Wednesday, December 04, 2013

The three phases of space-time

If life grows over your head, your closest pop psy magazine recommends dividing it up into small, manageable chunks. Physicists too apply this method in difficult situations. Discrete approximations – taking a system apart into chunks – are enormously useful to understand emergent properties and to control misbehavior, such as divergences. Discretization is the basis of numerical simulations, but can also be used in an analytic approach, when the size of chunks is eventually taken towards zero.

Understanding space and time in the early universe is such a difficult situation where gravity is misbehaved and quantum effects of gravity should become important, yet we don’t know how to deal with them. Discretizing the system and treating it similar to other quantum systems is the maybe most conservative approach one can think of, yet it is challenging. Normally, discretization is used for a system within space and time. Now it is space and time themselves that are being discretized. There is no underlying geometry as reference on which to discretize.

Causal Dynamical Triangulations (CDT), pioneered by Loll, Ambjørn and Jurkiewicz, realizes this most conservative approach towards quantum gravity. Geometry is decomposed into triangular chunks (or their higher-dimensional versions respectively) and all possible geometries are summed over in a path integral (after Wick-rotation) with the weight given by the discretized curvature. The curvature is encoded in the way the chunks are connected to each other. The term ‘causal’ refers to a selection principle for geometries that are being summed over. In the end, the continuum limit can be taken, so this approach in an by itself doesn’t mean that spacetime fundamentally is discrete, just that it can be approximated by a discretization procedure.

The path integral that plays the central role here is Feynman’s famous brain child in which a quantum system takes all possible paths, and observables are computed by suitably summing up all possible contributions. It is the mathematical formulation of the statement that the electron goes through both slits. In CDT it’s space-time that goes through all allowed chunk configurations.

Evaluating the path integral of the triangulations is computationally highly intensive, but simple universes can now be simulated numerically. The results that have been found during the last years are promising: The approach produces a smooth extended geometry that appears well-behaved. This doesn’t sound like much, but keep in mind that they didn’t start with anything resembling geometry! It’s discrete things glued together, but it reproduces a universe with a well-behaved geometry like the one we see around.

Or does it?

The path integral of CDT contains free parameters, and most recently the simulations found that the properties of the universe it describes depend on the value of the parameters. I find this very intriguing because it means that, if space-time's quantum properties are captured by CDT, then space-time has various different phases, much like water has different phases.

In the image below you see the phase-diagram of space-time in CDT.

CDT phase diagram of space-time.
After Fig 5 in 1302.2173

The parameter κ is proportional to the inverse of Newton’s constant, and the parameter Δ quantifies the (difference in the) abundance of two different types of chunks that space-time is built up of. The phase marked C in the upper left, with the Hubble image, is where one finds a geometry resembling our universe. In the phase marked A to the right space-time falls apart into causally disconnected pieces. In the phase marked B at the bottom, space-time clumps together into a highly connected graph with a small diameter that doesn’t resemble any geometry. The numerical simulations indicate that the transition between the phases C and A is first order, and between C and B it’s second order.

In summary, in phase A everything is disconnected. In phase B everything is connected. In phase C you can share images of your lunch with people you don’t know on facebook.

Now you might say, well, but the parameters are what they are and facebook is what it is. But in quantum theory, parameters tend to depend on the scale, that is the distance or energies by which a system is probed. Physicists say “constant’s run”, which just rephrases the somewhat embarrassing statement that a constant is not constant. Since our universe is not in thermal equilibrium and has cooled down from a state of high temperature, constants have been running, and our universe can thus have passed through various phases in parameter space.

Of course it might be that CDT in the end is not the right way to describe the quantum properties of gravity. But I find this a very interesting development, because such a geometric phase transition might have left observable traces and brings us one step closer to experimental evidence for quantum gravity.

You can find a very good brief summary of CDT here, and the details eg in this paper.

Images used in the background of the phase-diagram, are from here, here and here.


  1. I have seen the CDT approach discussed a lot recently and though I favor geometry visualization this has the same sort of limitation as questions on the origin and extent of life over the universe. Do we see
    Life as Shannon's chemical information hardware or Von Neumann's (see Paul Davies) also top down universal constructor software?
    We seem to live in some geometrical region not clearly discrete nor continuous each of us a quasifinite universe.
    I do not feel "quantum" in QM gravity covers the concept. I call it quasication. For those who can decode such geometry there is a general algorithm. Then again Principia presented its case in the older geometry to describe Newton's calculus.

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  4. A second order phase transition has critical opalescence. That gives spontaneous small-scale fluctuations without requiring seeds to have them survive against evolving enormous scale fluctuations (the far end of the cosmic background radiation scale).
    1:10 launch. Methanol-cyclohexane going immiscible with temperature rise
    Liquid carbon dioxide warmed under pressure. Same result in either direction.

    Too strongly coupling gravitation wrecks the universe. Too weakly differentiating its building blocks wrecks the universe. Progress! Delta-kappa is 60% the separation of Yukawa alpha-lambda. A theta-iota theory is the answer. Why does physics have such difficulties with these things? "8^>)

  5. " ... in phase A everything is disconnected. In phase B everything is connected."

    I think that's a problem for CDT. It assumes a priori what it is trying to prove.

    Quantum mechanics, conventionally, takes the same tack. A measure on the unconstrained interval (- oo, + oo) assumes a totally disconnected space, and after measurement the space is multiply connected -- as if the measurer (observer)determined the configuration. One can't get everywhere continuous spacetime from this essentially linear schema; general relativity and all the evidence in favor of it would have to be abandoned.

    I would rather instead abandon the idea of a quantized spacetime, and start with an everywhere simply connected space. The negative term that adds curvature to local spacetime would then describe quanta as elements of a global topological continuum rather than discrete primordial elements.

    I think Joy Christian got it right.


  6. Can a space-time phase transition of this sort save us from the necessity of inflation?

  7. /* if space-time is fundamentally discrete, it has various different phases, much like water has different phases */

    This is the consequence of all emergent systems composed of discrete objects. In dense aether model the vacuum and particle interior represents an energetic continuum - the particles are just formed more dense foam, than the vacuum.


  8. The result brings in mind p-adic physics: the idea about space-time having real and p-adic regions with p-adic regions serving as space-time correlates of cognition ("thought bubbles").

    p-Adic topology is totally disconnected as the technical term goes: two open balls are either disjoint or same. This makes definition of manifold concept difficult. p-Adics however form a continuum in the sense that differential calculus makes sense. Integral calculus only if one defines p-adic manifold using maps to real manifolds as chart maps: real manifold and its p-adic cognitive representations would form pairs.

    In Topological Geometrodynamics framework p-adic space-time sheets serve as correlates for cognition and all number fields are glued along rationals to larger superstructure meaning generalisation of physics.

  9. Matti, Why do we conclude something in the models can only form pairs (from your view) ? In such lines in the solar spectrum QM seems to resolve the anomalies. And the first few differentiations seem enough to consider for the bulk of system mass. This does bring up a Leibniz background to the models and his Monads as the richer complication as consciousness. Is there an intrinsic property of arithmetical numbers at play here close to the topology? Over an abstract interval such information can contain all the information we imagine as emerging in steps and converging boundry integration of parts in the holographic conceptions. For we can play a 2D chess game on a 3D board, but it is more difficult to play than extending chess logically with parts and field as nD and the convex structures are smashed in our discrete view. It is not clear to me a cyclic universe would have unique or multiple solutions, a "best of possible worlds". Nor that our equations must come in strictly odd or even shadow pairs. Can a comet off disc fall occasionally by its own disturbence? Entanglement and chirality (polarity) does not tell us enough.

  10. Arun,

    Maybe! The idea that a pre-geometric phase might have been a totally connected graph (or at least a very connected graph) with a small diameter was previously discussed in some of Fotini's papers on 'quantum graphity' and I think that was one of their motivations. It addresses the horizon problem in an obvious way. But inflation is very successful and there are many achievements that have to be reproduced. One of them is eg the spectral index, which (I believe) should be possible to extract somehow from the properties of the phase transition. Best,


  11. Bee,
    Even with a course grained understanding this seems an amazing intellectual endeavor. It must be very exciting to begin with some gossamer abstraction, breathe life into it with a pulse of intensive simulation and then have it take on personal idiosyncrasy and exhibit changes of life. Likely one would need to guard one’s clarity from enthrallment with the wonder of it all.

    So, if each tetrahedron spans two discrete times do they capture in micro the larger, least energy transitions of state of the universe. That is, do the struts of the tetrahedrons represent energy transitions and the hubs represent… what? … information?

    So, if each tetrahedron spans two discrete times do they capture in micro the larger, least energy transitions of state of the universe. That is, do the struts of the tetrahedrons represent energy transitions and the hubs represent… what? … information?

  12. To Edgar Otto:

    I assume that you mean by pairs pairs of real and p-adic manifolds. The observation is that the notion of manifold defined in terms of chart maps does not work in p-adic context if charts are p-adic. Total disconnectedness is the problem.

    One could however use real charts and map the p-adic points space-time points to real ones by some variant of canonical identification. Canonical identification is a continuous map but does not respect differential structure: image of smooth p-adic surface is not smooth real surface. One would not be able to define various induced fields since gradients of imbedding space coordinates would be singular.

    The manner to resolve the question is to map only some discrete subset of points of p-adic space-time surface to real ones. This brings in some resolution.

    There is also the requirement that symmetries are respected -at least in some resolution. One could also demand that rational points correspond to each other as such: this means that the map respects symmetries in some resolution. Without cutoff the map would be extremely discontinuous.
    Below this cutoff continuity is respected down to the discretisation cutoff. Continuity and symmetries get both their own piece of cake.

    One obtains discrete set of points as real chart map. The conditions that these points belong to a preferred extremal of Kaehler action allows to assign to these points more or less unique real preferred extremal. p-Adic preferred extremal is thus mapped to more or less unique real preferred extremal.

    One can say that p-adic surface as correlated of mind is cognitive representation of real surface as correlate of matter. Einstein's geometrization program thus applies also to "thought bubbles". This allows also to speak about topological invariants of p-adic surfaces as analogs of real ones.

    1. Matti, I have another reply I will post eventually for your take on these field and geometry problems. But it id hard to discuss such connectivity issued while my thin link to the net by smart phone had "data connectivity problems "

  13. I posted a few corrections to your blog post here:

  14. Bee, CDT remembers me (even those weird discrete chunks of "space" and "time") tropical curves and surfaces. Of course, I can be wrong... :P

  15. Hi Bee,

    if I'm not mistaken CDT applies is restricted to globally hyperbolic spacetimes, don't you think that this constraint is too strong?

  16. Unknown,

    Yes, I do because I find not globally hyperbolic cases quite interesting. Alas, I think the issue is that in CDT it's an all-or-nothing choice and, looking around, the nothing option seems to agree better with what I see. Best,


  17. Bee:One of them is eg the spectral index, which (I believe) should be possible to extract somehow from the properties of the phase transition

    I think that is an interesting concept and one which I find might have some play in the gravitational rainbow? Hehe...and why not such a colorful world.



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