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Does Ads/CFT correspondence apply to LQG Kodama state?

  1. May 5, 2015 #1
    first paper

    Quantum gravity with a positive cosmological constant
    Lee Smolin
    59 pages

    "A quantum theory of gravity is described in the case of a positive cosmological constant in 3+1 dimensions. Both old and new results are described, which support the case that loop quantum gravity provides a satisfactory quantum theory of gravity. These include the existence of a ground state, discoverd by Kodama, which both is an exact solution to the constraints of quantum gravity and has a semiclassical limit which is deSitter spacetime. The long wavelength excitations of this state are studied and are shown to reproduce both gravitons and, when matter is included, quantum field theory on deSitter spacetime. Furthermore, one may derive directly from the Wheeler-deWitt equation, Planck scale, computable corrections to the energy-momentum relations for matter fields. This may lead in the next few years to experimental tests of the theory.
    To study the excitations of the Kodama state exactly requires the use of the spin network representation, which is quantum deformed due to the cosmological constant. The theory may be developed within a single horizon, and the boundary states described exactly in terms of a boundary Chern-Simons theory. The Bekenstein bound is recovered and the N bound of Banks is given a background independent explanation.
    The paper is written as an introduction to loop quantum gravity, requiring no prior knowledge of the subject. The deep relationship between quantum gravity and topological field theory is stressed throughout. "

    second paper

    Generalizing the Kodama State I: Construction
    Andrew Randono
    First part in two part series, 20 pages

    The Kodama State is unique in being an exact solution to all the ordinary constraints of canonical quantum gravity that also has a well defined semi-classical interpretation as a quantum version of a classical spacetime, namely (anti)de Sitter space. However, the state is riddled with difficulties which can be tracked down to the complexification of the phase space necessary in its construction.
    from the paper

    In this respect the Kodama state is unique. Not only is the state an
    exact solution to all the constraints of canonical quantum gravity, a rarity in itself, but it also has a well defined physical interpretation as the quantum analogue of a familiar classical spacetime, namely de Sitter or anti-de Sitter space depending on the sign of the cosmological constant[1, 2, 3]. Thus, the state is a candidate for the fulfillment of one of the distinctive advantages of a non-perturbative approach over perturbative techniques: the former has the potential to predict the purely quantum mechanical ground state on which perturbation theory can be based. In addition, the Kodama state has many beautiful mathematical properties relating the seemingly disparate fields of abstract knot theory and quantum field theory on a space of connections[4].

    In particular, the exact form of the state is known in both the connection representation where it is the exponent of the Chern-Simons action, and in the q-deformed spin network representation where it is a superposition of 2...


    Since Kodama state semiclassical limit is (anti)de Sitter space, LQG + Kodama state gives you Ads/CFT correspondence.

    LQG + Kodama respects the holographic principle via Ads/CFT correspondence.
  2. jcsd
  3. May 5, 2015 #2


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    I think I said this in some other thread, but I'm not sure the OP has seen it: http://arxiv.org/abs/0905.3627. The authors are studing LQG, but they mysteriously find an AdS/CFT formula popping up.
  4. May 7, 2015 #3
    ive not seen that paper.

    the above researchers claim that LQG+Kodama gives (anti)DS.

    ==quote Randono paper A page 3==

    1.1 Problems
    Despite all of these positive attributes of the Kodama state, the state is
    plagued with problems. Among these are the following:

    Non-normalizability: The Kodama state is not normalizable under
    the kinematical inner product, where one simply integrates |Ψ|2 over all values of the complex Ashtekar connection. The state is not known to be normalizable under a physical inner product defined by, for example, path integral methods. Linearized perturbations around the state are known to be non-normalizable under a linearized inner product[7].

    CPT Violation: The states are not invariant under CPT[8]. This is particularly poignant objection in view of the CPT theorem of perturbative quantum field theory, which connects CPT violation with Lorentz violation. It is not known if the result carries over to non-perturbative quantum field theory, but it has yet to be demonstrated that the Kodama state does not predict Lorentz violation.

    Negative Energies: It has been argued by analogy with a similar non-perturbative Chern-Simons state of Yang-Mills theory that the Kodama state necessarily contains negative energy sectors[8]. If the energy of one sector of the state is strictly positive, the CPT inverted state will necessarily contain negative energy sectors.

    Non-Invariance Under Large Gauge Transformations: Although the state is invariant under the small gauge transformations generated by the quantum constraints, it is not invariant under large gauge transformations where it changes by a factor related to the winding number of the map from the manifold to the gauge group. However, it has been argued that the non-invariance of the Kodama state under large gauge transformations give rise to the thermal properties of de Sitter spacetime2[9]. Thus, non-invariance under large gauge transformations could be a problem or a benefit, but it is deserving of mention.

    Reality Constraints: The Lorentzian Kodama state is a solution to the quantum constraints in the Ashtekar formalism where the connection is complex. To obtain classical general relativity one must implement reality conditions which ensure that the metric is real. It is an open problem as to how to implement these constraints on a general state. Generally it is believed that the physical inner product will implement the reality constraints, but this could change the interpretation of the state considerably.

    LQG+ Kodama gives AdS but has those problems.

    since AdS is dual to CFT in one less dimension, does CFT address those problems? just map those problems into the corresponding CFT
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