Classical limit of spinfoams on arbitrary...

Classical limit of spinfoams on arbitrary triangulations

Claudio Perini

Institute for Gravitation and the Cosmos, Penn State University

ILQGS - Valentine’s Day 2012

C. Perini (Penn State University) ILQGS - Valentine’s Day 2012 1 / 25

References

Curvature in spinfoams, Magliaro-CP, Mar 2011 (CQG Highlight)

Regge gravity from spinfoams, Magliaro-CP, May 2011 (to app. CQG)

Emergence of gravity from spinfoams, Magliaro-CP, Aug 2011 (EPL)

Einstein-Regge equations in spinfoams, CP, Oct 2011 (JPCS)

The flipped limit of EPRL spinfoam, CP, Feb 2012 (to app.)

Main inspiration

Lorentzian spin foam amplitudes: graphical calculus and asymptotics,Barrett-Dowdall-Fairbairn-Hellmann-Pereira

On the semiclassical limit of 4d spin foam models, Conrady-Freidel

Related directions

Asymptotics of Spinfoam Amplitude on Simplicial Manifold: Lorentzian Theory, Han-Zhang

Perturbative quantum gravity with the Immirzi parameter, Benedetti-Speziale


Plan of the talk

The theory

LQG kinematics

LQG dynamics in Spinfoam language

truncated amplitudes vs. continuum limit

Low-energy behavior

semiclassicality: the flipped limit

stationary phase in two diff. schemes

emergence of classical geometry

application: structure of graviton propagator

To be done: homeworks


The theory

(tentative, of course)


LQG in a nutshell

KINEMATICS: spin-networks

Gauge-invariant H = ⊕ΓHΓ does not depend on a lattice continuum theory

HΓ = L2(SU(2)#links/SU(2)#nodes) comp. to Yang-Mills (1)

Pictorially, Γ =quanta of space with adjacency relations

s(l)

t(l)

l

Use Peter-Weyl decomposition in SU(2) irreps:

HΓ = ⊕jl ⊗N HN , HN = Inv⊗l⊃N Hjl (2)

Choosing Bloch SU(2) coh. states for Hjl have (overcomplete) basis |Γ, jl, ~nNl〉 of spin-nets


LQG in a nutshell (2)

KINEMATICS: geometry operators and semiclassical interpretation

Based on flux-holonomy algebra: Ll =∫l∗ E ∈ su(2), gl = Pe

∫l A ∈ SU(2)

{~Ll, gl′} = γ~ ~τgl′δll′ ~τ su(2) gen. γ Barbero-Immirzi parameter (3)

Quantization: • HΓ

• flux −→ ~Llf(gl) = γ~d

d~α

∣∣∣∣α=0

f(e~α·~τgl)

• holonomy −→ glf(gl) = glf(gl)

The gauge-invariant Barbieri-Penrose operator acts on each HN (N = s(l) = s(l′))

Gll′ = ~Ll · ~Ll′

l = l′ is area operator: Al =

√~Ll · ~Ll Al |Γ, j, ~n〉 = γ~

√jl(jl + 1) |Γ, j, ~n〉

l 6= l′ is rel. to angle operator. But non commuting, thus tetrahedron only in s.c. sense!


LQG in a nutshell (3)

DYNAMICS: spin-foams (2-complex, t-evolution of spin-network with branching points)4. COVARIANT FORMULATION: SPIN FOAM

Figure 5: A spin foam seen as evolution of spin-networks

colored with the irreducible representations and intertwiners inherited from spin-networks, namely it is aspin foam.

The underlying discreteness discovered in LQG is crucial to have a well-defined physical scalar product.Indeed the functional integral for gravity is replaced by a discrete sum over quantum amplitudes associatedto combinatorial objects with a foam-like structure. A spin foam represents a possible quantum historyof the gravitational field. Boundary data in the path integral are given by quantum states of 3-geometry,namely spin-networks.

Deriving Spin Foam Models from the canonical theory can be a difficult task; for this reason they arederived directly from the Lagrangian of General Relativity. The most common setting is a discretizationof the topological BF field theory: given a simplicial decomposition of spacetime one defines a discretizeversion of BF theory and then quantize it. The result of the quantization is a recipe for spin foamamplitudes on a fixed triangulation of spacetime. This is a bad feature, because we expect from thecanonical theory that the Feynmann path integral contains, besides a sum over quantum numbers, also asum over 2-complexes. The most common tool used to recover the sum over 2-complexes (usually a sumover triangulations) is an auxiliary field theory called Group Field Theory (GFT) [78, 79, 80, 81]. TheSFM/GFT duality permits the identification

Z[Γ] = ZGFT[Γ] , (135)

where the GFT amplitude ZGFT[Γ] is defined inside a Feynman expansion

ZGFT =∑

Γ

λv[Γ]

sym[Γ]ZGFT[Γ] . (136)

The GFT sum (136) is over a certain class of triangulations, and possibly also over spacetime topologies,and v[Γ] is the number of vertices in the triangulation Γ. The physical origin of the sum over graphscan be understood as follows: in a realistic model for quantum gravity, the sum over 2-complexes, or atleast over simplicial triangulations, must be present in order to capture the infinite number of degrees offreedom of General Relativity: General Relativity is a field theory with local degrees of freedom. This isalso what we expect from the canonical theory, from expression like (134).

34

Can be thought as dual to spacetime cellular decomposition: v, e, f dual of v∗, e∗, f∗

Transition amplitude for the boundary spin-network:

W (jext, ~next) =∑

jint,~nint

∏f

F (j)∏e

E(j, ~n)∏v

V (j, ~n) (4)

Can be interpreted both as a set of Feynman rules for the vertices and propagators of the2-complex, or as a Misner-Hawking path integral over virtual geometries


Decoration of oriented 2-complex

e

f

s(e, f)

t(e, f)

f jf

(ev) gev = g−1ve ∈ SL(2,C)(ef) ~nef

Intuitively:

jf area of f∗

~nef unit normal to f∗ in the frame of e∗

gev parallel transport from v to e, gv′v = gv′egev for the full edge


The simple form of the dynamics (EPRL/FK)

W (jext, ~next) =∑jint

∫dg

∫d~nint

∏f

(2jf + 1)Pf (5)

Face amplitude:

Pf = tr−→∏

e⊂fPef (6)

Pef = gt(e,f),eY |jf , ~nef 〉〈jf , ~nef |Y †g†s(e,f),e

(half for ext. faces) (7)

Y is the embedding of SU(2) irreps j into lowest weight k = j of SL(2,C) irreps (j, γj)

H(j,γj) =⊕k≥jH(j,γj)k (8)

Path integral-like representation (Freidel-Conrady, Barrett . . . , CP-Magliaro)


∫dg

∫d~nint exp(S(j, g, ~n)) (9)

the action S is sum of local terms (omitting extra variables zvf ∈ CP1)


Truncated amplitudes vs continuum limit (taken from Rovelli’s paper)

amplitude W (σ∞)~→0−−−→ eiS∞

N→∞

−−−−→

N→∞

−−−−→

truncated amplitude W (σN )~→0−−−→ eiSN


Low-energy behavior


Concepts of semiclassicality: the flipped limit

Area spectrum:

a = γ~√j(j + 1) ∼ γ~ j (10)

1. large quantum numbers j →∞, large spacetime atoms2. alternatively, keep the size fixed but disregard quantum geometry

flipped limit j →∞, γ → 0, γj = fix

Dynamics breaks equivalence. Restoring the Planck constant,

W =

∫e

1~S

0+ 1γ~S

′(11)

1. is the usual classical limit ~→ 0 (oscillations same magnitude)2. the second (Holst) term oscillates more rapidly

can also combine 1. and 2., similar to earlier Bojowald’s proposal in LQC.

In the semiclassical region the exponent is rapidly oscillating suppressed unless stationaryphase: classical paths are generated from constructive interference


Setup of the stationary phase

Def. the partial amplitude W (jext, ~next; jint), still no sum over internal spins


W (jext, ~next; jint) (12)

PRO well-defined problem, rigorous results,

CONS partial information on equations of motion

Intuition: eom of partial amplitude analogous to variation of the connection in first order gravity

The classical limit of the partial amplitude can be studied in the asymptotic expansion

1. Wγ(αjint, αjext), α→∞ (Conrady-Freidel)2. Wγ/α(αjint, αjext), α→∞ (CP-Magliaro)

Equations of motion

1. ReS = δgS = δ~nS = 0

2. ReS′ = δgS′ = δ~nS′ = 0

Result: the eom are the same 1.=2. ! for most of following consider the two cases at once


Eom in terms of 4D geometry variables

From (jf , gev , ~nef ) define the space-like bivectors in R1,3

E1

E2

E1 ^ E2

Aef (v) := afgve ∗ (Ne ∧ Ñe)

have used a null decomposition in terms of

N = (1, ~n), Ñ = (1,−~n) (13)

Eom

Transport Af (v) := Aef (v) = Ae′f (v) v ⊂ e, e′

Closure∑f⊃e sgn(v, e, f)Af (v) = 0

plus, by construction

Simplicity Af (v) ∧Af (v) = 0Cross-simplicity Af (v) ∧Af ′ (v) = 0 f, f ′ ⊃ e

These four constraints define a bivector geometry.

Result: bivector geometries are the stationary phase configurations of path integral


Emergence of classical geometry: the 4-simplex (Barrett et al.)

Given oriented Lorentzian 4-simplex (σv , µv), its area bivectors

Af (σv , µv) := µv af ∗Ns(f,v) ∧Nt(f,v)|Ns(f,v) ∧Nt(f,v)|

(14)

are nondegenerate (n.d.) and satisfy bivector geometry constraints.

Theorem (Barrett and Crane). Given a n.d. bivector geometry Af (v) there exists a uniqueoriented 4-simplex (σv , µv) in R1,3, up to inversion xµ → −xµ and translation, such that

Af (v) = Af (σv , µv) (15)

and µv is independent from ambiguity in σv .

Useful property: parity P : ~x→ −~x flips µv . Indeed

PAf (v) = PAf (σv , µv) = Af (Pσv ,−µv) (16)

Consequences: given a arbitrary n.d. stationary configuration (jf , gev , ~nef ), there is always aP-related with µv constant on the 2-complex.

Call such configuration global orientation consistent


Spinfoam symmetries

Symmetries of the action and of the solution space

Local Lorentz invariance at vertices:

gve → hvgve (17)

Spin lift SO+(1, 3)→ SL(2,C):gve → −gve (18)

Unit vectors reparametrization at edges:

~nef → ue~nef (19)gve′ → uegve′ (20)

where ue ∈ SU(2) commutes with gs(e),e and gt(e),e and e′ 6= e.

Symmetries of the solution space but not of the action

Local parity (see previous slide) at vertices:

gve → (g†ve)−1 (21)

Global parity of the unit vectors:~nef → −~nef (22)


Emergence of classical geometry: glueing of 4-simplices

Apply Barrett-Crane theorem to two neighboring spinfoam vertices v, v′. Reconstruct

(σv , µv), (σv′,µ′ )

Theorem.

there is a Poincaré transformation s.t. τe → τ ′e, N → −N ′

its rotation part i.e. the O(1, 3) Levi-Civita holonomy is computed as (g in adjoint)

Uv′v :=

{µe gv′egev , µv = µv′

µe gv′ePgev , µv 6= µv′(23)

µv =reconstructed orientations, µe = pvpv′ =inversion ambiguity in the reconstruction

this theorem a n.d. solution of eom determine a classical Regge geometry

this theorem+symmetries solutions in same symmetry class determine the same geometry

Observe: the spinfoam on-shell gv′v is not always the Levi-Civita connection. There are in generalP , T insertions at the edge. Can use P to turn any solution into a global orientation consistent,so always in first case of (23).


Curvature

The concept of curvature in simplicial spacetime is the one of Regge calculus: curvature lives onhinges. 4-simplices are individually flat but can be glued to have deficit angle.

✓1

✓2 ✓3

⇥

Figure: 2D analogy

Hinge=point (2D), line segment (3D), triangle (4D)

Nontrivial parallel transport around a loop, boundary of face f

Uf (v) := Uvvn . . . Uv′v (24)

For µv = const. = 1

Uf (v) = (Πe⊂fµe) gf (v) = e∗Af (v)Θf (mod flip of f∗ and inversion) (25)

relates simply to deficit angle Θ of Regge calculus.


On-shell nondegenerate action

For µv = 1 configuration the stationary action gives, in the two schemes

1. S(a, g, ~n)|on-shell = S0(a, g, ~n)|on-shell = i∑f afΘf (Regge action)

2. S′(a, g, ~n)|on-shell = 0

In second case, the total action is S = S0 + 1γS′, and thus get the same result as 1.

For µv = −1 we get the c.c., thus minus the Regge actionFor µv not constant (related by parity) we get only a generalized Regge action

For jf Regge-like, in large spin and flipped limit have the asymptotic expansion

W ∼ Aei~SRegge + p.r. + deg. + q-corr.

A=slowly varying prefactor different in the two schemes, p.r.= parity-related. The q-corr. are

1. ~ corrections quantum gravity corrections

2. γ~ corrections quantum geometry γ-corrections::::comp.

:::::single

::::term e

i~SEH (gµν)


Towards Einstein equations

ei~SRegge is quantum gravity path integral with Holst-Palatini action, and on-shell connection

Einstein eqns must be obtained studying full eom (spin variations)

Get extra eom in both schemes, but now they reveal difference

1. δjf S = 0 Θf = 0 flatness problem (Mamone-Rovelli, Bonzom)

2. δjf S′ = 0 Θ∗f = 0 automatically satisfied, torsionless

Suggests postulate 2. (instead of 1.) to get rid of the flatness problem.

first restrict to Regge geometries, then vary the action wrt spins⇓

Regge equations ∼ Ricci flatness correct!

But what is the physical origin of γ → 0? After all, γ is a fixed parameter. . .Not clear to me, but recall are using truncated effective theory and not in the continuum regime.

Could check if γ runs to zero (?) under coarse-graining, whatever it means


Graviton propagator (Bianchi-CP-Magliaro)

D(2)12 = a

30(R

(2)12 + γX12 + γ

2Y12) +O(~a20)

Scales as a30 as expected for correlations of objects with dimensions of area square

No component is suppressed, solves Barrett-Crane difficulties

Matches with standard QFT and Regge calculus, up to quantum geometry γ-corrections

Most interesting feature:

The γ corrections are parity-breaking ! γX + γ2Y = ei2π3 (γPX + γ2PY )

L.H. 6= R.H. chiral gravitons?

Potentially important prediction, must be studied further.


To be done(homeworks)

Devised a new semiclassical expansions: the flipped limit

Have shown how classical geometry emerges from constructive interference in path integral

Besides the ‘good’ eiS/~ term, parity-related terms and degenerate contribution: show thesedon’t affect semiclassical observables (in known simple examples it is the case!)

How does the amplitude behave in the limit of large 2-complexes?

Do the semiclassical results for the truncated theory still hold?

Check the robustness of graviton propagator structure, compute next orders

Include matter in the analysis


To be done(homeworks)

Devised a new semiclassical expansions: the flipped limit

Have shown how classical geometry emerges from constructive interference in path integral

Besides the ‘good’ eiS/~ term, parity-related terms and degenerate contribution: show thesedon’t affect semiclassical observables (in known simple examples it is the case!)

How does the amplitude behave in the limit of large 2-complexes?

Do the semiclassical results for the truncated theory still hold?

Check the robustness of graviton propagator structure, compute next orders

Include matter in the analysis

I know how to do it!


Classical limit of spinfoams on arbitrary...

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