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What's new in Lego Path gravity?

  1. Nov 5, 2007 #1


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    CDT, which could be called Lego Path gravity, is a very simple and, in its own terms, remarkably successful approach to quantum spacetime dynamics.

    The recent Loll paper, called The Emergence of Spacetime, or Quantum Gravity on your Desktop, is something to read if you want to keep tabs on current CDT work.

    There are two basic ideas:
    the Legoblock idea
    Use a huge number of identical spacetime building blocks----little chunks of 4D looking like higherdimension pyramids or more precisely higherdimension tetrahedrons. Every shape of geometry can be approximated with these spacetime "Legoblocks" depending on how they are stuck together. There is no surrounding space.

    the Path Integral idea
    A spacetime is like a path, or history of evolution, between the geometry that space has at the start and at the finish. What the spacetime does is fill in: showing how the geometry evolves. It shows all the intermediate shapes. It is like the path in a Feynman path integral. In this case, a path thru the land of all possible spatial geometries.

    One should be able to calculate a probability or amplitude for each way that geometry can evolve---for each path. In fact, following Feynman's example one can just write down a GRAVITATIONAL PATH INTEGRAL plugging in the Einstein-Hilbert where the action term belongs. IF ONE CAN DO THE INTEGRAL then one should be able to calculate all sorts of things straightforwardly---expectation values, amplitudes to get from here to there

    For concreteness, have a look at Loll's review paper. The gravitational path integral is equation (1).

    To evaluate any path integral you have to integrate over the realm of all possible paths the particle could take, so you need a measure on pathspace. In our case, to evaluate a GRAVITATIONAL path integral you need a measure on the realm of all possible spacetimes (all possible paths from initial to final geometry.)

    Incidentally, to make things simple they take the initial and final spatial geometries to be trivial---essentially just one-point spaces.

    To make it possible to do the integral, what the researchers do is restrict the realm of all possible spacetime geometries down to a REPRESENTATIVE SAMPLE. It's often called "regularizing". They restrict down to ALL THOSE SPACETIMES YOU CAN BUILD WITH LEGOBLOCKS.

    And then they let the Path Integral happen inside a computer (Monte Carlo style).

    It is quite a clever approach. A small number of researchers do it: Renate Loll, Jan Ambjorn, a few others. They call their approach CAUSAL DYNAMICAL TRIANGULATIONS

    From time to time they bring out surprising results. Now just this week Renate Loll posted an overview paper, written for non-specialists, which she gave as invited speaker at a big international conference at Sydney in July. It explains the whole thing very clearly, and it also tells us to expect a new paper to appear by Ambjorn Goerlich Jurkiewicz and Loll.

    This thread is to discuss the new CDT paper, if anyone wants to. I will get some quotes and paraphrase some stuff from the paper.

    A big point that both Jan Ambjorn made in his invited talk at Loops '07 in June and Renate Loll makes here is that if they generate a whole lot of spacetimes in the computer and average them up they get the four-sphere S4.
    That which is to deSitter space as euclidean is to lorentzian.

    They've also started including matter, so it is not just pure geometry, and there is some discussion of that.
    Last edited: Nov 6, 2007
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  3. Nov 6, 2007 #2


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    Something to notice: the approach does not postulate that these little identical 4D legoblocks EXIST.
    they just provide a systematic way of sampling the realm of possible geometries, without too much overcounting

    It is an extremely economical or MINIMALIST approach. It assumes as little extra objects and structure as possible.

    there is no assumption of DISCRETENESS either. One calculates using these assemblages of legoblocks, but then one lets their SIZE GO TO ZERO.

    1. no extra junk is postulated to exist
    2. economy----the approach travels light
    3. non-discreteness

    there is no preferred geometry at the outset. the spacetime assemblages are not immersed in anything, they have no higherdimensional space surrounding them, they have no boundary (except at the initial and final points)

    4. the approach is independent of any chosen geometry

    Interestingly, in circumstances like this the DIMENSION of the space around given point at a given distance scale is a measurable quantity which is not fixed by assumption and does not need to be an integer. DIMENSION IS A QUANTUM OBSERVABLE WHICH CAN TAKE ON FRACTIONAL VALUES, like 1.5 and 2.6. There are several intuitive meanings to give to the idea of dimensionality, including e.g. the relation between distance and volume: stepping from block to adjacent block, count how many blocks you can contact within a certain number of steps. In this CDT approach, with its minimum of assumptions YOU CAN'T EVEN TAKE THE DIMENSIONALITY FOR GRANTED. Even though you build with 4D blocks you don't necessarily get routine dimensionality.

    It was something of a triumph when Ambjorn and Loll reported in 2004 that a 4D macroscopic spacetime did actually emerge, using their approach. And indeed it was discovered that the dimension was less than that down at a microscopic level.

    I shall get some quotes from Loll's new paper as soon as time permits. Have to go out for now.
    Last edited: Nov 7, 2007
  4. Nov 11, 2007 #3


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    sample quote from the Loll paper

    ===sample from http://arxiv.org/abs/0711.0273. ===
    Is there an approach to quantum gravity which is conceptually simple, relies on very few fundamental physical principles and ingredients, emphasizes geometric (as opposed to algebraic) properties, comes with a definite numerical approximation scheme, and produces robust results, which go beyond showing mere internal consistency of the formalism? The answer is a resounding yes: it is the attempt to construct a nonperturbative theory of quantum gravity, valid on all scales, with the technique of so-called Causal Dynamical Triangulations.

    Most remarkable at this stage is perhaps the fully dynamical emergence of a classical background (and solution to the Einstein equations) from a nonperturbative sum over geometries, without putting in any preferred geometric background at the outset. In addition, there is concrete evidence for the presence of a fractal spacetime foam on Planckian distance scales.

    The availability of a computational framework provides built-in reality checks of the approach, whose importance can hardly be overestimated.

    A central aim of any theory attempting to address the problem of quantum gravity is to derive spacetime as is from an underlying, dynamical quantum principle. By “spacetime” we mean spacetime together with its geometric properties on all scales, from the largest, cosmological scales to the smallest... “Geometric” here is to be understood in a suitably generalized sense at very short distances, where we expect the classical description of spacetime in terms of a differentiable manifold with a smooth Lorentzian metric to be no longer adequate.

    The short-distance structure of a particular model of quantum gravity we will be discussing in what follows is incompatible with such a smooth assignment, but nevertheless possesses more ‘primitive’ metric properties, allowing us to measure - in the sense of quantum theory- certain lengths and volumes.

    The reason for the expected breakdown of classicality near the Planck scale is the dominance of large quantum fluctuations, as expressed in the nonrenormalizability of the perturbative quantization of gravity.

    One reason why it is so difficult to get a handle on the quantum dynamics of the fluctuations is that they affect what in [usual] quantum field theories ..is just part of the fixed background structure, namely, spacetime itself.

    The failure of the perturbative approach to quantum gravity in terms of linear fluctuations around a fixed background metric implies that the fundamental dynamical degrees of freedom of quantum gravity at the Planck scale are definitely not gravitons.

    At this stage, we do not yet know what they are...

    ..hope that...theories of quantum gravity will lead to definite predictions of observable consequences of a nontrivial microstructure of spacetime. Potential examples of this include cumulative effects imprinted on highly energetic photons or neutrinos reaching us [over cosmological distances] [1], and a quantum-gravitational origin of the vacuum energy of empty space, the so-called dark energy.

    ... let us set out our mission statement, which is

    To look for a consistent theory of quantum gravity, which describes the dynamical behaviour of spacetime geometry on all scales and reproduces Einstein’s theory of general relativity on large scales. At the same time, it should also predict new observable phenomena.

    With all due respect to past efforts and achievements, the search for such a theory has yet to succeed, be it in “pure gravity” or so-called unified approaches (these days usually based on string theory). Despite occasional claims to the contrary, there is at this stage no compelling evidence whether or not all fundamental interactions have to be unified at the Planck scale. At any rate, it is clear we are dealing with a formidable problem, where technical difficulties often appear entangled with conceptual ones.

    In the face of these longstanding difficulties, physicists have drawn sometimes far-reaching conclusions about what should be done:

    • Does the theory need new ingredients, either in the form of new symmetries, or in terms of new, fundamental objects? A prime example of the former is (the hitherto unobserved) supersymmetry, and examples of the latter include fundamental strings, loops and higher-dimensional membranes.

    • Do we need to question or modify the fundamental principles of quantum theory and/or general relativity? An example of the former is the suggestion – motivated by considerations originating in quantum gravity – that there may be a deterministic theory underlying standard quantum theory [2].

    • Most radically, do we need to change our notion of what constitutes a physical theory? Have we reached the end of the road in terms of “reductionist” physics, despite its past successes in high-energy physics? Do we have to take up “landscaping” or metatheorizing?

    Many theorists feel a growing unease about these choices. Aren’t there any loopholes in the arguments that lead us down these highly speculative roads?

    Last edited: Nov 11, 2007
  5. Nov 11, 2007 #4


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    What she is saying here corroborates the work of Reuter and Percacci, and also i would say the efforts of Garrett Lisi.

    By a combination of careful reasoning and the example of Triangulation's successes, she is arguing for the conservative MINIMALIST position.

    1. You don't have to pretend that fanciful objects exist like string or brane or extra D. We can make a background independent QFT that includes geometry without any drastic novelties.

    2. You don't have to change the basic principles of quantum mechanics either.

    3. We don't have to give up on the basic drive of science to understand why the world is what it is using falsifiable predictive theories and progressively reducing the play of free parameters in them. We don't have to follow Lenny Susskind into Anthropic Landscape Land

    The thing to notice is that for Loll THE LITTLE TRIANGLE LEGOBLOCKS DON'T HAVE TO EXIST they are just a way to do ordinary field theory!

    the triangulation method is just a way to do a PATH INTEGRAL like from 1950, using a modified EINSTEIN HILBERT ACTION like from 1915, and have it be about spatial geometry (instead of fixing a prior geometry), and have it not blow up, and to be able to check stuff by COMPUTER CALCULATION.

    It is that simple. It is extremely conservative. The fact that she and Ambjorn and the others made it work is profoundly significant.

    It also tends to support what Reuter and Percacci and their people are saying which is that ordinary Einstein gravity is RENORMALIZABLE with ordinary field theory using a slightly modified Einstein-Hilbert that they found as a UV fix point of the flow on action-space.

    That is, background independent QFT (not depending on any prior metric geometry) is possible with already-tried methods and is renormalizable after all---so they didn't have to fiddlefutz around with imaginary paraphernalia for 20 or 30 years.

    And nobody has to change the rules of quantum mechanics, or give up on the 400 year old tradition of Baconian science and go landscaping.

    I think what Loll and Reuter's work is is this: it is not the end of the road it is a GREEN LIGHT.

    Loll says gravity (= the geometry field) is well behaved if you describe the geometry by an assemblage of identical blocks

    Reuter says gravity (= the geometric field) is well behaved if you describe it with a metric distance-function, and moreover the constants in the action RUN as they do in other field theory----the fix point is just the BARE action that applies at extreme UV.

    What someone like Garrett Lisi needs, and what it looks to me like Carlo Rovelli is trying to do, is to show that
    gravity (= geometry) is well behaved if you describe it with a principle-bundle CONNECTION

    So what Loll and Reuter are doing is giving a green light signal which is go-for-the-connection.

    A connection is just a way of stitching the tangentspaces at each point of the manifold together---so you can take the derivative not just at a single point and do parallel transport
    the tangentspace at a single point has the information about taking derivatives at that point
    so relating tangent vectors at two different points by transport is much the same thing as connecting how you take derivative at the two separate points.

    if you can connect the business of taking derivatives at this point, with the same business at the other point, and connect all the points this way, then that is tantamount to describing the overall shape or geometry of the manifold. And you can say what geodesics are. Here's a wikipedia article

    So a connection is as good a way to describe the shape of the manifold as a metric distance-function is. And you can FORMULATE 1915 GR gravity in terms of connections---Ashtekar did this in 1987 or so. So if connections are just as good as metrics, and you can make a QFT of the metric (like Reuter says) then you should be able to make a QFT of the connection.

    It's just basing the background independent field theory on different but equivalent degrees of freedom.

    So suppose Rovelli gets the green light, what does that mean? what I think is that for him the SPIN NETWORKS DON'T HAVE TO EXIST in a material sense, they are just ways of testing the connection. The configuration space is the connections and a spinnetwork is just a function defined on connections----analogous to a wave function.
    If you have a powerful microscope and look down at a small bit of space you don't expect to see little spin networks, or a patch of a big comprehensive spin network. those things are mathematical objects defined on the space of all possible connections (=all possible geometries).
    quantum mechanics is the mathematical theory of those functions, whether they are single particle wave functions defined on the real line, or geometric field wave functions defined on geometric configuration space.

    so we are not talking about material existence but about how to define functions on the space of possible connections (=geometries)

    if anybody has a better way to define a function on connection-space, a better way than with a spinnetwork, they should say (and we could tell one of Rovelli's group when they come around)

    What I am saying is that Renate Loll is on one hill, and Garrett is on a different hill, and Carlo is laboring in the valley between them.

    I value Loll's vision. It is conservative and minimalist and persuasively argued and supported by her work. But her work doesnt describe geometry using connections, and Lisi seems to need a connection handle on shape (so he can weld the standard particles onto it).
    Last edited: Nov 11, 2007
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