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Landmark paper by Smolin pulls together recent QG research

  1. Oct 14, 2015 #1

    marcus

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    http://arxiv.org/abs/1510.03858
    The thermodynamics of quantum spacetime histories
    Lee Smolin
    (Submitted on 13 Oct 2015)
    We show that the simplicity constraints, which define the dynamics of spin foam models, imply, and are implied by, the first law of thermodynamics, when the latter is applied to causal diamonds in the quantum spacetime. This result reveals an intimate connection between the holographic nature of gravity, as reflected by the Bekenstein entropy, and the fact that general relativity and other gravitational theories can be understood as constrained topological field theories.
    To state and derive this correspondence we describe causal diamonds in the causal structure of spin foam histories and generalize arguments given for the near horizon region of black holes by Frodden, Gosh and Perez and Bianchi. This allows us to apply a recent argument of Jacobson to show that if a spin foam history has a semiclassical limit described in terms of a smooth metric geometry, that geometry satisfies the Einstein equations.
    These results suggest also a proposal for a quantum equivalence principle.
    39 pages, 6 figures
     
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  3. Oct 14, 2015 #2

    marcus

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    It's a major step forward, partly, I think, because it pulls together so many loose ends. I'll highlight a few indicative passages.
    It comes at the 20th anniversary of Ted Jacobson's seminal and provocative 1995 paper showing GR as the "Einstein Equation of State", deriving GR from thermodynamics of unknown "atoms of geometry".
    ==quote==
    We show that the simplicity constraints, which define the dynamics of spin foam models, imply, and are implied by, the first law of thermodynamics, when the latter is applied to causal diamonds in the quantum spacetime. This result reveals an intimate connection between the holographic nature of gravity, as reflected by the Bekenstein entropy, and the fact that general relativity and other gravitational theories can be understood as constrained topological field theories.
    To state and derive this correspondence we describe causal diamonds in the causal structure of spin foam histories and generalize arguments given for the near horizon region of black holes by Frodden, Gosh and Perez and Bianchi. This allows us to apply a recent argument of Jacobson to show that if a spin foam history has a semiclassical limit described in terms of a smooth metric geometry, that geometry satisfies the Einstein equations.
    These results suggest also a proposal for a quantum equivalence principle.
    ==endquote==
    The range of people mentioned in the acknowledgements is suggestive:
    ** It is a pleasure to thank Bianca Dittrich, Laurent Freidel, Ted Jacobson, Rob Myers, Alejandro Perez, Aldo Riello, Carlo Rovelli and Wolfgang Wieland for comments and discussion. **
    It's also worth glancing at the bibliography which brings together string theory, AdS/CFT, shape dynamics, thermodynamics and much else.

    Not mincing words, Smolin proposes calling his equation (68) the First Law of Quantum Spacetime. It is analogous to the first law of Thermodynamics, transposing from classical thermodynamics to quantum geometry.

    Highlighting some parts of the conclusions:
    ==quote==
    By extending results of [18, 19, 36], we have shown that, given suitable conditions, the linear simplicity constraint of spin foam models, (14) implies the first law of quantum spacetime (68). This is initially expressed in terms of a micro canonical entropy, given by the ensemble at fixed area. But if we go to the canonical ensemble at fixed temperature we can follow [19] to compute the temperature. and from that, the entropy and assign the causal diamond to an equilibrium state at fixed temperature. This then implies the relationship between the area and the Boltzmann entropy (81), with the correct 1/4 , independent of the Immirzi parameter.

    We have also showed that the first law of thermodynamics implies the simplicity constraint.

    We further showed that if there exists a semiclassical limit (which we do not prove) this implies the thermodynamic first law, (86). This, in turn, implies the Einstein equations, as shown by Jacobson in [29].

    These results establish that there is a close connection between the holographic behaviour of quantum gravity and the fact that general relativity is closely related to a topological field theory. Indeed, this is precisely the connection anticipated in [10] and [14], The fact that general relativity is a constrained topological field theory is then the root of the holographic nature of gravity. Indeed, this has been since then a central feature of loop quantum gravity[14], which has been developed in different ways in [37]. It is fitting that this connection between the holographic and topological aspects of gravity is deepened by the simplicity constraints, which were also first used in works of Barrett and Crane[35].

    There is one big question that these results raise, which is that if general relativity, which is a time reversible theory, corresponds to equilibrium statistical mechanics, what is the time irreversible extension of general relativity that corresponds to non-equilibrium statistical mechanics? In particular, might it be one of the known irreversible extensions of general relativity[52]?
    ==endquote==
     
    Last edited: Oct 15, 2015
  4. Oct 15, 2015 #3
    if a spin foam history has a semiclassical limit described in terms of a smooth metric geometry, that geometry satisfies the Einstein equations.

    if there exists a semiclassical limit (which we do not prove) this implies the thermodynamic first law

    if
     
    Last edited by a moderator: Oct 15, 2015
  5. Oct 15, 2015 #4

    marcus

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    Yes, Smolin certainly makes that clear. He actually bold-faces the word "if" at several places in the article---to make sure the reader understands the logical status. Good for you, Kodama, for noticing this and pointing it out! :oldbiggrin:
    The paper has many interesting results, not only this one (which is already quite significant, I think.)
     
  6. Oct 15, 2015 #5
    why doesnt smolin first prove each and every if
     
  7. Oct 18, 2015 #6

    marcus

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    In mathematics (and I guess this goes for theoretical physics as well) theorems normally have conditions. Like "if A then B" or "if A, B, and C, hold, then D."

    This calls attention to the condition(s) and can serve as a challenge to other researchers to discover under what circumstances the conditions hold.

    Personally I like the way the exposition is organized in this paper! If you read it you see that Smolin is exploring different ways to significantly redefine what a "spin foam model" is!

    He introduces causal structure, following up on some original 2014 work by Wolfgang Wieland and by Marina Cortes and himself---which in turn goes back to Causal Sets (and earlier proposals by Fotini Markopoulou) and a variant called Energetic Causal Sets. He defines what he means by a spin foam with a Wieland structure.

    This opens up a lot of possibilities for what a simplicial QG model, or a causal spinfoam, could be. How it could be defined---what one could discover about a particular type of model. Basically it is an invitation to other researchers (like Bianca Dittrich and her co-workers, like Wolgang Wieland, and others) to explore. To me, as a non-expert on-looker, it looks like an opening to a very fertile area of new research.

    The idea of a "generalized boost" is intriguing. And bringing the idea of "causal diamond" into spinfoam context. I like the way he connects to the work of Eugenio Bianchi, and of Perez, Frodden, et al. The paper is 39 pages, it proves a lot of stuff, and it's full of new ideas. Still in draft form, in a sense, may go through some revision and correction, but basically complete.
     
    Last edited: Oct 19, 2015
  8. Oct 18, 2015 #7

    marcus

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    This passage on page 9 of the paper can serve to clarify what I just suggested, and motivate the introduction of causal structure in the spinfoam context.
    ==quote Smolin page 9==
    Having control of these causal structures will allow us to define and study causal diamonds in spin foam histories. We give a general description of causal diamonds in these discrete causal structures, this allows us to define observables for spin foam models in terms of expectation values of currents defined on the boundaries of causal diamonds.

    The thermodynamics of these quantum spacetimes is then defined by vacuum expectation values on the spatial boundary of the causal diamond, where by vacuum we mean the case that the currents on the null boundaries vanish.

    Physical quantities relevant to the low energy or semiclassical limit will, as Dittrich has emphasized, have to be defined following a process of coarse graining and renormalization. Also, as she emphasizes, physical observables are to be defined by coarse graining boundaries[39]. It is then important to show that the connection between the simplicity constraints and the first law emerges in a way that is largely independent of these processes and the details of how they are carried out. To accomplish this we give a very brief sketch of these processes. The key point is that the linear simplicity constraints, acting on boundary observables, apply just as well to coarse grained and renormalized quantities, because they are linear.
    ...
    ==endquote==

    I've highlighted the reference to important work by Dittrich and co-workers on the low energy limit in spinfoam QG context. It's worth noticing that this has involved Dittrich et al in reformulating spinfoam QG. One has to be clear now which spinfoam model one means, or else I suppose, work as Smolin does, at an appropriate level of generality and draw conclusions about any model satisfying such-and-such conditions. In any case here are some Dittrich et al papers to illustrate what I mean.

    http://arxiv.org/abs/1401.6441
    A new vacuum for Loop Quantum Gravity
    Bianca Dittrich, Marc Geiller
    (Submitted on 24 Jan 2014)
    We construct a new vacuum for loop quantum gravity, which is dual to the Ashtekar-Lewandowski vacuum. Because it is based on BF theory, this new vacuum is physical for (2+1)-dimensional gravity, and much closer to the spirit of spin foam quantization in general. To construct this new vacuum and the associated representation of quantum observables, we introduce a modified holonomy-flux algebra which is cylindrically consistent with respect to the notion of refinement by time evolution suggested in [1]. This supports the proposal for a construction of a physical vacuum made in [1,2], also for (3+1)-dimensional gravity. We expect that the vacuum introduced here will facilitate the extraction of large scale physics and cosmological predictions from loop quantum gravity.
    11 pages, 5 figures

    http://arxiv.org/abs/1409.1450
    The continuum limit of loop quantum gravity - a framework for solving the theory
    Bianca Dittrich
    (Submitted on 4 Sep 2014)
    The construction of a continuum limit for the dynamics of loop quantum gravity is unavoidable to complete the theory. We explain that such a construction is equivalent to obtaining the continuum physical Hilbert space, which encodes the solutions of the theory. We present iterative coarse graining methods to construct physical states in a truncation scheme and explain in which sense this scheme represents a renormalization flow. We comment on the role of diffeomorphism symmetry as an indicator for the continuum limit.
    19 pages, 1 figure. Draft chapter for a volume edited by A. Ashtekar and J. Pullin, to be published in the World Scientific series "100 Years of General Relativity"

    http://arxiv.org/abs/1506.08571
    A new realization of quantum geometry
    Benjamin Bahr, Bianca Dittrich, Marc Geiller
    72 pages 6 figures
     
    Last edited: Oct 18, 2015
  9. Oct 19, 2015 #8
    Once again Lee Smolin has produced here a wonderful paper. As you say the references alone give you a pretty good overview of Quantum Gravity. The only minor issue I have is that there isn't any reference to Penrose and his work which is weird given his output on spacetime and causality. Nevertheless this paper surely merits much study.
     
  10. Oct 19, 2015 #9
    "Finally, these results suggest a form of the quantum equivalence principle, which has been long sought[7]. In flat spacetime an accelerating observer sees a region of spacetime limited by an horizon, which is generated by a two surface, W, fixed by a boost. We can generalize the notion of a boost to mean any evolution of a region of quantum or classical spacetime that fixes a two surface, W. This two-surface W divides a spacial slice of the universe into two parts. In flat spacetime these are maximally entangled with each other. We can posit that this is also true in a dynamical quantum spacetime. As this generalizes a property of flat spacetime it is appropriate to call it a version of the equivalence principle. The result is that an observer inside the causal diamond sees a thermal state, given by ρW = e −Hboost(W)/TU (10) where Hboost(W) is the generator of the boosts that fix W. As will be explained, because of refoliation invariance in the interior of the causal diamond, this is unique." p-8

    So is the case where a stationary observer sees space-time expand also an example of a "generalized boost" in a causal diamond?
     
  11. Oct 19, 2015 #10

    marcus

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    David and Jim, thanks for the comment and interesting question! It happens I'm just on point of going out for the morning (doctor appointment etc...) and can't reply! Will get back this afternoon. In the meantime maybe someone else would like to respond.
     
  12. Oct 19, 2015 #11

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    Good question! In fact if I understand correctly Smolin's analysis does NOT cover that case. The area of the waist of each causal diamond is unchanged in a generalized boost. This is part of the definition of the generalized boost.
     
  13. Oct 20, 2015 #12
    does this mean Loop Quantum Gravity will be taken more serious now?
     
  14. Oct 20, 2015 #13
    I see where that is stated but I thought it was only meant to say that one causal diamond is considered in order to derive the hypothetical effect on the Hamiltonian evolving just two space time events. A change to the waist would imply events inside the diamond - which makes it less like a causal diamond.

    In the section on Generalized Boosts I don't see a prohibition on picturing the model as - causal LQG events generated by the appearance of "a waist".

    But my more general question is really - how would expansion be accommodated in this model? Doesn't it have to be? Or is the expansion of space time considered to be outside the purview of LQG somehow?
     
    Last edited: Oct 20, 2015
  15. Oct 20, 2015 #14
    In section 3.6 he defines the "Elementary Causal Diamond" as 3 time slices - giving one slice to the interior, which is interesting. This is what I meant in post #13

    In eq.45 he expresses the area of the waist - I know it's not the example used to describe the model but couldn't an initial change to that term as the starting point be imagined as the driver of evolution of the space-time foam?

    What "causes" the Pachner moves to happen (evolving space-time) in the absence of a pre-existing black hole or an accelerating observer?
     
    Last edited: Oct 20, 2015
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