DOCSLIB.ORG

Fixed Points of Quantum Gravity

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Fixed Points of Quantum Gravity PHYSICAL REVIEW LETTERS week ending VOLUME 92, NUMBER 20 21 MAY 2004 Fixed Points of Quantum Gravity Daniel F. Litim* Theory Division, CERN, CH-1211 Geneva 23 and School of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, United Kingdom (Received 15 December 2003; published 19 May 2004) Euclidean quantum gravity is studied with renormalization group methods. Analytical results for a nontrivial ultraviolet fixed point are found for arbitrary dimensions and gauge fixing parameters in the Einstein-Hilbert truncation. Implications for quantum gravity in four dimensions are discussed. DOI: 10.1103/PhysRevLett.92.201301 PACS numbers: 04.60.–m, 11.10.Hi, 11.15.Tk Classical general relativity is acknowledged as the R2 g interactions [8] or noninteracting matter fields theory of gravitational interactions for distances suffi- [13]. Further indications for the existence of a fixed point ciently large compared to the Planck length. At smaller are based on dimensionally reduced theories [14] and on length scales, quantum effects are expected to become numerical studies within simplicial gravity [15]. For phe- important. The quantization of general relativity, how- nomenological applications, see [16]. ever, still poses problems. It has long been known that In this Letter, we study fixed points of quantum gravity quantum gravity is perturbatively nonrenormalizable, in the approach put forward in [6], amended by an ade- meaning that an infinite number of parameters have to quate optimization [4,5]. The main new result is the be fixed to renormalize standard perturbation theory. It existence of a nontrivial ultraviolet fixed point in the has been suggested that Einstein gravity may be non- Einstein-Hilbert truncation in dimensions higher than perturbatively renormalizable, a scenario known as 4, a region which previously has not been accessible. asymptotic safety [1]. This requires the existence of a Analytical results for the flow and its fixed points are nontrivial ultraviolet fixed point with at most a finite given for arbitrary dimension. The optimization ensures number of unstable directions. Then it would suffice to the maximal reliability of the result in the present trun- adjust a finite number of parameters, ideally taken from cation, thereby strengthening earlier findings in four di- experiment, to make the theory asymptotically safe. mensions. Implications of these results are discussed. Nonperturbative renormalizability has already been es- The Exact Renormalization Group is based on a mo- tablished for a number of field theories [2]. mentum cutoff for the propagating degrees of freedom The search for fixed points in quantum field theory and describes the change of the scale-dependent effective calls for a renormalization group study. A particularly action ÿk under an infinitesimal change of the cutoff scale useful approach is given by the Exact Renormalization k. Thereby it interpolates between a microscopic action in Group, based on the integrating out of momentum modes the ultraviolet and the full quantum effective action in the from a path integral representation of the theory [3]. The infrared, where the cutoff is removed. In its modern strength of the method is its flexibility when it comes to formulation, the renormalization group flow of ÿk with approximations. General optimization procedures are the logarithmic scale parameter t lnk is given by available [4], increasing the domain of validity and the @ ÿ 1Tr ÿ 2 Rÿ1@ R: (1) convergence of the flow. Consequently, the reliability of t k 2 k t results based on optimized flows is enhanced [5]. The trace stands for a sum over indices and a loop Explicit flow equations for Euclidean quantum gravity integration, and R (not to be confused with the Ricci have been constructed by Reuter [6], using background scalar) is an appropriately defined momentum cutoff at field techniques [6–8]. Diffeomorphism invariance under the momentum scale q2 k2. For quantum gravity, we local coordinate transformations is controlled by modi- consider the flow (1) for ÿkg in the Einstein-Hilbert fied Ward identities, similar to those known for non- truncation, retaining the volume element and the Ricci Abelian gauge theories [9]. In general, methods originally scalar as independent operators. Apart from a classical developed for gauge theories [10], with minor modifica- gauge fixing, the effective action is given by Z tions, can now be applied to quantum gravity. p 1 d So far most studies have been concerned with thep ÿk d x gÿR g2k: (2) g 16Gk Einstein-Hilbertp truncation based on the operators and gR g in the effective action, where g is the deter- In (2) we introduced the gravitational coupling constant minant of the metric tensor g and R g the Ricci scalar. Gk and the cosmological constant k. The ansatz (2) In four dimensions, the high energy behavior of quantum differs from the Einstein-Hilbert action in d Euclidean gravity is dominated by a nontrivial ultraviolet fixed dimensions in that the couplings have turned into point [7,11,12], which is stable under the inclusion of scale-dependent functions. Denoting G and the 201301-1 0031-9007=04=92(20)=201301(4)$22.50 2004 The American Physical Society 201301-1 PHYSICAL REVIEW LETTERS week ending VOLUME 92, NUMBER 20 21 MAY 2004 unrenormalized Newtonian coupling and cosmological g d ÿ 2g modulo subleading corrections. This en- constant at some reference scale k ,andZN;k the tails 0. Then the flow is trivially solved by gk dÿ2 ÿ2 wave function renormalization factor for the Newtonian g k= and k k= , the canonical scaling coupling, we introduce dimensionless renormalized cou- of the couplings as implied by (3). Here, denotes the plings as momentum scale where the canonical scaling regime is reached. In the infrared limit g=g k=d2 g kdÿ2G kdÿ2Zÿ1 GG; kÿ2 : (3) bound k k N;k k k becomes increasingly small for any dimension. Their flows follow from (1) by an appropriate projection In the remaining part, we discuss the fixed points of (4) onto the operators in (2). A scaling solution of the flow (1) and their implications. A first understanding is achieved in the truncation (2) corresponds to fixed points for the in the limit of vanishing cosmological constant, where couplings (3). Explicit momentum cutoffs have been provided for the 1 ÿ 4dg d ÿ 2g g : (6) fluctuations in the metric field in the Feynman gauge [6] 1 2 2 ÿ dg and for its component fields in a traceless transverse The flow (6) displays two fixed points: the Gaussian one at decomposition in a harmonic background field gauge g 0 and a non-Gaussian one at g 1= 4d.Inthe with the gauge fixing parameter [7]. In either case the vicinity of the nontrivial fixed point for large k,the tensor structure of the regulator is fixed, while the scalar gravitational coupling behaves as G g =kdÿ2.This part is left free. Here, we employ the optimized cutoff k behavior is similar to asymptotic freedom in Yang- R k2 ÿ q2 k2 ÿ q2 for the scalar part [4,5]. In opt Mills theories. The anomalous dimension of the graviton the setup (1)–(3), we have computed the flow equation for (5) remains finite and 0 for all g between the infrared arbitrary and arbitrary dimension. To simplify the and the ultraviolet fixed point, because g g < notation, a factor 1= is absorbed into the definition k g 0 for all d>ÿ2 and g g . At the fixed point, (3). Below we present explicit formulas only for the limit bound universal observables are the eigenvalues of the stability !1where the results take their simplest form. The matrix at criticality, i.e., the critical exponents. The uni- general case is discussed elsewhere. The functions are versal eigenvalue is given by @ g=@gj ÿ. We find P @ 1 ; d ÿ 2 t P 4 d 2g 2 ÿ d; 2d (7) 2 (4) G NG d 2 d ÿ 2gP2 g @tg d ÿ 2 g ; P2 4 d 2g for the Gaussian (G) and the non-Gaussian (NG) fixed points, respectively. The eigenvalues at criticality have with polynomials P1;2 ; g; d, opposite signs, the Gaussian one being infrared attractive 3 2 2 3 and the non-Gaussian one being ultraviolet attractive. P1 ÿ16 4 4 ÿ 10dg ÿ 3d g d g They are degenerate in two dimensions. Away from the 2 ÿ 3 ÿ 4 10dg d g d g 1 fixed points, the flow (6) can be solved analytically for d 2 d d ÿ 16g 8dg ÿ 3g; arbitrary scales k. With the initial condition g at k , P 8 2 ÿ ÿ dg2: the solution gk for any k is 2 ÿ1= ÿ1= d d=2ÿ1 gk G g ÿ gk NG k A numerical factor cd ÿ 2 2 4 originating : (8) mainly from the momentum trace in (1) is scaled into g g ÿ g the gravitational coupling g ! g=c , unless indicated d Figure 1 shows the crossover from the infrared to the otherwise. The graviton anomalous dimension is given by ultraviolet fixed point in the analytic solution (8) in four d ÿ 2 d 2g : (5) 1 2 6 d ÿ 2g ÿ 2 ÿ 2 g It vanishes for vanishing gravitational coupling, in two 4 and minus two dimensions, or for diverging .Ona nontrivial fixed point the vanishing of g implies 2 2 ÿ d and reflects the fact that the gravitational coupling IR UV is dimensionless in 2d. This behavior leads to modifica- 0 η tions of the graviton propagator at large momenta, e.g., [7]. The flows (4) are finite except on the boundary -2 2 gbound 2 ÿ 1 = 2d ÿ 4 derived from 1= 0.
Load more

Recommended publications
  • Ultraviolet Fixed Point Structure of Renormalizable Four-Fermion Theory in Less Than Four Dimensions*
    160 Ultraviolet Fixed Point Structure of Renormalizable Four-Fermion Theory in Less Than Four Dimensions* Yoshio Kikukawa Department of Physics, Nagoya University, Nagoya 464-01,Japan Abstract We study the renormalization properties of the four-fermion theory in less than four dimen­ sions (D < 4) in 1/N expansion scheme. It is shown that /3 function of the bare coupling has a nontrivial ultraviolet fixed point with a large anomalous dimension ( 'Yti;.p = D - 2) in a similar manner to QED and gauged Nambu-Jona-Lasinio (NJL) model in ladder approximation. The anomalous dimension has no discontinuity across the fixed point in sharp contrast to gauged NJL model. The operator product expansion of the fermion mass function is also given. Introduction Recently the possibility that QED may have a nontrivial ultraviolet(UV) fixed point has been paid much attention from the viewpoints of "zero charge" problem in QED and raising condensate in technicolor model. Actually such a possibility was pointed out in ladder approximation in which the cutoff Schwinger-Dyson equation for the fermion self-energy possesses a spontaneous-chiral­ symmetry-breaking solution for the bare coupling larger than a non-zero value ( ao =eij/ 411" > ?r/3 = ac)- We can make this solution finite by letting ao have a cutoff dependence in such a way that ao( I\) _,. ac + 0 (I\ - oo ), ac being identified as the critical point with scaling behavior of essential-singularity type. At the critical point, fermion mass operator ;j;'lj; has a large anomalous dimension 'Yti;.p = 1,
    [Show full text]
  • The Real Problem with Perturbative Quantum Field Theory
    The Real Problem with Perturbative Quantum Field Theory James Duncan Fraser Abstract The perturbative approach to quantum field theory (QFT) has long been viewed with suspicion by philosophers of science. This paper offers a diag- nosis of its conceptual problems. Drawing on Norton's ([2012]) discussion of the notion of approximation I argue that perturbative QFT ought to be understood as producing approximations without specifying an underlying QFT model. This analysis leads to a reassessment of common worries about perturbative QFT. What ends up being the key issue with the approach on this picture is not mathematical rigour, or the threat of inconsistency, but the need for a physical explanation of its empirical success. 1 Three Worries about Perturbative Quantum Field Theory 2 The Perturbative Formalism 2.1 Expanding the S-matrix 2.2 Perturbative renormalization 3 Approximations and Models 4 Perturbative Quantum Field Theory Produces Approximations 5 The Real Problem 1 Three Worries about Perturbative Quantum Field Theory On the face of it, the perturbative approach to quantum field theory (QFT) ought to be of great interest to philosophers of physics.1 Perturbation theory has long 1I use the terms `perturbative approach to QFT' and `perturbative QFT' here to refer to the perturbative treatment of interacting field theories that is invariable found in QFT textbooks and forms the basis of much work in high energy physics. I am not assuming that this `approach' 1 played a special role in the QFT programme. The axiomatic and effective field theory approaches to QFT, which have been the locus of much philosophical at- tention in recent years, have their roots in the perturbative formalism pioneered by Feynman, Schwinger and Tomonaga in the 1940s.
    [Show full text]
  • Arxiv:1102.4624V1 [Hep-Th] 22 Feb 2011 (Sec
    Renormalisation group and the Planck scale Daniel F. Litim∗ Department of Physics and Astronomy, University of Sussex, Brighton, BN1 9QH, U.K. I discuss the renormalisation group approach to gravity, its link to S. Weinberg's asymptotic safety scenario, and give an overview of results with applications to particle physics and cosmology. I. INTRODUCTION Einstein's theory of general relativity is the remarkably successful classical theory of the gravitational force, charac- −11 3 2 terised by Newton's coupling constant GN = 6:67×10 m =(kg s ) and a small cosmological constant Λ. Experimen- tally, its validity has been confirmed over many orders of magnitude in length scales ranging from the sub-millimeter regime up to solar system size. At larger length scales, the standard model of cosmology including dark matter and dark energy components fits the data well. At shorter length scales, quantum effects are expected to become impor- tant. An order of magnitude estimate for the quantum scale of gravity { the Planck scale { is obtained by dimensional p 3 −33 analysis leading to the Planck length `Pl ≈ ~GN =c of the order of 10 cm, with c the speed of light. In particle physics units this translates into the Planck mass 19 MPl ≈ 10 GeV : (1) While this energy scale is presently out of reach for earth-based particle accelerator experiments, fingerprints of Planck-scale physics can nevertheless become accessible through cosmological data from the very early universe. From a theory perspective, it is widely expected that a fundamental understanding of Planck scale physics requires a quantum theory of gravity.
    [Show full text]
  • The Search for the Universality Class of Metric Quantum Gravity
    universe Article The Search for the Universality Class of Metric Quantum Gravity Riccardo Martini 1 , Alessandro Ugolotti 2 and Omar Zanusso 3,* 1 Okinawa Institute of Science, Technology Graduate University, Kunigami District, Okinawa 904-0495, Japan; [email protected] 2 Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany; [email protected] 3 Università di Pisa and INFN—Sezione di Pisa, Largo Bruno Pontecorvo 3, I-56127 Pisa, Italy * Correspondence: [email protected] Abstract: On the basis of a limited number of reasonable axioms, we discuss the classification of all the possible universality classes of diffeomorphisms invariant metric theories of quantum gravity. We use the language of the renormalization group and adopt several ideas which originate in the context of statistical mechanics and quantum field theory. Our discussion leads to several ideas that could affect the status of the asymptotic safety conjecture of quantum gravity and give universal arguments towards its proof. Keywords: quantum gravity; asymptotic safety 1. Introduction Citation: Martini, R.; Ugolotti, A.; A well-known fact is that in four dimensions a quantum theory of Einstein–Hilbert Zanusso, O. The Search for the gravity displays perturbatively non-renormalizable divergences starting at two loops [1,2]. Universality Class of Metric This happens because Newton’s constant has negative mass dimension, which in turn Quantum Gravity. Universe 2021, 7, requires a re-interpretation of the perturbative series as an effective expansion in inverse 162. https://doi.org/10.3390/ powers of the Planck mass, making the theory an effective IR model, rather than a UV universe7060162 complete one [3].
    [Show full text]
  • Arxiv:1707.04217V2 [Hep-Th] 28 Mar 2018
    More asymptotic safety guaranteed Andrew D. Bond1, ∗ and Daniel F. Litim1, y 1Department of Physics and Astronomy, U Sussex, Brighton, BN1 9QH, U.K. We study interacting fixed points and phase diagrams of simple and semi-simple quantum field theories in four dimensions involving non-abelian gauge fields, fermions and scalars in the Veneziano limit. Particular emphasis is put on new phenomena which arise due to the semi- simple nature of the theory. Using matter field multiplicities as free parameters, we find a large variety of interacting conformal fixed points with stable vacua and crossovers inbetween. Highlights include semi-simple gauge theories with exact asymptotic safety, theories with one or several interacting fixed points in the IR, theories where one of the gauge sectors is both UV free and IR free, and theories with weakly interacting fixed points in the UV and the IR limits. The phase diagrams for various simple and semi-simple settings are also given. Further aspects such as perturbativity beyond the Veneziano limit, conformal windows, and implications for model building are discussed. arXiv:1707.04217v2 [hep-th] 28 Mar 2018 ∗ [email protected] y [email protected] 2 Contents I. Introduction 3 II. Fixed points of gauge theories 4 A. Fixed points in perturbation theory 4 B. Gauge couplings 6 C. Yukawa couplings 7 D. Scalar couplings 8 E. Universal scaling exponents 9 III. Minimal models 11 A. Semi-simple gauge theory 11 B. Free parameters and Veneziano limit 13 C. Perturbativity to leading order 14 D. Anomalous dimensions 16 E. Running couplings beyond the leading order 17 IV.
    [Show full text]
  • 2 and Dark Matter Confront Asymptotic Safety
    Minimal models for g − 2 and dark matter confront asymptotic safety Kamila Kowalska∗ and Enrico Maria Sessolo† National Centre for Nuclear Research Pasteura 7, 02-093 Warsaw, Poland Abstract We use the framework of asymptotic safety above the Planck scale to constrain the parameter space of simple models of new physics that can accommodate the measured value of the anomalous magnetic moment of the muon and the relic density of dark matter. We couple parametrically to the trans-Planckian quantum physics a set of SU(2)L×U(1)Y invariant extensions of the Standard Model, each comprising an inert scalar field and one pair of colorless fermions that communicate to the muons through Yukawa-type interactions. The presence of an interactive UV fixed point in the system of gauge and Yukawa couplings imposes a set of boundary conditions at the Planck scale, which allow one to derive unique phenomenological predictions in each case and distinguish the different representations of the gauge group from one another. We apply to the models constraints from the h ! µµ signal strength at ATLAS and CMS, direct LHC searches for electroweak production with leptons and missing energy in the final state, and the dark matter relic density. We find that they further restrict the available parameter space. arXiv:2012.15200v2 [hep-ph] 27 May 2021 ∗[email protected] †[email protected] 1 Contents 1 Introduction2 2 Minimal models for the lepton g − 24 2.1 Lagrangian of the models . .7 2.2 Dark matter . 10 3 Trans-Planckian fixed points 11 3.1 General notions .
    [Show full text]
  • Effective and Fundamental Quantum Fields at Criticality
    Effective and fundamental quantum fields at criticality Dissertation zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat) vorgelegt dem Rat der Physikalisch-Astronomischen Fakult¨at der Friedrich-Schiller-Universit¨at Jena von Dipl.-Phys. Michael Scherer geboren am 2. November 1981 in Furtwangen Gutachter: 1. Prof. Dr. Holger Gies, Jena 2. Prof. Dr. Jan Pawlowski, Heidelberg 3. Prof. Dr. Litim, Sussex Tag der Disputation: 28. Oktober 2010 Effektive und fundamentale Quantenfelder an der Kritikalit¨at Zusammenfassung Die funktionale Renormierungsgruppe in der Formulierung von Wetterich wird als ge- eignete nicht-st¨orungstheoretische Methode f¨ur die qualitative und quantitative Unter- suchung universeller Ph¨anomene in Quantenfeldtheorien verwendet. Es werden Fluss- gleichungen f¨ur eine Klasse chiraler Yukawa-Modelle mit und ohne Eichbosonen abgeleitet und deren Fixpunktstruktur untersucht. Die vierdimensionalen chiralen Yukawa-Modelle dienen als Spielzeug-Modelle f¨ur den Higgs-Sektor des Standardmodells. Eine Balance bosonischer und fermionischer Fluktuationen erm¨oglicht asymptotisch sichere Fixpunkte in der untersuchten N¨aherung, was eine Interpretation dieser Theorie als fundamentale Theorie erlaubt und das Trivialit¨ats-Problem l¨ost. Außerdem erhalten wir Vorhersagen f¨ur die Higgs- und die Topquark-Masse unseres Spielzeugmodells. In drei Dimensio- nen berechnen wir die kritischen Exponenten, die neue Universalit¨atsklassen definieren. Damit liefern wir quantitative Vorhersagen f¨ur Systeme stark korrelierter chiraler Fermio- nen, die mit anderen nicht-st¨orungstheoretischen Methoden ¨uberpr¨uft werden k¨onnen. In einem Yukawa-System nicht-relativistischer zweikomponentiger Fermionen wird der Renormierungsgruppen-Fluss ebenfalls durch einen Fixpunkt dominiert, was zu Univer- salit¨at im BCS-BEC Crossover f¨uhrt. Wir entwickeln die Methode der funktionalen Renormierung hinzu einem quantitativen Niveau und berechnen unter anderem die kri- tische Temperatur ¨uber den gesamten Crossover.
    [Show full text]
  • Where to Look for Solving the Gauge Hierarchy Problem?
    Physics Letters B 718 (2012) 573–576 Contents lists available at SciVerse ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Where to look for solving the gauge hierarchy problem? C. Wetterich Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, D-69120 Heidelberg, Germany article info abstract Article history: A mass of the Higgs boson close to 126 GeV may give a hint that the standard model of particle physics Received 19 October 2012 is valid up to the Planck scale. We discuss perspectives for the solution of the gauge hierarchy problem Accepted 8 November 2012 at high scales. Scenarios with an ultraviolet fixed point have predicted a Higgs boson mass very close to Available online 9 November 2012 126 GeV if the fixed point value of the quartic scalar coupling is small. In this case the top quark pole Editor: A. Ringwald mass should be close to 172 GeV. © 2012 Elsevier B.V. All rights reserved. The ATLAS and CMS Collaborations at the LHC have announced fields being straightforward.) The relevant terms in the effective evidence for a mass of the Higgs boson in the range of 125– potential for ϕ and χ are 126 GeV [1,2]. A mass in the vicinity of 126 GeV has been pre- 1 † 2 † 2 dicted [3] within models of non-perturbative renormalizability of U = λ ϕ ϕ + γ ϕ ϕ χ + Uχ (χ), (1) gravity [4,5] which lead to a small value of the quartic scalar cou- 2 pling near the Planck scale. In this Letter we ask if a Higgs boson with dimensionless couplings λ and γ .(Termsm2ϕ†ϕ or νϕ†ϕχ mass around 126 GeV, if confirmed, could give hints for the energy can be absorbed by a redefinition of χ .) We choose conventions scale where a possible solution of the gauge hierarchy problem for χ such that its expectation value equals the reduced Planck could be found.
    [Show full text]
  • Exploring New Windows Into Fundamental Physics at High
    Exploring New Windows into Fundamental Physics at High Energies Published by Proefschriftmaken phone: +31(0)883746777 De Limiet 26 e-mail: [email protected] 4131 NR Vianen http://www.Proefchriftmaken.nl The Netherlands Cover illustration: Planck mapped the strength (color) and polarization (texture) of ra- diation from galactic dust. This is a photo from ESA/PLANCK COLLABORATION. One can find this photo from http://science.sciencemag.org/content/347/6222/595/tab- figures-data. Adrian Cho has used this figure. ISBN: 978-94-6380-312-0 NUR: 925 Copyright c 2019 Leihua Liu. All rights reserved. Exploring New Windows into Fundamental Physics at High Energies Nieuwe Invalshoeken in de Fundamentele Fysica van Hoge Energiee¨n (met een samenvatting in het Nederlands) Proefschrift ter verkrijging van de graad van doctor aan de Universiteit Utrecht op gezag van de rector magnificus, prof. dr. H.R.B.M. Kummeling, ingevolge het besluit van het college voor promoties in het openbaar te verdedigen op woensdag 24 april 2019 des middags te 12.45 uur door Leihua Liu geboren 9 september 1985 te Anhua, China Promotor: Prof. dr. S.J.G. Vandoren Copromoter: Dr. T. Prokopec This thesis is dedicated to my parents and wife, for their endless love and support. Contents List of Tables . ix List of Figures . xi 1 Introduction 1 1.1 A big picture of the Universe . 1 1.2 Why inflation? . 3 1.3 Inflationary model building . 4 1.4 The nature of gravity . 5 1.5 Outline of the thesis . 6 2 Standard cosmological model 7 2.1 A brief history of cosmology .
    [Show full text]
  • Renormalization Group - Wikipedia, the Free Encyclopedia 3/17/12 5:27 PM Renormalization Group from Wikipedia, the Free Encyclopedia
    Renormalization group - Wikipedia, the free encyclopedia 3/17/12 5:27 PM Renormalization group From Wikipedia, the free encyclopedia In theoretical physics, the renormalization group (RG) refers to a mathematical apparatus that allows systematic investigation of the changes of a physical system as viewed at different distance scales. In particle physics, it reflects the changes in the underlying force laws (codified in a quantum field theory) as the energy scale at which physical processes occur varies, energy/momentum and resolution distance scales being effectively conjugate under the uncertainty principle (cf. Compton wavelength). A change in scale is called a "scale transformation". The renormalization group is intimately related to "scale invariance" and "conformal invariance", symmetries in which a system appears the same at all scales (so-called self-similarity). (However, note that scale transformations are included in conformal transformations, in general: the latter including additional symmetry generators associated with special conformal transformations.) As the scale varies, it is as if one is changing the magnifying power of a notional microscope viewing the system. In so- called renormalizable theories, the system at one scale will generally be seen to consist of self-similar copies of itself when viewed at a smaller scale, with different parameters describing the components of the system. The components, or fundamental variables, may relate to atoms, elementary particles, atomic spins, etc. The parameters of the theory typically describe the interactions of the components. These may be variable "couplings" which measure the strength of various forces, or mass parameters themselves. The components themselves may appear to be composed of more of the self-same components as one goes to shorter distances.
    [Show full text]
  • Arxiv:1904.07042V1 [Gr-Qc] 15 Apr 2019
    Quantum spacetime and the renormalization group: Progress and visions Antonio D. Pereira1, 2, ∗ 1Instituto de F´ısica, Universidade Federal Fluminense, Campus da Praia Vermelha, Av. Litor^anea s/n, 24210-346, Niter´oi,RJ, Brazil 2Institute for Theoretical Physics, University of Heidelberg, Philosophenweg 12, 69120 Heidelberg, Germany The quest for a consistent theory which describes the quantum microstructure of spacetime seems to require some departure from the paradigms that have been followed in the construction of quan- tum theories for the other fundamental interactions. In this contribution we briefly review two approaches to quantum gravity, namely, asymptotically safe quantum gravity and tensor models, based on different theoretical assumptions. Nevertheless, the main goal is to find a universal con- tinuum limit for such theories and we explain how coarse-graining techniques should be adapted to each case. Finally, we argue that although seemingly different, such approaches might be just two sides of the same coin. I. INTRODUCTION The construction of the Standard Model (SM) of particle physics under the perturbative and continuum quantum- field theoretic dogmas led to a successful theory beyond dispute. The detection of the Higgs [1] and the absence of new physics at the LHC so far have crowned the SM as a very accurate quantum description of the fundamental interactions but gravity. However, there are some puzzles which are not addressed by the SM as, e.g., neutrino masses and dark matter. Moreover, the SM is not a fundamental quantum field theory valid up to arbitrarily short scales due to the existence of a Landau pole being thus valid up to some cutoff scale, [2{5].
    [Show full text]
  • Asymptotic Safety of Scalar Field Theories
    Asymptotic safety of scalar field theories Daniel F. Litim1 and Matthew J. Trott1, 2 1Department of Physics and Astronomy, University of Sussex, BN1 9QH, Brighton, UK 2 SUPA, School of Physics and Astronomy, University of St Andrews, KY16 9SS, St Andrews, UK We study 3d O(N) symmetric scalar field theories using Polchinski's renormalisation group. In the infinite N limit the model is solved exactly including at strong coupling. At short dis- tances the theory is described by a line of asymptotically safe ultraviolet fixed points bounded by asymptotic freedom at weak, and the Bardeen-Moshe-Bander phenomenon at strong sex- tic coupling. The Wilson-Fisher fixed point arises as an isolated low-energy fixed point. Further results include the conformal window for asymptotic safety, convergence-limiting poles in the complex field plane, and the phase diagram with regions of first and second order phase transitions. We substantiate a duality between Polchinski's and Wetterich's versions of the functional renormalisation group, also showing that that eigenperturbations are identical at any fixed point. At a critical sextic coupling, the duality is worked out in detail to explain the spontaneous breaking of scale symmetry responsible for the generation of a light dilaton. Implications for asymptotic safety in other theories are indicated. Contents I. Introduction 2 II. Renormalisation group 3 A. Polchinski equation3 B. Wetterich equation5 C. Duality 6 III. Running couplings 8 A. Local flows 9 B. Global flows 10 C. Interacting fixed points 11 IV. Infrared fixed point 13 A. Wilson-Fisher fixed point 13 B. Bootstrap for scaling exponents 14 C.
    [Show full text]