DOCSLIB.ORG

Spin Foam : Modeling Quantum Space Time

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Spin Foam : Modeling Quantum Space Time Spin foam : Modeling quantum space time Laurent Freidel Perimeter Institute MG12 Paris 2009 Background independent Gravity There has been a wealth of new developments in the recent years in the covariant formulation of loop quantum gravity (spin foam formulation) Aim of this talk: •To review briefly the general principles of Loop quantum gravity •To present some of the new developments in spin foam models: The construction of a discrete path integral for non perturbative quantum gravity •To present the remarkable convergence between the two approaches The key points behind these approaches are: The Gauge principle Diffeomorphism symmetry and background independence Non perturbative dynamics Background independent Gravity One of the basic problems we want to address is: How can we describe the elementary degrees of freedom of quantum gravity in a context where there is no background spacetime? One of the basic element is that the UV divergences that one encounters in quantizing GR are of a different nature : perturbation theory breaks down at the place where we do not know how to define spacetime In quantum gravity the nature of short distance quantum spacetime geometry cannot be postulated, it should be determined dynamically What is needed is a radical change of our concept of space and time These problems can be addressed in the context of Loop quantum Gravity and Spin foams: No transplanckian dof. The key is the symmetry: Diffeomorphism invariance plays a key role. •Redundancy of the description no obs. def. of spacetime points •It determines the dynamics of the gravitational field and coupling to matter 7 153F903957A7C27ED077364961 γab Ψ(Quantumγ) Gravity 7 7 7 7 ˆ Canonical approach 7 153F903957A7C27ED077364961At the quantumH levelΨ153F903957A7C27ED077364961( γthe) = description 0 of gravity is in term of states and γab operators and the153F903957A7C27ED077364961 dynamics is encoded in constraint equations qab 153F903957A7C27ED077364961153F903957A7C27ED077364961Hˆ ΨM (γ) = 0 Ψ(γ) 5 In usual description states are functional of a spacelike metric qab γγab Ψ(q) ab a b Hˆ Ψab(γ) = 0 The dynamics γ= Symmetry= ηijE i=E operatorialj constraints : Ψ(qˆ) ˆΨΨ((IγγJ)) IJ HΨ(q) = 0 5 55 Space diffeo constraints and timeHB reparametrisation:ΨM (γ) = 0X Wheeler de Witt eq. i f Hˆ Ψ(q) = 0 Aa ab →a b Hˆ Ψ (q) = 0 HHˆˆγΨΨ((γ=q))η =ij =E 0i 0Ej M 77 ˆ Operators are neededIJ toa extractIJ physicaliXf information fromHIIΨ JtheseJM (abq) states =II 0JJ a b B Ei Xˆ Aa BB q X=XfηfijEi Ej 153F903957A7C27ED077364961Hf ΨM (q) = 0 ab →→ a b → Covariant153F903957A7C27ED077364961 approacha q = ηijEi Eji 2 M E∂i M = Σ ab a b XXff γγababAa The dynamicsA(S )=is encodedγ#PXf inj γTransition(j += 1)ηijE amplitudesE i 2 i j A A(S)=γ#P j(j + 1) a ΨΨ((γγ))a One wish to rigously define the pathΨiM integral(γ) = Given M ∂M =EiΣ γ < 1 A M ∂Ma = Σ M ∂M! = Σ γa<!1 Eˆˆi A(S)=HHΨΨ(γ(qγ#))2 = = 0 0j(j + 1) a iS (g) Ψ (2γ) =P EΨM (q) M ΨMMˆ(ˆγ) = ΨΨMM((γq) = iDg e A(S)=γH#HPΨΨMMj((q(γ)j) =+ = 0 1)0 γ <!1 Boundary state g Σ2=γq ab a b γγ <!=iSiS1MηMij((Egg))E A(S)=! γ|#P j(j + 1) DDggee i j iSM (g) Ψ (q) Dg e iM The perturbative expansion around asymptoticaly flat configuration:gg ΣΣΨM (q)A agraviton propagator and scatteringg amplitude.γ <!1 !! || |Σ aq ! q Ei Ψ (q) No definition is yet available in WdWM theory: Spin foam, AdS/CFTHˆ Ψ?2 (Γ,j)? AHˆ(SΨ)=(Γ,jγ#)?P j(j + 1) q =?0 ? =?0γ ?<!1 S∆ = Ψ (Aq)f (#)Θf ∆ S∆ = Af (M#)Θf ∆ f f q " " S = S A Θ ∆ S∆ = ∆ S4S 4AS f Θf ∆f f f " ""f " IJ IJ g gYMB BB BIJ YM ∗ ∧∗ IJ ∧ # # k k f + f + i i− i i− Z∆ =Z∆ =djf dAjfv(jf ,iAev)(jf ,ie) jf ,ie j ,iv v " "f $e $ + + + ++ + + +ie i Av(jf ,ie)= 15j(j ,i )15j(j ,i ) f + e Av(jf ,ie)= f15j(e j ,i f)15j(e j ,i i)i− f + f e f e i i− i+i e − i+i e " "− $ $ ie ie γ γ eiSregge + c.c eiSregge + c.c LQG and spin foam In order to resolve the difficulty with the usual metric formulation one uses new variables (gauge fields) The expression of the gauge principe and diffeomorphism symmetry leads to the construction of new type of boundary states: spin network states LQG. A fundamental geometrical cutoff is incorporated in these states. The challenge is to formulate the dynamics in terms of these states and give a definition of the transition amplitude with these boundary states Spin foam model Loop Quantum gravity Spin Foam Canonical Dynamics Covariant 7 7 7 153F903957A7C27ED077364961 153F903957A7C27ED077364961 qab 153F903957A7C27ED077364961 Loop Quantum Gravity qab Ψ(q) γab Ψ(q) ˆ Ψ(γ) HTheΨ(q) gauge = 0 principle: 7 Hˆ Ψ(q) = 0 ˆ Geometryˆ has two functions:HΨM (q) = 0 Hˆ ΨM (q) = 0 153F903957A7C27ED077364961 HΨ(γ) = 0 ab a b ab a b A metric q ab or a frameq field= ηij E i Etoj give us rods and clocks q = ηijEi Ej • Hˆ ΨM (γ) = 0 Ψ(q) i Ai ab a b Aa a γ = ηijEi Ej A mediumHˆ Ψ(q ) which = 0 allows parallela transport from one point to anothera a • E Ei connection Ai i Hˆ ΨM (q) =a 0 2 A(S)=γ#2 j(j + 1) A(S)=γ#P j(j + 1) qab = η EaEb P ij i j γ <!1 γ <!1 We canAi use the unifying language of gauge theory that appears in a ΨM (q) the descriptionEa of Electromagnetism,ΨM (q) the strong and weak interaction + principle ofi general covariance q 2 q A(S)=γ#P j(j + 1) ˆ These principles leads to a unique proposal for a set of variablesH thatΨ(Γ ,jdescribe)? γ <!1 Hˆ Ψ(Γ,j)? the geometry of space at the quantum level =?0 ? Ψ (q) =?0 ? M S∆ = Af (#)Θf ∆ S = A (#)Θ ∆ f q ∆ f f " ˆ f HΨ(Γ,j)? " S∆ = S4S Af Θf ∆ =?0 ? f S∆ = S4S Af Θf ∆ " " S∆ = Af (#)Θf ∆ f IJ " " gYM B BIJ f ∗ ∧ " IJ # gYM B BIJ k S∆ = S4S Af Θf ∆ f + ∗ ∧ i i− f # " " k f + Z = d A (j ,i ) i i− ∆ jf v f e IJ g B B jf ,ie v YM ∗ ∧ IJ " $ Z∆ = djf Av(jf ,ie) # + + + + i k A (j ,i )= 15j(j ,i )15j(j ,i ) f e f + jf ,ie v v f e f e f e i+i i i− " $ − i+i e − + + + + ie " $ Z∆ = Avd(jjff ,ie)=Av(jf ,ie)15j(j ,i )15j(j ,i ) f + f e f e i i− jf ,ie v i i+i e e " $ "− $ + + + + ie γ Av(jf ,ie)= 15j(jf ,ie )15j(jf ,ie ) fi+i ie − i+i e iSregge "− $ e + c.c γ ie iSregge γ e + c.c eiSregge + c.c 2 (je,Zv) ze = je + iθe z1 z2 z3 z4 T ∗G = GC 2 Sj z Φ =2j ln(1 + z 2) j | | j, ω j, z = (1 +ω ¯z)2j = eΦj (z,w¯) ! | " SL(2, C) # ,Z | " 2 Xi = jiNi N =1 Hv = jiNi =0 i ! 1 N a,Nb = %ab N c { } j c ZZ" Ψ(z)= Ψ z ω z ! | " | " Φ(z,z¯) ω = ∂∂¯Φ , ∂¯Ψ(z) = 0 7 2 1 Φ(z,z¯) Ψ = dzd¯z√ωe− ! Ψ z z Ψ || || ! | "! | " "P 153F903957A7C27ED077364961 ∆(Re(z))∆(Im(z)) γab j1 j2 j3 j4 Ψ(γ) i 1 i jk Γa % jkωa Hˆ Ψ≡(γ2) = 0 1 1 g kvg k− ge kvgek− e =(vv") (vv!) (vv!) v! v! The →gaugeHˆ Ψ Mprinciple(γ)→ = 0 i i0 ab Ka ωaa b γ = ηij≡E E At the canonical level one chose a spacelikei i slicej i jkS of spacetime and de % jkΓ e =0 we can formulate the hamiltonian dynamics− i of ∧gravity as an SU(2) gauge theory Aa 1 i Ω = a A E Electric field: pull back of the frame fieldκγ E on S∧ i i"Σ 1 Magnetic Field: Connection allowingE aparallel= % transportej ek%abc i 2 ijk a b i i i Ashtekar-Barbero connection: Aa = Γa + γKa Extrinsic curvature 3 σ Spin connection Immirzi parameter ab a bi i det(γ)γ = EKiabE= Kaieb a j a j (3) Conjugate variables: SU(2) Ei (x1),Ab(jy) =k κγ δb δi δ (x y) E{i = %ijke }e − 2 ∧ SU(2) index ∼ γ =1 Wave functions are gauge invariant functional of A σ =1 F (A)=F (A) ∗ tµ = Nn+ N µ S = dt Ea Ai H(A, E, N) i Lt a − ! !Σ a i H(A, E, N)=N Ca + NC + AtGi i b Ca = FabEi κ C = − F ijEaEb √γ ab i j G = Ea i −∇a i A a Ψ(ya)= dXAeiS(X ,y ) A ! " ∂ S P (X, y) a ≡ a P A ∃ a A ab δS = P ∂ a S + P ∂ A S = η P P V (y) y X a b − A ∂XA P =0 ( + V (y)) Ψ(y) = 0 −!y Ai = Γi Ki a a − a γ, σ [Ea(x),Aj(y)] = iκγ δbδjδ3(x y) i b a i − ψ(A) 1 h (A)=exp A(t)dt SU(2) e ←− ∈ #!0 $ A(t) iσ Aa e˙µ ≡ a µ ∂th(t)=A(t)h(t) h(0) = 0 g 1 1 A A = gAg− +dgg− → 1 h g(1)h g− (0) e → e a Ei The gauge principle Wilson Line: Given a loop in a SU(2) representation one can integrate the connection along the loop (holonomy) and construct a simple gauge invariant functional.
Load more

Recommended publications
  • The Feynman Propagator for Quantum Gravity: Spin Foams, Proper Time, Orientation, Causality and Timeless-Ordering
    Brazilian Journal of Physics, vol. 35, no. 2B, June, 2005 481 The Feynman Propagator for Quantum Gravity: Spin Foams, Proper Time, Orientation, Causality and Timeless-Ordering D. Oriti Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK, EU Received on 19 December, 2004 We discuss the notion of causality in Quantum Gravity in the context of sum-over-histories approaches, in the absence therefore of any background time parameter. In the spin foam formulation of Quantum Gravity, we identify the appropriate causal structure in the orientation of the spin foam 2-complex and the data that characterize it; we construct a generalised version of spin foam models introducing an extra variable with the interpretation of proper time and show that different ranges of integration for this proper time give two separate classes of spin foam models: one corresponds to the spin foam models currently studied, that are independent of the underlying orientation/causal structure and are therefore interpreted as a-causal transition amplitudes; the second corresponds to a general definition of causal or orientation dependent spin foam models, interpreted as causal transition amplitudes or as the Quantum Gravity analogue of the Feynman propagator of field theory, implying a notion of ”timeless ordering”. 1 Introduction emerge only in a semiclassical approximation [3]. Regarding the issue of causality, there seem to be two basic attitude one This paper discusses the issue of causality in non-perturbative can take: one can hold that we cannot speak of causality in quantum gravity, and more precisely in the spin foam formu- absence of time and therefore just as time itself also a notion lation of the theory.
    [Show full text]
  • Loop Quantum Cosmology, Modified Gravity and Extra Dimensions
    universe Review Loop Quantum Cosmology, Modified Gravity and Extra Dimensions Xiangdong Zhang Department of Physics, South China University of Technology, Guangzhou 510641, China; [email protected] Academic Editor: Jaume Haro Received: 24 May 2016; Accepted: 2 August 2016; Published: 10 August 2016 Abstract: Loop quantum cosmology (LQC) is a framework of quantum cosmology based on the quantization of symmetry reduced models following the quantization techniques of loop quantum gravity (LQG). This paper is devoted to reviewing LQC as well as its various extensions including modified gravity and higher dimensions. For simplicity considerations, we mainly focus on the effective theory, which captures main quantum corrections at the cosmological level. We set up the basic structure of Brans–Dicke (BD) and higher dimensional LQC. The effective dynamical equations of these theories are also obtained, which lay a foundation for the future phenomenological investigations to probe possible quantum gravity effects in cosmology. Some outlooks and future extensions are also discussed. Keywords: loop quantum cosmology; singularity resolution; effective equation 1. Introduction Loop quantum gravity (LQG) is a quantum gravity scheme that tries to quantize general relativity (GR) with the nonperturbative techniques consistently [1–4]. Many issues of LQG have been carried out in the past thirty years. In particular, among these issues, loop quantum cosmology (LQC), which is the cosmological sector of LQG has received increasing interest and has become one of the most thriving and fruitful directions of LQG [5–9]. It is well known that GR suffers singularity problems and this, in turn, implies that our universe also has an infinitely dense singularity point that is highly unphysical.
    [Show full text]
  • Aspects of Loop Quantum Gravity
    Aspects of loop quantum gravity Alexander Nagen 23 September 2020 Submitted in partial fulfilment of the requirements for the degree of Master of Science of Imperial College London 1 Contents 1 Introduction 4 2 Classical theory 12 2.1 The ADM / initial-value formulation of GR . 12 2.2 Hamiltonian GR . 14 2.3 Ashtekar variables . 18 2.4 Reality conditions . 22 3 Quantisation 23 3.1 Holonomies . 23 3.2 The connection representation . 25 3.3 The loop representation . 25 3.4 Constraints and Hilbert spaces in canonical quantisation . 27 3.4.1 The kinematical Hilbert space . 27 3.4.2 Imposing the Gauss constraint . 29 3.4.3 Imposing the diffeomorphism constraint . 29 3.4.4 Imposing the Hamiltonian constraint . 31 3.4.5 The master constraint . 32 4 Aspects of canonical loop quantum gravity 35 4.1 Properties of spin networks . 35 4.2 The area operator . 36 4.3 The volume operator . 43 2 4.4 Geometry in loop quantum gravity . 46 5 Spin foams 48 5.1 The nature and origin of spin foams . 48 5.2 Spin foam models . 49 5.3 The BF model . 50 5.4 The Barrett-Crane model . 53 5.5 The EPRL model . 57 5.6 The spin foam - GFT correspondence . 59 6 Applications to black holes 61 6.1 Black hole entropy . 61 6.2 Hawking radiation . 65 7 Current topics 69 7.1 Fractal horizons . 69 7.2 Quantum-corrected black hole . 70 7.3 A model for Hawking radiation . 73 7.4 Effective spin-foam models .
    [Show full text]
  • De-Sheng Li. Pati-Salam Model in Curved Space-Time from Square Root Lorentz Manifold
    Physical Science & Biophysics Journal MEDWIN PUBLISHERS ISSN: 2641-9165 Committed to Create Value for Researchers Pati-Salam Model in Curved Space-Time from Square Root Lorentz Manifold De-Sheng Li1,2,3* Review Article 1College of Physics, Mechanical and Electrical Engineering, Jishou University, China Volume 5 Issue 2 2Interdisciplinary Center for Quantum Information, National University of Defense Technology, Received Date: July 07, 2021 China Published Date: August 11, 2021 3 Institute of High Energy Physics, Chinese Academy of Sciences, China DOI: 10.23880/psbj-16000186 *Corresponding author: De-Sheng Li, College of Physics, Mechanical and Electrical Engineering, Jishou University, Jishou 416000, P. R. China, Email: [email protected] Abstract There is a UU(4') × (4) -bundle on four-dimensional square root Lorentz manifold. Then a Pati-Salam model in curved space-time (Lagrangian) and a gravity theory (Lagrangian) are constructed on square root Lorentz manifold based on self-parallel transportation principle. An explicit formulation of Sheaf quantization on this square root Lorentz manifold is shown. Sheaf quantization is based on superposition principle and construct a linear Sheaf space in curved space-time. The in Standard Model (SM) of particle physics and Einstein gravity are found in square root metric and the connections of bundle. transition amplitude in path integral quantization is given which is consistent with Sheaf quantization. All particles and fields ±c Thegravity interactions theory is betweenEinstein-Cartan particles/fields kind with are torsion. described There by are Lagrangian new particles, explicitly. right There handed are neutrinos, few new physics dark photon, in this Fiona,model. TheX 01212 and YYYYY,,,,**.
    [Show full text]
  • Quantum Gravity, Field Theory and Signatures of Noncommutative Spacetime
    HWM–09–5 EMPG–09–10 Quantum Gravity, Field Theory and Signatures of Noncommutative Spacetime1 Richard J. Szabo Department of Mathematics Heriot-Watt University Colin Maclaurin Building, Riccarton, Edinburgh EH14 4AS, U.K. and Maxwell Institute for Mathematical Sciences, Edinburgh, U.K. Email: [email protected] Abstract A pedagogical introduction to some of the main ideas and results of field theories on quantized spacetimes is presented, with emphasis on what such field theories may teach us about the problem of quantizing gravity. We examine to what extent noncommutative gauge theories may be regarded as gauge theories of gravity. UV/IR mixing is explained in detail and we describe its relations to renormalization, to gravitational dynamics, and to deformed dispersion relations arXiv:0906.2913v3 [hep-th] 5 Oct 2009 in models of quantum spacetime of interest in string theory and in doubly special relativity. We also discuss some potential experimental probes of spacetime noncommutativity. 1Based on Plenary Lecture delivered at the XXIX Encontro Nacional de F´ısica de Part´ıculas e Campos, S˜ao Louren¸co, Brasil, September 22–26, 2008. Contents 1 Introduction 1 2 Spacetime quantization 3 2.1 Snyder’sspacetime ............................... ...... 3 2.2 κ-Minkowskispacetime. .. .. .. .. .. .. .. .. .. ... 4 2.3 Three-dimensional quantum gravity . .......... 5 2.4 Spacetime uncertainty principle . ........... 6 2.5 Physicsinstrongmagneticfields . ......... 7 2.6 Noncommutative geometry in string theory . ........... 8 3 Field theory on quantized spacetimes 9 3.1 Formalism....................................... ... 9 3.2 UV/IRmixing ..................................... 10 3.3 Renormalization ................................. ..... 12 4 Noncommutative gauge theory of gravity 14 4.1 Gaugeinteractions ............................... ...... 14 4.2 Gravity in noncommutative gauge theories .
    [Show full text]
  • Spin Foam Vertex Amplitudes on Quantum Computer—Preliminary Results
    universe Article Spin Foam Vertex Amplitudes on Quantum Computer—Preliminary Results Jakub Mielczarek 1,2 1 CPT, Aix-Marseille Université, Université de Toulon, CNRS, F-13288 Marseille, France; [email protected] 2 Institute of Physics, Jagiellonian University, Łojasiewicza 11, 30-348 Cracow, Poland Received: 16 April 2019; Accepted: 24 July 2019; Published: 26 July 2019 Abstract: Vertex amplitudes are elementary contributions to the transition amplitudes in the spin foam models of quantum gravity. The purpose of this article is to make the first step towards computing vertex amplitudes with the use of quantum algorithms. In our studies we are focused on a vertex amplitude of 3+1 D gravity, associated with a pentagram spin network. Furthermore, all spin labels of the spin network are assumed to be equal j = 1/2, which is crucial for the introduction of the intertwiner qubits. A procedure of determining modulus squares of vertex amplitudes on universal quantum computers is proposed. Utility of the approach is tested with the use of: IBM’s ibmqx4 5-qubit quantum computer, simulator of quantum computer provided by the same company and QX quantum computer simulator. Finally, values of the vertex probability are determined employing both the QX and the IBM simulators with 20-qubit quantum register and compared with analytical predictions. Keywords: Spin networks; vertex amplitudes; quantum computing 1. Introduction The basic objective of theories of quantum gravity is to calculate transition amplitudes between configurations of the gravitational field. The most straightforward approach to the problem is provided by the Feynman’s path integral Z i (SG+Sf) hY f jYii = D[g]D[f]e } , (1) where SG and Sf are the gravitational and matter actions respectively.
    [Show full text]
  • Causal Dynamical Triangulations in Four Dimensions
    Jagiellonian University Faculty of Physics, Astronomy and Applied Computer Science Causal Dynamical Triangulations in Four Dimensions by Andrzej G¨orlich Thesis written under the supervision of Prof. Jerzy Jurkiewicz, arXiv:1111.6938v1 [hep-th] 29 Nov 2011 presented to the Jagiellonian University for the PhD degree in Physics Krakow 2010 Contents Preface 5 1 Introduction to Causal Dynamical Triangulations 9 1.1 Causaltriangulations. 11 1.2 The Regge action and the Wick rotation . 14 1.3 The author’s contribution to the field . ..... 17 2 Phase diagram 21 2.1 Phasetransitions ................................ 24 2.2 Relation to Hoˇrava-Lifshitz gravity . ........ 28 3 The macroscopic de Sitter Universe 31 3.1 Spatialvolume .................................. 31 3.2 Theminisuperspacemodel. 37 3.3 Thefourdimensionalspacetime. 38 3.4 GeometryoftheUniverse . 45 4 Quantum fluctuations 53 4.1 Decomposition of the Sturm-Liouville matrix . ........ 58 4.2 Kineticterm ................................... 60 4.3 Potentialterm .................................. 61 4.4 Flow of the gravitational constant . ..... 63 5 Geometry of spatial slices 67 5.1 Hausdorffdimension ............................... 67 5.2 Spectraldimension ............................... 70 5.3 The fractal structure of spatial slices . ........ 71 6 Implementation 75 6.1 Parametrization of the manifold . ..... 75 6.2 MonteCarloSimulations. 81 6.3 MonteCarloMoves................................ 85 3 4 CONTENTS Conclusions 91 A Derivation of the Regge action 93 B Constrained propagator 99 Bibliography 101 The author’s list of publications 107 Acknowledgments 109 Preface To reconcile classical theory of gravity with quantum mechanics is one of the most chal- lenging problems in theoretical physics. Einstein’s General Theory of Relativity, which supersedes the Newton’s law of universal gravitation, known since 17th century, is a geo- metric theory perfectly describing gravitational interactions observed in the macroscopic world.
    [Show full text]
  • Amplitudes in the Spin Foam Approach to Quantum Gravity
    Amplitudes in the Spin Foam Approach to Quantum Gravity by Lin-Qing Chen A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Doctor of Philosophy in Physics Waterloo, Ontario, Canada, 2017 c Lin-Qing Chen 2017 Examining Committee Membership The following served on the Examining Committee for this thesis. The decision of the Examining Committee is by majority vote. External Examiner Jesus Fernando Barbero Gonzalez Prof. Supervisor Lee Smolin Prof. Co-supervisor Laurent Freidel Prof. Internal Member Niayesh Afshordi Prof. Committee Member Bianca Dittrich Prof. Internal-external Examiner Eduardo Martin-Martinez Prof. ii This thesis consists of material all of which I authored or co-authored: see Statement of Con- tributions included in the thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. iii Statement of Contributions The construction of the holomorphic Spin Foam model and the calculation of Pachner Moves have been done in collaboration with Andrzej Banburski and Jeff Hnybida, which was supervised by Laurent Freidel. The asymptotics is based on work in collaboration with Andrzej Banburski. The rest of the original results are based solely on my own work. The publications related with this thesis: L. Q. Chen, “Bulk amplitude and degree of divergence in 4d spin foams,” Phys. Rev. D 94, no. 10, 104025 (2016) A. Banburski and L. Q. Chen, “Simpler way of imposing simplicity constraints,” Phys. Rev. D 94, no. 10, 104003 (2016) A.
    [Show full text]
  • Spin Foams and Noncommutative Geometry
    SPIN FOAMS AND NONCOMMUTATIVE GEOMETRY DOMENIC DENICOLA, MATILDE MARCOLLI, AND AHMAD ZAINY AL-YASRY Abstract. We extend the formalism of embedded spin networks and spin foams to in- clude topological data that encode the underlying three-manifold or four-manifold as a branched cover. These data are expressed as monodromies, in a way similar to the encoding of the gravitational field via holonomies. We then describe convolution alge- bras of spin networks and spin foams, based on the different ways in which the same topology can be realized as a branched covering via covering moves, and on possible composition operations on spin foams. We illustrate the case of the groupoid algebra of the equivalence relation determined by covering moves and a 2-semigroupoid algebra arising from a 2-category of spin foams with composition operations corresponding to a fibered product of the branched coverings and the gluing of cobordisms. The spin foam amplitudes then give rise to dynamical flows on these algebras, and the existence of low temperature equilibrium states of Gibbs form is related to questions on the existence of topological invariants of embedded graphs and embedded two-complexes with given prop- erties. We end by sketching a possible approach to combining the spin network and spin foam formalism with matter within the framework of spectral triples in noncommutative geometry. Contents 1. Introduction 2 2. Spin networks and foams enriched with topological data 3 2.1. Spin networks 3 2.2. Three-manifolds and cobordisms as branched coverings 4 2.3. Wirtinger relations 6 2.4. Covering moves for embedded graphs 7 2.5.
    [Show full text]
  • Spin Foam Models of String Theory ∗
    Spin Foam Models of String Theory ∗ A. Mikovi´c† Departamento de Matem´atica Universidade Lus´ofona de Humanidades e Tecnologias Av. do Campo Grande, 376, 1749 – 024 Lisbon, Portugal Abstract We review briefly the spin foam formalism for constructing path integrals for the BF and related theories. Then we describe how the path integral for the string theory on a group manifold can be defined as a two-dimensional spin foam state sum. 1. Introduction The spin foam models represent a way to define path integrals for gauge theories as a sum over the irreducible representations (irreps) of the gauge group of the amplitudes for the colored two-complex associated to a triangu- lation of the spacetime manifold, hence the name spin foam. The prototype spin foam model was the Ponzano-Regge model of 3d Euclidian gravity [1], where the partition function is defined as a sum over the SU(2) spins of the product of the 6j symbols associated to the tetrahedra of a triangulation. Boulatov then showed that the PR model can be derived by path-integral quantization of the 3d SU(2) BF theory [2], and soon after Ooguri showed that the path integral for the 4d SU(2) BF theory can be represented as a sum over the spins of the amplitudes for the colored dual two-complex associated with the manifold triangulation [3]. Indenpendently from these developements, mathematicians Turaev and Viro formulated an invariant for a three-manifold which as an expression was the same as the PR parti- tion function, but insted of the SU(2) group they used the quantum SU(2) group at a root of unity [4].
    [Show full text]
  • The Holst Spin Foam Model Via Cubulations
    Home Search Collections Journals About Contact us My IOPscience The Holst spin foam model via cubulations This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2012 New J. Phys. 14 103054 (http://iopscience.iop.org/1367-2630/14/10/103054) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 194.94.224.254 The article was downloaded on 18/06/2013 at 06:57 Please note that terms and conditions apply. New Journal of Physics The open–access journal for physics The Holst spin foam model via cubulations Aristide Baratin1, Cecilia Flori2 and Thomas Thiemann3 1 Max Planck Institute for Gravitational Physics, Albert-Einstein-Institute, Am Muhlenberg¨ 1, 14476 Potsdam, Germany 2 Perimeter Institute for Theoretical Physics, 31 Caroline Street N, Waterloo, ON N2L 2Y5, Canada 3 Institute for Theoretical Physics III, University of Erlangen-Nurnberg,¨ Staudtstrasse 7, 91058 Erlangen, Germany E-mail: [email protected], cfl[email protected] and [email protected] New Journal of Physics 14 (2012) 103054 (30pp) Received 16 April 2012 Published 31 October 2012 Online at http://www.njp.org/ doi:10.1088/1367-2630/14/10/103054 Abstract. Spin foam models are an attempt at a covariant or path integral formulation of canonical loop quantum gravity. The construction of such models usually relies on the Plebanski formulation of general relativity as a constrained BF theory and is based on the discretization of the action on a simplicial triangulation, which may be viewed as an ultraviolet regulator.
    [Show full text]
  • Arxiv:1811.06425V1 [Hep-Ph] 15 Nov 2018 Sapsil Ouint H Irrh Rbe Ftehgsms.How Mass
    A modern guide to θ-Poincar´e Andrea Addazi & Antonino Marcian`o1 Department of Physics & Center for Field Theory and Particle Physics, Fudan University, 200433 Shanghai, China Abstract Motivated by the recent interest in underground experiments phenomenol- ogy (see Ref. [1]), we review the main aspects of one specific non-commutative space-time model, based on the Groenewold-Moyal plane algebra, the θ- Poincar´espace-time. In the θ-Poincar´escenario, the Lorentz co-algebra is deformed introducing a non-commutativity of space-time coordinates. In such a theory, a new quantum field theory in non-commutative space-time can be reformulated. Tackling on several conceptual misunderstanding and technical mistakes in the literature, we will focus on several issues such: i) the construction of fields theories in θ-Poincar´e; ii) the unitarity of the S- matrix; iii) the violation of locality, iv) the violation of the spin statistic theorem and the Pauli principle; v) the observables for underground exper- iments. 1 Introduction & Conclusions It is commonly retained that no any bounds to quantum gravity theories may be inferred from current or next future experiments. The naive motivation is that the energy-scales reached by current experiments, such as collider physics, are far below the characteristic scale of quantum gravity, the Planck scale, for several orders of mag- nitude. For example, it is worth to remind that the Large Hadron Collider (LHC) can probe energy scales of about 1 10 TeV or so, which is 15th 16th order of magnitude − − 16 arXiv:1811.06425v1 [hep-ph] 15 Nov 2018 down to the Planck scale (10 TeV).
    [Show full text]