantum Perturbation Theory for two Non–Resonant Oscillators with Quartic Interaction Luca Salasnich Dipartimento di Matematica Pura ed Applicata, Universit`a di Padova, Via Belzoni 7, I–35131 Padova, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Padova, Via Marzolo 8, I–35131 Padova, Italy Istituto Nazionale per la Fisica della Materia, Unit`a di Milano, Via Celoria 16, I–20133 Milano, Italy Abstract. We study the classical and quantum perturbation theory for two non–resonant oscillators coupled by a nonlinear quartic interaction. In particular we analyze the question of quantum corrections to the torus quantization of the classical perturbation theory (semiclassical mechanics). We obtain up to the second order of perturbation theory an explicit analytical formula for the quantum energy levels, which is the semiclassical one plus quantum corrections. We compare the ”exact” quantum levels obtained numerically to the semiclassical levels studying also the effects of quantum corrections. Key words: Classical Perturbation Theory; Quantum Mechanics; General Mechanics 1 Introduction Nowadays there is considerable renewed interest in the transition from classical mechanics to quantum mechanics, a powerful motivation behind that being the problem of the so–called quantum chaos [1–3]. An important aspect is represented by the semiclassical quantization formula of the (regular) energy levels for quasi–integrable systems [4–6], the so–called torus quantization, initiated by Einstein [7] and completed by Maslov [8]. It has been recently shown [9,10] that, for perturbed non–resonant harmonic oscillators, the algorithm of classical perturbation theory can be used to formulate the quantum mechanical perturbation theory as the semiclassically quantized classical perturbation theory equipped with the quantum corrections in powers of ¯h ”correcting” the classical Hamiltonian that appears in the classical algorithm.
    [Show full text]
  • Perturbation Theory and Exact Solutions
    PERTURBATION THEORY AND EXACT SOLUTIONS by J J. LODDER R|nhtdnn Report 76~96 DISSIPATIVE MOTION PERTURBATION THEORY AND EXACT SOLUTIONS J J. LODOER ASSOCIATIE EURATOM-FOM Jun»»76 FOM-INST1TUUT VOOR PLASMAFYSICA RUNHUIZEN - JUTPHAAS - NEDERLAND DISSIPATIVE MOTION PERTURBATION THEORY AND EXACT SOLUTIONS by JJ LODDER R^nhuizen Report 76-95 Thisworkwat performed at part of th«r«Mvchprogmmncof thcHMCiattofiafrccmentof EnratoniOTd th« Stichting voor FundtmenteelOiutereoek der Matctk" (FOM) wtihnnmcWMppoft from the Nederhmdie Organiutic voor Zuiver Wetemchap- pcigk Onderzoek (ZWO) and Evntom It it abo pabHtfMd w a the* of Ac Univenrty of Utrecht CONTENTS page SUMMARY iii I. INTRODUCTION 1 II. GENERALIZED FUNCTIONS DEFINED ON DISCONTINUOUS TEST FUNC­ TIONS AND THEIR FOURIER, LAPLACE, AND HILBERT TRANSFORMS 1. Introduction 4 2. Discontinuous test functions 5 3. Differentiation 7 4. Powers of x. The partie finie 10 5. Fourier transforms 16 6. Laplace transforms 20 7. Hubert transforms 20 8. Dispersion relations 21 III. PERTURBATION THEORY 1. Introduction 24 2. Arbitrary potential, momentum coupling 24 3. Dissipative equation of motion 31 4. Expectation values 32 5. Matrix elements, transition probabilities 33 6. Harmonic oscillator 36 7. Classical mechanics and quantum corrections 36 8. Discussion of the Pu strength function 38 IV. EXACTLY SOLVABLE MODELS FOR DISSIPATIVE MOTION 1. Introduction 40 2. General quadratic Kami1tonians 41 3. Differential equations 46 4. Classical mechanics and quantum corrections 49 5. Equation of motion for observables 51 V. SPECIAL QUADRATIC HAMILTONIANS 1. Introduction 53 2. Hamiltcnians with coordinate coupling 53 3. Double coordinate coupled Hamiltonians 62 4. Symmetric Hamiltonians 63 i page VI. DISCUSSION 1. Introduction 66 ?.
    [Show full text]
  • Conformal Invariance and Quantum Integrability of Sigma Models on Symmetric Superspaces
    Physics Letters B 648 (2007) 254–261 www.elsevier.com/locate/physletb Conformal invariance and quantum integrability of sigma models on symmetric superspaces A. Babichenko a,b a Racah Institute of Physics, the Hebrew University, Jerusalem 91904, Israel b Department of Particle Physics, Weizmann Institute of Science, Rehovot 76100, Israel Received 7 December 2006; accepted 2 March 2007 Available online 7 March 2007 Editor: L. Alvarez-Gaumé Abstract We consider two dimensional nonlinear sigma models on few symmetric superspaces, which are supergroup manifolds of coset type. For those spaces where one loop beta function vanishes, two loop beta function is calculated and is shown to be zero. Vanishing of beta function in all orders of perturbation theory is shown for the principal chiral models on group supermanifolds with zero Killing form. Sigma models on symmetric (super) spaces on supergroup manifold G/H are known to be classically integrable. We investigate a possibility to extend an argument of absence of quantum anomalies in nonlocal current conservation from nonsuper case to the case of supergroup manifolds which are asymptotically free in one loop. © 2007 Elsevier B.V. All rights reserved. 1. Introduction super coset PSU(2, 2|4)/SO(1, 4) × SO(5). Hyperactivity in at- tempts to exploit integrability and methods of Bethe ansatz as a Two dimensional (2d) nonlinear sigma models (NLSM) on calculational tool in checks of ADS/CFT correspondence (see, supermanifolds, with and without WZ term, seemed to be ex- e.g., reviews [12] and references therein), also supports this in- otic objects, when they appeared in condensed matter physics terest, since both spin chains (on the gauge theory side) and 2d twenty years ago as an elegant calculational tool in problems NLSM (on the ADS side) appearing there, usually have a su- of self avoiding walks [1] and disordered metals [2].
    [Show full text]
  • Perturbation Theory in Celestial Mechanics
    Perturbation Theory in Celestial Mechanics Alessandra Celletti Dipartimento di Matematica Universit`adi Roma Tor Vergata Via della Ricerca Scientifica 1, I-00133 Roma (Italy) ([email protected]) December 8, 2007 Contents 1 Glossary 2 2 Definition 2 3 Introduction 2 4 Classical perturbation theory 4 4.1 The classical theory . 4 4.2 The precession of the perihelion of Mercury . 6 4.2.1 Delaunay action–angle variables . 6 4.2.2 The restricted, planar, circular, three–body problem . 7 4.2.3 Expansion of the perturbing function . 7 4.2.4 Computation of the precession of the perihelion . 8 5 Resonant perturbation theory 9 5.1 The resonant theory . 9 5.2 Three–body resonance . 10 5.3 Degenerate perturbation theory . 11 5.4 The precession of the equinoxes . 12 6 Invariant tori 14 6.1 Invariant KAM surfaces . 14 6.2 Rotational tori for the spin–orbit problem . 15 6.3 Librational tori for the spin–orbit problem . 16 6.4 Rotational tori for the restricted three–body problem . 17 6.5 Planetary problem . 18 7 Periodic orbits 18 7.1 Construction of periodic orbits . 18 7.2 The libration in longitude of the Moon . 20 1 8 Future directions 20 9 Bibliography 21 9.1 Books and Reviews . 21 9.2 Primary Literature . 22 1 Glossary KAM theory: it provides the persistence of quasi–periodic motions under a small perturbation of an integrable system. KAM theory can be applied under quite general assumptions, i.e. a non– degeneracy of the integrable system and a diophantine condition of the frequency of motion.
    [Show full text]
  • On the Spectrum of PP-Wave Matrix Theory
    Preprint typeset in JHEP style - HYPER VERSION hep-th/0207034 AEI-2002-047 On the Spectrum of PP-Wave Matrix Theory Nakwoo Kim and Jan Plefka Max-Planck-Institut f¨ur Gravitationsphysik Albert-Einstein-Institut Am M¨uhlenberg 1, D-14476 Golm, Germany kim,[email protected] Abstract: We study the spectrum of the recently proposed matrix model of DLCQ M-theory in a parallel plane (pp)-wave background. In contrast to matrix theory in a flat background this model contains mass terms, which lift the flat directions of the potential and renders its spectrum discrete. The supersymmetry algebra of the model groups the energy eigenstates into supermultiplets, whose members differ by fixed amounts of energy in great similarity to the representation of supersymmetry in AdS spaces. There is a unique and exact zero-energy groundstate along with a multitude of long and short multiplets of excited states. For large masses the arXiv:hep-th/0207034v3 10 Jul 2002 quantum mechanical model may be treated perturbatively and we study the leading order energy shifts of the first excited states up to level two. Most interestingly we uncover a protected short multiplet at level two, whose energies do not receive perturbative corrections. Moreover, we conjecture the existence of an infinite series of similar protected multiplets in the pp-wave matrix model. Contents 1. Introduction 1 2. The Model and its Quantization 2 3. Supersymmetry Algebra and Structure of the Spectrum 4 4. The Perturbative Energy Spectrum and Protected States 8 5. Conclusions and Outlook 12 A. The Supersymmetry Algebra 13 B.
    [Show full text]
  • Normalization of the Ground State of the Supersymmetric Harmonic
    Letters in High Energy Physics LHEP xx, xxx, 2018 Normalization of the Ground State of the Supersymmetric Harmonic Oscillator Ahmed Ayad School of Physics, University of the Witwatersrand, Private Bag 3, WITS-2050, Johannesburg, South Africa Abstract Supersymmetry plays a main role in all current thinking about superstring theory. Indeed, many remark- able properties of string theory have been explained using supersymmetry as a tool. So far, there has been no unbroken supersymmetry observed in nature, and if nature is described by supersymmetry, it must be broken. Supersymmetry may be broken spontaneously at any order of perturbation theory or dynamically due to nonperturbative effects. To examine the methods of supersymmetry breaking, special attention is given to discuss the normalization of the ground state of the supersymmetric harmonic oscillator. This study explains that perturbation theory gives incorrect results for both the ground-state wave function and the energy spectrum and it fails to give an explanation to the supersymmetry breaking. Keywords: supersymmetry breaking, perturbation theory, fermionic as well as bosonic fields using the usual quantum nonperturbative effects mechanical operators method. Then, we consider supersym- metric classical mechanics and study generalized classical Pois- son brackets to Dirac brackets and quantization rules in or- 1. INTRODUCTION der to introduce the superalgebra. Furthermore, the formal- ism of supersymmetry in quantum mechanics is introduced. Supersymmetry, often abbreviated SUSY, was first introduced Subsequently, we give a general background in the concepts in 1971 by Gelfand and Likhtman, Raymond, and Neveu and and methods of supersymmetric quantum mechanics. This fol- Schwartz, and later, it was rediscovered by other groups [1, 2].
    [Show full text]
  • Combinatorial Algorithms for Perturbation Theory and Application on Quantum Computing Yudong Cao Purdue University
    Purdue University Purdue e-Pubs Open Access Dissertations Theses and Dissertations 12-2016 Combinatorial algorithms for perturbation theory and application on quantum computing Yudong Cao Purdue University Follow this and additional works at: https://docs.lib.purdue.edu/open_access_dissertations Part of the Computer Sciences Commons, and the Quantum Physics Commons Recommended Citation Cao, Yudong, "Combinatorial algorithms for perturbation theory and application on quantum computing" (2016). Open Access Dissertations. 908. https://docs.lib.purdue.edu/open_access_dissertations/908 This document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact [email protected] for additional information. Graduate School Form 30 Updated PURDUE UNIVERSITY GRADUATE SCHOOL Thesis/Dissertation Acceptance This is to certify that the thesis/dissertation prepared By Yudong Cao Entitled Combinatorial Algorithms for Perturbation Theory and Application on Quantum Computing For the degree of Doctor of Philosophy Is approved by the final examining committee: Sabre Kais Chair Mikhail Atallah Co-chair David Gleich Ahmed Sameh Samuel Wagstaff To the best of my knowledge and as understood by the student in the Thesis/Dissertation Agreement, Publication Delay, and Certification Disclaimer (Graduate School Form 32), this thesis/dissertation adheres to the provisions of Purdue University’s “Policy of Integrity in Research” and the use of copyright material. Approved by Major Professor(s): Sabre Kais Approved by: William
    [Show full text]
  • Statistical Field Theory University of Cambridge Part III Mathematical Tripos
    Preprint typeset in JHEP style - HYPER VERSION Michaelmas Term, 2017 Statistical Field Theory University of Cambridge Part III Mathematical Tripos David Tong Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 OBA, UK http://www.damtp.cam.ac.uk/user/tong/sft.html [email protected] { 1 { Recommended Books and Resources There are a large number of books which cover the material in these lectures, although often from very different perspectives. They have titles like \Critical Phenomena", \Phase Transitions", \Renormalisation Group" or, less helpfully, \Advanced Statistical Mechanics". Here are some that I particularly like • Nigel Goldenfeld, Phase Transitions and the Renormalization Group A great book, covering the basic material that we'll need and delving deeper in places. • Mehran Kardar, Statistical Physics of Fields The second of two volumes on statistical mechanics. It cuts a concise path through the subject, at the expense of being a little telegraphic in places. It is based on lecture notes which you can find on the web; a link is given on the course website. • John Cardy, Scaling and Renormalisation in Statistical Physics A beautiful little book from one of the masters of conformal field theory. It covers the material from a slightly different perspective than these lectures, with more focus on renormalisation in real space. • Chaikin and Lubensky, Principles of Condensed Matter Physics • Shankar, Quantum Field Theory and Condensed Matter Both of these are more all-round condensed matter books, but with substantial sections on critical phenomena and the renormalisation group. Chaikin and Lubensky is more traditional, and packed full of content.
    [Show full text]
  • Time–Independent Perturbation Theory
    Appendix A Time{Independent Perturbation Theory References Davydov - Quantum Mechanics, Ch. 7. ² Morse and Feshbach, Methods of Theoretical Physics, Ch. 9. ² Shankar, Principles of Quantum Mechanics, Ch. 17. ² Cohen-Tannoudji, Diu and LaloÄe, Quantum Mechanics, vol. 2, Ch. 11. ² T-Y. Wu, Quantum Mechanics, Ch. 6. ² A.1 Introduction Another review topic that we discuss here is time{independent perturbation theory because of its importance in experimental solid state physics in general and transport properties in particular. There are many mathematical problems that occur in nature that cannot be solved ex- actly. It also happens frequently that a related problem can be solved exactly. Perturbation theory gives us a method for relating the problem that can be solved exactly to the one that cannot. This occurrence is more general than quantum mechanics {many problems in electromagnetic theory are handled by the techniques of perturbation theory. In this course however, we will think mostly about quantum mechanical systems, as occur typically in solid state physics. Suppose that the Hamiltonian for our system can be written as = + 0 (A.1) H H0 H where is the part that we can solve exactly and is the part that we cannot solve. H0 H0 Provided that we can use perturbation theory; that is, we consider the solution of H0 ¿ H0 the unperturbed Hamiltonian and then calculate the e®ect of the perturbation Hamil- H0 tonian . For example, we can solve the hydrogen atom energy levels exactly, but when H0 we apply an electric or a magnetic ¯eld, we can no longer solve the problem exactly.
    [Show full text]