DOCSLIB.ORG

Advanced Algorithms for the Simulation of Gauge Theories With

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Advanced Algorithms for the Simulation of Gauge Theories With Fachbereich Physik WUB-DIS 2001-7 BUGH Wuppertal D-42097 Wuppertal Germany Ph.D. Thesis Advanced Algorithms for the Simulation of Gauge Theories with Dynamical Fermionic Degrees of Freedom Wolfram Schroers November 2, 2001 arXiv:hep-lat/0304016v1 25 Apr 2003 Abstract The topic of this thesis is the numerical simulation of quantum chromodynamics including dynamical fermions. Two major problems of most simulation algorithms that deal with dynamical fermions are (i) their restriction to only two mass-degenerate quarks, and (ii) their limitation to relatively heavy masses. Realistic simulations of quantum chromodynamics, however, require the inclusion of three light dynamical fermion flavors. It is therefore highly important to develop algorithms which are efficient in this situation. This thesis is focused on the implementation and the application of a novel kind of algorithm which is expected to overcome the limitations of older schemes. This new algorithm is named Multiboson Method. It allows to simulate an arbitrary number of dynamical fermion flavors, which can in principle have different masses. It will be shown that it exhibits better scaling properties for light fermions than other methods. Therefore, it has the potential to become the method of choice. An explorative investigation of the parameter space of quantum chromodynamics with three flavors finishes this work. The results may serve as a starting point for future realistic simulations. iv Contents List of Figures ix List of Tables xi 1 Introduction 3 2 Quantum Field Theories and Hadronic Physics 7 2.1 PhenomenologyofStrongInteractions . .......... 8 2.2 Classical Field Theories . .... 12 2.3 Quantization ....................................... ... 13 2.3.1 Non-relativistic Quantum Mechanics . .... 14 2.3.2 The Axioms of Relativistic Quantum Field Theories . ... 14 2.3.3 ThePathIntegral ................................. .. 18 2.3.4 Ensembles........................................ 25 2.4 GaugeTheories..................................... .... 26 2.5 QuantumChromodynamics . .. .. .. .. .. .. .. .. .. .. .. .. ..... 28 2.5.1 Running Coupling and Energy Scales . .. 29 2.5.2 FactorizableProcesses .. .. .. .. .. .. .. .. .. .. .. .. .... 30 2.5.3 LatticeQCD ...................................... 31 2.6 Discretization...................................... .... 32 2.6.1 Scalar Fields . 32 2.6.2 GaugeFields ...................................... 33 2.6.3 D-Theory ........................................ 34 2.6.4 Fermion Fields . 36 2.6.5 Yang-Mills Theory . 41 3 Numerical Methods 43 3.1 Monte-CarloAlgorithms ............................... ..... 43 3.1.1 MarkovChains ..................................... 44 3.2 Autocorrelation.................................... ..... 46 3.2.1 Autocorrelation Function . ... 46 3.2.2 Exponential Autocorrelation Time . .... 47 3.2.3 Integrated Autocorrelation Time . ..... 48 3.2.4 Scaling Behavior . 49 3.2.5 ShortTimeSeries ................................... 50 3.3 MeasuringHadronMasses.. .. .. .. .. .. .. .. .. .. .. .. ...... 53 3.4 Bosonic Sampling algorithms . .... 54 3.4.1 Metropolis Algorithm . 55 3.4.2 HeatbathAlgorithm ................................. 56 3.4.3 Overrelaxation .................................... 61 3.5 Fermionic Sampling Algorithms . .... 61 3.5.1 Sampling with the Wilson Matrix . 62 3.5.2 Hybrid Monte-Carlo Algorithm . .. 62 v Contents 3.5.3 Multiboson Algorithms . 64 3.6 Matrix Inversion Algorithms . ...... 68 3.6.1 Static Polynomial Inversion . ... 70 3.6.2 Conjugate-GradientIteration . ...... 72 3.6.3 GMRESAlgorithm................................... 74 3.6.4 Stabilized Bi-Conjugate Gradient Algorithm . ..... 75 3.7 Eigenvalue Algorithms . .... 75 4 Tuning of Multiboson Algorithms 79 4.1 Optimizing the Polynomial Approximation . ..... 79 4.1.1 Tuning the Quadratically Optimized Polynomials . ... 80 4.1.2 Algorithm for Polynomials . 84 4.1.3 Computing the Reweighting Correction . .... 85 4.2 TuningtheDynamicalParameters . ....... 87 4.2.1 Practical Determination of Autocorrelations . ........ 87 4.2.2 Acceptance Rates vs. Polynomial Approximation Quality . ....... 92 4.2.3 Dynamical Reweighting Factor . ... 99 4.2.4 UpdatingStrategy ................................. 102 4.3 ImplementationSystems. .. .. .. .. .. .. .. .. .. .. .. .. ...... 108 4.3.1 APE Platform ..................................... 108 4.3.2 ALiCE Cluster..................................... 109 4.3.3 Accuracy Considerations and Test Suites . ...... 109 4.3.4 ArchitecturesandEfficiency. .... 112 4.4 Summary .......................................... 113 5 Comparison of Dynamical Fermion Algorithms 115 5.1 Simulation Runs at Different Parameters . ........ 115 5.2 Efficiency of Multiboson Algorithms . ..... 116 5.2.1 Tuning the MB-HB Algorithm . 116 5.2.2 Tuning the MB-OR Algorithm . 117 5.2.3 Direct Algorithmic Comparison . 118 5.2.4 Scaling Behavior of Algorithms . 120 5.3 Summary .......................................... 122 6 Exploring the Parameter Space with Three Degenerate DynamicalFlavors 123 6.1 TheNon-ZeroTemperatureCrossover . .......... 123 6.2 TheChiralLimit...................................... 124 6.3 ExplorativeStudies.................................. ..... 126 6.3.1 The Case β =5.3.................................... 126 6.3.2 The Case β =5.2.................................... 127 6.4 SummaryandOutlook................................. .... 138 7 Summary and Outlook 141 A Notations and Conventions 143 A.1 DiracMatrices ...................................... 144 vi Contents B Groups and Algebras 147 B.1 GroupsandRepresentations. ........ 147 B.2 TheU(1)Group ..................................... 149 B.3 The SU(N)Groups....................................... 149 B.3.1 TheSU(2)Group................................... 149 B.3.2 TheSU(3)Group................................... 150 B.4 ThePoincar´eGroup ................................. ..... 150 B.5 Spin-StatisticsTheorem .............................. ...... 152 B.6 GrassmannAlgebras ................................. ..... 152 B.6.1 Definitions ....................................... 153 B.6.2 Derivatives ....................................... 154 B.6.3 Integration ...................................... 154 C Local Forms of Actions Used 157 C.1 GeneralExpressions ................................ ...... 157 C.2 LocalFormsofVariousActions . ....... 158 C.2.1 PureGaugeFields ................................... 158 C.2.2 Lattice Fermion Fields . 159 D Logistics for Running Large Numerical Productions 167 D.1 DesignoftheDatabase............................... ...... 168 Bibliography 173 vii Contents viii List of Figures 2.1 Fundamental representations of the SU(3)F flavor group. The left graph shows the quark triplet (u,d,s) and the right graph shows the anti-quark triplet (¯u,d¯,¯s). 10 ′ 2.2 Pseudoscalar meson octet together with the singlet (the η state) as classified by the parameters of the SU(3)F group. 10 2.3 Vector meson octet together with the singlet (the φ state) as classified by the parameters of the SU(3)F group. 11 2.4 Baryon octet as classified by the parameters of the SU(3)F group. ............ 11 2.5 Relations of different axiomatic frameworks for quantum field theory............ 17 2.6 Relations between the different kinds of n-point functions consisting of vacuum expectation values of products of field operators. The figure has been taken from [28]. 18 2.7 Different methods for obtaining prediction in QCD. ......... 32 2.8 Spectrum of the hopping matrix, D(y, x) in Eq. (2.82), in the limit β 0......... 39 → 2.9 Spectrum of the hopping matrix, D(y, x) in Eq. (2.82), in the limit β ........ 40 → ∞ 4.1 Histogram of the 512 smallest eigenvalues of Q˜2(y, x). ................... 80 4.2 Histogram of the 512 largest eigenvalues of Q˜2(y, x). .................... 80 α=1 4.3 Test function λ Pn1=20(λ) for a quadratically optimized polynomial. 81 4.4 Norms R and R vs. the lower interval limit ǫ..................... 82 | 20| k 20k 4.5 Norms R180 and R180 as defined in Eq. (4.2) vs. the lower interval limit ǫ....... 82 | | k ˆ160k ˜2 4.6 Residual norms for R200(Q ) vs. the lower interval limit ǫ.................. 84 α=1 4.7 Polynomial λ P20(λ) which has been obtained by applying the GMRES algorithm to a thermalized gauge field configuration together with the corresponding quadratically optimized polynomial. 84 4.8 Residual vector norm for both the GMRES and the quadratically optimized polynomials. 85 2 2 4.9 Individual correction factors as computed using the 512 lowest eigenvalues of Q˜ (y, x) for the quadratically optimized polynomial P180(Q˜ ). 86 4.10 Similar to Fig. 4.9, but with the cumulative correction factor from the 512 lowest eigenvalues. 86 4.11 PlaquettehistoryofHMCrun. ....... 88 4.12 Normalized autocorrelation function and integrated autocorrelation time vs. the cutoff for the HMC plaquette history. 88 4.13 Variance σB (Plaquette) vs. the bin size B using the Jackknife method for the HMC plaquette history. 89 4.14 Order-1-lag-30 differenced series of the HMC plaquette history. .............. 89 4.15 Correlation function Γ (1) (t)togetherwithitsintegralvs.thecutoff. 90 AD30 A) 4.16 Integrated autocorrelation time as obtained from the lag-differencing method for varying lags l. 90 4.17 Similar to Fig. 4.15, but with a differencing lag l =23.................... 90 4.18 Time series of the average plaquette from the TSMB algorithm. ............. 91 4.19 Autocorrelation function and the corresponding integral as a function of the cutoff for the plaquette history from the TSMB run. 91 4.20 Jackknife
Load more

Recommended publications
  • Simulating Quantum Field Theory with a Quantum Computer
    Simulating quantum field theory with a quantum computer John Preskill Lattice 2018 28 July 2018 This talk has two parts (1) Near-term prospects for quantum computing. (2) Opportunities in quantum simulation of quantum field theory. Exascale digital computers will advance our knowledge of QCD, but some challenges will remain, especially concerning real-time evolution and properties of nuclear matter and quark-gluon plasma at nonzero temperature and chemical potential. Digital computers may never be able to address these (and other) problems; quantum computers will solve them eventually, though I’m not sure when. The physics payoff may still be far away, but today’s research can hasten the arrival of a new era in which quantum simulation fuels progress in fundamental physics. Frontiers of Physics short distance long distance complexity Higgs boson Large scale structure “More is different” Neutrino masses Cosmic microwave Many-body entanglement background Supersymmetry Phases of quantum Dark matter matter Quantum gravity Dark energy Quantum computing String theory Gravitational waves Quantum spacetime particle collision molecular chemistry entangled electrons A quantum computer can simulate efficiently any physical process that occurs in Nature. (Maybe. We don’t actually know for sure.) superconductor black hole early universe Two fundamental ideas (1) Quantum complexity Why we think quantum computing is powerful. (2) Quantum error correction Why we think quantum computing is scalable. A complete description of a typical quantum state of just 300 qubits requires more bits than the number of atoms in the visible universe. Why we think quantum computing is powerful We know examples of problems that can be solved efficiently by a quantum computer, where we believe the problems are hard for classical computers.
    [Show full text]
  • Geometric Algorithms
    Geometric Algorithms primitive operations convex hull closest pair voronoi diagram References: Algorithms in C (2nd edition), Chapters 24-25 http://www.cs.princeton.edu/introalgsds/71primitives http://www.cs.princeton.edu/introalgsds/72hull 1 Geometric Algorithms Applications. • Data mining. • VLSI design. • Computer vision. • Mathematical models. • Astronomical simulation. • Geographic information systems. airflow around an aircraft wing • Computer graphics (movies, games, virtual reality). • Models of physical world (maps, architecture, medical imaging). Reference: http://www.ics.uci.edu/~eppstein/geom.html History. • Ancient mathematical foundations. • Most geometric algorithms less than 25 years old. 2 primitive operations convex hull closest pair voronoi diagram 3 Geometric Primitives Point: two numbers (x, y). any line not through origin Line: two numbers a and b [ax + by = 1] Line segment: two points. Polygon: sequence of points. Primitive operations. • Is a point inside a polygon? • Compare slopes of two lines. • Distance between two points. • Do two line segments intersect? Given three points p , p , p , is p -p -p a counterclockwise turn? • 1 2 3 1 2 3 Other geometric shapes. • Triangle, rectangle, circle, sphere, cone, … • 3D and higher dimensions sometimes more complicated. 4 Intuition Warning: intuition may be misleading. • Humans have spatial intuition in 2D and 3D. • Computers do not. • Neither has good intuition in higher dimensions! Is a given polygon simple? no crossings 1 6 5 8 7 2 7 8 6 4 2 1 1 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 2 18 4 18 4 19 4 19 4 20 3 20 3 20 1 10 3 7 2 8 8 3 4 6 5 15 1 11 3 14 2 16 we think of this algorithm sees this 5 Polygon Inside, Outside Jordan curve theorem.
    [Show full text]
  • Introduction to Computational Social Choice
    1 Introduction to Computational Social Choice Felix Brandta, Vincent Conitzerb, Ulle Endrissc, J´er^omeLangd, and Ariel D. Procacciae 1.1 Computational Social Choice at a Glance Social choice theory is the field of scientific inquiry that studies the aggregation of individual preferences towards a collective choice. For example, social choice theorists|who hail from a range of different disciplines, including mathematics, economics, and political science|are interested in the design and theoretical evalu- ation of voting rules. Questions of social choice have stimulated intellectual thought for centuries. Over time the topic has fascinated many a great mind, from the Mar- quis de Condorcet and Pierre-Simon de Laplace, through Charles Dodgson (better known as Lewis Carroll, the author of Alice in Wonderland), to Nobel Laureates such as Kenneth Arrow, Amartya Sen, and Lloyd Shapley. Computational social choice (COMSOC), by comparison, is a very young field that formed only in the early 2000s. There were, however, a few precursors. For instance, David Gale and Lloyd Shapley's algorithm for finding stable matchings between two groups of people with preferences over each other, dating back to 1962, truly had a computational flavor. And in the late 1980s, a series of papers by John Bartholdi, Craig Tovey, and Michael Trick showed that, on the one hand, computational complexity, as studied in theoretical computer science, can serve as a barrier against strategic manipulation in elections, but on the other hand, it can also prevent the efficient use of some voting rules altogether. Around the same time, a research group around Bernard Monjardet and Olivier Hudry also started to study the computational complexity of preference aggregation procedures.
    [Show full text]
  • 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.
    [Show full text]
  • Zero-Point Energy in Bag Models
    Ooa/ea/oHsnif- ^*7 ( 4*0 0 1 5 4 3 - m . Zero-Point Energy in Bag Models by Kimball A. Milton* Department of Physics The Ohio State University Columbus, Ohio 43210 A b s t r a c t The zero-point (Casimir) energy of free vector (gluon) fields confined to a spherical cavity (bag) is computed. With a suitable renormalization the result for eight gluons is This result is substantially larger than that for a spher­ ical shell (where both interior and exterior modes are pre­ sent), and so affects Johnson's model of the QCD vacuum. It is also smaller than, and of opposite sign to the value used in bag model phenomenology, so it will have important implications there. * On leave ifrom Department of Physics, University of California, Los Angeles, CA 90024. I. Introduction Quantum chromodynamics (QCD) may we 1.1 be the appropriate theory of hadronic matter. However, the theory is not at all well understood. It may turn out that color confinement is roughly approximated by the phenomenologically suc­ cessful bag model [1,2]. In this model, the normal vacuum is a perfect color magnetic conductor, that iis, the color magnetic permeability n is infinite, while the vacuum.in the interior of the bag is characterized by n=l. This implies that the color electric and magnetic fields are confined to the interior of the bag, and that they satisfy the following boundary conditions on its surface S: n.&jg= 0, nxBlg= 0, (1) where n is normal to S. Now, even in an "empty" bag (i.e., one containing no quarks) there will be non-zero fields present because of quantum fluctua­ tions.
    [Show full text]
  • 18. Lattice Quantum Chromodynamics
    18. Lattice QCD 1 18. Lattice Quantum Chromodynamics Updated September 2017 by S. Hashimoto (KEK), J. Laiho (Syracuse University) and S.R. Sharpe (University of Washington). Many physical processes considered in the Review of Particle Properties (RPP) involve hadrons. The properties of hadrons—which are composed of quarks and gluons—are governed primarily by Quantum Chromodynamics (QCD) (with small corrections from Quantum Electrodynamics [QED]). Theoretical calculations of these properties require non-perturbative methods, and Lattice Quantum Chromodynamics (LQCD) is a tool to carry out such calculations. It has been successfully applied to many properties of hadrons. Most important for the RPP are the calculation of electroweak form factors, which are needed to extract Cabbibo-Kobayashi-Maskawa (CKM) matrix elements when combined with the corresponding experimental measurements. LQCD has also been used to determine other fundamental parameters of the standard model, in particular the strong coupling constant and quark masses, as well as to predict hadronic contributions to the anomalous magnetic moment of the muon, gµ 2. − This review describes the theoretical foundations of LQCD and sketches the methods used to calculate the quantities relevant for the RPP. It also describes the various sources of error that must be controlled in a LQCD calculation. Results for hadronic quantities are given in the corresponding dedicated reviews. 18.1. Lattice regularization of QCD Gauge theories form the building blocks of the Standard Model. While the SU(2) and U(1) parts have weak couplings and can be studied accurately with perturbative methods, the SU(3) component—QCD—is only amenable to a perturbative treatment at high energies.
    [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]
  • Casimir Effect on the Lattice: U (1) Gauge Theory in Two Spatial Dimensions
    Casimir effect on the lattice: U(1) gauge theory in two spatial dimensions M. N. Chernodub,1, 2, 3 V. A. Goy,4, 2 and A. V. Molochkov2 1Laboratoire de Math´ematiques et Physique Th´eoriqueUMR 7350, Universit´ede Tours, 37200 France 2Soft Matter Physics Laboratory, Far Eastern Federal University, Sukhanova 8, Vladivostok, 690950, Russia 3Department of Physics and Astronomy, University of Gent, Krijgslaan 281, S9, B-9000 Gent, Belgium 4School of Natural Sciences, Far Eastern Federal University, Sukhanova 8, Vladivostok, 690950, Russia (Dated: September 8, 2016) We propose a general numerical method to study the Casimir effect in lattice gauge theories. We illustrate the method by calculating the energy density of zero-point fluctuations around two parallel wires of finite static permittivity in Abelian gauge theory in two spatial dimensions. We discuss various subtle issues related to the lattice formulation of the problem and show how they can successfully be resolved. Finally, we calculate the Casimir potential between the wires of a fixed permittivity, extrapolate our results to the limit of ideally conducting wires and demonstrate excellent agreement with a known theoretical result. I. INTRODUCTION lattice discretization) described by spatially-anisotropic and space-dependent static permittivities "(x) and per- meabilities µ(x) at zero and finite temperature in theories The influence of the physical objects on zero-point with various gauge groups. (vacuum) fluctuations is generally known as the Casimir The structure of the paper is as follows. In Sect. II effect [1{3]. The simplest example of the Casimir effect is we review the implementation of the Casimir boundary a modification of the vacuum energy of electromagnetic conditions for ideal conductors and propose its natural field by closely-spaced and perfectly-conducting parallel counterpart in the lattice gauge theory.
    [Show full text]
  • Perturbative Algebraic Quantum Field Theory at Finite Temperature
    Perturbative Algebraic Quantum Field Theory at Finite Temperature Dissertation zur Erlangung des Doktorgrades des Fachbereichs Physik der Universität Hamburg vorgelegt von Falk Lindner aus Zittau Hamburg 2013 Gutachter der Dissertation: Prof. Dr. K. Fredenhagen Prof. Dr. D. Bahns Gutachter der Disputation: Prof. Dr. K. Fredenhagen Prof. Dr. J. Louis Datum der Disputation: 01. 07. 2013 Vorsitzende des Prüfungsausschusses: Prof. Dr. C. Hagner Vorsitzender des Promotionsausschusses: Prof. Dr. P. Hauschildt Dekan der Fakultät für Mathematik, Informatik und Naturwissenschaften: Prof. Dr. H. Graener Zusammenfassung Der algebraische Zugang zur perturbativen Quantenfeldtheorie in der Minkowskiraum- zeit wird vorgestellt, wobei ein Schwerpunkt auf die inhärente Zustandsunabhängig- keit des Formalismus gelegt wird. Des Weiteren wird der Zustandsraum der pertur- bativen QFT eingehend untersucht. Die Dynamik wechselwirkender Theorien wird durch ein neues Verfahren konstruiert, das die Gültigkeit des Zeitschichtaxioms in der kausalen Störungstheorie systematisch ausnutzt. Dies beleuchtet einen bisher un- bekannten Zusammenhang zwischen dem statistischen Zugang der Quantenmechanik und der perturbativen Quantenfeldtheorie. Die entwickelten Methoden werden zur ex- pliziten Konstruktion von KMS- und Vakuumzuständen des wechselwirkenden, mas- siven Klein-Gordon Feldes benutzt und damit mögliche Infrarotdivergenzen der Theo- rie, also insbesondere der wechselwirkenden Wightman- und zeitgeordneten Funktio- nen des wechselwirkenden Feldes ausgeschlossen. Abstract We present the algebraic approach to perturbative quantum field theory for the real scalar field in Minkowski spacetime. In this work we put a special emphasis on the in- herent state-independence of the framework and provide a detailed analysis of the state space. The dynamics of the interacting system is constructed in a novel way by virtue of the time-slice axiom in causal perturbation theory.
    [Show full text]
  • Electromagnetic Field Theory
    Electromagnetic Field Theory BO THIDÉ Υ UPSILON BOOKS ELECTROMAGNETIC FIELD THEORY Electromagnetic Field Theory BO THIDÉ Swedish Institute of Space Physics and Department of Astronomy and Space Physics Uppsala University, Sweden and School of Mathematics and Systems Engineering Växjö University, Sweden Υ UPSILON BOOKS COMMUNA AB UPPSALA SWEDEN · · · Also available ELECTROMAGNETIC FIELD THEORY EXERCISES by Tobia Carozzi, Anders Eriksson, Bengt Lundborg, Bo Thidé and Mattias Waldenvik Freely downloadable from www.plasma.uu.se/CED This book was typeset in LATEX 2" (based on TEX 3.14159 and Web2C 7.4.2) on an HP Visualize 9000⁄360 workstation running HP-UX 11.11. Copyright c 1997, 1998, 1999, 2000, 2001, 2002, 2003 and 2004 by Bo Thidé Uppsala, Sweden All rights reserved. Electromagnetic Field Theory ISBN X-XXX-XXXXX-X Downloaded from http://www.plasma.uu.se/CED/Book Version released 19th June 2004 at 21:47. Preface The current book is an outgrowth of the lecture notes that I prepared for the four-credit course Electrodynamics that was introduced in the Uppsala University curriculum in 1992, to become the five-credit course Classical Electrodynamics in 1997. To some extent, parts of these notes were based on lecture notes prepared, in Swedish, by BENGT LUNDBORG who created, developed and taught the earlier, two-credit course Electromagnetic Radiation at our faculty. Intended primarily as a textbook for physics students at the advanced undergradu- ate or beginning graduate level, it is hoped that the present book may be useful for research workers
    [Show full text]
  • Lo Algebras for Extended Geometry from Borcherds Superalgebras
    Commun. Math. Phys. 369, 721–760 (2019) Communications in Digital Object Identifier (DOI) https://doi.org/10.1007/s00220-019-03451-2 Mathematical Physics L∞ Algebras for Extended Geometry from Borcherds Superalgebras Martin Cederwall, Jakob Palmkvist Division for Theoretical Physics, Department of Physics, Chalmers University of Technology, 412 96 Gothenburg, Sweden. E-mail: [email protected]; [email protected] Received: 24 May 2018 / Accepted: 15 March 2019 Published online: 26 April 2019 – © The Author(s) 2019 Abstract: We examine the structure of gauge transformations in extended geometry, the framework unifying double geometry, exceptional geometry, etc. This is done by giving the variations of the ghosts in a Batalin–Vilkovisky framework, or equivalently, an L∞ algebra. The L∞ brackets are given as derived brackets constructed using an underlying Borcherds superalgebra B(gr+1), which is a double extension of the structure algebra gr . The construction includes a set of “ancillary” ghosts. All brackets involving the infinite sequence of ghosts are given explicitly. All even brackets above the 2-brackets vanish, and the coefficients appearing in the brackets are given by Bernoulli numbers. The results are valid in the absence of ancillary transformations at ghost number 1. We present evidence that in order to go further, the underlying algebra should be the corresponding tensor hierarchy algebra. Contents 1. Introduction ................................. 722 2. The Borcherds Superalgebra ......................... 723 3. Section Constraint and Generalised Lie Derivatives ............. 728 4. Derivatives, Generalised Lie Derivatives and Other Operators ....... 730 4.1 The derivative .............................. 730 4.2 Generalised Lie derivative from “almost derivation” .......... 731 4.3 “Almost covariance” and related operators ..............
    [Show full text]
  • Perturbative Algebraic Quantum Field Theory
    Perturbative algebraic quantum field theory Klaus Fredenhagen Katarzyna Rejzner II Inst. f. Theoretische Physik, Department of Mathematics, Universität Hamburg, University of Rome Tor Vergata Luruper Chaussee 149, Via della Ricerca Scientifica 1, D-22761 Hamburg, Germany I-00133, Rome, Italy [email protected] [email protected] arXiv:1208.1428v2 [math-ph] 27 Feb 2013 2012 These notes are based on the course given by Klaus Fredenhagen at the Les Houches Win- ter School in Mathematical Physics (January 29 - February 3, 2012) and the course QFT for mathematicians given by Katarzyna Rejzner in Hamburg for the Research Training Group 1670 (February 6 -11, 2012). Both courses were meant as an introduction to mod- ern approach to perturbative quantum field theory and are aimed both at mathematicians and physicists. Contents 1 Introduction 3 2 Algebraic quantum mechanics 5 3 Locally covariant field theory 9 4 Classical field theory 14 5 Deformation quantization of free field theories 21 6 Interacting theories and the time ordered product 26 7 Renormalization 26 A Distributions and wavefront sets 35 1 Introduction Quantum field theory (QFT) is at present, by far, the most successful description of fundamental physics. Elementary physics , to a large extent, explained by a specific quantum field theory, the so-called Standard Model. All the essential structures of the standard model are nowadays experimentally verified. Outside of particle physics, quan- tum field theoretical concepts have been successfully applied also to condensed matter physics. In spite of its great achievements, quantum field theory also suffers from several longstanding open problems. The most serious problem is the incorporation of gravity.
    [Show full text]