DOCSLIB.ORG

String Theory and Critical Dimensions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
String Theory and Critical Dimensions String theory and critical dimensions George Jorjadze Free University of Tbilisi Zielona Gora - 24.01.2017 GJ| String theory and critical dimensions 1/18 Contents Introduction Particle dynamics String dynamics Particle in a static spacetime Particle quantization in static gauge String dynamics in static gauge String quantization in static gauge Connection to the covariant quantization Conclusions GJ| String theory and critical dimensions Contents 2/18 Introduction The present knowledge of particle physics is summarized by the standard model (SM), which combines the electroweak and strong interactions without real unification. However, this interplay is necessary because some particles take part in both interactions. The Lagrangian of SM has the structure 1 µν ¯ µ 2 2 2 2 L = 4 Tr (FµνF ) + (iγ Dµ − M) + jDφj − λ(jφj − a ) ; with Higgs field φ and Yang-Mills type fields Fµν = @µAν − @nAµ + g[Aµ;Aν] ;Dµ = @µ − eAµ ; which combine 12 force carriers: 8 gluons, photon and 3 gauge bossons. Matter fields in SM are represented by: 12 leptons (e; νe), (µ, νµ), (τ; ντ ); together with antiparticles, 36 quarks (u; d), (c; s), (t; b), in 3 colors and with antiparticles. Totally, there are about 60 elementary particles and 20 parameters in SM. GJ| String theory and critical dimensions Introduction 3/18 Though SM is in excellent agreement with the experimental data, it is believed that SM is only a step towards the complete fundamental theory. The main drawback of SM is the absence of the gravity forces in it. The effects of gravity are quite negligible in particle physics, however they are crucial in cosmology and in the study of the early universe. At the same time, there is no consistent theory of quantum gravity itself. Therefore, construction of a quantum theory that would include both gravity and the other forces is fundamentally necessary. The most promising candidate for the unification of gravity with the other forces is superstring theory. In this theory all particles, including graviton, are combined in one geometrical object - string and particles are treated as its excited modes. Since all particles arise from string excitation, all particles and all forces are naturally incorporated into a single theory. GJ| String theory and critical dimensions Introduction 4/18 The fundamental theory should have a certain degree of uniqueness. A theory with adjustable dimensionless parameters, like SM, is not unique. String theory has only one dimensionful parameter - string length. It is the first sign that string theory is rather unique. Another sign of the uniqueness of string theory is the fact that it fixes the spacetime dimension by symmetry principles. Namely, the quantum realization of spacetime symmetries fixes the spacetime dimension, which is 26 for bosonic strings and it is 10 for superstrings. This numbers are called critical dimensions. Quantization of a bosonic string and a realization of the critical dimension by a new method is the main topic of the seminar. We use the static gauge approach, which is most natural from the physical point of view, however it creates hard operator ordering ambiguities. We show how one can avoid the quantization problems and how to calculate the critical dimension in a simple way. GJ| String theory and critical dimensions Introduction 5/18 Particle dynamics The geometric action Z Z p µ ν S = −m ds = −M dτ −gµνx_ x_ : The Polyakov action Z g x_ µx_ ν λ M 2 S = dτ µν − : 2λ 2 The Dirac action Z λ S = dτ p x_ µ − gµν p p + M 2 : µ 2 µ ν Connection between the velocity and momentum µ µν x_ = λg pν : The mass-shell condition µ ν 2 2 gµνx_ x_ + λ M = 0 : GJ| String theory and critical dimensions Particle dynamics 6/18 String dynamics The Nambu-Goto action 1 Z p S = − dτdσ (_x · x0)2 − (_x)2 (x0)2 : 2π The Polyakov action 1 Z p S = − dτdσ h hαβ (@ x · @ x) : 4π a β The Dirac action 1 Z Z S = dτ dσ (p · x_) − λ (p2 + (x0)2) − λ (p · x0) : 2π 1 2 Constraints p2 + (x0)2 = 0 ; p · x0 = 0 : GJ| String theory and critical dimensions String dynamics 7/18 Particle in a static space-time The action of a particle in the first order formulation Z h λ i S = dτ p x_ µ − gµνp p + M 2 : µ 2 µ ν The mass shell condition µν 2 g pµpν + M = 0 : 0 The static gauge x + p0 τ = 0 : The reduced action Z h 1 i S = dτ p x_ n − (p )2 : n 2 0 −p0 = E > 0 is the particle energy. GJ| String theory and critical dimensions Particle in a static space-time 8/18 Static space-time metric tensor g00 (x) 0 gµν = ; g00 < 0 : 0 gmn(x) Space-time coordinates xµ = (x0 ; x) : f(x) Time component g00 (x) = −e : The squared energy 2 mn 2 f(x) E = h (x) pmpn + M e : This function is associated with the Hamiltonian of a non-relativistic particle moving in the potential M 2 ef(x) in a curved background with metric tensor −f(x) hmn = e gmn(x) ; and hmn(x) is the inverse metric. GJ| String theory and critical dimensions Particle in a static space-time 9/18 Particle quantization in static gauge Scalar product with a covariant measure Z N p ∗ h 2j 1i = d x h(x) 2(x) 1(x) ; where h(x) = det hmn(x) : i The momentum operators pn = −i@n − 4 @n log h : The energy square operator 2 2 f(x) E = −∆h + aRh(x) + M e ; where ∆h is the Laplace-Beltrami operator and Rh is the scalar curvature. How to fix a? [DeWitt, Bastianeli,...] GJ| String theory and critical dimensions Particle quantization in static gauge 10/18 String dynamics in static gauge Open string action in the first order formulation Z Z π dσ _ µ µ 0 0µ 0µ S = dτ PµX − λ1(PµP + XµX ) − λ2(PµX ) : 0 π µ 0 0µ 0µ Virasoro constraints PµP + XµX = 0 ; PµX = 0 : 0 0 Static gauge X + P0τ = 0 ; P0 = 0 : The reduced action Z Z π dσ 1 _ k 2 S = dτ PkX − P0 : 0 π 2 Free-field Hamiltonian 1 Z π dσ H = P~ 2 + X~ 0 2 : 2 0 π GJ| String theory and critical dimensions String dynamics in static gauge 11/18 The free fields on the (τ; σ) strip 1 1 Xk(τ; σ) = φ k(τ + σ) + φ k(τ − σ) : 2 2 k k k k P an −inz Mode expansion φ (z) = q + p z + i n6=0 n e : Canonical Poisson brackets k l kl k l kl fp ; q g = δ ; fam; ang = im δ δm+n : The generators of conformal transformations 1 1 Z 2π dz 1 X L = eimz φ~0(z)2 = ~a ~a ; m 2 2π 2 m−n n 0 n=−∞ provide the Witt algebra fLm;Lng = i (m − n) Lm+n : The constraints read Lm = 0; m 6= 0 : GJ| String theory and critical dimensions String dynamics in static gauge 12/18 The dynamical integrals Z π dσ Z π dσ P µ = Pµ ;J µν = (Pµ Xν − Pν Xµ) : 0 π 0 π Reduced dynamical integrals X ak al P k = pk ;J kl = pkql − pkql + i −n n ; n n6=0 p i X ak P 0 = p0 = 2L ;J 0k = p0qk + n L : 0 p0 n −n n6=0 Deformed boosts 0 1 i X X ak J 0k = J 0k + f (p0) n L :::L : p0 @ j n −n1 −nj A j≥2 n1;:::;nj Poisson brackets of the form 0k k fJ ;Lmg = Am Lm : GJ| String theory and critical dimensions String dynamics in static gauge 13/18 String quantization in static gauge The ground state k k k a0j~p i = p j~p i ; anj~p i = 0 ; n > 0 : 1 2 P Virasoro generators L0 = 2 ~p + N, N = n>0 ~a−n ~an : The Virasoro algebra D−1 3 [Lm;Ln] = (m − n)Lm+n + 12 (m − m)δm+n : The physical states L1jΨi = 0 ;L2jΨi = 0 : The energy operator p0 = p~p 2 + 2(N − a) : The boost operators N i X X ak J 0k = p0qk + f (n1;::;nj )(p0)L ··· L n : p0 −n1 −nj n n=1 (n1;:::;nj ) GJ| String theory and critical dimensions String quantization in static gauge 14/18 The action of boosts on the first excited level ! i pk i f (1) J 0kj~p; 1i = qkp0 − + L ak j~p; 1i : 2p0 p0 −1 1 0k 0k 0l Phys. conditions L1J j~p; 1i = 0 ; [J ; J ]j~p; 1i = i j~p; 1i : The solution: f (1) = 1 ; a = 1 : The second level calculations ! i pk i i f (2) i f (1;1) qkp0 − + L ak + L ak + L L ak j~p; 2i ; 2p0 p0 −1 1 2 p0 −2 2 2 p0 −1 −1 2 0k 0k with conditions L1J j~p; 2i = 0;L2J j~p; 2i = 0 ; yield 1 e f (1;1) = − ; f (2) = ; e = 2(p0)2 ;D = 26 : e + 1 e + 1 GJ| String theory and critical dimensions String quantization in static gauge 15/18 Connection to the covariant quantization The physical states in covariant quantization L^mjj phi = 0 ; for m > 0; and (L^0 − 1)jj phi = 0 ; 1 L^ = L − L0 ; with L0 = a0 a0 : m m m m 2 m−n n The physical states of static gauge quantization become physical states of the covariant quantization.
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]
  • Scale Without Conformal Invariance in Membrane Theory
    Scale without conformal invariance in membrane theory Achille Mauria,∗, Mikhail I. Katsnelsona aRadboud University, Institute for Molecules and Materials, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands Abstract We investigate the relation between dilatation and conformal symmetries in the statistical mechanics of flex- ible crystalline membranes. We analyze, in particular, a well-known model which describes the fluctuations of a continuum elastic medium embedded in a higher-dimensional space. In this theory, the renormalization group flow connects a non-interacting ultraviolet fixed point, where the theory is controlled by linear elas- ticity, to an interacting infrared fixed point. By studying the structure of correlation functions and of the energy-momentum tensor, we show that, in the infrared, the theory is only scale-invariant: the dilatation symmetry is not enhanced to full conformal invariance. The model is shown to present a non-vanishing virial current which, despite being non-conserved, maintains a scaling dimension exactly equal to D − 1, even in presence of interactions. We attribute the absence of anomalous dimensions to the symmetries of the model under translations and rotations in the embedding space, which are realized as shifts of phonon fields, and which protect the renormalization of several non-invariant operators. We also note that closure of a sym- metry algebra with both shift symmetries and conformal invariance would require, in the hypothesis that phonons transform as primary fields, the presence of new shift symmetries which are not expected to hold on physical grounds. We then consider an alternative model, involving only scalar fields, which describes effective phonon-mediated interactions between local Gaussian curvatures.
    [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 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.
    [Show full text]
  • The Quantized Fermionic String
    Class 2: The quantized fermionic string Canonical quantization Light-cone quantization Spectrum of the fermionic string, GSO projection . The super-Virasoro constraints, which allow to eliminate the ghosts, are implemented and analyzed in essentially the same way as in the bosonic string. One new feature is the existence of two sectors: bosonic and fermionic, which have to be studied separately. To remove the tachyon one has to perform the so-called GSO projection, which guarantees space-time supersymmetry of the ten-dimensional theory. There are two possible space-time supersymmetric GSO projections which result in the Type IIA and Type IIB superstring. Summary The fermionic string is quantized analogously to the bosonic string, although now we'll find the critical dimension is 10 . One new feature is the existence of two sectors: bosonic and fermionic, which have to be studied separately. To remove the tachyon one has to perform the so-called GSO projection, which guarantees space-time supersymmetry of the ten-dimensional theory. There are two possible space-time supersymmetric GSO projections which result in the Type IIA and Type IIB superstring. Summary The fermionic string is quantized analogously to the bosonic string, although now we'll find the critical dimension is 10 The super-Virasoro constraints, which allow to eliminate the ghosts, are implemented and analyzed in essentially the same way as in the bosonic string. To remove the tachyon one has to perform the so-called GSO projection, which guarantees space-time supersymmetry of the ten-dimensional theory. There are two possible space-time supersymmetric GSO projections which result in the Type IIA and Type IIB superstring.
    [Show full text]
  • Lectures on the Renormalization Group in Statistical Physics
    Lectures on the renormalization group in statistical physics Riccardo Fantoni∗ Dipartimento di Scienze Molecolari e Nanosistemi, Universit`aCa’ Foscari Venezia, Calle Larga S. Marta DD2137, I-30123 Venezia, Italy (Dated: April 10, 2017) We review some of the ideas of the renormalization group in the statistical physics of classical and quantum fluids theory. The origin, the nature, the basis, the formulation, the critical exponents and scaling, relevance, irrelevance, and marginality, universality, and Wilson’s concept of flows and fixed point in a space of Hamiltonians. CONTENTS I. Notation 1 II. The origin of RG 2 III. The decay of correlation functions 3 IV. The challanges posed by critical phenomena 4 V. The critical exponents 7 A. The classical exponent values 7 B. The Ising exponent values 8 C. Exponent relations 8 VI. The Gaussian model and the upper critical dimension 9 VII. The task of RG 9 VIII. The basis and formulation 10 Acknowledgments 14 References 14 In a recent Review of Modern Physics, M. E. Fisher (Fisher, 1998) presented, to a wide audience, the ideas of the Renormalization Group (RG) theory behind statistical mechanics of matter physics and Quantum Field Theory (QFT). We will also follow the lectures given by N. Goldenfeld (Goldenfeld, 1992) at the University of Illinois at Urbana- Champaign in 1992. Despite its name the theory is not really about a group but about a semigroup since the set of transformations involved is not necessarily invertible. The theory is thought as one of the underlying ideas in the theoretical structure of QFT even if the roots of RG theory has to be looked upon the theory of critical phenomena of the statistical mechanics of matter physics.
    [Show full text]
  • 2. My First Path Integral
    2. My First Path Integral It’s now time to understand a little better how to deal with the path integral βF[m(x)] Z = m(x) e− (2.1) D Z Our strategy – at least for now – will be to work in situations where the saddle point dominates, with the integral giving small corrections to this result. In this regime, we can think of the integral as describing the thermal fluctuations of the order parameter m(x) around the equilibrium configuration determined by the saddle point. As we will see, this approach fails to work at the critical point, which is the regime we’re most interested in. We will then have to search for other techniques, which we will describe in Section 3. Preparing the Scene Before we get going, we’re going to change notation. First, we will change the name of our main character and write the magnetisation as m(x) φ(x) ! If you want a reason for this, I could tell you that the change of name is in deference to universality and the fact that the field could describe many things, not just magnetisa- tion. But the real reason is simply that fields in path integrals should have names like φ. (This is especially true in quantum field theory where m is reserved for the mass of the particle.) We start by setting B =0;we’llturnB back on in Section 2.2.Thefreeenergyis then 1 1 1 F [φ(x)] = ddx ↵ (T )φ2 + ↵ (T )φ4 + γ(T )( φ)2 + ... 2 2 4 4 2 r Z Roughly speaking, path integrals are trivial to do if F [φ(x)] is quadratic in φ,and possible to do if the higher order terms in F [φ(x)] give small corrections.
    [Show full text]
  • Critical Points of Coupled Vector-Ising Systems. Exact Results
    Critical points of coupled vector-Ising systems. Exact results Gesualdo Delfino1,2 and Noel Lamsen1,2 1SISSA, Via Bonomea 265, 34136 Trieste, Italy 2INFN sezione di Trieste, 34100 Trieste, Italy Abstract We show that scale invariant scattering theory allows to exactly determine the critical points of two-dimensional systems with coupled O(N) and Ising order pameters. The results are obtained for N continuous and include criticality of loop gas type. In particular, for N =1 we exhibit three critical lines intersecting at the Berezinskii-Kosterlitz-Thouless transition point of the Gaussian model and related to the Z4 symmetry of the isotropic Ashkin-Teller model. For N = 2 we classify the critical points that can arise in the XY-Ising model and provide exact answers about the critical exponents of the fully frustrated XY model. arXiv:1902.09901v1 [cond-mat.stat-mech] 26 Feb 2019 When a statistical mechanical system possesses two order parameters, phase transitions associated with each of them can take place at different points of the phase diagram. It is possible, however, that the two types of ordering set in at the same point, and that this gives rise to novel critical behavior with new critical exponents. The example of a vector order parameter for O(N) symmetry coupled to a scalar (Ising) order parameter for Z2 symmetry is paradigmatic of the combination of continuous and discrete symmetries and was addressed since the early days of the perturbative expansion in 4 ε dimensions [1]. The case N = 2 (XY- − Ising model) [2] has been higly debated in two dimensions also because it shares the ground state degeneracy of the fully frustrated (FF) XY model [3] describing a Josephson-junction array in a magnetic field [4].
    [Show full text]
  • Introduction to Superstring Theory
    Introduction to Superstring Theory Carmen N´u~nez IAFE (UBA-CONICET) & PHYSICS DEPT. (University of Buenos Aires) VI ICTP LASS 2015 Mexico, 26 October- 6 November 2015 . Programme Class 1: The classical fermionic string Class 2: The quantized fermionic string Class 3: Partition Function Class 4: Interactions . Outline Class 1: The classical fermionic string The action and its symmetries Gauge fixing and constraints Equations of motion and boundary conditions Oscillator expansions . The spectra of bosonic strings contain a tachyon ! it might indicate the vacuum has been incorrectly identified. The mass squared of a particle T is the quadratic term in the 2 @2V (T ) j − 4 ) action: M = @T 2 T =0 = α0 = we are expanding around a maximum of V . If there is some other stable vacuum, this is not an actual inconsistency. Why superstrings? . The mass squared of a particle T is the quadratic term in the 2 @2V (T ) j − 4 ) action: M = @T 2 T =0 = α0 = we are expanding around a maximum of V . If there is some other stable vacuum, this is not an actual inconsistency. Why superstrings? The spectra of bosonic strings contain a tachyon ! it might indicate the vacuum has been incorrectly identified. If there is some other stable vacuum, this is not an actual inconsistency. Why superstrings? The spectra of bosonic strings contain a tachyon ! it might indicate the vacuum has been incorrectly identified. The mass squared of a particle T is the quadratic term in the 2 @2V (T ) j − 4 ) action: M = @T 2 T =0 = α0 = we are expanding around a maximum of V .
    [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]
  • Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory
    Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory Roberto Fern´andez J¨urg Fr¨ohlich Alan D. Sokal February 8, 2012 Mafalda cartoon “No dejes para ma˜nana” c Quino (Joaqu´ınLavado). Reprinted with permission. Foreword Simple random walks — or equivalently, sums of independent random vari- ables — have long been a standard topic of probability theory and mathemat- ical physics. In the 1950’s, non-Markovian random-walk models, such as the self-avoiding walk, were introduced into theoretical polymer physics, and grad- ually came to serve as a paradigm for the general theory of critical phenomena. In the past decade, random-walk expansions have evolved into an important tool for the rigorous analysis of critical phenomena in classical spin systems and of the continuum limit in quantum field theory. Among the results obtained by random-walk methods are the proof of triviality of the ϕ4 quantum field theory in space-time dimension d ( ) 4, and the proof of mean-field critical behavior 4 ≥ for ϕ and Ising models in space dimension d ( ) 4. The principal goal of the ≥ present monograph is to present a detailed review of these developments. It is supplemented by a brief excursion to the theory of random surfaces and various applications thereof. This book has grown out of research carried out by the authors mainly from 1982 until the middle of 1985. Our original intention was to write a research paper. However, the writing of such a paper turned out to be a very slow process, partly because of our geographical separation, partly because each of us was involved in other projects that may have appeared more urgent.
    [Show full text]
  • Path Integral Approach to Birth-Death Processes on a Lattice for Memory Effects [22]
    J. Physique 46 (1985) 1469-1483 SEPTEMBRE 1985, 1469 Classification Physics Abstracts 05.40 - 68.70 - 02.50 Path integral approach to birth-death processes on a lattice L. Peliti Dipartimento di Fisica, Universita « La Sapienza », Piazzale Aldo Moro 2, I-00185 Roma and Gruppo Nazionale di Struttura della Materia, Unità di Roma, Roma, Italy (Reçu le 8 mars 1985, accepté le 22 mai 1985) Résumé. 2014 On donne une formulation par intégrales de chemin du formalisme de Fock pour objets classiques, premièrement introduit par Doi, et on l’applique à des processus généraux de naissance et mort sur réseau. L’intro- duction de variables auxiliaires permet de donner une forme Markovienne aux lois d’évolution des chemins aléa- toires avec mémoire et des processus irréversibles d’agrégation. Les théories des champs existantes pour ces pro- cessus sont obtenues dans la limite continue. On discute brièvement des implications de cette méthode pour leur comportement asymptotique. Abstract 2014 The Fock space formalism for classical objects first introduced by Doi is cast in a path integral form and applied to general birth-death processes on a lattice. The introduction of suitable auxiliary variables allows one to formulate random walks with memory and irreversible aggregation processes in a Markovian way, which is treatable in this formalism. Existing field theories of such processes are recovered in the continuum limit. Impli- cations of the method for their asymptotic behaviour are briefly discussed. the which has a renormalizable 1. Introduction. process, perturbation theory around some upper critical dimension D,. A certain number of models of irreversible aggregation For the case of random walks with memory such processes, which lead to the formation of objects field theories have been introduced on a heuristic possessing some sort of scale invariance, have been basis [12, 13, 16].
    [Show full text]
  • Introduction to String Theory A.N
    Introduction to String Theory A.N. Schellekens Based on lectures given at the Radboud Universiteit, Nijmegen [Word cloud by www.worldle.net] Contents 1 Current Problems in Particle Physics7 1.1 Problems of Quantum Gravity.........................9 1.2 String Diagrams................................. 11 2 Bosonic String Action 15 2.1 The Nambu-Goto action............................ 15 2.2 The Free Boson Action............................. 16 2.3 World sheet versus Space-time......................... 18 2.4 Symmetries................................... 18 2.5 Conformal Gauge................................ 20 2.6 The Equations of Motion............................ 20 2.7 Conformal Invariance.............................. 22 3 String Spectra 23 3.1 Mode Expansion................................ 23 3.1.1 Closed Strings.............................. 23 3.1.2 Open String Boundary Conditions................... 24 3.1.3 Open String Mode Expansion..................... 25 3.1.4 Open versus Closed........................... 25 3.2 Quantization.................................. 26 3.3 Negative Norm States............................. 27 3.4 Constraints................................... 27 3.5 Mode Expansion of the Constraints...................... 27 3.6 The Virasoro Constraints............................ 29 3.7 Operator Ordering............................... 29 3.8 Commutators of Constraints.......................... 30 3.9 Computation of the Central Charge...................... 31 3.10 The Virasoro Algebra.............................. 32 3.11
    [Show full text]