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A Loop quantum gravity and diffeomorphism group

  1. Mar 29, 2016 #1

    A. Neumaier

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    How is loop quantum gravity related to the principle of relativity, in particular to the diffeomorphism invariance of general relativity?
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  3. Mar 29, 2016 #2


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    the thing to focus on is a relatively recent formulation of Lqg called "covariant Lqg". If I google that, I get:
    and also a free downloadable draft version of a recent book:
    and also the Cambridge UP page for the book "Covariant Loop Quantum Gravity"

    It should clarify what you asked about, namely the relation between covariant (or spinfoam "path histories") formulation of Lqg and the principle of diffeomorphism invariance for theories formulated on a 4d continuum.
    Last edited: Mar 29, 2016
  4. Mar 30, 2016 #3


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    Is diffeomorphism-invariance in LQG an emergent property due to quantum consistency, as e.g. U(1) gauge symmetry is in QED (i.e. it enables one to remove ghosts)?
  5. Mar 30, 2016 #4


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    You might be interested in a series of work by Jonathan Engle. Here is the most recent.

    He is one of the original EPRL authors (EPRL formulation became covariant Lqg)
    and he has given careful attention to this issue "how related to diffeo invariance..."

    Here e.g. is an earlier paper:
    Spin foams
    Jonathan Engle
    (Submitted on 19 Mar 2013)
    The spin foam framework provides a way to define the dynamics of canonical loop quantum gravity in a spacetime covariant way, by using a path integral over histories of quantum states which can be interpreted as `quantum space-times'. This chapter provides a basic introduction to spin foams aimed principally at beginning graduate students and, where possible, at broader audiences.
    Comments: 32 pages, 14 figures, 2 tables, to appear as a chapter of "The Springer Handbook of Spacetime," edited by A. Ashtekar and V. Petkov (Springer-Verlag, at Press)
  6. Apr 1, 2016 #5
    Reading from my notes from Smolin's Problem of Time course (lecture 6c, available from PIRSA) the diffeomorphism constraint in LQG is constructed from 4 components of the Gauss' Law constraint. These constraints arise from the Hamiltonian which is constructed from the Plebanksi action, which rewrites the usual Einstein-Hilbert action in terms of frame fields rather than the metric.

    During quantisation the Diffeo constraint reduces the number of states (Wilson loops) in the Hilbert space from an infinite to a finite number, one state per diffeomorphism equivalence class.
  7. Apr 1, 2016 #6


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    Nice comment. As a convenience to us, if not too much of a nuisance, would you give a link to the lecture video? Or the slides pdf, if available?
  8. Apr 2, 2016 #7
    The lecture (6c) is here:- http://pirsa.org/08020048/

    Construction of the Plebanski action through to Ashketar is in 6a and 6b. Issues surrounding quantisation, including finding Diffeo invariant states of Ashketar is in 6d.

    The full series is here:- http://pirsa.org/C08003

    I recommend downloading the PDF with each lecture as the video quality can be a tad blurry at times.
  9. Apr 2, 2016 #8


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    So it's a Jan 2008 lecture series.
    The formulation has changed some since, which could matter. There is EPRL and variations like Jon Engle's.
    I don't know if it would make a difference to you but you might be interested in looking at some of the more recent papers I linked to.
    e.g. in post #4
    Now the standard textbook on LQG is "Covariant LQG" (2014) by Rovelli and Vidotto. There is a free draft version one can download or one can buy the book. I gave a link. It's different from what was current in 2008.
    Last edited: Apr 2, 2016
  10. Apr 4, 2016 #9
    Thanks for that Marcus. I splashed out and ordered a copy and also downloaded it. I really enjoy reading anything by Rovelli.

    The theorem, called LOST, that reduces the diffeo invariant states to one unique state is quite generally applicable, though I am not sure how covariant LQG handles it specifically.

    LOST is discussed in this interesting paper by Ashtekar where he talks about 'diffeomorphism covariance' which is a term I'm not familar with so not sure if it relates. (I see I spelt his name wrong again in my previous post!)


    "Some surprising implications of background independence in canonical quantum gravity" by Abhay Ashtekar

    "There is a precise sense in which the requirement of background independence suffices to uniquely select the kinematics of loop quantum gravity (LQG). Specifically, the fundamental kinematic algebra of LQG admits a unique diffeomorphism invariant state. Although this result has been established rigorously, it comes as a surprise to researchers working with other approaches to quantum gravity. The goal of this article is to explain the underlying reasons in a pedagogical fashion using geometrodynamics, keeping the technicalities at their minimum. This discussion will bring out the surprisingly powerful role played by diffeomorphism invariance (and covariance) in non-perturbative, canonical quantum gravity."

    " The first goal of this article is to show, by a careful analysis of the WDW theory, that there is in fact no tension. We will see that there are apparently surprising results also in the WDW theory; diffeomorphism invariance is a much stronger requirement that one might have first thought. Results can seem counter intuitive if one does not carefully distinguish between the kinematical algebra and the algebra of diffeomorphism invariant variables. The second goal is to provide intuition for the origin of the two features of the kinematics of LQG which are not shared by familiar Minkowskian quantum field theories —the non-separability of the kinematical Hilbert space and the non-existence of a local connection operator Aia(x). We will see that they can be traced back to the diffeomorphism covariance of LQG. Finally, it will be instructive to compare the role of gauge invariance in the Maxwell theory with that of diffeomorphism invariance. We will find that, because of its inherent non-locality, diffeomorphism invariance is much more powerful than gauge invariance." (my bold)

    There appear to be some who find the LOST theorem to be too restrictive:-
    " The first part of today’s paper explains all this and shows that the difference in the treatments can be summarised by saying that the usual Fock space quantisation of the string uses a Hilbert space built upon a covariant state whereas the loopy approach insists on invariance of that state which is a much stronger requirement.
    My point is that covariance is the property which is physically required (and in fact states in the classical field theory are covariant but not invariant) and thus statements like the LOST theorem have too strict assumtions."

    Both of these papers are over 5 years old so this issue may already be resolved. I couldn't find anything more recent though.
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