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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. -
Arxiv:1910.05796V2 [Math-Ph] 26 Oct 2020 5
Towards a conformal field theory for Schramm-Loewner evolutions Eveliina Peltola [email protected] Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60, D-53115 Bonn, Germany We discuss the partition function point of view for chordal Schramm-Loewner evolutions and their relation- ship with correlation functions in conformal field theory. Both are closely related to crossing probabilities and interfaces in critical models in two-dimensional statistical mechanics. We gather and supplement previous results with different perspectives, point out remaining difficulties, and suggest directions for future studies. 1. Introduction 2 2. Schramm-Loewner evolution in statistical mechanics and conformal field theory4 A. Schramm-Loewner evolution (SLE) 5 B. SLE in critical models – the Ising model6 C. Conformal field theory (CFT) 7 D. Martingale observables for interfaces 9 3. Multiple SLE partition functions and applications 12 A. Multiple SLEs and their partition functions 12 B. Definition of the multiple SLE partition functions 13 C. Properties of the multiple SLE partition functions 16 D. Relation to Schramm-Loewner evolutions and critical models 18 4. Fusion and operator product expansion 20 A. Fusion and operator product expansion in conformal field theory 21 B. Fusion: analytic approach 22 C. Fusion: systematic algebraic approach 23 D. Operator product expansion for multiple SLE partition functions 26 arXiv:1910.05796v2 [math-ph] 26 Oct 2020 5. From OPE structure to products of random distributions? 27 A. Conformal bootstrap 27 B. Random tempered distributions 28 C. Bootstrap construction? 29 Appendix A. Representation theory of the Virasoro algebra 31 Appendix B. Probabilistic construction of the pure partition functions 32 Appendix C. -
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).......................... -
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 . -
Arxiv:0807.3472V1 [Cond-Mat.Stat-Mech] 22 Jul 2008 Conformal Field Theory and Statistical Mechanics∗
Conformal Field Theory and Statistical Mechanics∗ John Cardy July 2008 arXiv:0807.3472v1 [cond-mat.stat-mech] 22 Jul 2008 ∗Lectures given at the Summer School on Exact methods in low-dimensional statistical physics and quantum computing, les Houches, July 2008. 1 CFT and Statistical Mechanics 2 Contents 1 Introduction 3 2 Scale invariance and conformal invariance in critical behaviour 3 2.1 Scaleinvariance ................................. 3 2.2 Conformalmappingsingeneral . .. 6 3 The role of the stress tensor 7 3.0.1 An example - free (gaussian) scalar field . ... 8 3.1 ConformalWardidentity. 9 3.2 An application - entanglement entropy . ..... 12 4 Radial quantisation and the Virasoro algebra 14 4.1 Radialquantisation.............................. 14 4.2 TheVirasoroalgebra .............................. 15 4.3 NullstatesandtheKacformula . .. 16 4.4 Fusionrules ................................... 17 5 CFT on the cylinder and torus 18 5.1 CFTonthecylinder .............................. 18 5.2 Modularinvarianceonthetorus . ... 20 5.2.1 Modular invariance for the minimal models . .... 22 6 Height models, loop models and Coulomb gas methods 24 6.1 Heightmodelsandloopmodels . 24 6.2 Coulombgasmethods ............................. 27 6.3 Identification with minimal models . .... 28 7 Boundary conformal field theory 29 7.1 Conformal boundary conditions and Ward identities . ......... 29 7.2 CFT on the annulus and classification of boundary states . ......... 30 7.2.1 Example................................. 34 CFT and Statistical Mechanics 3 7.3 -
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 di↵erent 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 • Agreatbook,coveringthebasicmaterialthatwe’llneedanddelvingdeeperinplaces. 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 • Abeautifullittlebookfromoneofthemastersofconformalfieldtheory.Itcoversthe material from a slightly di↵erent 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. Shankar covers modern methods of QFT, with an easygoing style suitable for bedtime reading. Anumberofexcellentlecturenotesareavailableontheweb.Linkscanbefoundon the course webpage: http://www.damtp.cam.ac.uk/user/tong/sft.html. -
Ads/CFT Correspondence
AdS/CFT Correspondence GEORGE SIOPSIS Department of Physics and Astronomy The University of Tennessee Knoxville, TN 37996-1200 U.S.A. e-mail: [email protected] Notes by James Kettner Fall 2010 ii Contents 1 Path Integral 1 1.1 Quantum Mechanics ........................ 1 1.2 Statistical Mechanics ........................ 3 1.3 Thermodynamics .......................... 4 1.3.1 Canonical ensemble . 4 1.3.2 Microcanonical ensemble . 4 1.4 Field Theory ............................. 4 2 Schwarzschild Black Hole 7 3 Reissner-Nordstrom¨ Black Holes 11 3.1 The holes . 11 3.2 Extremal limit . 13 3.3 Generalizations . 15 3.3.1 Multi-center solution . 15 3.3.2 Arbitrary dimension . 16 3.3.3 Kaluza-Klein reduction . 17 4 Black Branes from String Theory 19 5 Microscopic calculation of Entropy and Hawking Radiation 23 5.1 The hole . 23 5.2 Entropy . 25 5.3 Finite temperature . 27 5.4 Scattering and Hawking radiation . 29 iii iv CONTENTS LECTURE 1 Path Integral 1.1 Quantum Mechanics Set ~ = 1. Given a particle described by coordinates (q, t) with initial and final coor- dinates of (qi, ti) and (qf , tf ), the amplitude of the transition is ∗ hqi, ti | qf , tf i = hψ |qf , tf i = ψ (qf , tf ) . where ψ satisfies the Schrodinger¨ equation ∂ψ 1 ∂2ψ i = − . ∂t 2m ∂q2 Feynman introduced the path integral formalism Z −iS hqi, ti | qf , tf i = [dq] e q(ti)=qi,q(tf )=qf R where S = dtL is the action and L = L(q, q˙) is the Lagrangian, e.g., 1 L = mq˙2 − V (q) 2 We convert to imaginary time via the Wick rotation τ = it so 1 L → − mq˙2 − V (q) ≡ −L 2 E where the derivative is with respect to τ, LE is the total energy (which is bounded Rbelow), and the subscript E stands for Euclidean. -
On Mass Interaction Principle
On Mass Interaction Principle Chu-Jun Gu∗ Institute of Theoretical Physics, Grand Genius Group, 100085, Beijing, China 1 2 Abstract This paper proposes mass interaction principle (MIP) as: the particles will be subjected to the random frictionless quantum Brownian motion by the collision of space time particle (STP) prevalent in spacetime. The change in the amount of action of the particles during each collision is an integer multiple of the Planck constant h. The motion of particles under the action of STP is a quantum Markov process. Under this principle, we infer that the statistical inertial mass of a particle is a statistical property that characterizes the difficulty of particle diffusion in spacetime. Starting from this principle, this article has the following six aspects of work: First, we derive the mass-diffusion coefficient uncertainty and the quantized commutation, and derive the most basic coordinate-momentum uncertainty and time-energy uncertainty of quantum mechanics, and then clearly reveal the particle-wave duality, which are properties exhibited by particles collided by STP. Second, we created the three decompositions of particles velocity. The comprehensive property of three velocities deduced the equation of motion of the particle as Schrödinger equation, and made a novel interpretation of Heisenberg’s uncertainty principle and Feynman’s path integral expression. And reexamine the quantum measurement problem, so that the EPR paradox can be explained in a self-consistent manner. Third, we reinterpret the physical origin of quantum spins. Each spacetime random impact not only gives the particle of matter the action of a Planck constant , but also produces the quantum fluctuation properties of the material particles. -
Arxiv:2006.02298V1 [Gr-Qc] 3 Jun 2020
Lorentzian quantum cosmology in novel Gauss-Bonnet gravity from Picard-Lefschetz methods Gaurav Narain a∗ and Hai-Qing Zhang a,b† a Center for Gravitational Physics, Department of Space Science, Beihang University, Beijing 100191, China. b International Research Institute for Multidisciplinary Science, Beihang University, Beijing 100191, China Abstract In this paper we study some aspects of classical and quantum cosmology in the novel-Gauss- Bonnet (nGB) gravity in four space-time dimensions. Starting with a generalised Friedmann- Lemaˆıtre-Robertson-Walker (FLRW) metric respecting homogeneity and isotropicity in arbitrary space-time dimension D, we find the action of theory in four spacetime dimension where the limit D 4 is smoothly obtained after an integration by parts. The peculiar rescaling of Gauss-Bonnet → coupling by factor of D 4 results in a non-trivial contribution to the action. We study the system − of equation of motion to first order nGB coupling. We then go on to compute the transition probability from one 3-geometry to another directly in Lorentzian signature. We make use of combination of WKB approximation and Picard-Lefschetz (PL) theory to achieve our aim. PL theory allows to analyse the path-integral directly in Lorentzian signature without doing Wick rotation. Due to complication caused by non-linear nature of action, we compute the transition amplitude to first order in nGB coupling. We find non-trivial correction coming from the nGB coupling to the transition amplitude, even if the analysis was done perturbatively. We use this result to investigate the case of classical boundary conditions. arXiv:2006.02298v1 [gr-qc] 3 Jun 2020 ∗ [email protected] † [email protected] 1 I. -
String Theory and Related Subjects
String theory and related subjects Marios Petropoulos Centre de Physique Théorique, Ecole Polytechnique ICFP Theoretical Physics What is string theory? • Worldwide scientific activity • Branch of mathematical/theoretical physics • high-energy particle physics • gravity/cosmology • statistical mechanics/condensed matter • geormetry and more mathematical topics A few words on particle physics ... • Since 1931 (Berkeley) particles are produced in accelerators and colliders • at high energy: relativistic c • at microscopic scales: quantum ħ • Particles and interactions studied with relativistic quantum mechanics: Quantum Field Theory • Culminated in the 70’s with a gauge theory: the Standard Model Note: QFT is a tool used in statistical mechanics and condensed matter ... and gravity • Described since 1916 by a classical field theory named General Relativity • Verified with high precision in the solar system and beyond: mercury perihelion precession, Shapiro effect, pulsars, GPS • Successfully applied to cosmology: Big-Bang and further evolution • Hard to reconcile with ħ: non-renormalizable QFT String theory historically • Prehistory : dual models (1967-1974) • History : theory of gravity (1974-1984) • Modern ages : theory of everything (1984-2000) • Maturity : toolbox like and beyond QFT (2000-) String theory ideally • The point-like object is abandoned • The techniques include QFT • The output is the possibility for various particles and interactions to emerge ... • ... in particular gauge theories & quantum gravity defined in 10 space-time dimensions String theory practically • Quantum gravity • supergravity • various aspects of black holes such as black-hole thermodynamics ... • Grand-unified theory • analyse compactifications: geometry • comply with SM: phenomenology, supersymmetry String theory practically • Cosmology • initial singularity • cosmological evolution • More generally • develop techniques beyond QFT, dualities ... • understand (and prove?) holography: integrability • apply holographic correspondence to QCD, condensed-matter systems, superfluids .. -
Indecomposability in Field Theory and Applications to Disordered Systems and Geometrical Problems Vasseur Romain
Indecomposability in field theory and applications to disordered systems and geometrical problems Vasseur Romain To cite this version: Vasseur Romain. Indecomposability in field theory and applications to disordered systems andge- ometrical problems. Mathematical Physics [math-ph]. Université Pierre et Marie Curie - Paris VI, 2013. English. tel-00876155 HAL Id: tel-00876155 https://tel.archives-ouvertes.fr/tel-00876155 Submitted on 23 Oct 2013 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. UNIVERSITE´ PIERRE ET MARIE CURIE - PARIS VI et INSTITUT DE PHYSIQUE THEORIQUE´ - CEA/SACLAY Ecole´ Doctorale de Physique de la R´egion Parisienne - ED 107 Th`ese de doctorat Sp´ecialit´e: Physique Th´eorique Indecomposability in field theory and applications to disordered systems and geometrical problems pr´esent´ee par Romain Vasseur pour obtenir le grade de Docteur de l’Universit´ePierre et Marie Curie Th`ese pr´epar´ee sous la direction de Hubert Saleur et de Jesper Jacobsen Soutenue le 27 Septembre 2013 devant le jury compos´ede : Denis Bernard Examinateur -
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.