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PI video of new QG ideas from Eyo Ita, Chopin Soo, C-Y Chou

  1. Aug 2, 2014 #1

    marcus

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    http://pirsa.org/14070033/
    Physical Hilbert space for the affine group formulation of 4D, gravity of Lorentzian signature.
    Eyo Ita
    The authors have revealed a fundamental structure which has been hidden within the Wheeler-DeWitt (WDW) constraint of four dimensional General Relativity (GR) of Lorentzian signature in the Ashtekar self-dual variables. The WDW equation can be written as the commutator of two geometric entities, namely the imaginary part of the Chern-Simons functional Q and the local volume element V(x) of 3-space. Upon quantization with cosmological constant, the WDW equation takes on the form of the Lie algebra of the affine group of transformations of the straight line, with Q and V(x) playing the role of the generators for the Lie algebra. The generators are Hermitian, which addresses the issue of the implementation of the reality conditions of GR at the quantum level. Additionally, the irreducible unitary representations (IUR) implement the positivity of the spectrum of the volume operator V(x) at the quantum level This development has led to the existence of elements of the physical Hilbert space for four dimensional gravity of Lorentzian signature, the full theory, in the form of irreducible, unitary representations of the affine group of transformations of the straight line. The affine Lie algebraic structure of the WDW equation remains intact even in the presence of nongravitational fields. This feature has led to the extension of the affine group formulation to elements of the physical Hilbert space for gravity coupled to the full Standard Model of particle physics, quantized on equal footing. Work on the physical interpretation of the states with respect to gauge-diffeomorphism invariant observables, and spacetime geometries solving the Einstein equations is in progress. The journal reference for these results are as follows: - The first result has been published in CQG 30 (2013) 065013 - The second result has just been published in Annals of Physics Journal Vol.343, pages 153-163, April 2014
    31 July 2014

    http://arxiv.org/abs/1306.1489
    Affine group formulation of the Standard Model coupled to gravity
    Ching-Yi Chou, Eyo Ita, Chopin Soo
    (Submitted on 6 Jun 2013)
    This work demonstrates that a complete description of the interaction of matter and all forces, gravitational and non-gravitational, can in fact be realized within a quantum affine algebraic framework. Using the affine group formalism, we construct elements of a physical Hilbert space for full, Lorentzian quantum gravity coupled to the Standard Model in four spacetime dimensions. Affine algebraic quantization of gravitation and matter on equal footing implies a fundamental uncertainty relation which is predicated upon a non-vanishing cosmological constant.
    9 pages, published in Annals of Physics (April 2014)
    http://inspirehep.net/record/1237226
    http://inspirehep.net/author/profile/E.E.Ita.1

    http://arxiv.org/abs/arXiv:1207.7263
    Affine group representation formalism for four dimensional, Lorentzian, quantum gravity
    Chou Ching-Yi, Eyo Ita, Chopin Soo
    (Submitted on 30 Jul 2012)
    Within the context of the Ashtekar variables, the Hamiltonian constraint of four-dimensional pure General Relativity with cosmological constant, Λ, is reexpressed as an affine algebra with the commutator of the imaginary part of the Chern-Simons functional, Q, and the positive-definite volume element. This demonstrates that the affine algebra quantization program of Klauder can indeed be applicable to the full Lorentzian signature theory of quantum gravity with non-vanishing cosmological constant; and it facilitates the construction of solutions to all of the constraints. Unitary, irreducible representations of the affine group exhibit a natural Hilbert space structure, and coherent states and other physical states can be generated from a fiducial state. It is also intriguing that formulation of the Hamiltonian constraint or Wheeler-DeWitt equation as an affine algebra requires a non-vanishing cosmological constant; and a fundamental uncertainty relation of the form (ΔV/⟨V⟩)ΔQ ≥ 2πΛL2Planck (wherein V is the total volume) may apply to all physical states of quantum gravity.
    13 page, published in Classical and Quantum Gravity (March 2013)
    http://inspirehep.net/record/1124330
     
    Last edited: Aug 2, 2014
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  3. Aug 3, 2014 #2

    marcus

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    I don't feel qualified to evaluate or recommend in this case. The work by Soo and Ita is in part based on published but not widely known papers by Laszlo B. Szabados which might be worth checking out.
    http://arxiv.org/abs/arXiv:0711.4009
    A note on the Hamiltonian constraint in canonical GR
    Laszlo B. Szabados
    (Submitted on 26 Nov 2007 (v1), last revised 25 Mar 2008 (this version, v2))
    The Hamiltonian constraint of the coupled Einstein-Yang-Mills-Higgs system with a cosmological constant is shown to be a pure Poisson bracket of a dimensionless functional on the phase space and the volume of the three-space. One of its potential consequences, a restriction on the eigenstates of the volume operator in a class of canonical quantum gravity theories, is also pointed out.
    6 pages, published in Class. Quant. Grav. 25 (2008)

    http://arxiv.org/abs/gr-qc/0110106
    On the role of conformal three-geometries in the dynamics of General Relativity
    Laszlo B Szabados
    (Submitted on 24 Oct 2001)
    It is shown that the Chern-Simons functional, built in the spinor representation from the initial data on spacelike hypersurfaces, is invariant with respect to infinitesimal conformal rescalings if and only if the vacuum Einstein equations are satisfied. As a consequence, we show that in the phase space the Hamiltonian constraint of vacuum general relativity is the Poisson bracket of the imaginary part of this Chern-Simons functional and Misner's time (essentially the 3-volume). Hence the vacuum Hamiltonian constraint is the condition on the canonical variables that the imaginary part of the Chern- Simons functional be constant along the volume flow. The vacuum momentum constraint can also be reformulated in a similar way as a (more complicated) condition on the change of the imaginary part of the Chern-Simons functional along the flow of York's time.
    15 pages, published in Class.Quant.Grav.19 (2002)
     
  4. Aug 3, 2014 #3

    marcus

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    I think one or two earlier papers by Chopin Soo may be essential to understanding the direction this research is going:
    http://arxiv.org/abs/gr-qc/0703074
    Three-geometry and reformulation of the Wheeler-DeWitt equation
    Chopin Soo
    (Submitted on 13 Mar 2007)
    A reformulation of the Wheeler-DeWitt equation which highlights the role of gauge-invariant three-geometry elements is presented. It is noted that the classical super-Hamiltonian of four-dimensional gravity as simplified by Ashtekar through the use of gauge potential and densitized triad variables can furthermore be succinctly expressed as a vanishing Poisson bracket involving three-geometry elements. This is discussed in the general setting of the Barbero extension of the theory with arbitrary non-vanishing value of the Immirzi parameter, and when a cosmological constant is also present. A proposed quantum constraint of density weight two which is polynomial in the basic conjugate variables is also demonstrated to correspond to a precise simple ordering of the operators, and may thus help to resolve the factor ordering ambiguity in the extrapolation from classical to quantum gravity. Alternative expression of a density weight one quantum constraint which may be more useful in the spin network context is also discussed, but this constraint is non-polynomial and is not motivated by factor ordering. The article also highlights the fact that while the volume operator has become a preeminient object in the current manifestation of loop quantum gravity, the volume element and the Chern-Simons functional can be of equal significance, and need not be mutually exclusive. Both these fundamental objects appear explicitly in the reformulation of the Wheeler-DeWitt constraint.
    10 pages

    http://arxiv.org/abs/gr-qc/0512025
    Further simplification of the constraints of four-dimensional gravity
    Chopin Soo
    (Submitted on 5 Dec 2005 (v1), last revised 13 Mar 2007 (this version, v2))
    The super-Hamiltonian of 4-dimensional gravity as simplified by Ashtekar through the use of gauge potential and densitized triad variables can furthermore be succinctly expressed as a Poisson bracket between the volume element and other fundamental gauge-invariant elements of 3-geometry. This observation naturally suggests a reformulation of non-perturbative quantum gravity wherein the Wheeler-DeWitt Equation is identical to the requirement of the vanishing of the corresponding commutator. Moreover, this reformulation singles out spin network states as the preeminent basis for expansion of all physical states.
    3 pages. Original article replaced by brief conference report. Published in Proc. 7th. Asia-Pacific Int. Conf. on Gravitation and Astrophysics (ICGA7)
     
  5. Aug 3, 2014 #4

    marcus

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    Cho-Pin Soo's interest in the Chern-Simons functional goes way back, e.g. to this 1994 paper
    http://inspirehep.net/record/373263
    http://arxiv.org/abs/gr-qc/9405015
    The Chern-Simons Invariant as the Natural Time Variable for Classical and Quantum Cosmology
    Lee Smolin, Chopin Soo
    (Submitted on 6 May 1994)
    We propose that the Chern-Simons invariant of the Ashtekar-Sen connection is the natural internal time coordinate for classical and quantum cosmology. The reasons for this are a number of interesting properties of this functional, which we describe here.
    1)It is a function on the gauge and diffeomorphism invariant configuration space, whose gradient is orthogonal to the two physical degrees of freedom, in the metric defined by the Ashtekar formulation of general relativity.
    2)The imaginary part of the Chern-Simons form reduces in the limit of small cosmological constant, Λ, and solutions close to DeSitter spacetime, to the York extrinsic time coordinate.
    3)Small matter-field excitations of the Chern-Simons state satisfy, by virtue of the quantum constraints, a functional Schroedinger equation in which the matter fields evolve on a DeSitter background in the Chern-Simons time. We then n propose this is the natural vacuum state of the theory for Λ≠0.
    4)This time coordinate is periodic on the configuration space of Euclideanized spacetimes, due to the large gauge transformations, which means that physical expectation values for all states in non-perturbative quantum gravity will satisfy the KMS condition, and may then be interpreted as thermal states.
    5)Forms for the physical hamiltonians and inner product which support the proposal are suggested, and a new action principle for general relativity, as a geodesic principle on the connection superspace, is found.
    32 pages, gr-qc/9405015, CGPG-94/4-1 (revised and extended, Oct. 94)

    Here's his profile:
    http://inspirehep.net/author/profile/C.Soo.2
    39 papers, interesting range of topics
     
  6. Aug 3, 2014 #5

    marcus

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    So maybe the recent work of Soo and Ita is not primarily based (as I said in post #2) so much on the 2002 and 2008 papers of Laszlo Szabatos but actually more on the earlier papers of Soo (and Smolin in that one case), going back to 1994. That 1994 S&S paper was published in Nuclear Physics B (1995)
     
  7. Aug 4, 2014 #6

    marcus

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    Another Soo Ita paper came out today:
    http://arxiv.org/abs/1408.0710
    Exact solutions of the Wheeler-DeWitt equation and the Yamabe construction
    Eyo Ita, Chopin Soo
    (Submitted on 30 Jul 2014)
    Exact solutions of the Wheeler-DeWitt equation of the full theory of four dimensional gravity of Lorentzian signature are obtained. They are characterized by Schrödinger wavefunctionals having support on 3-metrics of constant spatial scalar curvature, and thus contain two full physical field degrees of freedom in accordance with the Yamabe construction. These solutions are moreover Gaussians of minimum uncertainty and they are naturally associated with a rigged Hilbert space. In addition, in the limit the regulator is removed, exact 3-dimensional diffeomorphism and local gauge invariance of the solutions are recovered.
    13 Pages

    As a reminder, Eyo Ita gave an impressive and well-organized talk at Perimeter last week:
    http://pirsa.org/14070033/
    Physical Hilbert space for the affine group formulation of 4D, gravity of Lorentzian signature.
    The authors have revealed a fundamental structure which has been hidden within the Wheeler-DeWitt (WDW) constraint of four dimensional General Relativity (GR) of Lorentzian signature in the Ashtekar self-dual variables. The WDW equation can be written as the commutator of two geometric entities, namely the imaginary part of the Chern-Simons functional Q and the local volume element V(x) of 3-space. Upon quantization with cosmological constant, the WDW equation takes on the form of the Lie algebra of the affine group of transformations of the straight line, with Q and V(x) playing the role of the generators for the Lie algebra. …
    ...The journal reference for these results are as follows: - The first result has been published in CQG 30 (2013) 065013 - The second result has just been published in Annals of Physics Journal Vol.343, pages 153-163, April 2014
    31 July 2014

    I have tried to condense and excerpt the abstracts to give the flavor of the research without taking too much space. Here are the papers that the PI video by Ita was presenting:

    http://arxiv.org/abs/1306.1489
    Affine group formulation of the Standard Model coupled to gravity
    Ching-Yi Chou, Eyo Ita, Chopin Soo
    (Submitted on 6 Jun 2013)
    This work demonstrates that a complete description of the interaction of matter and all forces, gravitational and non-gravitational, can in fact be realized within a quantum affine algebraic framework... Affine algebraic quantization of gravitation and matter on equal footing implies a fundamental uncertainty relation which is predicated upon a non-vanishing cosmological constant.
    9 pages, published in Annals of Physics (April 2014)
    http://inspirehep.net/record/1237226
    http://inspirehep.net/author/profile/C.Soo.2 (National Cheng Kung University--Taiwan)
    http://inspirehep.net/author/profile/E.E.Ita.1 (US Naval Academy--Annapolis)

    http://arxiv.org/abs/arXiv:1207.7263
    Affine group representation formalism for four dimensional, Lorentzian, quantum gravity
    Chou Ching-Yi, Eyo Ita, Chopin Soo
    (Submitted on 30 Jul 2012)
    Within the context of the Ashtekar variables, the Hamiltonian constraint of four-dimensional pure General Relativity with cosmological constant, Λ, is reexpressed as an affine algebra with the commutator of the imaginary part of the Chern-Simons functional, Q, and the positive-definite volume element... It is also intriguing that formulation of the Hamiltonian constraint or Wheeler-DeWitt equation as an affine algebra requires a non-vanishing cosmological constant; and a fundamental uncertainty relation of the form (ΔV/⟨V⟩)ΔQ ≥ 2πΛL2Planck (wherein V is the total volume) may apply to all physical states of quantum gravity.
    13 page, published in Classical and Quantum Gravity (March 2013)
    http://inspirehep.net/record/1124330
     
    Last edited: Aug 4, 2014
  8. Aug 4, 2014 #7

    julian

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    Just to add a bit see self-dual Palantitni action on wiki which I just added proof of Ashtekar formulism.
     
  9. Aug 4, 2014 #8

    marcus

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    Last edited: Aug 4, 2014
  10. Jul 26, 2015 #9

    MTd2

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    Eyo sent me a message:

    ************************
    You should inform Marcus in the Physics Forums about the new ITQGpaper- would like to get his take

    Cheers,

    Eyo


    Yes Dan, certainly.

    At this stage we have just a DOI,
    10.1093/ptep/109ptv, and the published version in PTEP journal should be out in print any day now.
    The arXiv version of the paper is gr-qc/1501.06282 titled "Intrinsic Time Quantum Geometrodynamics", which is very close to the accepted version. My co-authors are Chopin Soo and Hoi-Lai Yu.

    Cheers,
     
  11. Aug 9, 2015 #10

    marcus

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    MTd2 thanks for keeping us posted about Eyo Eyo Ita's publications! I cannot comment substantively. I do not understand how redshift z can serve as time because it goes backwards, it gets larger as you go back in time. Or they may mean the SCALE FACTOR, which increases with time and is equal to 1/(z+1).
    Also I do not understand where planck's constant h-bar went. It seems to disappear. Maybe it is somehow "eaten" by intrinsic time or by some preferred unit in which intrinsic time is measured.:oldbiggrin:
    Anyway I don't have a good enough understanding to comment on the substance of the paper. However I can note that the arXiv version is
    http://arxiv.org/abs/1501.06282

    AND THE INSPIRE VERSION IS
    http://inspirehep.net/record/1341029?ln=en

    and the Inspire version should be updated to show that the paper has been published by PTEP

    There are a great many references to "Cotton-York" tensor and this reference seems to be important:
    [4] N. O ́ Murchadha, C. Soo and H.-L. Yu, ‘Intrinsic time gravity and the Lichnerowicz–York equation’, Class. Quantum Grav.30, 095016 (2013)

    MTd2 I suspect that if you want to understand the new Ito Soo Yu paper you pointed to, it will be necessary to study this one:
    http://arxiv.org/abs/1208.2525
    Intrinsic time gravity and the Lichnerowicz-York equation
    Niall Ó Murchadha, Chopin Soo, Hoi-Lai Yu
    (Submitted on 13 Aug 2012)
    We investigate the effect on the Hamiltonian structure of general relativity of choosing an intrinsic time to fix the time slicing. 3-covariance with momentum constraint is maintained, but the Hamiltonian constraint is replaced by a dynamical equation for the trace of the momentum. This reveals a very simple structure with a local reduced Hamiltonian. The theory is easily generalised; in particular, the square of the Cotton-York tensor density can be added as an extra part of the potential while at the same time maintaining the classic 2 + 2 degrees of freedom. Initial data construction is simple in the extended theory; we get a generalised Lichnerowicz-York equation with nice existence and uniqueness properties. Adding standard matter fields is quite straightforward.
    4 pages
    http://inspirehep.net/record/1127320?ln=en
     
    Last edited: Aug 9, 2015
  12. Aug 10, 2015 #11

    MTd2

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    Hi Marcus, here is your reply:

    ****************************************


    Thanks Dan,

    Appreciate your relaying the message. I took a look at the queries on redshift and Planck's constant, and also discussed these with my collaborators. Would like to provide the following info:

    - Intrinsic time T is always proportional to ln V, where V is the total (global) spatial volume of the universe and this is true whether considering the full theory or minisuperspace. So indeed in inhomogeneous models V would still be the 3DdI (3D diffeomorphism invariant) clock, while you cannot directly correlate z to the scale factor and volume. Luckily in our universe, z is a pretty good approximation for scale factor and volume.
    Using T_0/T propto ln(1+z) is the same as saying V_0/V propto 1+z. Very large z means vol. V at that value of z is much smaller than current volume. Also z>0 implies redshifted COMPARED TO NOW.
    Astronomers are interested in comparing past to NOW.

    In any instant of the future, you the astronomer of the future can do the same and set that date as 0 and discover the present (past of the future) will have a z larger than 0, and so on. So you can see that redshifts ALWAYS happen when you compare any future time with any past tie as long as the universe is expanding.

    Extrapolating in this scheme of things, z approaching -1 simply means a time in the future in which the volume now is negligible compared to the asymptotic future infinity. This is consistent, also because only intrinsic time INTERVAL is physically meaningful, not absolute time. So you always need to compare with something.

    - Regarding Planck's constant. When you write down the Schrodinger equation for the universe which governs its evolution with respect to intrinsic time, i\hbar{\delta\psi}/{\delta{T}}=H_{Phys}\psi, with T the intrinsic time and H_{Phys} the physical Hamiltonian, it turns out that Planck's constant \hbar cancels out from both sides of the equation, and you can really regard everything as dimensionless.

    One of the main results of ITQG is that the commutation relations amongst the momentric variables (related to the gravitational momentum) exhibit an independent SU(3) structure at each spatial point. You can see this by writing the momentric (up to a numerical factor of 2) as the (dimensionless) SU(3) generators times \hbar\delta(0), then the commutation relations are just the SU(3) Lie algebra (Gell-Mann matrices and eight-fold way). This SU(3) structure provides a natural regularization of the theory, and gravity can be seen fundamentally as an SU(3) structure with its associated representation theory.

    Now the Physical Hamiltonian H_{Phys} contains the WDW Kinetic operator (which is just the Laplacian for SU(3), devoid of operator-ordering ambiguities), which involves two powers of the momentric, hence two powers of Planck's constant, under a square root.
    Taking that square root yields one power of \hbar, which cancels out the \hbar on the left hand side of the Schrodinger equation. The above analysis still holds in the presence of interaction potentials (for example, the Cotton-York potential), since the kinetic operator transforms by similarity transformation in order to accomodate the potential terms.

    Hope this helps.

    Regards,

    Eyo
     
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