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The Immirzi parameter- the Kodama State - Revisited

  1. Sep 22, 2007 #1


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    The Immirzi parameter- the Kodama State - Revisited
    In Search of Quantum de Sitter Space: Generalizing the Kodama State
    Althought this paper is about Generalizing the Kodama State, Andrew Randono
    does explain some basics of the Immirzi parameter and the tetra.
    Here is a partial list of the “judges”.

    The Faculty of the Graduate School of The University of Texas at Austin
    Abhay Ashtekar, Carlo Rovelli, Richard Matzner, Lee Smolin, and my undergraduate advisor Allen Everett
    Ted Jacobson, Don Witt, Chopin Soo, Stephon Alexander Laurent Freidel, Ed

    Previous discussion of the Kodama State at https://www.physicsforums.com/showthread.php?t=143886
    Randono perfects Kodama state
    Here is what wiki says http://en.wikipedia.org/wiki/Loop_quantum_gravity#The_Kodama_state
    …. Eyo Eyo Ita published papers that build on Randono's generalized Kodama state, and argue that a generalized Kodama state can be built that can couple to matter and the Hamiltonian constraint can reproduce the dynamics of general relativity, resulting in a finite, full quantum gravity…
    Here are the listed publications of Eyo Eyo Ita
    Some quotes from Andrew Randono
    p. 3 …. the Immirzi parameter,
    β. When this parameter is taken to be β = −i, the original Kodama state can
    be constructed, but modern formulation of Loop Quantum Gravity take the
    parameter to be real precisely in order to avoid complications from complexification.
    At present, the parameter is believed to have physical ramifications, in
    that it modulates the size of quanta of space at the Planck length, and its value is believed to be fixed by a matching of the semi-classical derivation of black hole entropy with the full quantum gravitational derivation.

    NOTE: I have also done it by letting the Immirzi parameter be equal to the minimum length – the diameter of a sphere (mini black hole)

    p. 8 Since the equivalence principle is almost certain to break down at very small length scales, choosing the metric or the tetrad to be dynamical is like choosing an answer to which came first, the chicken or the egg? As we will see later, there are good theoretical reasons for taking the tetrad to be the more fundamental object.

    NOTE: Review the info that I have supplied on why you need to have 12 dynamic plaquetes and why it creates the double tetra.

    p. 15
    Thus, we see that the low-energy signature of Einstein-Cartan theory is an
    axial-axial spin-current interaction in the effective field theory that is a relic
    of the spin-torsion interaction.

    NOTE: To make things easier to visualize, just imagine 12 (2d) coins doing a dynamic dance around a central point in 3d. (Not in a black hole but in vacuum)

    The boundary term is the topological Nieh-Yan class. We see from this that the
    Immirzi parameter will measure the width of torsional fluctuations in the path
    integral[13]. The parameter has a dramatic effect in the quantum theory—in
    particular, we will see in the quantum theory that area will be quantized in
    multiples of βl2 P l. Thus, at the quantum level, the parameter fine tunes the
    discretization scale.

    p. 47
    … we will see, implementing the Gauss and diffeomorphism constraints is much
    easier when the phase space consists of a connection as a dynamical position variable.

    Note: Imagine those 12 coins and how their faces would be orientated with reference to the center of their “dance”. The center is within the space that is traced by the coins. The coins trace out a double tetra.

    p. 57
    In total, then, the spin network states form an orthonormal basis of states that span the kinematical Hilbert space HDG. In addition, there is a duality between the spin network state and functionals on the space of generalized connections, A. This duality is displayed elegantly by the Kodama state, and it allows for a semi-classical interpretation in a quantum theory whose elementary building blocks are Planck scale discrete geometries.

    Note: As I have stated before, in https://www.physicsforums.com/showthread.php?t=185087 , Topological order - string-net condensation + loop quantum gravity, cubic packing is not the right picture. The picture must be hex. packing.
    LQG and particle physics
    Xiao-Gang Wen and Michael Levin are two solid-state physicists who have attempted to model elementary particles such as electrons and photons as resulting from a discrete lattice structure of spacetime in analogy to phonons in solid state physics. In the paper "Photons and electrons as emergent phenomena" they attempt to model elementary particles as emergent properties of a String-net condensation in analogy to phonons in solid state physics, and LQG's spin networks have the properties necessary to reproduce the Standard Model as the result of the collective behavior of a group of spin network.[13][14] This approach differs from the preon approach in that Wen and Levin see particles as an emergent property of quantum spacetime, rather than built up of smaller substructure as is the case with both preon and string theory.

    Those who understand the Kodama State can add their observations.
    Let the experts speak. I’m listening.

  2. jcsd
  3. Sep 29, 2007 #2


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    Maybe we can get some other experts into the discussion by including the following paper.
    A New Spin Foam Model for 4d Gravity
    Authors: Laurent Freidel, Kirill Krasnov
    (Submitted on 13 Aug 2007)
    " One important outcome of our approach is that it also allow us to introduce the Immirzi parameter into the framework of spin foam quantisation. "
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