DOCSLIB.ORG

Introduction to Quantum Field Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Introduction to Quantum Field Theory Introduction to Quantum Field Theory John Cardy Michaelmas Term 2010 { Version 13/9/10 Abstract These notes are intended to supplement the lecture course `Introduction to Quan- tum Field Theory' and are not intended for wider distribution. Any errors or obvious omissions should be communicated to me at [email protected]. Contents 1 A Brief History of Quantum Field Theory 2 2 The Feynman path integral in particle quantum mechanics 4 2.1 Imaginary time path integrals and statistical mechanics . 7 3 Path integrals in ¯eld theory 9 3.1 Field theory action functionals . 10 3.2 The generating functional . 11 3.3 The propagator in free ¯eld theory . 14 4 Interacting ¯eld theories 18 4.1 Feynman diagrams . 18 4.2 Evaluation of Feynman diagrams . 26 5 Renormalisation 29 5.1 Analysis of divergences . 29 5.2 Mass, ¯eld, and coupling constant renormalisation . 32 1 QFT1 2 6 Renormalisation Group 39 6.1 Callan-Symanzik equation . 40 6.2 Renormalisation group flows . 41 6.3 One-loop computation in ¸Á4 theory . 44 6.4 Application to critical behaviour in statistical mechanics . 46 6.5 Large N ............................ 50 7 From Feynman diagrams to Cross-sections 53 7.1 The S-matrix: analyticity and unitarity . 58 8 Path integrals for fermions 62 1 A Brief History of Quantum Field Theory Quantum ¯eld theory (QFT) is a subject which has evolved considerably over the years and continues to do so. From its beginnings in elementary particle physics it has found applications in many other branches of science, in particular condensed matter physics but also as far a¯eld as biology and economics. In this course we shall be adopting an approach (the path integral) which was not the original one, but became popular, even essential, with new advances in the 1970s. However, to set this in its context, it is useful to have some historical perspective on the development of the subject (dates are only rough). ² 19th C. Maxwell's equations { a classical ¯eld theory for electromag- netism. ² 1900: Planck hypothesises the photon as the quantum of radiation. ² 1920s/30s: development of particle quantum mechanics: the same rules when applied to the Maxwell ¯eld predict photons. However relativistic particle quantum mechanics has problems (negative energy states.) ² 1930s/40s: realisation that relativity + quantum mechanics, in which particles can be created and destroyed, needs a many-particle descrip- QFT1 3 tion where the particles are the quanta of a quantised classical ¯eld theory, in analogy with photons. ² 1940s: formulation of the calculation rules for quantum electrodynam- ics (QED) { Feynman diagrams; the formulation of the path integral approach. ² 1950s: the understanding of how to deal with the divergences of Feyn- man diagrams through renormalisation; QFT methods begin to be applied to other many-body systems eg in condensed matter. ² 1960s: QFT languishes { how can it apply to weak + strong interac- tions? ² 1970s: renormalisation of non-Abelian gauge theories, the renormal- isation group (RG) and asymptotic freedom; the formulation of the Standard Model ² 1970s: further development of path integral + RG methods: applica- tions to critical behaviour. ² 1970s: non-perturbative methods, lattice gauge theory. ² 1980s: string theory + quantum gravity, conformal ¯eld theory (CFT); the realisation that all quantum ¯eld theories are only e®ective over some range of length and energy scales, and those used in particle physics are no more fundamental than in condensed matter. ² 1990s/2000s: holography and strong coupling results for gauge ¯eld theories; many applications of CFT in condensed matter physics. Where does this course ¯t in? In 16 lectures, we cannot go very far, or treat the subject in much depth. In addition this course is aimed at a wide range of students, from exper- imental particle physicists, through high energy theorists, to condensed matter physicists (with maybe a few theoretical chemists, quantum com- puting types and mathematicians thrown in). Therefore all I can hope to do is to give you some of the basic ideas, illustrated in their most simple contexts. The hope is to take you all from the Feynman path integral, through a solid grounding in Feynman diagrams, to renormalisation and QFT1 4 the RG. From there hopefully you will have enough background to un- derstand Feynman diagrams and their uses in particle physics, and have the basis for understanding gauge theories as well as applications of ¯eld theory and RG methods in condensed matter physics. 2 The Feynman path integral in particle quantum mechanics In this lecture we will recall the Feynman path integral for a system with a single degree of freedom, in preparation for the ¯eld theory case of many degrees of freedom. Consider a non-relativistic particle of unit mass moving in one dimension. The coordinate operator isq ^, and the momentum operator isp ^. (I'll be careful to distinguish operators and c-numbers.) Of course [^q; p^] = ih¹. We denote the eigenstates ofq ^ by jq0i, thusq ^jq0i = q0jq0i, and hq0jq00i = ±(q0 ¡ q00). ^ 1 2 Suppose the hamiltonian has the form H = 2p^ + V (^q) (we can consider more general forms { see later.) The classical action corresponding to this is Z t h i f 1 2 S[q] = 2q_ ¡ V (q(t)) dt ti where q(t) is a possible classical trajectory, or path. According to Hamil- ton's principle, the actual classical path is the one which extremises S { this gives Lagrange's equations. The quantum amplitude for the particle to be at qf at time tf given that it was at qi at time ti is ¡iH^ (tf ¡ti)=¹h M = hqf je jqii : According to Feynman, this amplitude is equivalently given by the path integral Z I = [dq] eiS[q]=¹h which is a integral over all functions (or paths) q(t) which satisfy q(ti) = qi, q(tf ) = qf . Obviously this needs to be better de¯ned, but we will try to make sense of it as we go along. QFT1 5 t2 t1 Figure 1: We can imagine doing the path integral by ¯rst ¯xing the values of q(t) at times (t1; t2;:::). In order to understand why this might be true, ¯rst split the interval (ti; tf ) into smaller pieces (tf ; tn¡1; : : : ; tj+1; tj; : : : ; t1; ti) with tj+1 ¡ tj = ¢t. Our matrix element can then be written N factors z }| { ¡iH^ ¢t=¹h ¡iH^ ¢t=¹h M = hqf j e : : : e jqii (Note that we could equally well have considered a time-dependent hamil- tonian, in which case each factor would be di®erent.) Now insert a complete set of eigenstates ofq ^ between each factor, eg at time-slice tj insert Z 1 dq(tj)jq(tj)ihq(tj)j ¡1 so that Z Y ¡iH^ ¢t=¹h M = dq(tj)hq(tj+1)je jq(tj)i j R On the other hand, we can think of doing the path integral [dq] by ¯rst ¯xing the values fq(tj)g at times ftjg (see Fig. 1) and doing the integrals over the intermediate points on the path, and then doing the integral over the fq(tj)g. Thus Z Z R t Y j+1 1 2 (i=¹h) t ( 2 q_ ¡V (q(t)))dt I = dq(tj) [dq(t)] e j j Thus we can prove that M = I in general if we can show that Z R t ^ j+1 1 2 ¡iH¢t=¹h (i=¹h) t ( 2 q_ ¡V (q(t)))dt hq(tj+1)je jq(tj)i = [dq(t)] e j QFT1 6 for an arbitrarily short time interval ¢t. First consider the case when V = 0. The path integral is Z R t (i=2¹h) j+1 q_2dt [dq]e tj Let q(t) = qc(t) + ±q(t) where qc(t) interpolates linearly between q(tj) and q(tj+1), that is ¡1 qc(t) = q(tj) + (¢t) (t ¡ tj)(q(tj+1) ¡ q(tj)) and ±q(tj+1) = ±q(tj) = 0. Then 0 1 Z 2 Z tj+1 2 q(tj+1) ¡ q(tj) 2 q_ dt = (¢t) @ A + (±q_) dt tj ¢t and Z R t 2 Z R j+1 2 2 (i=2¹h) t q_ dt i(q(t )¡q(t )) =2¹h¢t (i=2¹h) (±q_) dt [dq]e j = e j+1 j [d(±q)]e The second factor depends on ¢t but not q(tj+1) or q(tj), and can be absorbed into the de¯nition, or normalisation, of the functional integral. The ¯rst factor we recognise as the spreading of a wave packet initially localised at q(tj) over the time interval ¢t. This is given by usual quantum mechanics as ¡ip^2¢t=2¹h hq(tj+1)je jq(tj)i (and this can be checked explicitly using the SchrÄodingerequation.) Now we argue, for V 6= 0, that if ¢t is small the spreading of the wave packet is small, and therefore we can approximate V (q) by (say) V (q(tj)). Thus, as ¢t ! 0, Z R t j+1 1 2 1 2 (i=¹h) t ( 2 q_ ¡V (q(t)))dt ¡i(¢t=¹h)( q^ +V (^q)) [dq]e j » hq(tj+1)je 2 jq(tj)i Putting all the pieces together, an integrating over the fq(tj)g, we obtain the result we want. As well as being very useful for all sorts of computations, the path integral also provides an intuitive way of thinking about classical mechanics as a R limit of quantum mechanics. Ash ¹ ! 0 in the path integral [dq]eiS[q]=¹h, the important paths are those corresponding to stationary phase, where ±S[q]=±q = 0.
Load more

Recommended publications
  • Arxiv:2009.05574V4 [Hep-Th] 9 Nov 2020 Predict a New Massless Spin One Boson [The ‘Lorentz’ Boson] Which Should Be Looked for in Experiments
    Trace dynamics and division algebras: towards quantum gravity and unification Tejinder P. Singh Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India e-mail: [email protected] Accepted for publication in Zeitschrift fur Naturforschung A on October 4, 2020 v4. Submitted to arXiv.org [hep-th] on November 9, 2020 ABSTRACT We have recently proposed a Lagrangian in trace dynamics at the Planck scale, for unification of gravitation, Yang-Mills fields, and fermions. Dynamical variables are described by odd- grade (fermionic) and even-grade (bosonic) Grassmann matrices. Evolution takes place in Connes time. At energies much lower than Planck scale, trace dynamics reduces to quantum field theory. In the present paper we explain that the correct understanding of spin requires us to formulate the theory in 8-D octonionic space. The automorphisms of the octonion algebra, which belong to the smallest exceptional Lie group G2, replace space- time diffeomorphisms and internal gauge transformations, bringing them under a common unified fold. Building on earlier work by other researchers on division algebras, we propose the Lorentz-weak unification at the Planck scale, the symmetry group being the stabiliser group of the quaternions inside the octonions. This is one of the two maximal sub-groups of G2, the other one being SU(3), the element preserver group of octonions. This latter group, coupled with U(1)em, describes the electro-colour symmetry, as shown earlier by Furey. We arXiv:2009.05574v4 [hep-th] 9 Nov 2020 predict a new massless spin one boson [the `Lorentz' boson] which should be looked for in experiments.
    [Show full text]
  • 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.
    [Show full text]
  • Oxford Physics Department Notes on General Relativity
    Oxford Physics Department Notes on General Relativity S. Balbus 1 Recommended Texts Weinberg, S. 1972, Gravitation and Cosmology. Principles and applications of the General Theory of Relativity, (New York: John Wiley) What is now the classic reference, but lacking any physical discussions on black holes, and almost nothing on the geometrical interpretation of the equations. The author is explicit in his aversion to anything geometrical in what he views as a field theory. Alas, there is no way to make sense of equations, in any profound sense, without geometry! I also find that calculations are often performed with far too much awkwardness and unnecessary effort. Sections on physical cosmology are its main strength. To my mind, a much better pedagogical text is ... Hobson, M. P., Efstathiou, G., and Lasenby, A. N. 2006, General Relativity: An Introduction for Physicists, (Cambridge: Cambridge University Press) A very clear, very well-blended book, admirably covering the mathematics, physics, and astrophysics. Excellent coverage on black holes and gravitational radiation. The explanation of the geodesic equation is much more clear than in Weinberg. My favourite. (The metric has a different overall sign in this book compared with Weinberg and this course, so be careful.) Misner, C. W., Thorne, K. S., and Wheeler, J. A. 1972, Gravitation, (New York: Freeman) At 1280 pages, don't drop this on your toe. Even the paperback version. MTW, as it is known, is often criticised for its sheer bulk, its seemingly endless meanderings and its laboured strivings at building mathematical and physical intuition at every possible step. But I must say, in the end, there really is a lot of very good material in here, much that is difficult to find anywhere else.
    [Show full text]
  • An Introduction to Quantum Field Theory
    AN INTRODUCTION TO QUANTUM FIELD THEORY By Dr M Dasgupta University of Manchester Lecture presented at the School for Experimental High Energy Physics Students Somerville College, Oxford, September 2009 - 1 - - 2 - Contents 0 Prologue....................................................................................................... 5 1 Introduction ................................................................................................ 6 1.1 Lagrangian formalism in classical mechanics......................................... 6 1.2 Quantum mechanics................................................................................... 8 1.3 The Schrödinger picture........................................................................... 10 1.4 The Heisenberg picture............................................................................ 11 1.5 The quantum mechanical harmonic oscillator ..................................... 12 Problems .............................................................................................................. 13 2 Classical Field Theory............................................................................. 14 2.1 From N-point mechanics to field theory ............................................... 14 2.2 Relativistic field theory ............................................................................ 15 2.3 Action for a scalar field ............................................................................ 15 2.4 Plane wave solution to the Klein-Gordon equation ...........................
    [Show full text]
  • NP As Minimization of Degree 4 Polynomial, Integration Or Grassmann Number Problem, and New Graph Isomorphism Problem Approac
    1 P?=NP as minimization of degree 4 polynomial, integration or Grassmann number problem, and new graph isomorphism problem approaches Jarek Duda Jagiellonian University, Golebia 24, 31-007 Krakow, Poland, Email: [email protected] Abstract—While the P vs NP problem is mainly approached problem, vertex cover problem, independent set problem, form the point of view of discrete mathematics, this paper subset sum problem, dominating set problem and graph proposes reformulations into the field of abstract algebra, ge- coloring problem. All of them stay in widely understood ometry, fourier analysis and of continuous global optimization - which advanced tools might bring new perspectives and field of discrete mathematics, like combinatorics, graph approaches for this question. The first one is equivalence of theory, logic. satisfaction of 3-SAT problem with the question of reaching The unsuccessfulness of a half century search for zero of a nonnegative degree 4 multivariate polynomial (sum the answer might suggest to try to look out of this of squares), what could be tested from the perspective of relatively homogeneous field - try to apply advances algebra by using discriminant. It could be also approached as a continuous global optimization problem inside [0; 1]n, of more distant fields of mathematics, like abstract for example in physical realizations like adiabatic quantum algebra fluent in working with the ring of polynomials, computers. However, the number of local minima usually use properties of multidimensional geometry, or other grows exponentially. Reducing to degree 2 polynomial plus continuous mathematics including numerical methods n constraints of being in f0; 1g , we get geometric formulations perfecting approaches for common problem of continuous as the question if plane or sphere intersects with f0; 1gn.
    [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]
  • 2D Potts Model Correlation Lengths
    KOMA Revised version May D Potts Mo del Correlation Lengths Numerical Evidence for = at o d t Wolfhard Janke and Stefan Kappler Institut f ur Physik Johannes GutenbergUniversitat Mainz Staudinger Weg Mainz Germany Abstract We have studied spinspin correlation functions in the ordered phase of the twodimensional q state Potts mo del with q and at the rstorder transition p oint Through extensive Monte t Carlo simulations we obtain strong numerical evidence that the cor relation length in the ordered phase agrees with the exactly known and recently numerically conrmed correlation length in the disordered phase As a byproduct we nd the energy moments o t t d in the ordered phase at in very go o d agreement with a recent large t q expansion PACS numbers q Hk Cn Ha hep-lat/9509056 16 Sep 1995 Introduction Firstorder phase transitions have b een the sub ject of increasing interest in recent years They play an imp ortant role in many elds of physics as is witnessed by such diverse phenomena as ordinary melting the quark decon nement transition or various stages in the evolution of the early universe Even though there exists already a vast literature on this sub ject many prop erties of rstorder phase transitions still remain to b e investigated in detail Examples are nitesize scaling FSS the shap e of energy or magnetization distributions partition function zeros etc which are all closely interrelated An imp ortant approach to attack these problems are computer simulations Here the available system sizes are necessarily
    [Show full text]
  • Full and Unbiased Solution of the Dyson-Schwinger Equation in the Functional Integro-Differential Representation
    PHYSICAL REVIEW B 98, 195104 (2018) Full and unbiased solution of the Dyson-Schwinger equation in the functional integro-differential representation Tobias Pfeffer and Lode Pollet Department of Physics, Arnold Sommerfeld Center for Theoretical Physics, University of Munich, Theresienstrasse 37, 80333 Munich, Germany (Received 17 March 2018; revised manuscript received 11 October 2018; published 2 November 2018) We provide a full and unbiased solution to the Dyson-Schwinger equation illustrated for φ4 theory in 2D. It is based on an exact treatment of the functional derivative ∂/∂G of the four-point vertex function with respect to the two-point correlation function G within the framework of the homotopy analysis method (HAM) and the Monte Carlo sampling of rooted tree diagrams. The resulting series solution in deformations can be considered as an asymptotic series around G = 0 in a HAM control parameter c0G, or even a convergent one up to the phase transition point if shifts in G can be performed (such as by summing up all ladder diagrams). These considerations are equally applicable to fermionic quantum field theories and offer a fresh approach to solving functional integro-differential equations beyond any truncation scheme. DOI: 10.1103/PhysRevB.98.195104 I. INTRODUCTION was not solved and differences with the full, exact answer could be seen when the correlation length increases. Despite decades of research there continues to be a need In this work, we solve the full DSE by writing them as a for developing novel methods for strongly correlated systems. closed set of integro-differential equations. Within the HAM The standard Monte Carlo approaches [1–5] are convergent theory there exists a semianalytic way to treat the functional but suffer from a prohibitive sign problem, scaling exponen- derivatives without resorting to an infinite expansion of the tially in the system volume [6].
    [Show full text]
  • Feynman Diagrams, Momentum Space Feynman Rules, Disconnected Diagrams, Higher Correlation Functions
    PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 Dr. Anosh Joseph, IISER Mohali LECTURE 05 Tuesday, January 14, 2020 Topics: Feynman Diagrams, Momentum Space Feynman Rules, Disconnected Diagrams, Higher Correlation Functions. Feynman Diagrams We can use the Wick’s theorem to express the n-point function h0jT fφ(x1)φ(x2) ··· φ(xn)gj0i as a sum of products of Feynman propagators. We will be interested in developing a diagrammatic interpretation of such expressions. Let us look at the expression containing four fields. We have h0jT fφ1φ2φ3φ4g j0i = DF (x1 − x2)DF (x3 − x4) +DF (x1 − x3)DF (x2 − x4) +DF (x1 − x4)DF (x2 − x3): (1) We can interpret Eq. (1) as the sum of three diagrams, if we represent each of the points, x1 through x4 by a dot, and each factor DF (x − y) by a line joining x to y. These diagrams are called Feynman diagrams. In Fig. 1 we provide the diagrammatic interpretation of Eq. (1). The interpretation of the diagrams is the following: particles are created at two spacetime points, each propagates to one of the other points, and then they are annihilated. This process can happen in three different ways, and they correspond to the three ways to connect the points in pairs, as shown in the three diagrams in Fig. 1. Thus the total amplitude for the process is the sum of these three diagrams. PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 ⟨0|T {ϕ1ϕ2ϕ3ϕ4}|0⟩ = DF(x1 − x2)DF(x3 − x4) + DF(x1 − x3)DF(x2 − x4) +DF(x1 − x4)DF(x2 − x3) 1 2 1 2 1 2 + + 3 4 3 4 3 4 Figure 1: Diagrammatic interpretation of Eq.
    [Show full text]
  • 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]
  • On the Emergence of Time in Quantum Gravity
    Imperial/TP/98–99/23 On the Emergence of Time in Quantum Gravity1 J. Butterfield2 All Souls College Oxford OX1 4AL and C.J. Isham3 The Blackett Laboratory Imperial College of Science, Technology & Medicine South Kensington London SW7 2BZ 13 December 1998 Abstract We discuss from a philosophical perspective the way in which the normal concept of time might be said to ‘emerge’ in a quantum the- ory of gravity. After an introduction, we briefly discuss the notion of arXiv:gr-qc/9901024v1 8 Jan 1999 emergence, without regard to time (Section 2). We then introduce the search for a quantum theory of gravity (Section 3); and review some general interpretative issues about space, time and matter (Section 4). We then discuss the emergence of time in simple quantum ge- ometrodynamics, and in the Euclidean approach (Section 5). Section 6 concludes. 1To appear in The Arguments of Time, ed. J. Butterfield, Oxford University Press, 1999. 2email: [email protected]; jeremy.butterfi[email protected] 3email: [email protected] 1 Introduction The discovery of a satisfactory quantum theory of gravity has been widely regarded as the Holy Grail of theoretical physics for some forty years. In this essay, we will discuss a philosophical aspect of the search for such a theory that bears on our understanding of time: namely, the senses in which our standard ideas of time, and more generally spacetime, might be not fundamental to reality, but instead ‘emergent’ as an approximately valid concept on sufficiently large scales. In taking up this topic, our aim in part is to advertise to philosophers of time an unexplored area that promises to be fruitful.
    [Show full text]