Loop Quantum Gravity a General-Covariant Lattice Gauge Theory Francesca Vidotto UNIVERSITY of the BASQUE COUNTRY

Home , Francesca Vidotto

Loop Quantum Gravity a general-covariant lattice gauge theory Francesca Vidotto UNIVERSITY OF THE BASQUE COUNTRY

Bad Honnef - August 2nd, 2018

Loop Quantum Gravity Francesca Vidotto THE GRAVITATIONAL FIELD

GENERAL RELATIVITY: background independence!

U(1) SU(2) SU(3)

SL(2,C)l

FIELDS ⟷ GAUGE SYMMETRIES

GRAVITY AS AN INTERACTING GAUGE FIELD

Loop Quantum Gravity Francesca Vidotto IN THIS TALK

GAUGES IN - diffeos - graph/lattice LOOP GENERAL RELATIVITY {- Lorentz {- group variables QUANTUM GRAVITY

GAUGE VARIABLES are the handle for the possible interactions of a system

Loop Quantum Gravity Francesca Vidotto GAUGE invariant or not? GAUGE SYMMETRIES

Rovelli, Why gauge? 2013

■ Local Yang-Mills gauge transformations ■ Local Lorentz transformations ■ Diffeomorphism gauge transformations

DIRAC A system is gauge invariant t if evolution is under-determined. ϕ~ (t) ϕ(t)

DETERMINISM ^t Classical physics is deterministic → consider only gauge invariant quantities as “PHYSICAL” ϕ

Loop Quantum Gravity Francesca Vidotto SYMMETRIES: GUIDE OR REDUNDANCY?

Rovelli, Why gauge? 2013

REDUNDANCY: EFFECTIVENESS OF GAUGE UNEXPLAINED! Arkani-Hamed, Cachazo, Kaplan (’10): gauge is just a complication! AdS/CFT: physics coded on the asymptotic boundary …

GAUGE INVARIANCE: INTERPRETATION, NOT COUPLING We interpret a physical system in terms of its gauge invariant objects Necessity of coupling the system via non-gauge-invariant variables!

COUPLING GAUGE SYSTEM µ EM: F = dA L = ¯( Aµ) µ⌫ Gravity: gµ⌫ L = pggµ⌫ T

Gauge-invariant coupling from gauge-variant variables Loop Quantum Gravity Francesca Vidotto ROVELLI’S SPACE FLEET

Rovelli, Why gauge? 2013

Gauge invariant observables: N-1 relative distances M-1 relative distances a = x x n =1,...,n 1 n n+1 n

N 1 N 1 1 2 1 2 L1 = (˙yn 1 y˙n) L1 = (˙xn 1 x˙ n) 2 2 n=1 n=1 X X 1 L = (˙y x˙ )2 int 2 1 N EXTRA DOF: more gauge-inv observables than in the single systems

Loop Quantum Gravity Francesca Vidotto GAUGE AND RELATIONALITY

Rovelli, Why gauge? 2013

RELATIONALITY The observables of a system are not its gauge-dependent quantities but rather the relation between them: Dirac observables are always relational In fact, we always deal with subsystems. HOLISM The observables of the coupled system are more than the sum of individual ones. Gauge-dependent variables contain information about how to couple systems.

MEASUREMENT VS PREDICTIONS Gauge variables are what we measure but we cannot predict (Partial Observables). A couple of gauge variables is gauge invariant and it allows predictions.

These insights applies directly to General relativity and Quantum Mechanics. To carry the space ﬂeet analogy for Yang-Mills theory requires some care! (The, 2013)

Loop Quantum Gravity Francesca Vidotto Examples

Time

General Relativity

Yang-mills Field

Loop Quantum Gravity Francesca Vidotto GENERAL RELATIVITY

BACKGROUND INDEPENDENCE Bodies are only localized with respect to one another. Bodies includes all dynamical objects, also the gravitational ﬁeld. Spacetime is built up by contiguity relations: the fact of being “next to one another”.

GRAVITY AS A GAUGE THEORY The proper time over a world line is a partial observable. Only if I have a second clock I can make predictions.

COUPLING TO A MATERIAL REFERENCE SYSTEM The components of the gravitational field with respect to the directions defined by the matter system are gauge-invariant quantities of the coupled system; but they are gauge-dependent quantities of the gravitational field, measured with respect to a given external frame.

VANISHING HAMILTONIAN Project on the physical space ignoring the partial observables, that are non-gauge-invariant.

Loop Quantum Gravity Francesca Vidotto BOUNDARY FORMALISM Oeckl, 2003 Time is pure gauge, the Hamiltonian constraint determine time evolution Gauge-constraint for the internal gauge

Democratisation of gauges: they all determine dynamical constraints (qn0 ,pn0 ,t0)

among partial observables measured at the boundaries of a process. ✓n0

These constraints code the full content of a dynamical theory:

Yang-Mills constraint determines variable change ✓n (qn,pn,t) wrt a change of the internal boundary frame. Diff constraint determines variable change wrt a change in the location of the spatial boundary reference frame. Hamiltonian constraint dertermines collective variable change wrt a change in the temporal location of the boundary (time of measurement) Indeterminacy ⟷ arbitrariness of the frame choice. Dynamics is the study of relations between partial observables that are gauge-dependent quantities of a system to which we can couple an apparatus.

Loop Quantum Gravity Francesca Vidotto LOOP QUANTUM GRAVITY

And God said ( , , ) H A W Hilbert Space: and there was Transition Amplitude: L N = L [SU(2) /SU(2) ] Wv =(PSL(2, ) Y v)(1I) H 2 SPACETIME C

Operator Algebra: i j 2 ij k La,Lb = iab⇤ ⇥k La ⇥ ⇤ ( , , ) deﬁnes a background independent quantum ﬁeld theory H A W Loop Quantum Gravity Francesca Vidotto Loop Quantum Gravity Kinematics

Loop Quantum Gravity Francesca Vidotto

•

HILBERT SPACE & OPERATOR ALGEBRA •

•

Abstract graphs: Γ={N,L}

jl •

•

Group variables: h l SU (2) •

{ ~ 2 s(l) l L su(2) • l 2 Graph Hilbert space: = L [SU(2)L/SU(2)N ] t(l) H 2 G The space admits a basis ,j ,v ll H ` n | i l QUANTUM GEOMETRY l Gauge invariant operator G ll = L l L l with Gll =0 · 0 l n Al X2 Penrose’s spin-geometry theorem (1971), and Minkowski theorem (1897)

h l “Holonomy of the Ashtekar-Barbero connection along the link”

L ⌅ = L i ,i =1 , 2 , 3 SU(2) generators l { l} gravitational ﬁeld operator (tetrad)

Loop Quantum Gravity Francesca Vidotto

L⌅ = L i ,i =1 , 2 , 3 SU(2) generators l { l} gravitational ﬁeld operator (tetrad) Gll

l l Gauge invariant operator G = L L with G =0 ll l · l ll0 l n Al X2 Penrose’s spin-geometry theorem (1971), and Minkowski theorem (1897)

Loop Quantum Gravity Francesca Vidotto REPRESENTING GEOMETRIES

Composite operators:

i i Area: A = L lLl. l ⇥ 2 2 i j k Volume: V R = V n ,V n = ijkLlL Ll” . 9 | l | n R Angle: LiLi l l0

hG¯ 35 Geometry is quantized: DISCRETE: eigenvalues are discrete ` = 10 m P c3 ⇠ FUZZY: the operators do not commute r quantum superposition (coherent states)

Lorentz invariance is compatible with quantum discreteness! (do not confuse with classical discreteness) Same as the angular momentum: the discrete spectrum does not break the rotational symmetry.

“Without a deep revision of classical notions it seems hardly possible to extend the quantum theory of gravity also to [the short-distance] domain.” Matvei Bronstein

Loop Quantum Gravity Francesca Vidotto

•

HILBERT SPACE & OPERATOR ALGEBRA •

•

Abstract graphs: Γ={N,L}

jl • • QUANTUM GEOMETRY •

Group variables: h l SU (2)

{ ~ 2 s(l) l L su(2) • l 2 Graph Hilbert space: L N t(l) = L2[SU(2) /SU(2) ] 4D H G The space admits a basis ,j ,v Extrinsic Curvature ll H | ` ni l l 1D Gauge invariant operator G = L L with G =0 Intrinsic Curvature 3D ll l · l ll0 2D l n Al X2 Penrose’s spin-geometry theorem (1971), and Minkowski theorem (1897)

h l “Holonomy“Holonomy ofof thethe Ashtekar-BarberoAshtekar-Barbero connection along thethe link”link” ⌅ i ⌅L l = L li ,i =1 , 2 , 3 SU(2)SU(2) generatorsgenerators Ll = {L },i=1, 2, 3 { l} gravitational ﬁeldﬁeld operatoroperator (tetrad)(tetrad) Gll

Loop Quantum Gravity Francescal Vidottol Gauge invariant operator G = L L with G =0 ll l · l ll0 l n Al X2 Penrose’s spin-geometry theorem (1971), and Minkowski theorem (1897)

Loop Quantum Gravity Francesca Vidotto

•

HILBERT SPACE •

•

v •

Abstract graphs: Γ={N,L}

jl • • hl SU(2) •

Group variables: 2 s(l)

{ ~ l L su(2) • l 2 Graph Hilbert space: = L [SU(2)L/SU(2)N ] t(l) H 2

The space admits a basis ,j`,vn H | i • Restrict the states to a fixed graph with a finite number N of nodes. • • This defines an approximated kinematics of the universe, inhomogeneous but truncated at a finite number of cells. • • • • • The graph captures the large scale d.o.f. obtained averaging the metric over the faces of a cellular decomposition formed by N cells. • •

The full theory can be regarded as an expansion for growing N. • • For instance FRW cosmology corresponds to the lower order where there is only a regular cellular decomposition: the only d.o.f. is given by the volume. . Different graphs can be useful to model different physical situations. • . • Loop Quantum Gravity Francesca Vidotto QUANTUM DISCRETENESS IS COMPATIBLE WITH LORENTZ INVARIANCE

Classical discreteness breaks Lorentz invariance. Quantum discreteness does not!

Example: ROTATIONAL INVARIANCE

Classical: a vector with only discrete components breaks SO(3).

Quantum: quantum vector with discrete eigenvalues is compatible with a SO(3) invariant theory.

Geometry is a quantum geometry! L L(β) A boost do not change the spectrum L(β) of geometrical quantities, only their L probability to be measured.

Loop Quantum Gravity Francesca Vidotto Lorentz invariance and quantum discreteness are compatible

=> Geometry is quantum geometry

Lorentz invariance and quantum discreteness are compatible

=> Geometry is quantum geometry CONVERGENCE BETWEEN QED & QCD PICTURES

All physical QFT are constructed via a truncation of the d.o.f. (cfr: particles in QED, lattice in QCD) All physical calculation are performed within a truncation. The limit in which all d.o.f. is then recovered is pretty different in QED and QCD:

+ + ....

→ → ....

QUANTUM GRAVITY Diff invariance ! [Rovelli, Ditt-invariance, 2011]

The lattice is not on spacetime, it is spacetime! Lattice = Feynaman diagram !!!

Lattice site = small region of space = excitations of the gravitational ﬁeld = quanta of space = quanta of the ﬁeld

Loop Quantum Gravity Francesca Vidotto 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 discretiza- tion 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 of the the continuous limit of the discretized theory, that is, if system and q the eigenstates of its q observables. The | i lim SN (q, t, q0,t0)=S(q, t, q0,t0). (9) transition amplitude STRUCTURE OFN THE THEORY !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 No criticalThe interest point of this structureViability is that of it the remains expansion: mean- obtained by reading out the Hamilton function from the Noingful infinite in renormalization di↵eomorphism invariantfirst systemsradiative corrections and o↵ ersare logarithmic an (Riello’12) quantum transition amplitude: Physicalexcellent scale: conceptual Planck length tool for dealingRegime with of validity background of the expansion: in- L L 1/R dependent physics. To see this, let’sPlanck first⌧ consider⌧ its gen- lim( i~) log W (q, t, q0,t0)=S(q, t, q0,t0). (5) eralization to finite dimensional parametrizedp systems. 0 ~! Loop Quantum Gravity Francesca Vidotto 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 LIMIT ℏ⟶0

→

DISCRETE NO DISCRETENESS

FUZZY NO FUZZYNESS

PROBABILISTIC A CLASSICAL FIELD ・

Loop Quantum Gravity Francesca Vidotto LIMIT ℏ⟶0

→

DISCRETE NO DISCRETENESS

FUZZY NO FUZZYNESS

PROBABILISTIC A CLASSICAL FIELD

Loop Quantum Gravity Francesca Vidotto Loop Quantum Gravity Dynamics

Loop Quantum Gravity Francesca Vidotto THE THEORY

And God said ( , , ) H A W Hilbert Space: and there was Transition Amplitude: L N = L [SU(2) /SU(2) ] Wv =(PSL(2, ) Y v)(1I) H 2 SPACETIME C

Operator Algebra: i j 2 ij k La,Lb = iab⇤ ⇥k La ⇥ ⇤ ( , , ) deﬁnes a background independent quantum ﬁeld theory H A W Loop Quantum Gravity Francesca Vidotto A REMINDER OF THE CLASSICAL THEORY

a (1,3) a Variables e = e a dx R and ! = !adx sl(2, C) 2 2

1 Action S[e, ]= B[e] F [] where B =(e e)⇤ + (e e) ^ ^ ^ Z

1 Boundary B oi = K i = e o e i and Bij = Li = eo ei gauge s.t. tetrads are diagonal ^ ^

ni =(1, 0, 0, 0)

Simplicity constraint K~ = L~

SL(2, C) SU(2) Lorentzian area A = eo ei = Ki = Li ! ^ ZR ZR ZR

Loop Quantum Gravity Francesca Vidotto NO HIDDEN DOF 1 Spinfoam dynamics = constrained BF theory ⇥,B=(e e)⇤ + (e e) ^ ^ ~ ~ t K + L =0

P 0 Constantly accelerated observer: 8⇡G =1

z Kz generators of boosts

E = aKz generator of proper time evolution P

Frodden-Gosh-Perez ‘11: dE = adA = dS T T black hole

Jacobson ‘95: if dE = adA holds, then for any point and any observer the Einstein’s equations follow. simplicity constraint + Lorentz invariance + general covariance = GR

Loop Quantum Gravity [Baez-Bunn ‘15 , Chirco,-Haggard,-Riello,-Rovelli, ‘14] Francesca Vidotto LIMIT ℏ⟶0

→

DISCRETE NO DISCRETENESS

FUZZY NO FUZZYNESS

PROBABILISTIC A CLASSICAL FIELD

Loop Quantum Gravity Francesca Vidotto FEYNMAN GRAPH SPINFOAM

Loop Quantum Gravity Francesca Vidotto A REMINDER OF THE CLASSICAL THEORY

a (1,3) a Variables e = e a dx R and ! = !adx sl(2, C) 2 2

1 Action S[e, ]= B[e] F [] where B =(e e)⇤ + (e e) ^ ^ ^ Z

1 Boundary B oi = K i = e o e i and Bij = Li = eo ei gauge s.t. tetrads are diagonal ^ ^

ni =(1, 0, 0, 0)

Simplicity constraint K~ = L~

SL(2, C) SU(2) Lorentzian area A = eo ei = Ki = Li ! ^ ZR ZR ZR

Loop Quantum Gravity Francesca Vidotto DYNAMICS

()= d W W jf v v ⇥f ⇥

“spinfoam”: two-complex with faces f and edges e colored with spins and intertwiners, bounded by

dj =2j +1 Wv Wv =(PSL(2, ) Y v)(1I) C σ : spinfoam Y : . Hj ⌅⇤ Hj ⇥ H(p=(j+1),k=j) SU(2) rep SL(2,C) rep

Loop Quantum Gravity Francesca Vidotto SPINFOAM AMPLITUDES [Engle-Pereira-Livine-Rovelli, Freidel-Krasnov ’08]

Probability amplitude P ()= W 2 | | ⇥| Amplitude associated to a state of a boundary of a 4d region

Superposition principle

Locality: vertex amplitude • • • Lorentz covariance

Unitary irreducible representations •

Simplicity constraint • Classical limit: GR

www.cpt.univ-mrs.fr/~rovelli/IntroductionLQG.pdf Loop Quantum Gravity Francesca Vidotto BOUNDARY FORMALISM

QUANTUM MECHANICS GENERAL RELATIVITY

Process ← Locality → Spacetime region State Boundary, space region

Spacetime is a process, a state is what happens at its boundary.

Boundary state = in ⌦ out Boundary

Amplitude of the process A = W ( ) Spacetime region

Loop Quantum Gravity Francesca Vidotto SPINFOAM AMPLITUDES [Engle-Pereira-Livine-Rovelli, Freidel-Krasnov ’08]

2 Probability amplitude P ()= W 4d | | ⇥| Amplitude associated to a state of a boundary of a 4d region

3d boundary Superposition principle W ⇥ = W () | ⇥ Locality: vertex amplitude W () Wv. v W =(P Y )(1I) Lorentz covariance v SL(2,C) v boundary graph Unitary irreducible representations

Simplicity constraint K~ = L~

Classical limit: GR Barrett, Dowall, Fairbain, Gomes, Hellmann, Alesci...’09

a spin network history

Loop Quantum Gravity Francesca Vidotto FROM QUANTUM TO CLASSICAL

FROM QUANTUM TO CLASSICAL The classical limit is ℏ ⟶ 0 , the limit for ∞ quanta is relevant for the continuous limit No thermodynamical limit is needed at that stage.

EMERGENCE OF SPACETIME IS STANDARD CLASSICAL EMERGENCE just as the electromagnetic ﬁeld emerges from photons

SPACETIME IN THE QUANTUM REGIME IS MADE OF QUANTA ■ there is no classical spacetime in the quantum regime ■ same as in Q.E.D. where there are photons

SPACETIME IN THE QUANTUM REGIME IS A QUANTUM PROCESS ■ states are deﬁned by the continuity relations between quanta ■ a spinfoam is a quantum interaction, but also a spacetime region

Loop Quantum Gravity Francesca Vidotto CONCLUSIONS

REALITY NOT MADE BY GAUGE-INVARIANT QUANTITIES ONLY gauge-variant quantities are not a mathematical redundancy: systems couple via gauge-dependent quantities gauge variables are components of relational observables which depend on more than a single component

LOOP QUANTUM GRAVITY

GAUGES IN - diffeos - graph/lattice LOOP GENERAL RELATIVITY {- Lorentz {- group variables QUANTUM GRAVITY

GRAPHS are a natural diff invariant structures no Lorentz invariance breaking lattice site = small region of space = excitations of the gravitational ﬁeld = quanta of space = quanta of the ﬁeld

KINEMATICS is based on group variables and gauge invariant states discrete fuzzy geometry

LORENTZIAN DYNAMICS obtained quantizing the simplicity constrain it maps the SU(2) boundary states into the Lorentzian bulk

Loop Quantum Gravity Francesca Vidotto

Gauge is ubiquitous. It is not unphysical redundancy of our mathematics. It reveals the relational structure of our world.

Loop Quantum Gravity Francesca Vidotto