On quantum gravity problem within geometric approach D.G. Pak - PowerPoint PPT Presentation

On quantum gravity problem within geometric approach D.G. Pak JINR, LTP Int. Workshop “Bogolyubov Readings Sept. 2010

Plan of talk * Preliminaries: geometric approach to quantum gravity problem I. Vacuum tunneling in Einstein gravity. II. Quantum gravity models with torsion Main principles, ideas Yang-Mills type gravity model Induced effective Einstein gravity III. Minimal model of Lorentz gauge gravity with torsion Basic idea, the model Dynamical content in Lagrange formalism Covariant quantization * Conclusions

Preliminaries Quantum Gravity – Great Puzzle – numerous approaches: Quantum Einstein gravity may exist non-perturbatively Classical geometric generalizations including torsion, Weyl fields, non-metriicty Canonical gravity approach Loop gravity Spin foam models Super-gravity, -strings, M-theory Extra-dimensions, brane worlds Non-commutative geometry Analogous models in condensed matter physics ……………………… Gravitation is manifestation of geometry and it does not exist as a fundamental force at all ----- an extreme viewpoint

I. Vacuum tunneling in Einstein gravity Definitions of vacuum: Λ = 0: = 0 → = η 1. R g flat space-time ijkl ij ij 1 Λ ≠ 0: R − + Λ = 0 → = 0 2. Rg g g (i) an absolute vacuum ij 2 ij ij ij ↓ Fubini-Study instanton CP 2 (ii) Any static vacuum solution describes vacuum-vacuum transition → 0 when → ±∞ g t mn Asymptotically flat metric 3, 1 χ = τ = 3. 0: Λ ฀ with space topology S 3 or RP 3 this type of vacuum metrics corresponds to asymptotically locally Euclidean instantons (ALE)

On Fubini-Study instanton on CP 2 2 4 + % % x x x x a Axial ABJ anomaly = ( δ − ), m n m n g 2 2 2 2 mn mn + ρ + ρ a a 1 = , % n x C x 5 µ µ * ∂ = ∂ ( ψγ γ ψ ) = a j ge RR m mn µ µ 5 a 4 ⎛ ⎞ 0 1 0 0 ⎜ ⎟ − 1 0 0 0 , ⎜ ⎟ = ⎜ C ⎟ mn 0 0 0 1 ⎜ ⎟ /Egichi&Freund, PRL’1976/ 0 0 − 1 0 ⎝ ⎠ 3 χ = 3, τ = 1, Λ = 2 2 An analog to BPST instanton a The anti-instanton is given by the same metric but with opposite orientation of the space-time, i.e. with inversed vielbein

Topological structure of the vacuum: the main assumption: * The vielbein (tetrad) e am is a fundamental variable of quantum gravity, not a metric tensor. π ( (1,3)) = π ( (3)) = ฀ SO SO The non-trivial topology is provided by 3 3 In Euclidean gravity the vacuum is classified by two integers (m, n) since SO(4)~SU(2)xSU(2) π ( (4)) = π ( (2) × (2)) = ⊗ ฀ ฀ SO SU SU 3 3 S. Hawking (1978): There is no vacuum tunneling due to infinite /Cargese lectures procs./ barrier between vacuums.

In search of instantons: a simple ansatz • A pure gauge vielbein and pure ( ) = ( ) δ = ( ) am ab bm am e x L x L x 0 gauge spin connection are: ∂ % ϕ = mcd ce m ed L L 0 In temporal gauge the topological vacuums 1 ∫ 3 = Tr d x ω ( ϕ ) mcd N are classified by Chern-Simons number 0 2 CS 16 CS π A simple hedgehog ansatz n x i i ωη ˆ = ( ) ρ δ = Θ x n e g e with an arbitrary trial function “ g” ma ma ma ρ produces a class of conformally ˆ 2 2 2 = / , tan ω = / , ρ = + x x r r t r t flat metrics η i Θ n is a generalization of ‘t Hooft matrices ma ma

3 ' g = 2 ( '' + ) = 0 For the vanishing Ricci scalar R g g ρ 2 λ one obtains a simple regular solution - ( ) ρ = + 1 g - Hawking wormhole : 2 ρ 2 4 λ 2 = ( δ ρ − 4 ) R x x 2 2 2 2 mn mn m n ρ ( ρ + λ ) Ricci and Weyl tensors are: = 0 → τ = 0 C ijkl The solution can be interpreted as instanton-antiinstanton pair in conformal gravity No other appropriate instanton solutions (asymptotically locally flat and with non-zero Hirzebruch signature) were found

In search of instantons: 1-1 correspondence between topological vacuums in Yang-Mills theory and gravity ϕ • The spin connection can be decomposed into rotation and boost parts mcd ( ) ϕ = Ω , B mcd m m (2) ≈ (3) ⊂ (1,3) SU SO SO Since π ( (1,3)) = π ( (3)) = π ( (2)) SO SO SU 3 3 3 one can construct vacuum spin connection ( ) ϕ ( ) = Ω ,0 vac e in terms of SU(2) gauge potential mcd m By this way it is not easy to retrieve the vacuum vielbein from the spin connection.

Explicit construction of non-trivial topological vacuums in SU(2) Yang-Mills theory /Baal&Wipf-2001, Cho-2006/ Introduce orthonormal α ˆ ˆ : = 0, α = 1,2,3 ∈ (2) n D n SU basis of SU(2) triplets i m i r ˆ ˆ [ , ] = × = 0 D D n F n One has the integrability condition m n i mn i r ˆ , Ω = − k Solution to these conditions gives a pure C n m m k gauge vacuum potential /Cho-2006/ 1 ˆ ˆ = − ε ( ⋅ ) k k C n n m ij i j 2 ˆ i Parameterizing the triplet n by angles of S 3 ~SU(2) 1 = sin γ ∂ α − sin α cos γ ∂ β , C m m m one obtains explicitly: 2 = cos γ ∂ α + sin α sin γ ∂ β , C m m m In 4d spherical coordinate system the 3 = cos α ∂ β + ∂ γ C radial coordinate hypersurfaces are m m m given by S 3 . So one can define the 1 σ = , i m i basis triple of left invariant dx C 2 m 1-forms on S 3 σ = 2 ε σ σ i ijk j k d Maurer-Cartan eqn.

( , , , ρ θ φ ψ ) Finally, the basis of pure gauge vielbein 1-forms in polar c.s. is defined as follows: 1 = ( d ρ ρσ , ) a i σ = ( , α β γ , ) i m i e dx C 0 2 m α θ φ ψ ( , , ), β θ φ ψ ( , , ), ( , , γ θ φ ψ ) where the angle functions 3 ( (2)) π SU define the homotopy classes To find instanton solutions one can apply a simple ansatz: ( ( ) , ( ) ) = ρ ρ ρ ρσ a i e g d g 0 i We will consider an ansatz corresponding τ = 1, i.e., we put α = θ β , = φ γ , = ψ to topological class with winding number The ansatz with two functions g 0 , g 3 2 2 2 4 = 1/ = − 1 / ρ (g 1 =g 2 =1) applied to Einstein eqn. produces g g a 3 0 the well-known Eguchi-Hanson instanton

Explicit proof of vacuum tunneling The space-like vielbein of E-H instanton = 1,2,3 = ( ) ρ ρσ = ( ) i i m i e g dx A x defines SU(2) gauge potential A i m : i m r r Passing to temporal gauge − 1 − 1 → + ∂ = 0 A UAU U U t t t in Cartesian coords. gives a system r ˆ ˆ 2 = exp[ ω ( , ) τ ( , )], = 1 ˆ i i U i r t f r t f of eqs. for gauge parameters ω , f rg ˆ ∫ 1, 3 ( ) ω + In asymptotic region t ±∞ one has the solution ฀ ฀ ฀ f dt c r 1 2 ρ where c 1 is determined by initial condition ω ( = −∞ = ) 0 t ˆ( n t = −∞ = ) (0,0,1) This implies transition from the trivial vacuum defined by t = +∞ to non-trivial vacuum with N CS =1 at defined by ⎛ ⎞ sin α ( )cos β ( ) r r where the functions , α β are defined by ⎜ ⎟ = ⎜ ˆ ˆ = − sin α ( )sin β ( ) n U n r r ⎟ =+∞ =+∞ =−∞ t t t r i ˆ ˆ ⎜ ⎟ ( , ω ) = exp[ α ( ) τ β ( )], i i U f r r cos α ( ) ⎝ ⎠ r =+∞ t 2 ˆ ( ) β = (sin β ( ),cos β ( ),0). i r r r e − S < = 1| = 0 > ฀ N N inst CS CS vac

Vacuum tunneling CP 2 via Fubini-Studi instanton Λ ≠ 0 \ ( t = ±∞ = ) 0 g mn χ = 3, τ = 1 via Eguchi-Hanson instanton Λ 0 ฀ χ = 2, τ = 1

What is strange in this vacuum tunneling? * 1979: Hawking’s claim was rather limited to asymptotic euclidean instanton. * 1979: Why others did not claim the vacuum tunneling? * 2008: Vacuum tunneling revisited /Y.M. Cho, Prog.Th.Phys.Suppl.,2008/ * ~1920s: Schwarzschild prefers RP 3 as more simple than S 3 * Indications to RP 3 topology of our space: --non-zero index I 3/2 of Dirac operator for E-H instanton; -- existence of the electrons. The more principal question is: Whether vielbein really represents a variable of quantum gravity? * Is the vielbein like a kinematic variable locally introduced on water surface? If this is so, what describes the microscopic structure of the space? * testing the quantum nature via gravitational Aharonov-Bohm effect: calculation of holonomy operator and experimental verification.

II. Quantum gravity models with torsion Why torsion (contortion, Lorentz connection)? * Equivalence principle, local Lorentz symmetry, gauge principle. If the vielbein is classic then the quantum fluctuation fluctn ϕ ( ) ⎯⎯⎯ → mcd e A of spin connection will create general Lorentz connection mcd * Einstein gravity as effective theory induced by quantum dynamics. Contortion (torsion) may provide the microscopic structure of the space * Existence of spin particles should imply torsion. A problem of non-existency of solution for the electron in Einstein gravity * Ideas from QCD: confinement, quantum condensate Torsion might be unobservable as a classic object like gluon in QCD--Quantum chromodynamics, there is no classical chromodynamics. * Contortion should possess properties of connection. Contortion as a part of Lorentz connection, not a tensor.