Rovelli-Santa Barbara.Key

Finite boundary observables and white holes

Carlo Rovelli

i. Finite boundary observables [cfr: Don]

iii. Concrete calculation of an observable: black to white hole tunnelling time, and Fast Radio Bursts Facets of the problem of quantum gravity Mars 31st Santa Barbara Facets of the problem of quantum gravity Mars 31st Santa Barbara 3WCPVWOTGIKQP V 3p3 ⇡m2v ⇠ r = r0 2m How big is a black hole? ⌧ Marios Christodoulou, Carlo Rovelli. Phys.Rev. D91 (2015) 6, 064046

r =2m

After a time m3: V m5 ⇠ v At the end of the 7 evaporation: V m 2 ⇠

The internal volume of a black hole is large even when its Area-Entropy is small: the number of internal degrees of freedom must be large in order not to violate local qft in its own domain of validity. … holography? A quantum theory of gravity cannot be a local qft:

1. Local observables are not gauge invariant in gravity: diff invariance 2. Space-time discreteness, or similar (cfr: both loops and strings) [cfr: Gabriele]

Local qft: degrees of freedom localised with respect to classical entities.

Quantum gravity: degrees of freedom localised with respect to one another, as in classical general relativity. i. General lessons from non perturbative QG ii. partial observables (“local observables and gauge invariance”)

Quantum theory gives the probabilities of the outcomes of a (local) interaction of a quantum system with another system.

General relativity describes the (local) interaction of spacetime regions with one another. i. General lessons from non perturbative QG ii. partial observables (“local observables and gauge invariance”)

Quantum theory gives the probabilities of the outcomes of a (local) interaction of a quantum system with another system.

General relativity describes the (local) interaction of spacetime regions with one another.

➜ Combine the two aspects i. General lessons from non perturbative QG ii. partial observables (“local observables and gauge invariance”)

Quantum theory gives the probabilities of the outcomes of a (local) interaction of a quantum system with another system.

General relativity describes the (local) interaction of spacetime regions with one another.

➜ Combine the two aspects

Quantum system Boundary = ➜ Hamilton function: S(q,t,q’,t’) Spacetime region i. General lessons from non perturbative QG ii. partial observables (“local observables and gauge invariance”)

Quantum theory gives the probabilities of the outcomes of a (local) interaction of a quantum system with another system.

General relativity describes the (local) interaction of spacetime regions with one another.

➜ Combine the two aspects

Quantum system Boundary = ➜ Hamilton function: S(q,t,q’,t’) Spacetime region

States: associated to 3d boundaries of spacetime regions. ➜ Boundary formalism Transition amplitudes: associated to 4d regions [cfr: Bianca]. t0 Hamilton function S(q, t, q0,t0)= dtL˜ (˙q (t˜),q (t˜)) t0,q0 q,t,q0,t0 q,t,q0,t0 Zt

i iH(t t0) S(q,t,q0,t0) Propagator W (q, t, q0,t0)= q e q0 e ~ h | | i⇠

i dtL t, q = D[q] e ~ Z R q(t)=q q(t0)=q0

m q(t) L[q, q˙]= q˙2 !2q2 Systems evolving in observable time 2 System evolving in 2 m q˙ 2 2 q(⌧),t(⌧) L0[q, q,˙ t, t˙]= ! tq˙ parameter time 2 t˙ ✓ ◆

˙ Hamilton function S(q, t, q0,t0, ⌧, ⌧ 0)= d⌧ L(˙qq,t,q0,t0 (⌧),qq,t,q0,t0 (⌧), tq,t,q0,t0 (⌧),tq,t,q0,t0 (⌧)) = S(q, t, q0,t0) Z i S(q,t,q0,t0) Propagator W (q, t, q0,t0)= q, t P q0,t0 e ~ h | | i⇠ i dtL(q,q,t,˙ t˙) = D[q, t] e ~ Z R q(⌧)=qq(⌧0)=q0 t(⌧)=tt(⌧0)=t0 '

iS[] Boundary functional W [', ⌃]= D e =' Z |⌃ Σ

7 '

iS[] Boundary functional W [', ⌃]= D e =' Z |⌃ Σ

But in a generally covariant theory: W [', ⌃]=W [']

Spacetime region Distance and time measurements

Particle detector

In a general relativistic theory, distance and time measurements are ﬁeld measurements like the other ones: they are determined by the boundary data of the problem. [cfr.: Jim]

7 iH(t t0) Transition amplitudes e 0 h | | i

Rep invariant P 0 = W 0 h | | i h | ⌦ i

Quantum gravity W h | boundaryi ➜ The boundary determines the here and now of the interaction whose correlations are predicted by quantum theory.

➜ The physical relative position of the points on the boundary is determined by the (quantum) state of geometry on the boundary itself

Introduction to Canonical Quantum gravity Loop Quantum Gravity Carlo Rovelli Francesca Vidotto, Carlo Rovelli CUP 2004 CUP 2014 Let’s make this concrete Covariant loop quantum gravity. Full deﬁnition.

n0 l 2 L N hl State space = L [SU(2) /SU(2) ] (h ) =lim n H l H H 3 !1

Kinematics Operators [cfr: Steve]: Boundary spin network (nodes, links)

Transition amplitudes W (hl)=N dhvf (hf ) A(hvf ) hf = hvf C C W =limW SU(2) v C v Z Yf Y C!1 Y Bulk Dynamics

j (j+1) j 1 Vertex amplitude A(h )= dg0 (2j + 1) D (h )D (g g ) 8⇡~G =1 vf e mn vf jmjn e e0 SL(2,C) j Z Yf X

e With cosmological constant: v

f SL(2,C) Chern-Simons Theory, a non-Planar Graph Operator, and 4D Loop Quantum Gravity with a Cosmological Constant C Hal M. Haggard, Muxin Han, Wojciech Kamiński, Aldo Riello. arXiv:1412.7546 spinfoam (vertices, edges, faces) On the structure of a background independent quantum theory: Hamilton function, transition amplitudes, classical limit and continuous limit

Carlo Rovelli Centre de Physique Th´eorique,Case 907, Luminy, F-13288 Marseille, EU (Dated: March 24, 2012) The Hamilton function is a powerful tool for studying the classical limit of quantum systems, which remains meaningful in background-independent systems. In quantum gravity, it clariﬁes the physical interpretation of the transitions amplitudes and their truncations.

I. SYSTEMS EVOLVING IN TIME gaussian point. Let L(qn,qn 1,tn,tn 1) be a discretization of the lagrangian. The multiple integral

Consider a dynamical system with configuration vari- i N dqn aL(qn,qn 1,tn,tn 1) able q , and lagrangian L(q, q˙). Given an initial con- WN (q, t, q0,t0)= e ~ n=1 (6) 2 C µ(qn) P figuration q at time t and a final configuration q0 at time Z t0,letq : be a solution of the equations of where µ(qn) is a suitable measure factor, tn =n(t0 t)/N q,t,q0,t0 ! C ⌘ motion such qq,t,q0,t0 (t)=q and qq,t,q0,t0 (t0)=q0. Assume na, and the boundary data are q0 = q and qN = q0, has for the moment this exists and is unique. The Hamilton two distinct limits. The continuous limit function is the function on ( )2 defined by C ⇥ lim WN (q, t, q0,t0)=W (q, t, q0,t0) (7) N t0 !1 gives the transition amplitude. While the classical limit S(q, t, q0,t0)= dt L(qq,t,q0,t0 , q˙q,t,q0,t0 ), (1) Zt lim( i~) log WN (q, t, q0,t0)=SN (q, t, q0,t0). (8) 0 namely the value of the action on the solution of the ~! equation of motion determined by given initial and final gives the Hamilton function of the classical dis- data. This function, introduced by Hamilton in 1834 [? cretized system, namely the value of the action N ] codes the solution of the dynamics of the system, has n=1 aL(qn,qn 1,tn,tn 1) on the sequence qn that ex- remarkable properties and is a powerful tool that remains tremizes this action at given boundary data. The dis- meaningful in background-independent physics. cretizationP is good if the classical theory is recovered as Let H be the quantum hamiltonian operator ofMain the results:the continuous (i) Classical limit of thelimit: discretized GR theory, that is, if system and q the eigenstates of its q observables. The (ii) Spacetime discreteness | i (iii)lim UVS Nand(q, t,IR q 0finite,t0)= amplitudesS(q, t, q0,t0). (9) transition amplitude N !1 i H(t0 t) Summarizing: W (q, t, q0,t0)= q0 e ~ q . (2) h | | i Exact quantum gravity General relativity 0 codes all the quantum dynamics. In a path integral for- transition amplitudes ~! Hamilton function ! mulation, it can be written as W (hl) S(q) q(t0)=q0 !1 i t0 !1 C

dtL(q,q˙) ! W (q, t, q0,t0)= D[q] e ~ t . (3) ! q(t)=q R LQG Regge Z 0 transition amplitudes ~! Hamilton function

Continuous limit W (hl) ! S(li ) In the limit in which ~ can be considered small, this can C b be evaluated by a saddle point approximation, and gives Classical limit ! ! i S(q,t,q0,t0) W (q, t, q0,t0) e ~ . (4) TABLE I. Continuous and classical limits ⇠ That is, the classical limit of the quantum theory can be TheRegime interest of validity of of thisthe expansion: structure is that it remains mean- obtained by reading out the Hamilton function from the ingful in di↵eomorphism invariant systems and o↵ers an quantum transition amplitude: excellent conceptual tool for dealing with background independent physics. To see this, let’s first consider its gen- lim( i~) log W (q, t, q0,t0)=S(q, t, q0,t0). (5) eralization to finite dimensional parametrized systems. 0 ~! The functional integral in (3) can be defined either by perturbation theory around a gaussian integral, or as a II. PARAMETRIZED SYSTEMS limit of multiple integrals. Let us focus on the second def- inition, useful in non-perturbative theories such as lattice I start by reviewing a few well-known facts about QCD and quantum gravity, which are not defined by a background independence. The system considered above 2

II. DISCRETIZATION finite in physical separations. Taking the discretized systems to its critical point can implement universality and Consider a harmonic oscillator with mass m and angu- wash away the e↵ect of the details of the discretization. lar frequency !. The action is This same behavior is present in field theory. It is natural to think that this pattern is universal. But things are dif- m dq 2 ferent when discretizing reparametrization-invariant sys- S = dt !2q2 . (1) 2 dt tems. Z ✓ ◆ ! Choose a fixed time interval t of interest and divide it into 2 a large number N of small steps of size a = t/N.The III. PARAMETRIZATION continuum theory will be recovered with N and II. DISCRETIZATION finite in physical separations. Taking the discretized sys- a 0, keeping the size t = Na of the time interval!1 fixed. Consider the system defined by the two variables q(⌧) tems to its critical point can implement universality and Discretize! the system on the time steps t = an,with and t(⌧), evolving in the evolution parameter ⌧, and gov- wash away the e↵nect of the details of the discretization. Consider a harmonic oscillator withinteger massnm=1and,...,N angu-.Thesystemisthendescribedbythe erned by the action lar frequency !. The action is This same behavior is present in field theory. It is natural variables qn = q(tn) and the action can be discretized as 2 to think that this pattern is universal. But things are dif-m q˙ 2 2 2 follows S = d⌧ ! tq˙ (7) m dq 2 2 ferent when discretizing reparametrization-invariant sys-2 t˙ S = dt ! q . (1) tems. 2 Z ✓ ◆ 2 dt ! m qn+1 qn 2 2 Z ✓ ◆ SN = a ! q . (2) where the dot indicate the derivative with respect to ⌧. 2 a n n ! Choose a fixed time interval t of interest and divide itX into ✓ ◆ It is immediate to see that this is physically fully equiva- a large number N of small steps ofA size standarda = t/N lattice.The procedure is thenIII. to define PARAMETRIZATION rescaled lent to the harmonic oscillator discussed in the previous m section. In fact, the equation of motion for q is continuum theory will be recovereddimensionless with N variablesand Qn = qn and ⌦ = a!,so !1 a~ a 0, keeping the size t = Na of thethat time the interval dimensionless fixed. actionConsider becomes the system defined by the two variables q(⌧) d q˙ 2 Discretize! the system on the time steps t = an,with and t(p⌧), evolving in the evolution parameter ⌧, and gov- = ! tq˙ (8) S n1 d⌧ t˙ 22 integer n =1,...,N.ThesystemisthendescribedbytheN = (Q Qerned)2 by⌦2Q the2 actionS (Q ) (3) 2 n+1 n n ⌘ N,⌦ n variables q = q(t ) and the action can~ be discretizedn as which gives immediately the harmonic oscillator equation n n X II.II. DISCRETIZATION DISCRETIZATION2 finitefinite in in physical physical separations. separations. Taking Taking the the discretized discretized sys- sys- m q˙ d22q/dt2 2 = !2q; while the equation for t is follows S = d⌧ ! tq˙ (7) tems to its critical point can implement universality and where all quantities on the r.h.s. are now dimensionless.2 t˙ tems to its critical point can implement universality and 2 Z ✓ ◆ wash2 away the e↵ect of the details of the discretization. m q q This action (or better, itsConsiderConsider analytical a a harmonic continuationharmonic oscillator oscillator in Eu- with with mass massmmandand angu- angu-d q˙wash away the e↵ect of the details of the discretization. n+1 n 2 2 + !2q2 =0, (9) SN = a clidean! time)qn . can be(2) studiedlarwhere frequency numerically the dot!. The indicate to action give the an is ap- derivative with respect to ⌧. ThisThis same same behavior behavior is is present present in in field field theory. theory. It It is is natural natural 2 a lar frequency !. The action is d⌧ t˙2 n ! ✓ to think that◆ this pattern is universal. But things are dif- X ✓ proximation◆ of the path integralIt is immediate to see that this is physically fully equiva- to think that this pattern is universal. But things are dif- 22 ferent when discretizing reparametrization-invariant sys- lent to the harmonicmm oscillatordqdq discussedwhich22 22 inis not the previous an independentferent equation: when discretizing it is simply reparametrization-invariant the 3 sys- A standard lattice procedure is then to define rescaledi SS == dtdt !! qq .. (1)(1) S[q(t)] iSN,⌦(Qn) tems. m ~ section. In fact,2 the equationdt of motionconservation for q is of energy2 tems. that follows from (8). The sys- dimensionless variables Qn = qn and ⌦D=[q(at!)],soe dQn e 2 dt(4) !! a~ ! ZZ ✓✓ ◆◆ that the dimensionless action becomes 0, butZ rather from the singleZ limit N . Let us seetem hasis not indeed conserved a large in gauge general. invariance, under arbitrary 2 !1d q˙ 2 2 II. DISCRETIZATIONp this more in detail.ChooseChoosefinite a ain fixed fixed physical time time separations. interval intervaltt=ofof interest interestTaking! tq˙ reparametrization theand and discretized divide divideConsider it it into into sys- now of(8) its the independent discretization variable of the⌧.Inthis, parametrized To take the continuum limit we have to send N ˙ , III. PARAMETRIZATION SN 1 2 2 2 d⌧ t!1 m! it is very similar to general relativity,III. which PARAMETRIZATION is equally Define as beforeaa dimensionless large largetems tonumber numberII. its critical DISCRETIZATION variablesNN ofof point small smallQn can steps steps= implement of ofq size sizen aa universalitysystem.== t/Nt/N.The.The The andfinite key observation in physical separations.is that while Taking the two the con- discretized sys- = (Qn+1 Qn) ⌦ butQn thisSN, is⌦ not(Qn) su (3)cient: we must also send ⌦ to its ~ II.~ DISCRETIZATION2 ⌘and T finite= !t in. physical Noticecontinuumcontinuum that separations. theorythese theory are will will Taking not be be defined recovered recovered the discretized usinginvariant with with sys-NtinuousN under equationsandand thetems reparametrization to of its motion critical (8) point of and its (9)can independent are implement degenerate universality and Consider an harmonicWhy this oscillator works? with critical“Ditt-invariance” mass valuem andn ⌦ angu-= 0, n(from since washwhichB⌦ Dittrich)= a away gives! and the immediatelya must e↵ect go of to the the zero details harmonic of the oscillator discretization.!1!1 equation X tems to itsa critical20, keeping2 point can the2 sizeimplementt = Na universalitypof thecoordinate time intervaland variables fixed. (Di↵Consider–invariance).Consider the the system system defined defined by by the the two two variables variablesqq((⌧⌧)) lar frequency !. The action is in thethe limit. time More step precisely,aConsider, whicha Thisd q/dt0, say would asame keeping harmonic we= want behavior be! the uselessq to; oscillator size while compute ist present in= the thisNa with equationtheof contextin mass thefield time theory. form andt interval(theis angu- It second is natural fixed. followwash fromaway the the first), e↵ect their of the discretization details of theare discretization. where all quantities on the r.h.s. are now dimensionless.wash away the!! e↵ect of the details of the discretization. andandtt((⌧⌧),), evolving evolving in in the the evolution evolution parameter parameter⌧⌧,, and and gov- gov- Consider a harmonic oscillator with mass m andpropagator angu-since a is notlar in frequency theDiscretizeDiscretizeto action, think! thethat. butthe The system systemrather this action pattern the on on is naturalthe the is universal. time time units steps stepsLet Butttnnindependent us things== discretizeanan,with,with are dif- equationsThis this system. same. behavior The As first before, is discretized present fix an in interval field equation theory. is It is natural This action (or better, its analytical2 continuationThis in Eu- same behavior is present in field2 theory. It is natural erned by the action lar frequency !. The action is given by the dynamicsintegerintegerferent itselfnn when=1=1 (in,...,N,...,N discretizing general.Thesystemisthendescribedbythe.Thesystemisthendescribedbythed relativityq˙ reparametrization-invariant2 these2 in ⌧,splititintoas before sys-Ntosteps thinkerned of that size by this thea and pattern action define is universal.⌧n = na, But things are dif- m dq 2 2 + ! q =0, (9) clidean time)S can= be studieddt numerically! q to. giveto an think(1) ap- that thisi pattern is universal.2 But things are dif- natural units arevariablesvariables providedH(tf qqt bynni)== theqq((ttn Planckn))d and and⌧ the the length).˙2 action action This can cantn be be= discretized discretizedt(⌧n) and asq asnferent= q( when⌧n). discretizing Consider the reparametrization-invariant discretized sys- 2 2 dt W (qf ,tf ; qi,ti)= qf etems.~ mqi dqt 2 2 22 Systems evolving in observable! time S = dt ✓ ! q◆ . (1) mm2 q˙q˙ 22 22 proximationm of thedq pathZ integral2✓2 ◆ yields theferent dimensionless whenh followsfollows| discretizing action| reparametrization-invarianti action systems.vn+1 = vn (tn+1 SS=t=n) ! qndd.⌧⌧ !(18)! tq˙tq˙ (7)(7) q(tf )=qf 2 dt S = dt ! q . (1) i ! 22 t˙˙ 2 dt tems. which is notZ an independentS✓[q(t)]◆ equation: it is simply the t Choose a fixed time intervali t !of interest and divide it into = D[q(t)]Ne ~ . (5) 22 qn+1 qn 2 ZZ ✓✓ ◆◆ Z ✓ ◆ S[q(t)] iSN,⌦(Qn) 2 m q q m ( ) t t D[q(t)] e ~ dQ e S 1 (4) (Q conservationQ ) mIII. of energy PARAMETRIZATIONq thatnn+1+1 followsqnn from22where2 (8).2 the The discrete sys- a velocity is2 nown+1 n 2 a large number NDiscretizeof small steps ofn size a =Nt/N.TheChoosenq a+1(t fixedi)=SSqiN timen== intervalaa t of interest2 andS divideN!!=qq it into.. a(2)(2) where the! dot indicate q the. derivative(10) with respect to ⌧. ! = Z N (T T )Q nn tn+1wheretn the dot indicaten the derivative with respect to ⌧. Choose a fixed time interval t of interest and divide it into tem has indeed22 an+1 large gaugen aan invariance, under!2 arbitrary III. PARAMETRIZATIONa ! continuumZ theory will be recoveredZ with N~ 2 anda largeTn number+1 Tn N of smallnn ✓ steps of size◆a = t/N.The! n ItaIt is is immediate immediateqn+1 to toqn see see that that this this is is physically physically fully fully equiva- equiva- Then this is!1 givenn by✓ reparametrizationIII. PARAMETRIZATIONXX of its✓ independent◆ ◆ variable X⌧.Inthis, a large numberaTo takeN0,of keeping the small continuum the steps size oft limit size= Naa weof= have thet/N time.The to send intervalN fixed.Xcontinuum, Consider theory the willN system be recovered defined with by theN two variablesand q(⌧) vn+1 (19) S (Q ,T ) (12) lentlent to to⌘ the thetn harmonic harmonic+1 tn oscillator oscillator discussed discussed in in the the previous previous continuum theory! will be recovered with N and !1 AAandit standard standard ist( very⌧), evolving lattice lattice similar procedure procedure in toN the general evolutionn isn is relativity, then then parameterNotice to to define!1 define which something⌧ rescaled, rescaled is and equally gov- important:Consider the the quantity system defineda drops by from the two variables q(⌧) Discretizebut this is the not system sucient: on the we time must steps alsotn send= an⌦,withtoa its 0, keeping the sizeiS⌘t = (NaQ )of the timemm interval fixed. section.section. In In fact, fact, the the equation equation of of motion motion for forqqisis !1 W (q ,t ; q ,t )=! dimensionlessdimensionless liminvariantdQ under variables variablese theN,⌦ reparametrizationQQnn ==(6) qqthisn andand of expression. its⌦⌦ independent== aa!!,so,so Indeed, the above reads a 0, keeping the size t = Na of the time interval fixed. f f Consideri Discretizei theerned system the by system the definedn action on by the the timen two steps variablesaa~ tn =qanand(⌧),with the fixedand sizet( time⌧), evolving step a is in replaced the evolution by the parameter variable ⌧, and gov- ! integercriticaln value=1,...,N⌦ = 0,.Thesystemisthendescribedbythe since ⌦ = a! and a Noticemust go that to zero the frequency⌦ 0 N has been absorbed in the nor-~ n Discretize the system on the time steps t = an,with and t(⌧), evolvingthatthatNcoordinate! the the in dimensionless dimensionlessZ the variables evolution action ( actionDi parameter↵ –invariance). becomes becomes⌧, and gov- dd q˙q˙ variablesin the limit.qn = Moreq(tn precisely,) and the say actionn we can want bemalization to discretized compute of theinteger as the dimensionless!1n =1,...,N.Thesystemisthendescribedbythe variables. Supposep now size time steperned (tn+1 by2t then). action What about the second22˙ equa- m qp˙2 m (qn+1 qn) 2 2 == !! tqtq˙ (8)(8) integer n =1,...,N.Thesystemisthendescribedbythe ernedm by the actionLet usm discretize this system. As before,2SN2 =tion? fix an The interval variation of the! action(tn+1 withtn respect)d⌧q t˙˙ . to(11)t gives followspropagator whereweQ want= toq computevariablesand QSSNN=q then =11 sameq(qtn)and transition andS = theis a action suitable amplituded⌧22 can22 be!22 as discretizedtq˙ as (7) d⌧n t 2 n 0 a i N == a f ((QQn+1 QQn)) ⌦˙⌦ QQ SSN,⌦2((QQn)) (3) (3)tn+1 tn m q˙ variables qn = q(tn) and the action can be discretized as ~ follows in ⌧,splititinto~ nN+1N2 stepsn oft size nan andeasily defineN,⌦ n⌧nn✓= na, ◆ 2 ˙ 2 normalization2 before factor for the~ measure.22 2 Z ✓ ⌘⌘ ◆ X S = d⌧ ! tq (7) System evolvingi in ~ m nn q˙ 2 2 whichwhich gives gives immediately immediately the the˙ harmonic harmonic oscillator oscillator equation equation follows m qn+1H(tf qtni) 2 2 p tn =pt(⌧n) and qn = q(˙⌧n). Consider the discretized 2 t W (qf ,tf ; qi,ti)= qf e ~ qi S = XXd⌧ ! tq 2 (7) 22 22 22 Z ✓ ◆ SN = parametera time The! factqn that. the(2) limitwhere isq(1)= obtainedq thef dot not indicate only˙ by the taking derivativeIn words, with respect the discretized to ⌧. dd actionq/dtq/dt = is= fully!! qq independent;; while while the the equation equation from for forttisis h | | i action m2 tqn+1 qn En+1 = En. (20) 2 a wheret(1)= allt quantities✓ on thei r.h.s.◆ are now2 2 dimensionless. n ✓2 q(tf )=qf N◆ but! also sendingwhere⌦ItStoN is its= immediateallf critical quantitiesZ a value to see on is thethat an essen- r.h.s.S this[q(⌧) is,t are(⌧ physically)]!a now.q This dimensionless.. elementary fully(2) equiva-where observation the dot is indicate the main the point derivative of this with respect to ⌧. m q q i ~ n n+1X n 2 2 !1W (qSf[,tq(tf)]; qi,ti)= 2D[q(⌧)]D[t(⌧)] ae . 22 S = a = ! q D. [q(t)](2)e ~ . (5) ThisThisq(0)= action actionqi (or (orn N better, better, q itsn its+1 analytical analyticalqn 2 continuation continuation! in in Eu- Eu- d q˙ N n tial defining featurewhere the of the dotlent discretization. indicate to them harmonic the derivative Near✓( oscillator the criti- with) ◆ discussed respectarticle.t toThat in⌧t Let. the is,us previous study energyIt theis is immediate consequences now conserved! to see of that thisd How this fact.q˙ is is it physically22 possible?22 fully equiva- A2 standard latticeDiscretizea procedure is then to define rescaled Zt(0)=ti X a 2 n+1 n 2 ++!! qq =0=0,, (9)(9) n q(ti)=qi ! clideanclideanSN time) time)= can cana be be studied studied numerically numerically! to to give giveq an an. ap- ap-(10) 22 ✓ ◆Z mcal value the correlationIt is immediate lengthssection. to see of In the that fact, discretized this the is equationtn physically+1 systemtn of fully motion equiva-The for mainq isn consequencelent to is the that harmonic the continuum oscillatordd⌧⌧ t˙t˙ limit discussed of in the previous dimensionlessX variables Qn = qn and ⌦ = a!,soA standard lattice2 procedure is then to(13) defineaIt rescaled is possible because the additional degree of freedom, a~ proximationproximation of ofn the the path path integral integrala ! ✓✓ ◆◆ A standardThen lattice this procedure is given is by then to definediverge rescaledThen in the thislent number is to given the of lattice byharmonic steps, oscillatorX so that they discussed remainm in thethe previous theorywhich is is not the givensection. position by the In of fact,double the time the limit equation steps,N is of now,a motion adjusted for q is that the dimensionless action becomes dimensionless variables Qn =d q˙ a qn and ⌦ = a!,so which is not an independent!1 equation:! it is simply the m p section. In fact, the equation of motion~ for q2 ˙is which is not an independent equation: it is simply the dimensionless variables Qn = qn and ⌦ = a!,so that theNotice dimensionless something actionii important: becomes= ! thetq quantitybya thedrops dynamics from(8) in order for the energy to be conserved. a~ SS[[qq((tt)])]˙ iSiSN,N,⌦⌦((QQnn)) conservation of energyd q˙ that follows from (8). The sys- SN 1 2 2 2 iS (Q ) DD[[qq((tt)])] ee~~ iSd⌧(Qtp,T ) dQdQnn ee (4)(4) conservation of energy that2 follows from (8). The sys- that the dimensionless=W (q action,t ;(qQ,t becomes)=Q lim) ⌦ QdQ e SN,W⌦((qQn,t); q (3)(6),t )=this lim expression.dQ dT Indeed,e N then n above(14) reads In other words, the continous parametrized= ! systemtq˙ adds (8) f f i n+1i n n n N,⌦ f n f i i d qn˙ n 2 !! temtem has has indeed indeed a ad large⌧ larget˙ gauge gauge invariance, invariance, under under arbitrary arbitrary ~ 2 p ⌦ 0 N ⌘ SN whichN1 Z givesZ immediately= ! tq˙2 the2 harmonicZZ2 oscillatora(8) degree equation of freedom which is fully gauge. The discrete n N ! Z = !1 Z(Qn+1 Qn) ⌦ Qn SN,⌦(Qn) (3) SN 1 X !1 2 2 d⌧ t˙ 2 2 reparametrizationreparametrization of of its its independent independent variable variable⌧⌧.Inthis,.Inthis, 2 2 2 ~ ToTod take take2q/dt the the=m continuum continuum! q;( whileqn+1 limit limit theqn equation) we we have have⌘2 forto to sendt sendnon-parametrizedis NN2 ,which, dynamics gives immediately breaks energy the harmonic conservation. oscillator equation = (Qn+1 Qn) ⌦ Qn SN,⌦(Qn) (3) !m S n= !m ! (t t ) q !1. (11) 2 where all quantitiesm on the⌘ r.h.s.m are nowwhere dimensionless.Q0 = qi andNXQN = qf , T0 = !ti andn+1 n n!12 itit2 is is very very2 similar similar to to general general relativity, relativity, which which is is equally equally ~ nwhere Q0 = a qi and QN = a qf and is awhich suitable gives~butbut immediately this this is is2 not not the su su harmonic~tcient:cient:n+1 wet we oscillatorn must must also alsoequation send sendBut the⌦⌦ totodiscrete its itsd q/dt parametrized= ! q;dynamics while the breaks equation the for gauget is ~ ~ TN = !t . There is no other limitn ✓ to take than2 N . ◆ XThisnormalization action (or factor better, for its the analytical measure. continuationN f2 in Eu-where2 criticalcritical all2 quantities value valueX⌦⌦ == on 0, 0, thed since since r.h.s.q˙⌦⌦== areaa!! nowandand dimensionless.aamustmust go go to to zero zero invariantinvariant under under the the reparametrization reparametrization of of its its independent independent • Systems evolving in time: Thed q/dt continuump= !limitq ;is while obtained the ptaking equation the number for+ !t 2 isofq!12 steps=0 to, infinityfreedom added(9) by the parametrizations and exploits it by where all quantitiesclidean time) on thep can r.h.s. be are studied now numerically dimensionless.p When to giveN anis ap-large,This action the average (or better, time its steps analytical are2 automati- continuation in Eu- coordinatecoordinate variables variables2 ( (DiDi↵↵–invariance).–invariance). The fact that the limit is obtained not only by and taking a couplinginin theIn theconstant words, limit. limit. to the Morea More critical discretized precisely, precisely,d value⌧ .t˙ action say say we we is want want fully to independent tofixing compute compute it so the the that from energy is conserved.d q˙ Concretely,2 2 the “po- This action (orproximation better, its of analytical the path continuationintegral in Eu-cally small. clidean time) can be2 studied✓ numerically◆ to give an ap- LetLet us us discretize discretize this this+ system.! system.q =0 As As, before, before, fix fix an an interval(9) interval N but also sending ⌦ to its critical value is an essen- propagatorpropagatora. Thisd elementaryq˙ 2 2 observation is the mainsitions” point of of the this intermediate timed⌧ stepst˙2 t are not gauge Below I study whether the classical+ ! q and=0, the quan- (9) ✓ n ◆ clidean time) can!1 be studied numerically• General to covariant give an systems: ap- The continuumproximationwhich limit is of is obtained the not˙2 path an taking independent integral just the number equation: of steps it to is infinity. simply the inin ⌧⌧,splititinto,splititintoNN stepssteps of of size sizeaaandand define define⌧⌧nn==nana,, tial defining featurei of the discretization. Near the criti- article.d⌧ Lett us study the consequences ofin this the fact. discrete theory: they are determined in order to S[q(t)] iStumN,⌦(Qn dynamics) given by such discretizationii of the ~ conservation✓ of energy◆ thatHH((ttf followstti)) from (8). The sys-whichttn is== nottt((⌧⌧n an)) and independent and qqn == qq((⌧ equation:⌧n).). Consider Consider it is simplythe the discretized discretized the proximation of the pathD integral[q(t)] e dQn e (4) ~ f i n n n n cal value the correlation lengths of the discretized system WW(The(qqff,t,tf mainf;;qqii,t,ti consequenceii)=)= qqff ee ~ is that theqqii continuumadjust the limit conservation of of energy. ! parametrized theorytem are has well indeed definedS a[q( large andt)] sensible. gauge invariance,iSN,⌦(Qn) under arbitrary action Z Z which is not anD independent[q(t)] e ~ h equation:h || dQ itn ise | simply| ii the (4) conservationaction of energy that follows from (8). The sys- diverge ini the number of lattice steps, so that they remain the theory is not given byqq((tt theff)=)=qq doubleff limit TheN integration,a of the discrete equations of motion of S[q(t)] iSN,⌦(Qn) conservationreparametrization of energy that follows of! its independent from (8). The variable sys-iiSS[[qq(⌧(t!1t)].Inthis,)] tem! has indeed a large gauge invariance, under arbitrary DTo[q( taket)] e ~ the continuumdQn limite we have to(4) send N , Z Z ~ qqnn+1+1 qqnn 2 == DD[[qq((tt)])] eethe~ parametrized.. (5)(5) systems can be performed( numerically,) 2 ! tem!1 has indeedit is a very large similar gauge invariance, to general under relativity, arbitrary which is equallyreparametrizationmm of its( independentaa ) variable22ttnn+1+1 ⌧tt.Inthis,nn 22 but this is not sucient: we must also send ⌦ toTo its take the continuum limit weqq((ttii have)=)=qqii to send N , SN = a ! q . (10) Z Z IV. CLASSICAL DYNAMICSZZ showing that they giveSN the= correcta dynamics.ttn+1 ttn See Figures! qnn . (10) invariant under the reparametrization of its!1 independentit is very similar22 to generaln+1 relativity,n whichaa is equally To take thecritical continuum value limit⌦ = we 0, since have⌦ to= senda! andN a must, goreparametrization to zerobut this is not of its su independentcient: we must variable also⌧.Inthis, send 1,⌦ 2to and its 3. Notice the irregularnn evolution aa of q (Figure !! ThenThencoordinate this this is is given given variables by by (Di↵ –invariance). XX n but this is notin the su limit.cient: More we precisely, must also say send we want⌦ !1to to its computeit is the verycritical similar value to⌦ = general 0, since relativity,⌦ = a! and whicha must is equally go1) to and zerot (Figureinvariant 2) as under functions the reparametrization of n:Thisreflectsthe of its independent The standard discretizationLet us discretize (2) of the this system system. breaks As before, fix an intervaln NoticeNotice something something important: important: the the quantity quantityaadropsdrops from from propagator invariantin the under limit. the More reparametrization precisely, say we of itswant independent to computearbitrary the dependencecoordinate on variables⌧ of the (Di parametrized↵ –invariance). system, critical value ⌦ = 0, since ⌦ = a! and a must go to zero iSiSN,N,⌦⌦((QQnn)) the intervalcoordinate of the variables physicalin ⌧W,splititintoW((qqf timef (,tDi,tff↵;;qtq–invariance).ii,tinto,tiiN)=)=Nsteps limsteps lim of of size equaldQdQa nandn ee define ⌧n =(6)(6)na,Letthisthis us expression. discretize expression. this Indeed, Indeed, system. the the As above above before, reads reads fix an interval in the limit. More precisely, say we want to compute the propagator ⌦⌦ 00 N gauge-fixed in the discretization (and determined by the i size. The discrete equation of motion, obtained! N minimiz- H(tf ti) tn = t(⌧n) and qn =NNq!(⌧n).Z Consider the discretizedin ⌧,splititintoN steps of size a and define ⌧ = na, ~ Let us discretize this system. As before,!1 fixZ an interval 22 n propagator W (qf ,tf ; qi,ti)= qf e qi !1 initial data). But then qnmmand tn (combine(qqn+1 qqn into)) the well- ing (2) with respectaction to qn is i n+1 n 22 22 h | | i in ⌧,splititintoN stepsm of sizeHa(tandf ti) definem ⌧n = na, tn = StS(NN⌧n==) and qn = q(⌧n). Consider!! ((ttnn+1 the+1 discretizedttnn))qq .. (11)(11) q(tf )=qf W (q ,t ; q ,t )=mq e ~ q m known solution of the harmonic oscillator equation, in nn i wherewheref f QQ0i0 ==i qqfii andandQQNN == i qqff andand isis a a suitable suitable 22 ttn+1 ttn i S[q(t)] aah~~ | | aai~~ action nn n+1 n H(tf ti) ~ tn = t(⌧n) and qn = q(⌧n2). Considerqn+1 qn 2 the discretizedNN ✓✓ ◆◆ W (q ,t ; q ,t )= q e ~ =q D[q(t)] e . (5) normalizationvn+1 = vn a factor! qn. forq(t thef )=q measure.f (15) their relative evolution (FigureX 3). f f i i f i normalizationm factor( for thea measure.) 2 tn+1i tn 2 X q(ti)=qi action S[q(t)] h | | i SN = pp a pp ! ~ Theqn di. ↵erence(10) between the discretizationqn+1 qn 2 of a stan- q(tf )=qf Z TheThe fact fact that that the the= limit limittn+1 is ist obtainedn obtainedD[q(t)] not note only only. by by(5) taking taking InIn words, words, the the discretized discretized action action is is fully fully independent independent from from i 2 a m ( a ) 2 tn+1 tn 2 S[q(t)] where I have defined the discreten velocity q(ti)=aqi ! ~ qn+1 qn 2 Z dard system andSN that= of a parametrizeda t t system! can be qn . (10) Then this= is given by D[q(t)] e . (5) mNN butbut( also alsoX sending sending) ⌦⌦t toto its its criticalt critical value value is is an an essen- essen- aa.. This This2 elementary elementaryn+1 observation observationn is is thea the main main point point of of this this !1 a 2 n+1 n 2 n a ! q(ti)=qi SN = !1a ! q . (10)directly seen in the equations of motion. In the first case, Z Thentialtial thisNotice defining defining is given somethingqtnn+1+1 feature feature bytnqn ofimportant: of the the discretization. discretization. the quantityn Near Neara drops the the criti- criti- from article.article.X Let Let us us study study the the consequences consequences of of this this fact. fact. 2 a iSN,⌦(Qn) vnn+1 a (16)! the equations of motion (15) contain explicitly the dis- Then this is givenW by(qf ,tf ; qi,ti)= lim dQn e (6) calcalXthis value value expression.⌘ the the correlation correlationa Indeed, lengths lengths the above of of the the reads discretized discretized system systemNoticeTheThe something main main consequence consequence important: the is is that that quantity the the continuum continuuma drops from limit limit of of ⌦ 0 N cretization size a. Therefore we are in fact dealing with a N ! divergediverge in in the the number number of of lattice lattice steps, steps,iS so soN, that that⌦(Qn they) they remain remainthisthe expression.the theory theory is is Indeed, not not given given the by byabove the the double reads double limit limitNN ,a,a !1 Z Notice somethingW (qf ,t important:f ; qi,ti)= the lim quantity2dQan dropse from (6) !1 ! Equation (15) gives the velocitym at the(q⌦n+1 next0 Nq time-stepn) 2 in one-parameter2 family of equations, which converge to the !1 ! iSN,⌦(Qn) this expression.S Indeed,= the aboveN ! reads ! (t t ) q . (11) W (qf ,tf ; qi,ti)= limm dQn e m terms(6) of the velocity atN the previous step!1 andZ of the dis-n+1 n n 2 where Q0 = ⌦ 0 qi and QN = qf and is a suitable 2 tn+1 tn continuum theory whenm the limit(qn+1a qn0) is appropriately a~ N a~ n 2 ! 2 2 N ! Z creteN impulse (the impulse ism the✓ force !q mtimes the taken.◆ Specifically,SN = if q (q ,v ,a) is the solution! (tn+1 of thetn) qn . (11) normalization!1 factor for the measure. wherem Q0 =(q qXi qand)2 QN = n qf and is a suitable 2 n 0 0t t p p n+1a~ n 2 a~ 2N n ✓ n+1 n ◆ Them fact that the limitm is obtained nottime only step byS takingaN).= As wellIn known, words, the because discretized of! the(tn action+1 approxima-tn is) fullyqn . independent(11)equations with from initial valuesX q0 andq1 = q0 + av0,then where Q0 = qi and QN = qf and is a suitable normalization2 t factort for the measure. a~ a~ tion involved in then discretization,pn+1 n the energyp the solution of the continuum theory is recovered as normalizationN factorbut for thealso measure. sending ⌦ to itsN critical value is an essen-TheXa fact.✓ This that elementary the limit is observation obtained not is the only◆ main by taking point of thisIn words, the discretized action is fully independent from ptial!1 defining featurep of the discretization. Near the criti- article. Let us study the consequences of this fact. The fact that the limit is obtained not only by taking In words,N the discretizedbutm also2 sending action2 ⌦2 isto fully its critical independent value is from an essen- a. This elementary observation is the main point of this !1En = v + ! q (17) q(t)=lim q t (q0,v0,a). (21) cal value the correlation lengths of the discretized systemtial definingThe feature mainn+1 consequence of then discretization. is that the Near continuum the criti- limitarticle. of Leta us0 studya the consequences of this fact. N but also sending ⌦ to its critical value is an essen- a. This elementary2 observation is the main point of this ! !1 diverge in the number of lattice steps, so that they remaincal valuethe the theory correlation is not given lengths by of the the double discretized limit systemN ,a The main consequence is that the continuum limit of tial defining feature of the discretization. Near the criti- article. Let us study the consequences of this fact. !1 ! diverge in the number of lattice steps, so that they remain the theory is not given by the double limit N ,a cal value the correlation lengths of the discretized system The main consequence is that the continuum limit of !1 ! diverge in the number of lattice steps, so that they remain the theory is not given by the double limit N ,a !1 ! 2

ii. A concrete calculation of an observable: Fast Radio Bursts and black to white hole tunnelling time 3WCPVWOTGIKQP

What happens here? A technical result in classical GR: The following metric is an exact vacuum solution, plus an ingoing and outgoing shell, of the Einstein I equations outside a ﬁnite spacetime region (grey).

ds2 = F (u, v)dudv + r2(u, v)(d✓2 + sin2✓d2) III II v u F (u ,v )=1,r(u ,v )= I I . Region I I I I I I 2 vI < 0.

3 32m r r r I F (u, v)= e 2m 1 e 2m = uv. Region II r 2m ⇣ ⌘ 1 uI uI Matching: rI (uI ,vI )=r(u, v) u(uI )= 1+ e 4m . vo 4m 3 ⇣ ⌘ 32m rq 2m Region III F (uq,vq)= e ,rq = vq uq. rq

Black hole fireworks: quantum-gravity effects outside the horizon spark black to white hole tunneling Hal M. Haggard, Carlo Rovelli The metric is determined by three constants: m, ✏, arXiv:1407.0989 Minkowski region

I Outgoing (null) shell

Past trapped horizon

White hole Quantum region III II

Schwarzschild region

Conventional black hole Future trapped horizon I Incoming (null) shell

Minkowski region Minkowski region

I Outgoing (null) shell

Past trapped horizon

White hole Quantum region III II

Schwarzschild region

Conventional black hole Future trapped horizon I Incoming (null) shell

Minkowski region I

I I

➜ I

➜

The Fingers Crossed The metric is determined by three constants:

- m is the mass of the collapsing shell. 1 m 3 - ✏ is the radius where quantum effect start on the shell: ✏ 3 lP . ⇠ mP Quantum pressure causes the bounce ✓ ◆

Planck stars Carlo Rovelli, Francesca Vidotto Int.J.Mod.Phys. D23 (2014) 12, 1442026 0 =

- is related to the distance from the horizonr where the theory is entirely classical u

t = 0 What does represent and what determines it?τ v R a = = r

r ⌧ r = 0 r internal = a Time dilation r = R ⇠ τ t m = 0 ⇠ v u 1 ms “A black hole is a short cuttothefuture”“A black hole isashort ⌧ R =2 R T: bouncelarge) time (very ⌧ m external ln( /m ⇠ m ) 2 ⇠ 10 9 years What do we expect of the bouncing time:

em Naive expectation from analogy with tunnelling in space ⇠ 3 m Page time. Requiring that AMPS ﬁrewall are avoided T = 8 ⇠ 2 1 m Minimal failure of local qft: RT >LPlanck > ⇠ < m ln m Calculation from LQG, ﬁrst contribution (too short!) ⇠ > :> Covariant loop quantum gravity. Calculation of T(m).

j, -k

j, k