Regge Quantum Gravity Aleksandar Mikovi c Lusofona University and - - PowerPoint PPT Presentation

Regge Quantum Gravity

Aleksandar Mikovi´ c Lusofona University and GFM - Lisbon University July 2014

Aleksandar Mikovi´ c Lusofona University and GFM - Lisbon University Regge Quantum Gravity

Regge quantum gravity

◮ Path integral quantization of GR with matter based on Regge

calculus such that a triangulation T(M) of the spacetime manifold M is taken as the fundamenal structure.

◮ If N is the number of cells of T(M), then for N ≫ 1, T(M)

looks like the smooth manifold M.

◮ It is not necessary to define the smooth limit N → ∞. Insted,

we need the large-N asymptotics of the observables.

◮ Semiclassical limit will be described by the effective action

Γ(L), which is computed by using the effective action equation from QFT, in the limit Lǫ ≫ lP.

◮ Inspired by the spin-cube models, which are generalizations of

the spin-foam models.

Aleksandar Mikovi´ c Lusofona University and GFM - Lisbon University Regge Quantum Gravity

Regge state-sum models

◮ The fundamental DOF are the edge-lengths Lǫ ≥ 0, and

Z =

• DE

E

• ǫ=1

dLǫ µ(L) exp

• iSRc(L)/l2

P

• ,

where SRc = −

F

• ∆=1

A∆(L)θ∆(L) + ΛcV4(L) , is the Regge action with a cosmological constant. DE ⊂ RE

+

such that the triangle inequalities hold.

◮ We choose the following PI measure

µ(L) = e−V4(L)/L4

0 ,

where L0 is a free parameter.

◮ We also introduce a classical CC length scale Lc such that

Λc = ±1/L2

c .

Aleksandar Mikovi´ c Lusofona University and GFM - Lisbon University Regge Quantum Gravity

Regge state-sum models

◮ We will use the effective action in order to determine the

quantum corrections.

◮ EA is different from the Wilsonian approach to quantization

which is used in QRC and CDT.

◮ In WQ approach

Z(g, λ) =

• DE

E

• ǫ=1

dLǫ µ(L) exp

• igSR(L)/l2

0 + iλV4(L)/l4

• ,

and one looks for points (g∗, λ∗) where Z ′′ diverges.

◮ In the vicinity of a critical point the correlation length diverges

⇔ transition from the discrete to a continuum theory.

◮ The semiclassical limit in WQ corresponds to the

strong-coupling region |g| ≥ 1 ⇒ can be studied only numerically.

Aleksandar Mikovi´ c Lusofona University and GFM - Lisbon University Regge Quantum Gravity

Effective action equation

◮ Let φ : M → R and S(φ) =

• M d4x L(φ, ∂φ) a QFT

flat-spacetime action. The effective action Γ(φ) is determined by the integro-differential equation eiΓ(φ)/ =

• Dh exp

i S(φ + h) − i

• M

d4x δΓ δφ(x)h(x)

• .

◮ A solution Γ(φ) ∈ C, so that a Wick rotation is used to obtain

Γ(φ) ∈ R: solve the Euclidean equation e−ΓE (φ)/ =

• Dh exp
• −1

SE(φ + h) + 1

• M

d4x δΓE δφ(x)h(x)

• ,

and then put x0 = −it in ΓE(φ).

◮ Wick rotation is equivalent to Γ(φ) → Re Γ(φ) + Im Γ(φ)

Aleksandar Mikovi´ c Lusofona University and GFM - Lisbon University Regge Quantum Gravity

Regge effective action

◮ In the case of a Regge state-sum model

eiΓ(L)/l2

P =

• DE (L)

dEx µ(L + x)eiSRc(L+x)/l2

P−i E ǫ=1 Γ′ ǫ(L)xǫ/l2 P ,

where l2

P = GN and DE(L) is a subset of RE obtained by

translating DE by a vector −L.

◮ DE(L) ⊆ [−L1, ∞) × · · · × [−LE, ∞) . ◮ Semiclassical solution

Γ(L) = SRc(L) + l2

PΓ1(L) + l4 PΓ2(L) + · · · ,

where Lǫ ≫ lP and |Γn(L)| ≫ l2

P|Γn+1(L)| .

Aleksandar Mikovi´ c Lusofona University and GFM - Lisbon University Regge Quantum Gravity

Perturbative solution

◮ Let Lǫ → ∞, then DE(L) → RE and

eiΓ(L)/l2

P ≈

• RE dEx µ(L + x)eiSRc(L+x)/l2

P−i E ǫ=1 Γ′ ǫ(L)xǫ/l2 P .

◮ The reason is DE(L) ≈ [−L1, ∞) × · · · × [−LE, ∞) so that

∞

−L

dx e−zx2/l2

P−wx = √π lP exp

• − 1

2 log z + l2

P

w2 4z +lP e−z¯

L2/l2

P

2√πz¯ L

• 1 + O(l2

P/z¯

L2) , where ¯ L = L + l2

P w 2z and Re z = −(log µ)′′. The non-analytic

terms in are absent since lim

L→∞ e−z¯ L2/l2

P = 0 ,

for exponentially damped measures.

Aleksandar Mikovi´ c Lusofona University and GFM - Lisbon University Regge Quantum Gravity

Perturbative solution

◮ For DE(L) = RE and µ(L) = const. the perturbative solution

is given by the EA diagrams Γ1 = i 2Tr log S′′

Rc ,

Γ2 = S2

3G 3 + S4G 2 ,

Γ3 = S4

3G 6 + S2 3S4G 5 + S3S5G 4 + S2 4G 4 + S6G 3 , ...

where G = i(S′′

Rc)−1 is the propagator and Sn = iS(n) Rc /n! for

n > 2, are the vertex weights.

◮ When µ(L) = const., the perturbative solution is given by

Γ(L) = ¯ SRc(L) + l2

P¯

Γ1(L) + l4

P¯

Γ2(L) + · · · , where ¯ SRc = SRc − il2

P log µ ,

while ¯ Γn is given by the sum of n-loop EA diagrams with ¯ G propagators and ¯ Sn vertex weights.

Aleksandar Mikovi´ c Lusofona University and GFM - Lisbon University Regge Quantum Gravity

Perturbative solution

◮ Therefore

Γ1 = −i log µ + i 2Tr log S′′

Rc

Γ2 = S2

3G 3 + S4G 2 + Res[l−4 P Tr log ¯

G] , Γ3 = S4

3G 6 + · · · + S6G 3 + Res[l−6 P Tr log ¯

G] +Res[l−6

P ¯

S2

3 ¯

G 3] + Res[l−6

P ¯

S4 ¯ G 2] , · · ·

◮ Since log µ(L) = O

• (L/L0)4

and for Lǫ > Lc , L0 >

• lP Lc

we get the following large-L asymptotics Γ1(L) = O(L4/L4

0) + log O(L2/L2 c) + log θ(L) + O(L2 c/L2)

and Γn+1(L) = O

• (L2

c/L4)n

+ L−2n

0c O

• (L2

c/L2)

• ,

where L0c = L2

0/Lc.

Aleksandar Mikovi´ c Lusofona University and GFM - Lisbon University Regge Quantum Gravity

QG cosmological constant

◮ For Lǫ ≫ lP and L0 ≫ √lP Lc the series

• n

l2n

P Γn(L)

is semiclassical.

◮ Let Γ → Γ/GN so that Seff = (Re Γ ± Im Γ)/GN ◮ One-loop CC

Seff = SRc GN ± l2

P

GNL4 V4 ± l2

P

2GN Tr log S′′

Rc + O(l4 P) ⇒

Λ = Λc ± l2

P

2L4 = Λc + Λqg .

◮ The one-loop cosmological constant is exact because there are

no O(L4) terms beyond the one-loop order.

Aleksandar Mikovi´ c Lusofona University and GFM - Lisbon University Regge Quantum Gravity

QG cosmological constant

◮ This is a consequence of the large-L asymptotics

log ¯ S′′

Rc(L) = log O(L2/¯

L2

c) + log θ(L) + O(¯

L2

c/L2)

¯ Γn+1(L) = O

• (¯

L2

c/L4)n

, where ¯ L2

c = L2 c

• 1 + il2

P(L2 c/L4 0)

−1/2.

◮ Note that

l2

P|Λqg| = 1

2 lP L0 4 ≪ 1 , because L0 ≫ lP is required for the semiclassical approximation.

◮ If Λc = 0, the observed value of Λ is obtained for L0 ≈ 10−5m

so that lPΛ ≈ 10−122.

Aleksandar Mikovi´ c Lusofona University and GFM - Lisbon University Regge Quantum Gravity

Smooth-spacetime limit

◮ Smooth spacetime is described by T(M) with E ≫ 1 ⇒

SR(L) ≈ 1 2

• M

d4x

• |g| R(g) ,

ΛV4(L) ≈ Λ

• M

d4x

• |g| = Λ VM ,

Tr log S′′

R(L) ≈

• M

d4x

• |g|
• a(LK)R2 + b(LK)RµνRµν

, where LK is defined by Lǫ ≥ LK ≫ lP .

◮ LK defines a QFT momentum UV cutoff /LK. LHC

experiments ⇒ LK < 10−19m ⇔ K > 1 TeV.

Aleksandar Mikovi´ c Lusofona University and GFM - Lisbon University Regge Quantum Gravity

Coupling of matter

◮ Scalar field on M

Sm(g, φ) = 1 2

• M

d4x

• |g| [gµν ∂µφ ∂νφ − U(φ)] .

◮ On T(M) we get

Sm = 1 2

• σ

Vσ(L)

• k,l

gkl

σ (L) φ′ k φ′ l − 1

2

• π

V ∗

π(L) U(φπ) ,

where φ′

k = (φk − φ0)/L0k. ◮ The total classical action

S(L, φ) = 1 GN SRc(L) + Sm(L, φ) .

Aleksandar Mikovi´ c Lusofona University and GFM - Lisbon University Regge Quantum Gravity

Coupling of matter

◮ The EA equation

eiΓ(L,φ)/l2

P =

• DE (L)

dEx

• RV dV χ exp
• i ¯

SRm(L + x, φ + χ)/l2

P

−i

• ǫ

∂Γ ∂Lǫ xǫ/l2

P − i

• π

∂Γ ∂φπ χπ/l2

P

• ,

where ¯ SRm = ¯ SRc + GNSm(L, φ).

◮ Perturbative solution

Γ(L, φ) = S(L, φ) + l2

PΓ1(L, φ) + l4 PΓ2(L, φ) + · · ·

is semiclassical for Lǫ ≫ lP, L0 ≫ lP and |√GN φ| ≪ 1. This can be checked in E = 1 toy model S(L, φ) = (L2 + L4/L2

c)θ(L) + L2θ(L)φ2(1 + ω2L2 + λφ2L2) .

Aleksandar Mikovi´ c Lusofona University and GFM - Lisbon University Regge Quantum Gravity

Coupling of matter

◮ Γ(L, φ) = Γg(L) + Γm(L, φ) ◮ Γm(L, φ) = V4(L) Ueff (φ) for constant φ and Ueff (0) = 0. ◮ Γg(L) = Γpg(L) + Γmg(L) and

Γmg(L) ≈ ΛmVM + Ωm(R, K) in the smooth-manifold approximation and K = 1/LK.

◮ Ωm = Ω1l2 P + O(l4 P) and

Ω1(R, K) = a1K 2

• M

d4x

• |g| R+

log(K/ω)

• M

d4x

• |g|
• a2R2+a3RµνRµν+a4RµνρσRµνρσ+a5∇2R
• +O(L2

K/L2) .

Aleksandar Mikovi´ c Lusofona University and GFM - Lisbon University Regge Quantum Gravity

CC problem solution

◮ The effective CC

Λ = Λc + Λqg + Λm , where Λm =

• γ

ved(γ, K) = l2

P K 4 f (K 2/ω2, λ l2 P)

is a sum of 1PI vacuum diagrams for Sm(η, φ) with the cutoff K.

◮ Postulate

Λc + Λm = 0 , then Λ = Λqg = l2

P

2L4 ≪ 1 l2

P

.

◮ Λc + Λm = 0 is possible because it is equivalent to

±1/L2

c + l2 P K 4 f (K 2/ω2, λ l2 P) = 0 .

Aleksandar Mikovi´ c Lusofona University and GFM - Lisbon University Regge Quantum Gravity

Conclusions

◮ L0 ≫ lP consistent with the semiclassical approximation. ◮ Exact SUSY ⇒ Λm = 0 and Λc = 0. If SUSY broken ⇒

Λm = −Λc = 0.

◮ For small Lǫ we need a non-perturbative solution of the EA

• equation. Will be relevant for inflation.

◮ Relation to the Wilsonian quantization. ◮ EA formalism only applicable for M = Σ × I. If

M = M0 ∪ (Σ × I) , ∂M0 = Σ , then EA can be interpreted as a de-Broglie-Bohm formulation for the Hartle-Hawking wavefunction.

◮ Regge SS model represents a simple QG theory consistent

with the observations.

Aleksandar Mikovi´ c Lusofona University and GFM - Lisbon University Regge Quantum Gravity

References

◮ A. Mikovi´

c and M. Vojinovi´ c, Cosmological constant in a Regge state-sum model of quantum gravity, arxiv:1407.1124.

◮ A. Mikovi´

c, Effective actions for Regge state-sum models of quantum gravity, arxiv:1402.4672.

◮ A. Mikovi´

c, Spin-cube models of quantum gravity, Rev. Math.

• Phys. 25 (2013) 10, 1343008.

◮ A. Mikovi´

c and M. Vojinovi´ c, Effective action and semiclassical limit of spin-foam models, Class. Quant. Grav. 28 (2011) 225004.

◮ A. Mikovi´

c and M. Vojinovi´ c, Effective action for EPRL/FK spin-foam models, J. Phys.: Conf. Ser. 360, (2012) 012049.

Aleksandar Mikovi´ c Lusofona University and GFM - Lisbon University Regge Quantum Gravity