Loop Quantum Gravity by sdfgsg234

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									                Loop Quantum Gravity
                      carlo rovelli



                                                              6                               sf
                                                  3
                                                                                      8
                                             !f

                                                                              q


                                                                                  7
                                                                          5
                                                      3           6           1               s1
                                                                              8




                                                          p



                                                                      5                       si
                                                      3
                                             !i                                           7




Carlo Rovelli         Loop Quantum Gravity                                            Granada 2010
The theory




            The theory is defined by the triple    (H, W, A)
                   H   is Hilbert space

                   W:H→C                  is a map that defines the dynamics

                   A     is an algebra of operators




 Carlo Rovelli                        Loop Quantum Gravity                    Granada 2010
The theory 1 H

 Hilbert space:

       ˜
       H=           HΓ               Γ : Abstract graph :
                                                                                       t(l)
                                                                                       l
                Γ                                                               s(l)

      Graph Hilbert space:    HΓ = L2 [SU (2)L /SU (2)N ]
                                                         −1
          Gauge transformations    ψ(Ul ) → ψ(Vs(l) Ul Vt(l) ),        Vn ∈ SU (2)N

           ˜
       H = H/ ∼ where ∼ defined identifying states on subgraph.




       The space    HΓ   admits a basis   |Γ, jl , in      labelled by a spin for each link
        and an intertwiner for each node. These states are called “spin network states”.




Carlo Rovelli                       Loop Quantum Gravity                                   Granada 2010
The theory II          A
 Operator algebra:

       Ll = {Li }, i = 1, 2, 3 left-invariant vector field for each link l :
              l
        “gravitational field operator (tetrad)”

        Ul : “Holonomy of the Ashtekar-Barbero connection along the link”.

       Composite operators:

             Area:          AΣ =           Li Li .
                                            l l
                                   l∈Σ
                                                           2
             Volume:        VR =          Vn ,        2
                                                     Vn   = |      Li Lj Lk |.
                                                                ijk l l l”
                                                           9
                                   n∈R

             Angle:         Li Li .
                             l l



      The spin network basis   |Γ, jl , in       diagonalizes the area and volume operators.




 Carlo Rovelli                           Loop Quantum Gravity                          Granada 2010
The theory III           W

     Dynamics:                |ψ =   |Γ, jl , in


                 W(ψ) =                       djf           Wv                  Wv
                               ∂σ=ψ       f            v
                                                                          σ : spinfoam

σ:   two-complex     ∆   with faces   f and edges e colored with spins jf and
 intertwiners e ,i     bounded by     Γ, jl , in . σ = (∆, jf , ie ) : “spinfoam”.

         dj = 2j + 1

                                  I)
         Wv = (PSL(2,C) ◦ Yγ ψv )(1

         Yγ : Hj −→ Hj ⊂ H(p=γ(j+1), k=j) .                                     Wv
                 SU (2) rep            SL(2, C) rep                        spinfoam vertex
                      SU (2) ⊂ SL(2, C)



 Carlo Rovelli                             Loop Quantum Gravity                          Granada 2010
                            Main conjecture:


           (H, W, A) defines a (background independent)
                        quantum field theory
                whose classical limit is general relativity




Carlo Rovelli                 Loop Quantum Gravity            Granada 2010
         In which sense this is a QFT ?

         In which sense it is background independent ?

         How do we compute transition amplitudes ? How do we extract physics ?

         What evidence do we have that its classical limit is General Relativity ?




Carlo Rovelli                       Loop Quantum Gravity                        Granada 2010
    In which sense this is a background independent QFT ?


  1. Notice the following structure in H :     There is a natural tensor map:

                            T :H⊗H→H

       defined by T (HΓ ⊗ HΓ ) = HΓ∪Γ

                                                                                                         ˆ
Therefore W defines also maps W2 : H → H (cfr: W(ψin ⊗ ψ fin ) = ψfin |e
                                                                                                       −iHt
                                                                                                                         |ψin )

                                                                                   6                                sf

   ➜ A two-complex is cobordism of graphs.
                                                                       3
                                                                                                           8
                                                                 !f

     A spinfoam is cobordism of spin networks.                                                     q


                                                                                                       7
                                                                                               5
                                                                           3                       1                s1

   ➜ A Hilbert space is associated to any connected graph
                                                                                       6

                                                                                                   8




   and an amplitude (a state) is associated to any two-complex                 p


   with the graphs as boundaries.                                          3
                                                                                           5                        si
                                                                  !i                                           7




   ➜ Compare: Atiyah Topological Quantum Field Theory



  Carlo Rovelli                     Loop Quantum Gravity                                                           Granada 2010
  In which sense this is a background independent QFT ?



       Key differences from Atiyah TQFT:

                 Boundary manifolds → graphs

                 Cobordism manifolds → Two complexes
                                                                     6                               sf
                                                         3
                                                                                             8
                                                  !f

                                                                                     q




                                            →
                                                                                         7
                                                                                 5
                                                             3           6           1               s1
                                                                                     8




                                                                 p



                                                                             5                       si
                                                             3
                                                   !i                                            7




                 Finite → Infinite dimensional boundary Hilbert spaces



Carlo Rovelli                     Loop Quantum Gravity                                                    Granada 2010
Physical interpretation of    HΓ :   the spin network states



  A canonical quantization of general relativity [See Ashtekar’s talk] leads to a space of
  states which is (up to technical details) precisely H .


                                                             Metric space, with
                                                             a triad field a Ashtekar-
                                                             Barbero connection.



                                         Ul
                               Ll


       This leads to   HΓ    with the operators    Ul and Ll .

       Diffeomorphism invariance: Imbedded graph → Abstract graph


 Carlo Rovelli                        Loop Quantum Gravity                              Granada 2010
   Physical interpretation of          HΓ    : the spin network states


 Basis |Γ, jl , vn     in   HΓ   that diagonalizes area and volume: spin network basis

     Nodes: discrete quanta of volume (chunks of space, atoms of space)
     with quantum number vn .
     Links: discrete quanta of area, with quantum number jl .


                                  jl


                                              →
                     vn




                A spin network state on a graph is a quantum state of geometry:
                   These are not states in space. These are states of space.


Carlo Rovelli                               Loop Quantum Gravity                         Granada 2010
   Physical interpretation of   HΓ    : the spin network states




                  jl
                       vn


                            →                           →


 Spin network diagonalize metric and have quantum spread extrinsic geometry

Coherent states : peaked in a given (discrete) intrinsic and extrinsic geometry
                                        [Thiemann, Speziale CR, Livine, Bianchi Magliaro Perini]

Triangulation interpretation: Regge or “twisted”                [Dittrich, Bonzom, Speziale Freidel, Livine]


Holomorphic representation: Basis of coherent states
                                       [Ashtekar Lewandowski Marolf Mourao, Bianchi Magliaro Perini]


Carlo Rovelli                        Loop Quantum Gravity                                     Granada 2010
Dynamics:            W
    Amplitude associated to a state   ψ      of a boundary of a 4d region
    Probability amplitude P (ψ) = | W |ψ |2




         4d region
                                                                                      Wv
     3d boundary            boundary graph           a spin network history    σ : spinfoam



      Superposition principle             W |ψ =              W (σ)
                                                        σ

       Locality: vertex amplitude         W (σ) ∼            Wv .
                                                         v

       Lorentz invariance                                        I)
                                        Wv = (PSL(2,C) ◦ Yγ ψv )(1                 Wv
                                                                              spinfoam vertex


 Carlo Rovelli                        Loop Quantum Gravity                              Granada 2010
Dynamics:        W                                                                                          ψ

                        SU (2)       SL(2,C)
  Natural immersion HΓ           ⊂ HΓ          :              SU (2) ⊂ SL(2, C)

     Yγ : Hj −→ Hj ⊂ H(p=γ(j+1), k=j) .             [Engle Pereira CR, Livine, Speziale, Freidel Krasnov,
                                                    Lewandowski Kaminski Kisielowski, 07-10]           Wv



i) If we replace Yγ with the identity, we obtain a TQFT which is well known: it is
the Ooguri quantization of the theory S[B, A] =            B∧F


                                                             1
ii) General relativity can be written as S[e, A] = ((e ∧ e) + e ∧ e) ∧ F
                                                                       ∗
                                                             γ
                          1
and B = (e ∧ e) + e ∧ e iff (in a fixed gauge): B ij − γ ij k B 0k = 0
                    ∗
                         γ


iii) Theorem [Ding CR 09]: on the image of Yγ :     ψ|B ij − γ         ij
                                                                            k   B 0k |φ = 0


 Carlo Rovelli                      Loop Quantum Gravity                                  Granada 2010
Dynamics:         W


       Asymptotic analysis


                                iSRegge              iSEinstein−Hilbert
                  Wv ∼ e                     ∼ e
                 [Barrett Dowdall Fairbairn Gomes Hellmann, Pereira]




    In the spin network basis W yields the cos of the action.
    In the holomorphic representation, only one of the two terms of the cos survives
   [Bianchi Magliaro Perini]




 Carlo Rovelli                            Loop Quantum Gravity                   Granada 2010
Dynamics:        W



                                              I)
                     Wv = (PSL(2,C) ◦ Yγ ψv )(1




 This natural vertex amplitude appear to yield the Einstein equations in the large
 distance classical limit: A natural group structure based on SU (2) ⊂ SL(2, C)
 appears to code the Einstein equations.



                     cfr :           = e γµ δ(p1 + p2 − k)
                                          AB




 Carlo Rovelli                     Loop Quantum Gravity                        Granada 2010
Dynamics:                                    W

                     6                               sf
         3




                                                          W |ψ =                                   Wv (ψv (σ)).
                                             8
 !f

                                     q
                                                                                   (2jf +1)

                                 5
                                         7
                                                                      σ       f                v
             3           6           1               s1
                                     8




                 p                                                                 I)
                                                          Wv = (PSL(2,C) ◦ Yγ ψv )(1
                             5                       si
             3
 !i                                              7




         σ : spinfoam



“Sum over histories” form of LQG: Dual interpretation:


      1. Discrete version of:                                          W (q) =             Dg eiSEH [g]
                                                                                    ∂g=q



      11. Sum over Feynman graphs:



 Carlo Rovelli                                              Loop Quantum Gravity                          Granada 2010
Dynamics:                                    W

                     6                               sf
         3




                                                          W |ψ =                                   Wv (ψv (σ)).
                                             8
 !f

                                     q
                                                                                   (2jf +1)

                                 5
                                         7
                                                                      σ       f                v
             3           6           1               s1
                                     8




                 p                                                                 I)
                                                          Wv = (PSL(2,C) ◦ Yγ ψv )(1
                             5                       si
             3
 !i                                              7




         σ : spinfoam



“Sum over histories” form of LQG: Dual interpretation:


      1. Discrete version of:                                          W (q) =             Dg eiSEH [g]
                                                                                    ∂g=q



      11. Sum over Feynman graphs:



 Carlo Rovelli                                              Loop Quantum Gravity                          Granada 2010
Dynamics:                                    W

                     6                               sf
         3




                                                          W |ψ =                                   Wv (ψv (σ)).
                                             8
 !f

                                     q
                                                                                   (2jf +1)

                                 5
                                         7
                                                                      σ       f                v
             3           6           1               s1
                                     8




                 p                                                                 I)
                                                          Wv = (PSL(2,C) ◦ Yγ ψv )(1
                             5                       si
             3
 !i                                              7




         σ : spinfoam



“Sum over histories” form of LQG: Dual interpretation:


      1. Discrete version of:                                          W (q) =             Dg eiSEH [g]
                                                                                    ∂g=q



      11. Sum over Feynman graphs:



 Carlo Rovelli                                              Loop Quantum Gravity                          Granada 2010
        Quantization methods:

    Canonical quantization à la Dirac of the ADM constraints of General Relativity in
    Ashtekar-Barbero variables.

            Holonomies as main variables
            Diff invariance → abstract graphs

    Discretization and lattice quantization (à la QCD).

            GR = BF+constraints → constrained are implemented weakly on the image of Yγ
            Lattice spacing independence

    Quantum geometry methods

          All these converge to the structure (H, W, A)


Carlo Rovelli                        Loop Quantum Gravity                         Granada 2010
   Physical assumptions:

                General Relativity (with standard matter couplings, in Ashtekar formulation)

                Standard quantum mechanics (modified to be general covariant)

                Diffeomorphism invariance fully implemented




Carlo Rovelli                            Loop Quantum Gravity                          Granada 2010
Extracting physics from the theory



                 There is no physics without approximations.


     Graph expansion: Restricting the theory to HΓ is a
  truncation of wavelengths short with respect to the total size
  of the region considered (cfr: cosmology) (cfr: lattice QCD).


    Vertex expansion: Similar to the vertex expansion in QED:
  number of “elementary processes considered in the transition
  amplitude”


     Large distance expansion: Large with respect to the Planck
  length (classical limit).                                        jl   1



 Carlo Rovelli                      Loop Quantum Gravity                    Granada 2010
Results: 1. graviton propagator

                                               q                                        q
                                            
                                      q          f q
                                            ¢¢f                                    
                                                                               q          f q
                                                                                    ¢¢f 
                                                                               
                        Graph    Γ5 = l ¢ f 5
                                      gg l ¢          £      .   Vertex        l ¢ f 5
                                                                               gg l ¢          £
        4d region
                                                5 5 £                                      5 £
                                        g ¢™ f £                                         5
                                          q &™™q                                 g ¢™ f £
                                          ¢
                                         g&        f£                              q
                                                                                   ¢
                                                                                  g& &™™q   f£


    Boundary state: ψL coherent state determined by the (intrinsic and extrinsic)
    geometry of the boundary of a flat 4-simplex.
    → Background info input dynamically via the boundary state.

    Amplitude:
                     Wmn = W |Lna · Lnb Lmc · Lmd |ψL
                      abcd
                                                                       c

    corresponds to the perturbative QFT’s graviton propagator

                    W abcd (xm , xn ) = 0|g ab (xn )g cd (xm )|0           c


                         Matches to first order ! [Bianchi Magliaro Perini]


 Carlo Rovelli                        Loop Quantum Gravity                                         Granada 2010
Results: 1. graviton propagator

                                               q                                        q
                                            
                                      q            q
                                            ¢¢f d m                                 
                                                                               q          f q
                                                                                    ¢¢f 
                                                 f                            
                        Graph    Γ5 = l ¢ f 5
                                      gg l ¢          £      .   Vertex        l ¢ f 5
                                                                               gg l ¢          £
        4d region                                 5 £                                      5 £
                                            ™5                                           5
                                          q       ™q
                                     b g ¢&™ f £ c                               g ¢™ f £
                                         n
                                          ¢
                                         g&        f£                              q
                                                                                  g&
                                                                                   ¢ &™™q   f£
                                                a

    Boundary state: ψL coherent state determined by the (intrinsic and extrinsic)
    geometry of the boundary of a flat 4-simplex.
    → Background info input dynamically via the boundary state.

    Amplitude:
                     Wmn = W |Lna · Lnb Lmc · Lmd |ψL
                      abcd
                                                                       c

    corresponds to the perturbative QFT’s graviton propagator

                    W abcd (xm , xn ) = 0|g ab (xn )g cd (xm )|0           c


                         Matches to first order ! [Bianchi Magliaro Perini]


 Carlo Rovelli                        Loop Quantum Gravity                                         Granada 2010
Results: II. cosmology

Triangulate a 3-sphere with two tetrahedra :
these capture the first d.o.f.’s in a mode
expansion of a cosmological metric

                         r       r   p
Dual graph:      ∆∗ =
                  2                                                   r        r z


Boundary state on ∆∗ ∪ ∆∗ coherent state ψz peaked on
                    2    2
                                                                          u
homogeneous isotropic geometries on the 3-sphere.

Amplitude W (z, z ) = W |ψz ⊗ ψz                                      r        r z

                              z 2 +z 2
        W (z, z ) ∼ zz e     − 2t
                                            [See Vidotto’s talk]


Reproduces the Friedmann dynamics:
- Is peaked on the classical solutions
- Satisfies a quantum constraint which reduces to the (gravitational
part of the) Friedmann hamiltonian for    →0

 Carlo Rovelli                       Loop Quantum Gravity                     Granada 2010
Open issues


             No UV divergences! Infrared divergences?
             [Speziale Perini CR, Bonzom, Smerlak, Rivasseau Gurau Oriti]


             Scaling by radiative corrections?

             Matter in spinfoam?

             Cosmological constant?

             Relation covariant canonical formalism’s dynamics?
             [Ashtekar Campigia Henderson, Wilson-Edwin Nelson,Vidotto CR]



    Observations are possible. Several suggestions, but:

           No empirical support yet

           No solid verifiable prediction yet


 Carlo Rovelli                              Loop Quantum Gravity             Granada 2010
Summary


      The theory is defined by the triple (H, W, A) .

      It is a generalization of a topological QFT in the sense of Atiyah

      Kinematics: quanta of space with quantized volume and area [see Ashtekar and Livine’s talks]
      Dynamics: transition amplitudes computed in expansions

      Indications supporting the conjecture that it is quantum GR:
           derivation from canonical quantization of GR
           derivation from discretization of GR and GR=BF+constraints.
           asymptotic of the vertex
           results on the low-energy limit : n-points functions, cosmology [see Vidotto’s talk]

      Main physical applications
          Loop Cosmology → Big bounce [see G. Mena-Marugan, Martin-Benito, Tanaka, Olmedo’s talks]
          Black hole entropy for real black holes [see Barbero, Diaz-Polo, Borja, talks]

     Loop Quantum Gravity provides a still incomplete, but clean and full-scale
      tentative quantum theory of space time and gravitation.


Carlo Rovelli                          Loop Quantum Gravity                              Granada 2010
Good news


   Main open problems 15 years ago:

          To construct a mathematically well defined background
           independent quantum field theory
          The “problem of time”
          Fate of GR singularities (Cosmological and Black Holes)
          Deriving a finite black hole entropy from first principle
          Curing ultraviolet divergences of standard field theories
          Computing quantum gravitational transition amplitudes.



                Most of the open problems in quantum gravity of 15 years ago
                     have a solution today in the context of loop gravity



Carlo Rovelli                          Loop Quantum Gravity                    Granada 2010

								
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