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Coarse graining and renormalization: the bottom up approach Bianca Dittrich, Perimeter Institute, Canada Asymptotic Safety Seminar, April 2014 Raussendorf [Picture by R. Raussendorf] Copyright: The M. C. Escher Company B. V. 1 arXiv:1209.4539 [gr-qc] B. Dittrich, J. Hnybida, arXiv:1312.5646 [gr-qc] Literature [33] “Holonomy spin foam models: definition and coarse graining,” Preprints Bahr, B. Dittrich, Hellmann, W. Kaminski [42] B. “Quantum groupF.spin net models: refinement limit and relation to spin foams,” Phys. Rev. D87 044048 (18pp) B. Dittrich, M. (2013) Martin-Benito, S. Steinhaus, arXiv:1208.3388 [gr-qc] arXiv:1312.0905[gr-qc] [45] “On the path integral measure for 4D discrete gravity,” Conceptual. [32] “On the space generalized fluxes for loop quantum gravity,” B. Dittrich, W. of Kaminski, S. Steinhaus, [41] “Time evolution as refining, coarse graining and entangling,” B. Dittrich, C. Guedes, D. Oriti to Dittrich, appear S. Steinhaus, B. Class. Quant. Grav. 30 (2013) 055008 (24pp) arXiv:1311.7565[gr-qc] [gr-qc] for loop quantum gravity,” [44] arXiv:1205.6166 “A new vacuum B. Dittrich, M.lattice Geiller, [40] “From “Topological theories from intertwiner dynamics,” [31] the discrete tofield the continuous – towards a cylindrically consistent dynamics,” arXiv:1401.6441[gr-qc] B. Dittrich, W. Kaminski, B. Dittrich, arXiv:1311.1798[gr-qc] New J. Phys. 14 (2012) 123004 (28pp) [43] arXiv:1205.6127 “Ising model[gr-qc] from intertwiners,” June 2013 Publication List – Bianca Dittrich B. Dittrich, J. Hnybida, [30] “How to construct di↵eomorphism symmetry on the lattice,” arXiv:1312.5646 [gr-qc] Refereed (Latest) Coarse graining resultsapproach for spin to foams / spin nets. arXiv:hep-th/0006199. R. Oeckl, “RenorB. Dittrich, [6] F. Markopoulou, “An algebraic coarse graining,” Preprints [6] F. Markopoulou, “An algebraic approach to coarse graining,” arXiv:hep-th/0006199. R. Oeckl, “RenorPoS (QGQGS 2011) 012 (23pp) malization of discrete models without Nucl. Phys. B 657 (2003) 107 [arXiv:gr-qc/0212047]. [42] “Quantum group spin netbackground,” models: refinement limit and relation to spin foams,” malization of discrete models without background,” Nucl. Phys. B 657 (2003) 107 [arXiv:gr-qc/0212047]. arXiv:1201.3840 [gr-qc] [39] “Coarse graining of spin net models: dynamics of intertwiners,” J. A. Zapata, “Loop quantisation from a lattice gauge theory perspective,” Class. Quant. Grav. 21 (2004) B. Dittrich, M. Martin-Benito, S. Steinhaus, J. A. Zapata, “LoopM. quantisation from a lattice gauge theory perspective,” Class. Quant. Grav. 21 (2004) B. Dittrich, Martin-Benito, E. Schnetter, [45] “On the path integral measure for discretestructure gravity,”of spin foams,” L115-L122, [arXiv:gr-qc/0401109]. [29] “Towards computational insights into the4D large–scale arXiv:1312.0905[gr-qc] L115-L122, [arXiv:gr-qc/0401109]. New J. Phys. W. 15 (2013) 103004 (38pp) B. Dittrich, Kaminski, S. Steinhaus, B. Dittrich, F.C. Eckert, [7] B. Bahr, arXiv:1306.2987[gr-qc] B. Dittrich, F. Hellmann and W. Kaminski, “Holonomy Spin Foam Models: Definition and Coarse [7] B. Bahr, B. Dittrich, F. Hellmann and W. (10pp) Kaminski, “Holonomy Spin Foam Models: Definition and Coarse J. Phys. Conf. Ser. 360 (2012) 012004 to appear [41] “Time evolution as refining, coarse graining and entangling,” New representation for LQG. Graining,” Phys. Rev. D 87 (2013) 044048 [arXiv:1208.3388 [gr-qc]]. B. Dittrich, F. Hellmann and W. KaminGraining,” Rev.S.D[gr-qc] 87 (2013) 044048 [arXiv:1208.3388 [gr-qc]]. B. Dittrich, F. Hellmann and W. KaminB.Phys. Dittrich, Steinhaus, [38] arXiv:1111.0967 “Bubble divergences andBoundary gauge symmetries in spin foams,” ski, “Holonomy Spin Foam Models: Hilbert spaces and Time Evolution Operators,” Class. Quant. [44] arXiv:1311.7565[gr-qc] “A new for loop quantum gravity,” ski, “Holonomy Spinvacuum Foam Models: Boundary Hilbert spaces and Time Evolution Operators,” Class. Quant. V. Bonzom, B. Dittrich, [28] integral measure and triangulation Grav. 30“Path (2013) 085005 [arXiv:1209.4539 [gr-qc]]. independence in discrete gravity,” B. Dittrich, M. Geiller, Grav. 30Phys. (2013) 085005 [arXiv:1209.4539 [gr-qc]]. Rev. S. D88 (2013) 124021 (23pp) B. Dittrich, Steinhaus [8] B. Dittrich, M. Martı́n-Benito and E. theories Schnetter,from “Coarse graining ofdynamics,” spin net models: dynamics of interarXiv:1401.6441[gr-qc] [40] Phys. “Topological lattice field intertwiner [8] B. Dittrich, M. Martı́n-Benito and E. Schnetter, “Coarse graining of spin net models: dynamics of interarXiv:1304.6632 [gr-qc] Rev. D85 (2012) 044032 (21pp) twiners,”B. New J. Phys. 15 (2013) 103004 [arXiv:1306.2987 [gr-qc]]. Dittrich, W.[gr-qc] Kaminski, twiners,”arXiv:1110.6866 New J. Phys. 15 (2013) 103004 [arXiv:1306.2987 [gr-qc]]. [43] “Ising model from intertwiners,” [37] “Dirac’s discrete hypersurface deformation algebras,” [9] B. Dittrich, M. Martı́n-Benito, S. Steinhaus, “The refinement limit of quantum group spin net models,” to arXiv:1311.1798[gr-qc] [9]Tensor B. Dittrich, M. Martı́n-Benito, S. Steinhaus, “The refinement limit of quantum [27]network “Coarse graining methods for spin net and(plus spin foam models,” B.Bonzom, Dittrich, Hnybida, method in condensed matter. many many more) group spin net models,” to V. B.J.Dittrich, appear appear B. Dittrich, F.C.Grav. Eckert, Martin-Benito arXiv:1312.5646 [gr-qc] Class. Quant. 30 M. (2013) 205013 (35pp) [10] M. Levin, C. J.P.Phys. Nave, “Tensor renormalization group approach to 2D classical lattice models,” Phys. Rev. New 14 (2012) 025008 (43pp), [10] M. Levin,arXiv:1304.5983 C. P. Nave, “Tensor [gr-qc] renormalization group approach to 2D classical lattice models,” Phys. Rev. Refereed Lett. 99 (2007) 120601,group [arXiv:cond-mat/0611687 “Quantum spin net models: [cond-mat.stat-mech]]. refinement limit and relation to spin foams,” Lett.[42] 99 arXiv:1109.4927 (2007) 120601,[gr-qc] [arXiv:cond-mat/0611687 [cond-mat.stat-mech]]. [36] “Constraint analysis for variational discrete systems,” B. Dittrich, M. Martin-Benito, S. Steinhaus, [11] Z. -C. Gu and X. -G.simplicial Wen, “Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Pro[26] “Canonical gravity,” [11] Z. -C. GuB.and X. -G. Wen, “Tensor-Entanglement-Filtering Renormalization Approach and Symmetry ProDittrich, P.A. Höhn, arXiv:1312.0905[gr-qc] tected Topological Order,” Phys. Rev. B 80 (2009) 155131 [arXiv:0903.1069 [cond-mat.str-el]]. [39]Topological “Coarse graining of spin net models: dynamics of intertwiners,” B. Dittrich, Order,” P.A. Höhn, tected Phys. Rev. B 80 (2009) 155131 [arXiv:0903.1069 [cond-mat.str-el]]. J. Math. Phys. 54 (2013) 093505 (43pp) Class. Quant. Grav. 29 (2012) 115009 (52pp), B. Dittrich, M. discrete Martin-Benito, Schnetter,Towards a cylindrically consistent dynamics,” New J. [12] B. Dittrich, “From the to the E. continuous: [41]arXiv:1108.1974 “Time evolution as refining, coarse graining entangling,” arXiv:1303.4294 [math-ph] [12] B. Dittrich, “From the discrete to the continuous: Towards aand cylindrically consistent dynamics,” New J. [gr-qc] J. Phys. (2013) 103004 (38pp) Phys. 14New (2012) 12300415 [arXiv:1205.6127 [gr-qc]]. Dittrich, S. Steinhaus, Phys. 14 B. (2012) 123004 [arXiv:1205.6127 [gr-qc]]. 2 Bottom-up design causal sets group field theory tensor models (causal) dynamical triangulations loop quantum gravity / spin foams emergent gravity string theory Your input of “fundamental excitation/ fundamental theory” continuum/ refinement limit: extract low energy/ large scale physics KEY PROBLEM ? Renormalization tools (auxiliary discrete?) building blocks 3 Overview Motivation. Spin foams and space time atoms. Coarse graining without a scale. The importance of refining states. Tensor network coarse graining algorithms. Application to spin foams / spin nets. Classifying (symmetry protected) phases. New representations and vacua in loop quantum gravity Expanding the theory around different vacua corresponding to different ways of refinement. 4 Why loop quantum gravity / spin foams? (My personal short list of reasons) •“minimal assumptions”: theories result from quantizing gravity (as a geometric theory), taking background independence seriously [gives you a higher probability to obtain a quantum theory of gravity] •choice of (connection) variables: originally motivated by “convenience”, •allowed first rigorous quantization of gravity (kinematical) •so far leads to the most advanced inclusion of diffeomorphism symmetry [kinematics: for instance LOST uniqueness theorem, but also BD, Geiller 14] [conceptually very important in particular for coarse graining / renormalization, key role in regaining gravity] [dynamics: we know what we are looking for! [for instance in the formulation BD, Steinhaus 13]] •(in particular spin foams): deep connection to topological field theories, (because there is) no Wick rotation involved [topological field theories are fixed points of renormalization flow] [Wick rotation leads to action unbounded from below: prevents useful continuum limit] 5 (not aBF fixed point) • Barrett Crane analogue model: =1 a b= bcoupling): = con=aquotient 1= b = cgrou • a high temp (strong •point) a = " (not a fixed a = 1, b = c = 0Crane • Barrett Crane analogue model: • Barrett analo = 1, b = c = 0 % • BF on • BF quotient on quotient group Z group = SZ % 2 a = " b BF Crane %= {α! } {Adt! (α! )} a =aa1,="= b•b= c= == 01%model: high temp (strong coupling): a b c = c a = b = 0, c = 1 a b c • Barrett analogue • Barrett Cra b (not fixed point) =a 1, bb= = 0=1,a10, a cbc= b"=c=b=c 1= 0 (not a afixed = =abbac0, == apoint) "= a (not a fixed point) =fixed c = 1(not Barrett Crane =a(not b= b a•c a = 1, b = c = 0a aanalogue =point) 1, b mod =m c % point) a fixed • Barrett Crane analogue a "=• bBarrett Crane • high temp (strong c atemp = analogue 1, (strong btemp = c =(strong 0 a fixed model: (not a fixed (not coupling): coup aa= a =point) b "=c•b high •= Cran •=•b Barrett analogue 1,high b c= cCrane = 0Barrett a = b c = 1 a = b baca= 01,bab = 0.1 × a0c== a=b0.1 fixed point) b cc= × a1 =0 b(not ccc= a== a0.1 b1= |L| " −1 • Barrett a== aanalogue 1, b× =apoint) 0 1, b = c = (not aa= fixed Crane model: b = 0 ! a = b c a = 1/2 1 3/2 0b3/2 c 1 (not fixed point) 0% a0.1 × a1/2 {α! } % % % BF %ba = a "= b % (not a fixed (not a fixed point) Crane analogue = •1, Barrett ba="=cb1/2 = 0 1c 3/2model: 0 p a = " b a = b 1/2 1 3/2 0 bfixed = 0 b%= 0(not0.1 ×= aa1, =point) b=0.1 c=×0a a b c ! a b = 0 0.1 × a % % a= 1/2 1point) 3/2 0analo =b 01(notc0.1 ab b× ×Crane a •0 Barrett 1/2b 3/2 aCrane = b analogue cmode == c00.1 a afixed • Barrett • Barrett Crane analogue 1/2 1 3/2 0 b = 0 0.1 × a 1b 0=0c 0.1 a 3/2 =b1,= =3/2 0× a a = b 1/2 c a =11, ba1/2 = = c 1, = b 0 = c = 0 × aa 1/2 1/2 1 3/2b =b 00= 0 0.10.1 1 (not fixed point) × a a = b c −1 fixed fixed point) 0 |L|b = 0 0.1(not " 3/2 1apoint) ×1/2 a a (not ! !−1 b = 0 0.1%0× a b = 0 0.1 × a 1/2 1 3/2 % % %1/2 % {α! α! } 1 3/2 0c a =1b 1/2 1/2 3/2 1 b = 0 0.1 × a a = b a = c b c ! % %a % 0 We keep the1/2ib! = 01 0.1 ×3/2 discretization / quantization/ 1/2 1 3/2 0 b = 0 0.1 × a reformulation into generalized b=0 b= 0.10 × a0.1 × a 1/2 1 3/2 1/2 1/2 1 1 3/2 3/2 0 0 lattice gauge theory %! % χ := R η What are spin foams? R{α!−1 α! } ! " %! δ(g ) years of ! RCe ∼ EF research: Reisenberger, Rovelli, BarrettCrane, Barbieri, Freidel-Krasnov, Engle-PereiraRovelli, Livine, ... " ! δ(g ) d = ˙ " # " $ g µ = (χ δ) α= α δ̃(α , ) exp(iS(geom))" Dgeom " R δ(g ) δ(g ) d % geom labels " # = δ α g % exp(iS(geom)) Dgeom geom labels δ(g α ). (0.175) A( geom labels) ! (0 exp(iS(geom)) Dgeom A( geom labels) 29 % 28 28 (0.176) A( geom labels) with local (i.e. factorizing) amplitude: geom labels 29 $ (0 α , A( geom labels) = " fund building blocks Af bb ( geom labels) 6 ψ{j} (A) = ψ{j} (A) ·!ηAL (A) χ{α! } exp(iS(geom)) Dgeom " % η Dgeom = ˙ :=exp(iS(geom)) R δ(g% α% ). labels) (0.175) (0.173) " (0.175) Space time atoms: relation to gravity action µ(ψ{j} ) = δ{j}, µ(ψ{j} ) = δ{j}, {Ad )} BF %t! (α!A( geom (0.176) exp(iS(geom)) Dgeom (0.175) (0.176) % A( = geom% labels) geom labels " A( geom labels) A % labels) " f bb ( geom labels) % A( geom (0.176) geom ηBFlabels = ˙ geom χ{α! } := R{Adt! (α δ(glabels) (0.173) (0.176) (0.174) A( % α% ). ! )} (χ δ̃(α µBF ) = " geom labels) (0.176) %) {αA( " !} fund building blocks % geom labels% geom η = ˙ := labels R{Ad α%f).bb ( geom labels) (0.173) (0.177) A( geomχ{α labels) =" t! (α! )} BF % !} " δ(g%A geom labels labels) (0.176) (0.177) % δ̃(α" (0.174) µBF labels) (χ{α! }A( ) =geom fund ! A( geom % )building blocks = A ( geom labels) f bb " " " labels) " A ( geom # " −1 $ A( geom labels) = " (0.177) f bb |L| % geom labels " ! δ̃(α ) (0.174) µ (χ ) = j1 A(j2jgeom j j j j R !−1 δ(g ) δ(g ) d g = δ α α , labels)% % (0.177) j 4 j4 {α j= %BF fund %A building % % blocks {α! }fund bb ( geom 1 j23labels) 5 !5 jα6! } 6 buildingf blocks ! 3 A( geom labels) = A ( geom labels) (0.177) ! " " " f bb % # $ % % % −1 fund building |L| " j1 j1j2 j2 j3jR jj4!−1 jj}55 jδ(g δ(g%! ) dblocks g% building = δ blocks α" % α% , 3 {α 4 α 6j6 % ) ! fund ! " " # labels) A( geom! labels) =" A ( geom (0.177) RCe ∼ EF $ % %! % f bb 29 |L| " −1 j3 j4 j5 j6 δ(g ) δ(g ! ) d g = δ α % α% , j10 ji1R1Ce j∼2 EF i5j3 j4 j5 jR6 {α!−1 fund% building %blocks % ! α! } 29 !% %! % sum over orientations of 29 j1 j2 j3 R j4Ce ∼j5EF j6 exp(iS(geom)) Dgeom (0.175) ! 3D 29 exp(iS(geom)) Dgeom % ! (0.175) 29 % exp(iS(geom)) 29A( geomDgeom labels) A( geom geom labels labels) ! space time atoms (0.175) (0.176) (0.176) % labels A(j) exp(iS ) + exp(−iS ) discr grav discr grav A( geom geom labels)∼A( =geom A ( geom labels) (0.177) f bb (0.176) j>>1 " labels) " 4D A( labels) geom labels) =labels A ( geom labels) (0.177) geom fund geom building f bbblocks A( geom = labels) (0.177) ! Af bb ( ! fund building blocks A( geom labels) = building (0.177) fund A( geom labels) blocks = " Af bb ( geom Af bblabels) ( geom labels) (0.177) j j j j j56 j6labels) fund = buildingfund (0.177) f bb ( geom labels) blocks building A blocks j1 j1j21 jj23 2jj43 j3j54 j6j45A(jgeom j1 j10 j10 i1 j2 j j3 j j4 jj1j5jj2 j6jj3 jj4 j5 j6 i 1 i2 3 4 5 6 1 i5 5 j10 i1 i5 fund building blocks Large jψ(j) (semi-classical) limit for single building blocks ∼ cos(S ) discr grav gives 29 discrete j>>1 GR action (for flat building blocks)! 29 29 [ Ponzano-Regge, Barrett et al, Conrady-Freidel, ... ] ψ(j) ∼ j>>1 cos(Sdiscr grav ) 7 Many space time atoms? Is there a phase describing smooth space time? Do we get General Relativity at “large scales”? What are the phases of spin foam theories (and in loop quantum gravity)? 8 Spin foams as lattice theories •there is an underlying (auxiliary) lattice: refinement limit with respect to this lattice •use tools developed in condensed matter, in particular tensor network renormalization. [Cirac,Levin, Nave, Gu, Wen,Verstraete,Vidal, ...] [‘Emergent gravity’] •Can deal with complex amplitudes: (we do not Wick rotate!) •However many conceptional differences: •result should be independent on choice of lattice (discretization independence) •diffeomorphism symmetry should emerge (at fixed points of renormalization flow) •(well supported) conjecture: discrete notion of diffeomorphism symmetry equivalent to discretization independence - should emerge in refinement limit [BD 08, Bahr, BD 09, Bahr, BD, Steinhaus 11, BD 12] . •There is no lattice scale, instead notion of scale included in dynamical variables. •Hope to flow to perfect discretization, mirroring exactly continuum theory at all scales at once. 9 Coarse graining without a scale 10 Fixed point model encodes all scales! •There is no lattice scale, instead notion of scale included in dynamical variables. •Hope to flow to perfect discretization, mirroring exactly continuum theory at all scales at once. View point: •Scale encoded in boundary state - not in renormalization step. •The initial model is an approximation (via discretization), which we hope to be valid in a small curvature regime [determined by choice of boundary states and number of building blocks]. •Models obtained by coarse graining improve this approximation and increase domain of validity. •Refinement limit corresponds to a ``perfect discretization’’. 11 Fixed point as a continuum theory 1 .0 1 .0 0 .5 a 3 If we do not have a lattice scale, how do we know that we reached the continuum limit? 0 .0 1 .0 0 .5 a2 0 .0 Answer: diagram for k = 8 with ↵0 = 0. The coloured dots indicate to which fixed point the respective w to: The green dots show the factorizing models, green for = 1 (area that starts at the Fixed points of renormalization flow lighter correspond toJ continuum limit. ) = (1, 0, 0)), darker for J = 2 (area that starts at the vertex (0,1,0)). Analogue BF theory is tarts at the(a) vertex (0,0,1)). The so-called ‘mixed’ (discretization fixed point is orange (area that starts at thecorrespond “local” amplitudes: topological independent) theories • to phases (b) “non-local” amplitudes (infinite bond dimension) correspond to phase transitions a2 1 .0 0 .8 0 .6 Ê Ê Ê ÊÊ Ê Ê Ê Ê 0 .2 0 .0 0 .0 Ê Ê Ê Ê 0 .4 Ê Ê Phase space: (some) parameters in initial model determine end points of coarse graining flow encoded in different colours Ê Ê Ê Ê ÊÊ Ê Ê 0 .2 0 .4 0 .6 0 .8 1 .0 a1 of the phase diagram for k = 8 with ↵0 = ↵3 = 0. The colouring is the same as in the previous the right corner corresponds to models flowing to the factorizing fixed point with J = 1, the upper ponds to models that flow to the factorizing fixed point with J = 2, the models at the bottom left 12 Fixed point as a continuum theory Fixed point model still looks discrete. How to reconstruct continuum theory? Answer (here a short summary, expanded upon in the next slides): •Build continuum theory as inductive limit: standard technique of loop quantum gravity [Ashtekar, Isham, Lewandowski 92+] •Based on embedding (or refining) coarser (boundary) states into Hilbert space describing finer boundary states • In this way any coarse (discrete looking) state can be understood as a state of the continuum Hilbert space. •(Cylindrically) consistency requirements on observables, amplitudes, inner product. • Surprisingly (condensed matter) tensor network renormalization methods are well adapted to this technique and can provide embedding maps - determined by dynamics of the system. [BD 12] •Allows to formulate continuum theory of spin foams based on discrete boundary states [BD, Steinhaus 13] 13 Inductive limit techniques and Tensor network coarse graining algorithms. 14 Boundary Hilbert spaces and transition amplitudes A A A A A A (local) amplitudes depending on (boundary) variables, represented by blue edges A A ! A A A A summing over (bulk) variables in path integral ONB A A A A boundary Hilbert space (tensor product over local sites) This represents a transition amplitude built from local amplitudes. The boundary Hilbert T T space has two components. A T A A A T This A represents a one-component boundary Hilbert space. A A ψ A A! ψ A! A! [generalized boundary formalism: Oeckl 03] 15 Embedding Hilbert spaces coarser boundary (with four sites) with attached boundary Hilbert space (tensor product over four sites) AAA AAA ψψψ finer boundary (16 sites) with attached boundary Hilbert space ψψψ (local) embedding map AAA AAA AA!A! ! AAA AAA ψψ2ψ22 AA!A! ! ψψ1ψ11 ψψ1ψ11 ψψ2ψ22 AAA AAA ψψ1ψ11 AA!A! ! embedding map identifies coarse states with states in finer boundary Hilbert space 16 Good embedding maps? What are the embedding maps good for? Answer: •efficiency! •do computation on the most coarse grained level possible •can represent continuum theory by discrete (boundary) data ➡ To be really efficient, embedding maps have to be adjusted to dynamics of the system: ➡Coarse Hilbert space should represent ``most typical states” (state describing smooth geometry / low energy) Whereas initial boundary states may rather describe fundamental excitations, and one expects a large number of these to describe smooth geometry. 17 projections ⇡b0 b : Cb0 ! Cb on the configuration spaces at our di in which the states are functions A A on the configuration cb ∼ cb! A quantum A A! Ainjections: A of the projections to define A family of functions {Sb }b∈B is cylindrically consistent on the inductive fa A A A A ψ (Cb , ιb ), if ψ2 ψ ψ1 ψ 0 0 ◆ : H ! H b bb b •embedding maps: ! (c)) Sb = ι∗bb! Sb! , i.e. Sb (c) = Sbb0 ! (ιbb ∀cb0∈(cCb0b) = b ( where b 7! •order boundaries into∗coarser and finer ψ ψ ψ1 ψ2 ! pullback of ι . This just implements that the value the fu where ιbb! is the The reader willbbnote that in the classical case we alsoofused in not depend onInthe on which to evaluate. therepresentative case we discussed hereone we chooses can however obtain projection A A A A graining was !based on variables. I.e. in the caseA that coarse A A A A A a refinement which doubled the number of fields, { i } to { i Ab : 1H C ) .and define the projection b !→ transformation = ( + ij ij i j •(transition) amplitudes encode dynamics: 2 Amplitudes encode the dynamics of the system ! ! 1 2 1 1 2 ! A [transition amplitudes if there are two boundary components] A ψ =( , ⇡ ( , , 2 , . . .) (0.147) 1 2 , . . .) . A(x1 , x2 , x3 , x4 ) = ! a(x1 , x2 , x3 , x4 , xbulk ) path integral •(transition) amplitudes for instance ιbb! : ψdefined !→ ψb!by where ψ (φ , γ , φ , . . .) = xψb (φ . 1 , φ2 , . . .) class b Such a projection satisfies ⇡ ◆ = Id where here the ◆ b (blue edges represent sum over variables) b where x are boundary data for the configuration spaces. A A ψ(x , x , x , x ) ψ(x1A , x! as x4 ) 2 , x3 ,linear A measure (which herewavewill be just understood fun is a boundary function amplitudes labelled by boundaries b is a boundary wave function •in fact we have a family of (transition) Hilbert spaces can defined byA providing a representation on A is be an (anti-)linear A functional A is an (anti-)linear functional on bdry Hilbert space H , Such a family of measures {µ} is cylindrically consistent ψ b 15 on1bdry Hilbert space H1if , ψ !2 ψ A(x1 , x2 , x3 , x4 ) =b0 b 1 12 ! a(x1 , x2 , x3 , x4 , xbulk ) xbulk 12 2 b! 0 1 class b b bb0 where x are boundary data 1 2 3 bulk 4 1 A(ψ) = A(xi )ψ̄(xi ) xi (0.148) ! A(ψ) = A(xi )ψ̄(xi ) b0 bbx0i b µb ( b ) = µ (◆ ( )) . b and b’. •Cylindrical consistency conditions relate amplitudes for different boundariesdefines (transition) amplitudes defines (transition) amplitudes State sum models associate amplitudes to space time regions with boundary (data) State sum models associate amplitudes to space time regions w Amplitude for a ‘larger’ region ψ1 for a ‘larger’ region glued from amplitudesψof2 smaller regions, Amplitude acts on ‘refined’ bdry Hilbert space H2 glued from amplitudes of smaller regions, acts on ‘refined’ bdry Hilbert space H2 The Ashtekar–Lewandowki measure [?, ?] for the kinematical gravity satisfies such consistency relations. One way to define inner product) would be via the path integrals acting as functi kinematical Hilbert spaces (technically on some dense subspac 18 Sb = ι"bb! Sb! i.e. Sb (c) = Sb! (ιbb! (c)) ∀c ∈ Cb Cylindrically consistent amplitudes A A ψ ψ Ab (ψb ) = Ab! (ιbb! (ψb )) A A Computing the amplitude for a coarse state should give the same result, as d variables Ab Ab state to a finer state •embedding coarse A A •using A the amplitude map for finer states. ψb A ιbb! (ψbψ ) ψ ψb ction A A ιbb! (ψb ) A A ψb ) itude over ables bb! Ab Ab! Ab ψb ψb ψb A A A A (ψb ) ιbb! (ψb ) Ab ψb Ab! Thus, computation for coarse states give already continuum results. ations, Cylindrically consistent amplitudes define the continuum limit of the theory. ly subparts A A erator A A ψ(x1 , x2 , x3 , x4 ) ιbb! (ψb ) is a boundary wave function Ab [BD 12, BD, Steinhaus 13] 19 What are good choices for the embedding maps (for the basis of boundary states)? 20 on to continuum Hamilton’s field actually exact!function evaluated on ‘edge wise’ linear The same procedure for squares with refined boundary data will in Motivation: transfer operator technique this approximation. A A A A The same procedure for squares with refined boundary data will in general give a correction to this approximation. uares with refined boundary data will in general to γ233 ,a. correction .. = 0 set give AA A AA AA A set γ233 , . . . = 0 A A A AA A AA A AA AA ! ! ! ONB T T A= T T A A TN A T A A T A A = TN A A insert id = A A T A A AA T 21 T A AA ONB |ψ"!ψ| insert id = T T ! A A A! ! A! A! Truncate by restricting ONB to the eigenvectors of T with the (in mod) A χ largest A A! eigenvalues. A! AA AA A A ! A ONB |ψ"!ψ| 21 T A Transition amplitude between two states !ψ|A|ψ2 " Transition betweenA A A amplitude A two states !ψ1 |A|ψ2 " T AA ONB A A ONB A AA A T Transition amplitude between two statesT !ψ|A|ψ2 " T insert id = ! ONB |ψ"!ψ| A = TN A A A 21 A A A Here coarse states correspond to low energy states. Results (typically) in non-local embedding maps (example: Fourier transform for free theories). A Explicit diagonalization difficult for large systems. A! A! A! A! 21 A = TN Localized embedding maps ! Truncate by restricting A A AA A AA A AA A A A A A A A AA A AA A AA A A A A A A A A AA A AA A AA A = id A A A A ! ! ! ONB !ONB ONB TN A = TN A ONB ! toTruncate the eigenvectors of T with the by restricting ONB ! ! ! χ largest (in A AA AA AA A Ato of T with the A eigenvectors A mod) eigenvalues. A A AA AA T amplitudethe function ONB A T A A A A χ largest (in mod) eigenvalues. Expect good approximation if ψ1 , ψ2 A areAin span A of these A eigenvectors. amplitude A A function A A= T A T N C2 good approximation ifAψ Expect =1 ,Tψ 2 C2 effective C areamplitude in2 span of these eigenvectors. T ! But: explicit diagonalization of T by difficult. Truncate restricting T T T T T ! amplitude function Truncate by restricting T Tamplitude ONB effective ON = id to the eigenvectors of T with t Localize truncations, But: explicit diagonalization of T difficult. Determined by (generalized) χ largest (in mod) eigenvalue only subparts to the eigenvectors of T with A AA A AA A AA diagonalize A A A the EV-decomposition. ofA transfer operator by χ largest (inTmod) eigenvalues.A A ! A Determined (generalized) Expect good approximation if ψ Ttruncations, T T T Localize TA A A A! A! are in span of these eigenvector effective amplitude T T only subparts EV-decomposition. diagonalize C2 a C2 c blocking of transfer operatorA C2 b A A A But: explicit diagonalization of T d Localize truncations, iteration procedure 22 diagonalize only subparts of transfer operator A A iteration AA A Aprocedure A! A! A! A A d f e A! A! AA! ! A! A! A! A! A! determine embedding maps H blocking H Determined by (generalized) embedding H EV-decomposition. = id embedding blocking embedding map after 3 iterations embedding A A A AA A AA A AA A A A A A= id = id = id = id 22 22 Relation to vacuum Here coarse states correspond to low energy states / states with few excitations from vacuum. Embedding maps depend on the dynamics of the system. However in General Relativity: What is vacuum? Hamiltonian is a constraint and hence zero (and we do not Wick rotate). Transfer matrix should be a projector if diffeomorphism symmetry is realized (with eigenvalues one and zero). NB: In 4D diffeomorphism symmetry is broken, eigenvalues take other values too. So we can hope that even for GR, unphysical states (eigenvalue zero) are truncated away. There is however a further mechanism that differentiates states even in GR: radial evolution. [BD, Steinhaus 13: Time evolution as refining, coarse graining and entangling] 23 Radial time evolution and coarse states • • • • • • • • • • igureConsider 6: Illustration of radial evolution in tensor networks: adding eight additional tensor (in a “radial time time evolution” from a ``smaller” to By a “larger” Hilbert space. ray) we perform one time evolution step. The boundary data grows exponentially from 4 to 48 . (We can easily construct such evolutions for systems based on simplicial discretizations, such as spin foams). The radial time evolution can be split into steps with time evolution operators Z R2map: This time evolution itself defines an embedding T (R1 , R2 ) = exp( Hr dr) . (6.1) The image of the time evolution defines coarser states in the larger Hilbert space. R1 his is a refining time evolution in the sense that the Hilbert space H2 at R2 will have more kinematical egrees of freedom than thedefinition Hilbert space H1 to at states R1 . The main idea coarse is to truncate theexcitations. degrees of freedom Conjecture: This leads describing grained H2 . Ideally one would performtime a singular valuecan decomposition T (R1 , R would givemaps. a maximal 2 ). This (Radial) evolution be used toofdefine useful embedding umber of dim(H1 ) singular values. Hence T (R1 , R2 ) will have a non–trivial co–image in H2 , which can [BD, Steinhaus 13] e projected out. Such a scheme might be indeed worthwhile to 14, investigate (for [also BD, Hoehn 11,13, Hoehn BD, Hoehn,further Jacobson wip]statistical systems), in order to btain an intuition about the truncations. One would expect that the reorganization of the degrees of eedom via the singular value decomposition will be highly non–local, as we have seen for the example section Thethis timedefinition evolution would considered exactlyto corresponds T (R1 , R2 ). maps. NB:2.1. Again leadthere in general non-localtoembedding Even such a classical investigation might be worthwhile: the radial evolution as a refining time volution will result in post constraints. Below we will introduce coarse graining time evolution steps, hat will implement a truncation, and should classically lead to pre–constraints. This suggest that for a ood choice of truncation, the pre–constraints should match to the post–constraints. 24 Tensor network renormalization methods ... rather using local embedding maps (to keep algorithms efficient) but different versions are possible. A A A! A A Contract initial amplitudes (sum over bulk variables). Obtain “effective amplitude” with more boundary variables. bare/initial amplitude depending on four variables A A A A A A A A A A! A A A A! A! A A Find an approximation (embedding map) that would 32 minimize the error as compared to full summation (dotted lines). For instance using singular value A! decomposition, keeping only the largest ones. Leads to field redefinition, and ordering of fields into more and less relevant. A A 32 Use embedding maps to define coarse grained 32 with the same (as initial) number of amplitude boundary variables. 25 Tensor network renormalization methods Iteration. Summation over bulk variables are truncated using the embedding maps. Associated embedding maps for boundary Hilbert space. 26 ψ ψ C2 A Fixed points give cylindrical consistent amplitudes A A A C2 A c 0.8 b 0.6 ψ ψ d A ω(g1 g2−1 ) g1 g A2 A A A H 0.2 H f e A 0.4 A Gauss ω̃(k) 0 Ab 10 15 A 20 25 ! AA k A 5 H Ab A A A ! 30 A A A A A ONB A A Ab Ab A A ψb ψ ψ ψbfor cylindrically consistent amplitudes b Condition ψb T 22 T A Ab b) ψb T Ab! Ab A T ψb ψb A ψb Ab A ψb satisfied for Ab! ιbb! (ψb ) A Ab A A! A! A 25 Ab! ψb A A Ab! A! A A ιbb! (ψb ) C2 C2 C2 = ψ(x1 , x2 , x3 , x4 )Ab ιbb! (ψb ) is a boundary wave function ιbb (ψb ) Ab ! A is an (anti-)linear functional C2 C2 27 Fixed points corresponding to topological field theories or interacting field theories •method works particularly well for convergence to a “gapped phase” (lead to topological field theories) •finitely many ground states separated from excited states: determine bond dimension (number of non-vanishing singular values) •phase transition: usually involve ‘infinite bond dimension’ •expect to find interacting theories / propagating degrees of freedom •truncation can be nevertheless quite good, and provides a method to determine phase transition (and sometimes even to calculate critical exponents) • However argument about radial time evolution suggest that better truncations (near phase transitions) might be possible if non-local embedding maps are incorporated. • Might even not need infinite bond dimension. 28 Applications: (analogue) spin foam models •4D spin foam models are very very complicated •devised analogue models capturing the essential dynamical ingredients of spin foams [similar to 4D lattice gauge and 2D spin system duality] •these spin nets can actually be interpreted as spin foams based on very special discretizations •unlike spin foams the spin net models are non-trivial in 2D •investigated the phase diagrams for such models (with quantum group structure) •these phase diagrams have a very rich structure (The details of this and the adaption of the tensor network methods take easily up another talk.) 29 Applications: (analogue) spin foam models 6.2.2 Lattice gauge theory/ Ising like systems Phase diagram for k = 8 For the quantum group with k = 8, we discuss the linear combintation of four fixe with P a maximal (even) spin 1 J 4, where we neglect J = 0 as argued above that J ↵J = 1, we have three free parameters. In figure 9 we show the full p coloured points indicating the fixed point they flow to. In figure 10 we show the Spin nets coupling 0 .0 a1 0 .5 confining phase ‘no space’ phase 1 .0 1 .0 0 .5 a 3 deconfining phase (topological phase) topological phase (gives 3D gravity!) 0 .0 0 .5 a2 0 .0 1 .0 Figure 9: Phase diagram for k = 8 with ↵0 = 0. The coloured dots indicate to initial models flow to: The green dots show the factorizing models, lighter green vertex (↵1 , ↵2 , ↵3 ) = (1, 0, 0)), darker for J = 2 (area that starts at the vertex blue (area that starts at the[ BD, vertex (0,0,1)). The so-called fixed point i Martin-Benito, Schnetter 2013,‘mixed’ BD, Martin-Benito, vertex (0,0,0)). Steinhaus 2013] Much richer phase structure! Interpretation: different phases describe uncoupled a2 space time atoms (green) and coupled space time 1 .0 atoms (orange,blue). 0 .8 0 .6 Ê Ê Ê ÊÊ Ê Ê Ê 30 Towards spin foam coarse graining •So far encouraging results for ‘spin foam analogue’ models. Conclusions: •Relevant parameters related to (SU(2)) intertwiners (also appearing in spin/anyon chains) leading to rich phase space structure. This is expected from the gravity dynamics. •Need to implement a weak version of discretization independence to uncover these rich phase spaces (escape the two lattice gauge theory phases). •Positive indication for finding a geometric phase in spin foams. •Are now looking at spin foam analogue and actual spin foam models in higher dimensions. •Need to extend tensor network tools (applicable to lattice gauge theories) to this end. [wip] 31 Each phase corresponds to a toplogical field theory. Defines embedding maps and associated vacua. New representations (Hilbert spaces) for Loop quantum gravity. 32 Loop quantum gravity •succeeded in constructing a quantum representation of geometric operators [Ashtekar, Lewandowski, Isham, Rovelli, Smolin, Corichi, ... 90’s] •spatial geometric operators (area, volume operators) have discrete eigenvalues •spin networks based on (finite) graphs give a basis in the (non-separable) LQG Hilbert space •despite this (discrete) basis the theory is defined in the continuum: via embedding maps •dynamics based on Hamiltonian constraints [Thiemann ’96] So far the theory is based on the Ashtekar-Lewandowski representation based on a (AL) vacuum describing • • zero volume spatial geometry maximal uncertainty in conjugated variable. 33 What is vacuum? •Ashtekar-Lewandowski representation /vacuum corresponds to confining phase for lattice gauge theories (i.e. embedding maps coincide). Question: Can we base the representation on a different vacuum corresponding to one of the other phases? 34 Loop quantum gravity vacua cyl Ĉ = I2,− Z∞,∞ [ιb1 ∞ (ψb1 ), ιb2 ,∞ (ψb2 )] = Zbcyl [ψ b1 ψb2 ] 1 ,b2 geometric variables: Ĉ = I2 − {A , E} = δ connection ψvac (A) ≡ 1 Ashtekar Lewandowski representation {A , E}- = δ (90’s) ψvac (A) ≡ 1 flux: spatial geometry , E≡0 peaked on degenerate (spatial) geometry maximal uncertainty in connection excitations: spin network states supported on graphs (representation) labels for edges , [BD, Geiller 2014, BD, Geiller to appear] E≡0 BF (topological) theory representation ψvac (EGauss ) ≡ 1 , F (A) ≡ 0 peaked on flat connections maximal uncertainty in spatial geometry 28 excitations: flux states supported on (d-1) D-surfaces (group) labels for faces 35 We applied this to BF (topological field theory), corresponding to unconfined phase. [BD, Geiller 14] But construction can be generalized to other topological field theories: 36 Excitations and vacuum (discretized) vacuum topological field state theory (dynamics) determine observables stable under refining time evolution (non-trivial step) continuum excitations Hilbert space ‘inductive limit’ construction for excitations 37 Advantages [BD, Geiller 2014, BD, Geiller to appear] •This new construction allows to expand loop quantum gravity around different vacua corresponding to the different phases (fixed points) for spin foams. • Ashtekar-Lewandowski representation: peaked on completely degenerate spatial geometry, maximal fluctuation in conjugated variable: difficult to built up states corresponding to smooth geometry •New representation [BD, Geiller 2014] : peaked on flat connection, maximal fluctuation in spatial geometry. Corresponds to the physical state in (2+1)D gravity and in general to condensate states with respect to Ashtekar-Lewandowski representation. •Facilitates construction of states corresponding to smooth geometries and should be useful for discussions of i.e. black hole entropy in loop quantum gravity [Sahlmann 2011]. •New representation much easier to interpret geometrically: new handle on the dynamics of the theory. [need substitute of Thiemanns (1996) quantization of Hamiltonian constraints] 38 coloured points indicating the fixed point they flow to. In figure 10 we show the inter 0 .0 Phase transitions? a1 0 .5 1 .0 1 .0 0 .5 a 3 0 .0 1 .0 0 .5 a2 0 .0 to expand quantum around different vacua •This new construction allowsFigure 9: Phaseloop diagram for k =gravity 8 with ↵ 0 = 0. The coloured dots indicate to whic initial models flow to: The green the factorizing models, lighter green for J corresponding to the different phases (fixed points) for dots spinshow foams. vertex (↵1 , ↵2 , ↵3 ) = (1, 0, 0)), darker for J = 2 (area that starts at the vertex (0,1,0 blue (area that starts at the vertex (0,0,1)). The so-called ‘mixed’ fixed point is ora vertex (0,0,0)). • What about phase transitions (‘non-trivial’ fixed points)? / propagating degrees of freedom. • Instead of topological theory expect conformal theory a 2 1 .0 • Expect such fixed point amplitudes to be non-local 0 .8 Ê non-local embedding maps Ê Ê ÊÊ 0 .6 Ê Ê 0 .4 Ê Ê Ê (also given ÊbyÊ time evolution?) Ê Ê Ê Ê Ê Ê symmetry there? Ê • Do we regain triangulation independence / diffeomorphism 0 .2 0 .0 0 .0 ÊÊ Ê Ê 0 .2 0 .4 0 .6 0 .8 1 .0 a1 39 Summary Coarse graining is about refining boundary states. Tensor network methods construct embeddings of coarse states, representing “low energy/coarse degrees of freedom”, into continuum Hilbert space. 40 Summary and Outlook: Quantum Space Time Engineering •We are on a good way to understand the ‘many body physics’ of spin foams and loop quantum gravity. •Corresponds to the ‘continuum limit’ of these models. •In the path integral approach (spin foams): mapping out the phase diagrams •connections to condensed matter physics / (new?) topological field theories and phases •challenge: going to higher dimensions •challenge: understanding symmetries of phase transition fixed points •In the canonical approach (loop quantum gravity): expanding around different vacua •facilitates investigation/construction of physical states describing extended/smooth geometries •each vacuum comes with its own set of excitations: investigate dynamics of these. 41 Thank you! 42