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Matter as excitations of spacetime lattice?

  1. Jun 17, 2010 #1
    It is a common theme among background independent quantum gravity theories that there should be some sort of discretization, or fuzziness, of the spacetime manifold occuring on Planckian scales.

    It has occured to me that if we take this discretization to consist of a lattice of sorts, might matter on this spacetime not be described via some sort of lattice excitation or vibration, in the same way that vibrations of a crystal lattice allows a particle description (i.e. the phonon)? Of course one would have to figure out, for example, how these excitations could induce a global curvature to the manifold, etc. But even so, it seems to me an interesting idea.

    I was just wondering if anyone here knows of any research that has been done regarding this idea, or has any insights they with to discuss. I would love to see if anyone has worked on this.

    Thanks!
    - Eric
     
  2. jcsd
  3. Jun 17, 2010 #2

    tom.stoer

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    One must not think about spacetime as a lattice; this would violate several symmetries, e.g. diff. inv. So the picture is definitly more complex.

    One idea is the loop quantum gravity (LQG) approach which is based on so-called spin networks. Edges connecting vertices carrying SU(2) degrees of freedom which are coupled at the vertices by intertwining operators. Edges and vertices are representing area and volume operators with quantized spectrum. LQG is a rather promising approach towards QG.

    There is an idea that these spin-networks can be generalized and that some kind of "braiding" and "twisting" of the generalized edges (ribbons) can occure. These topologically stable configurations can be combined to objects carrying algebraic properties similar to particles known from the standard model. Moves on these braids may correspond to interactions of elementary particles.

    Attention: this is a rather speculative idea - but fascinating as it may point towards particles entirely based on space-time.

    http://arxiv.org/abs/hep-th/0603022
    Quantum gravity and the standard model
    Sundance O. Bilson-Thompson, Fotini Markopoulou, Lee Smolin
    (Submitted on 3 Mar 2006 (v1), last revised 21 Apr 2007 (this version, v2))
    Abstract: We show that a class of background independent models of quantum spacetime have local excitations that can be mapped to the first generation fermions of the standard model of particle physics. These states propagate coherently as they can be shown to be noiseless subsystems of the microscopic quantum dynamics. These are identified in terms of certain patterns of braiding of graphs, thus giving a quantum gravitational foundation for the topological preon model proposed by one of us.
    These results apply to a large class of theories in which the Hilbert space has a basis of states given by ribbon graphs embedded in a three-dimensional manifold up to diffeomorphisms, and the dynamics is given by local moves on the graphs, such as arise in the representation theory of quantum groups. For such models, matter appears to be already included in the microscopic kinematics and dynamics.


    http://arxiv.org/abs/1002.1462
    Embedding the Bilson-Thompson model in an LQG-like framework
    Deepak Vaid
    (Submitted on 8 Feb 2010)
    Abstract: We argue that the Quadratic Spinor Lagrangian approach allows us to approach the problem of forming a geometrical condensate of spinorial tetrads in a natural manner. This, along with considerations involving the discrete symmetries of lattice triangulations, lead us to discover that the quasiparticles of such a condensate are tetrahedra with braids attached to its faces and that these braid attachments correspond to the preons in Bilson-Thompson's model of elementary particles. These "spatoms" can then be put together in a tiling to form more complex structures which encode both geometry and matter in a natural manner. We conclude with some speculations on the relation between this picture and the computational universe hypothesis.
     
  4. Jun 17, 2010 #3
    Some other sources:
    (Rovelli)
    http://relativity.livingreviews.org/Articles/lrr-2008-5/ [Broken]


    The video and slides of Carlo Rovelli's talk at Strings 2008 provides a good introductory overview of LQG.
    Here are the links:
    Video:
    http://cdsweb.cern.ch/record/1121957?ln=en
    Slides:
    http://indico.cern.ch/getFile.py/acc...s&confId=21917 [Broken]

    An early draft of Rovelli's book online free for anyone who doesn't have the published version:
    http://www.cpt.univ-mrs.fr/~rovelli/book.pdf

    Another good overview of the whole field of quantum geometry/gravity, Rovelli's chapter in Oriti's book:
    http://arxiv.org/abs/gr-qc/0604045

    Rovelli:
    Interestingly, LQG predicts that when areas and volumes of physical objects are measured they will come out in quantized levels, like the energy levels of a hydrogen atom. The area and volume operators have discrete spectrum. This is not put in by defining the theory on a lattice, it comes out as an advanced result, because the geometric operators corresponding to measurements are part of a quantum theory. Discrete spectra of operators is a kind of discreteness, and it turns out to be entirely compatible with Lorentz symmetry! Rovelli had a paper about that in 2002 or 2003. It's actually kind of neat how it works out. I will get the link for that in case you want to pursue it further.
    http://arxiv.org/abs/gr-qc/0205108
    Reconcile Planck-scale discreteness and the Lorentz-Fitzgerald contraction
    ////////////

    "A Planck-scale minimal observable length appears in many approaches to quantum gravity. It is sometimes argued that this minimal length might conflict with Lorentz invariance, because a boosted observer could see the minimal length further Lorentz contracted. We show that this is not the case within loop quantum gravity. In loop quantum gravity the minimal length (more precisely, minimal area) does not appear as a fixed property of geometry, but rather as the minimal (nonzero) eigenvalue of a quantum observable. The boosted observer can see the same observable spectrum, with the same minimal area. What changes continuously in the boost transformation is not the value of the minimal length: it is the probability distribution of seeing one or the other of the discrete eigenvalues of the area. We discuss several difficulties associated with boosts and area measurement in quantum gravity.
    http://arxiv.org/abs/gr-qc/0205108
     
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  5. Jun 17, 2010 #4

    apeiron

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    Isn't this now a fairly standard mental picture of things?

    It seems natural to take a top-down constraints-based approach as modelled in condensed matter physics. Spacetime is a "stuff" that can develop local knots or trapped resonances in soliton like fashion.

    But then it is hard to turn this intuitive vision into a mathematical model as maths needs to construct from the bottom-up. To compute, you need a discrete description. So in the move from intuition to the practical modelling, you have to use lattices, loops, strings, networks, infomation bits - something with essential "grain".

    So the situation would seem to be intuitively that grain, in the guise of particles and other local features, arises as a limit to a continuity of interactions. But practically, we have to come at the modelling from the other direction - assuming the grain to be what fundamentally exists, and then building up the continuous background that in fact gave rise to it.
     
  6. Jun 17, 2010 #5

    tom.stoer

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    @ericbrown86: These are nice LQG references, not for this ribbon-like braiding and twisting stuff.
     
    Last edited by a moderator: May 4, 2017
  7. Jun 17, 2010 #6

    tom.stoer

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    With the statement that this is a rather speculative idea - but fascinating as it may point towards particles entirely based on space-time I meant the ribbon idea, not spacetime discreteness. The latter one is somehow common to many approaches regarding QG, even if they differ in the details (effective two-dim. spacetime in the UV has been derived in CDT, asymptotic safety and Horava gravity; ST, AdS/CFT and LQG agree mostly on BH entropy; so there are some common features).

    But the main question is if this spacetime discreteness provides a structure from which matter degrees of freedom may emerge. This is certainly not standard. In standard spin networks the knotting of the graph doesn't matter, only its coloring is relevant. So standard LQG w/o ribbons is not sufficient.
     
  8. Jun 17, 2010 #7

    apeiron

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    Sorry, I missed that you were talking about braids as a particular approach.

    But you would agree with the general situation? Many would see locally fundamental features like particles as emergent in the fashion of solitons in condensed matter physics. Yet still the best approach to modelling such a reality remains one that starts with these local features simply existing as discrete entities, and then building upwards from them to a continuous fabric of reality.

    Braids for example would express certain degrees of freedom as a physical shape. It would take the shapes as having brute existence (not asking how they might in fact have emerged via global constraints) and then model how reality could be woven from the interactions the basic shapes make possible.
     
  9. Jun 17, 2010 #8
    What are elementary particles in NCG? Has Connes or others provided a planck scale picture of what SM in NCG?
     
  10. Jun 17, 2010 #9

    tom.stoer

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    As far as I understand (and I do not understand much about NCG) particles (or fields)emergy as some fluctuations in a fundamental dirac operator. I don't think that this is a "physical" picture.
     
  11. Jun 17, 2010 #10
    One interpretation of a particle is that it is a localized interaction of a field. Is this "field as fundamental" interpretation of particles compatible with things like strings and braiding or excitation of spacetime?
     
  12. Jun 18, 2010 #11

    tom.stoer

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    Are you referring to standard quantum field theory here?

    In string theory particles are something different (vibrating strings); in the low-energy effective models derived from strings a picture like standard quantum field theory emerges. So yes, there's compatibility.

    It is much too early to judge if and how standard quantum field theory and the above mentioned braids are related.
     
  13. Jun 18, 2010 #12
    standard quantum field theory - but I've wondered as much with preons, braiding, etc. And the interpretation of particles in NCG. What is spacetime fundamentally in NCG?
     
  14. Jun 18, 2010 #13

    tom.stoer

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    Fundamentally there is no spacetime in NCG.

    Usually one takes a manifold M and defines operators (Laplacian, Dirac, ...) on top of it. This results in a so-called spectral triple (A, H, D) with a C* algebra A, a Hilbert space H on which A acts and a differential operator D. One can then show that one can re-construct the metric properties on M by using eigenfunctions of D living in H.

    One can turn this round and start with a spectral triple (A, H, D) w/o underlying manifold! Again the reconstruction of the metric works, therefore something like a manifold emerges from (A, H, D).

    If this program is successfull the world is just a special (A, H, D)
     
  15. Jun 20, 2010 #14

    qsa

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    In my theory QSA space itself is defined by energy lines which the ends of one side accumulate to form particle position(Born’s rule) and the other end goes all over the universe. So in this theory space itself is defined by the particles, i.e. no energy lines defining particles no space defined.
    The theory unifies concepts of mass, charge, particle probability position, force/momentum (interactions), space and time all in one unified concept based on the random length of lines and their positions. The theory goes very well with Smolin’s comment that the concept of particles as end of lines should be studied.
    It is one example of what is possible. http://www.qsa.netne.net
    More info about QSA on http://fqxi.org/community/forum/topic/639
     
  16. Jun 21, 2010 #15

    qsa

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    Another example ... from marcus posts


    http://arxiv.org/abs/1006.2230

    On the geometrization of matter by exotic smoothness

    Torsten Asselmeyer-Maluga, Helge Rose
    17 pages
    (Submitted on 11 Jun 2010)
    "Clifford's hypothesis is investigated: A particle is made up of nothing but a distinct type of a space manifold, differing from the surrounding manifold of empty space. It is shown that this distinct space manifold representing matter differs from the surrounding vacuum by the exotic smoothness of its spacetime. The smoothness structure of spacetime can be described by a tree-like subset -- the Casson handle -- consisting of immersed discs and connecting tubes between them. The Weierstrass representation shows that the immersed discs are represented by spinors fulfilling the Dirac equation and leading to a mass-less Dirac term in the Einstein-Hilbert action. The connecting tubes between the discs realize an action term of a gauge field. Both terms are purly geometrical and characterized by the mean curvature of the components of the Casson handle. This gives a good support to Clifford's conjecture that matter is nothing more but an exotic kind of space."
     
  17. Jun 21, 2010 #16

    MTd2

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    Damn! I finally understood what Conne's Spectral Triple is about! I never understood even the basic idea! Thank you!

    Now, it makes sense to try to implement it with LQG, since both start without a manifold.
     
  18. Jun 21, 2010 #17


    Since the SM exists, and background independent GR exist, shouldn't the entire research be devoted to LQG+NCG?

    i.e loop non commutative geometry. LNG Noncommutative quantum geometery NQG noncommutative loop geometery NLG
     
    Last edited: Jun 21, 2010
  19. Jun 21, 2010 #18

    tom.stoer

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    In order to harmonize bot approaches they have to be cropped.

    LQG starts with a manifold and eventually arrives at spin networks. Spin networks are not present in NCG. NCG starts with a spectral triple which in itself is already an effective theory; can it be based on spin networks? LQG is compatible with aditionaly matter fields living on the vertices and links (additional coloring). NCG comes with its own idea how matter can emerge. LQG may provide braided SFs with "topologically emergent matter", but NGC wants to do that, too. NCG does not explain the nc space it si based (afaik), so it translates the unknown symmetry group of the SM to an unknwon nc space w/o explanation why this space.

    It is definitly a very interesting approach, but unfortunately both research programs are nearly complete; so technically they have not much in common, but regarding their features they have a huge overlap.
     
  20. Jun 21, 2010 #19

    MTd2

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    Oh, right. I just checked the sources. A spin network starts without a manifold since it lacks the concept of neighborhood, all that exists are vertexes that connects. The challenge is to find a set of rules that makes the wave function of spin foams, that is, spin networks evolving in space time, be the same of LQG, and so, be sure that gravity can be defined not only background independently, but also out of (almost) nothing.

    NCG seems to follow the same path, a spectral triple, which is almost nothing, into something real.

    So, I guess the cool thing is to start with a spectral triple spin network .
     
    Last edited: Jun 21, 2010
  21. Jun 21, 2010 #20

    tom.stoer

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    Afaik the spectral triple is based on an algebra of functions on top of "something"; this "something" need not be a manifold (but one can reconstruct certain aspects of a manifold like a metric using the spectral triple).

    So one needs at least "something", which could be a spin network. But I think the structure of a spin network is already "too rich" to be interesting. The NCG approach is nicest if the "something" has almost no structure.
     
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