String field from Loop/SpinfoamQG:how to build SM matter on discrete quantum geometry

  1. marcus

    marcus 24,508
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    String field from Loop SpinfoamQG:how to build SM matter on discrete quantum geometry

    note: EPRL is the current standard spinfoam formulation of Loop Quantum Gravity.

    http://arxiv.org/abs/1201.0525
    String Field Theory from Quantum Gravity
    Louis Crane
    (Submitted on 2 Jan 2012)
    Recent work on neutrino oscillations suggests that the three generations of fermions in the standard model are related by representations of the finite group A(4), the group of symmetries of the tetrahedron. Motivated by this, we explore models which extend the EPRL model for quantum gravity by coupling it to a bosonic quantum field of representations of A(4). This coupling is possible because the representation category of A(4) is a module category over the representation categories used to construct the EPRL model. The vertex operators which interchange vacua in the resulting quantum field theory reproduce the bosons and fermions of the standard model, up to issues of symmetry breaking which we do not resolve. We are led to the hypothesis that physical particles in nature represent vacuum changing operators on a sea of invisible excitations which are only observable in the A(4) representation labels which govern the horizontal symmetry revealed in neutrino oscillations. The quantum field theory of the A(4) representations is just the dual model on the extended lattice of the Lie group E6, as explained by the quantum Mckay correspondence of Frenkel Jing and Wang. The coupled model can be thought of as string field theory, but propagating on a discretized quantum spacetime rather than a classical manifold.
    15 pages

    Excerpt from introduction:"In the last few years, a new development [1] [2] [3] [4] has largely resolved the problems of the old BC model [5] for quantum gravity. It is now a natural task to study extensions of the EPRL model which would include realistic matter fields. It would be extremely desirable to find an algebraic extension of the EPRL model which was essentially unique or at least had a small number of possibilities and which gave us the standard model..."

    Excerpt from conclusions: "It was quite a surprise to have the string appear in this theory, which started from a completely different program. Rather than ending up with an almost infinite landscape as in Kaluza-Klein theories, we get an essentially unique theory, which relates fairly directly to the standard model.
    Embedding the string field in a discrete model for spacetime removes the difficulties that beset string field theories in a continuum.
    There is no longer the integration over worldsheet metrics which leads to bad behavior on moduli space in the Polyakov string; rather the dual models couple to the quantum geometry of the EPRL model itself."
     
    Last edited: Jan 4, 2012
  2. jcsd
  3. marcus

    marcus 24,508
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    Re: String field from Loop/SpinfoamQG:how to build SM matter on discrete quantum geom

    Reference [13] Frenkel Jing Wang is to this paper:
    http://arxiv.org/abs/math/9907166
     
  4. marcus

    marcus 24,508
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    Re: String field from Loop/SpinfoamQG:how to build SM matter on discrete quantum geom

    "It is now a natural task to study extensions of the EPRL model which would include realistic matter fields. It would be extremely desirable to find an algebraic extension of the EPRL model which was essentially unique...and which gave us the standard model..."

    So that's the program. I don't see any reason such a program couldn't succeed. He makes a start on it here and also describes what further work needs to be done.

    EPRL is a formulation of quantum geometry that makes a lot of sense and seems to yield GR in the appropriate limit. Some people like to emphasize what still needs to be worked on such as the relation of EPRL dynamics to Hamiltonian dynamics. Or they worry about the convergence of this or that limit process.

    That's fine and we shouldn't minimize the importance of attending to every possible imperfection. But Crane is saying that the quantum geometry is mature enough so that it is a "natural task" for some people to be thinking about putting standard matter into it.

    I think that's quite reasonable. Some people can work on getting EPRL absolutely right and absolutely equivalent to some Hamiltonian formulation (presumably yet to be invented.) And meanwhile other people don't have to wait but can go ahead and think about putting standard matter into the EPRL geometry as it is, even if not yet in final form.

    On page 11, at the beginning of section 12-Physical Consequences of the Model, Crane says:

    "The proposal that physical particles correspond to different vacua of an underlying quantum field of tetrons is a major departure. Could it have observable consequences?"

    So that's something to try to understand. A field with a discrete group of symmetries, living on a discrete geometry. In EPRL the dynamical evolution of geometry is represented by a labeled two-complex---analogous to a graph but in one higher dimension. The two complex ("foam") is BOUNDED BY A LABELED GRAPH that represents the geometric measurements we have made or will make---lengths, angles, areas, etc. and which implement the idea of a STATE of geometry. So a graph bounds a foam and the foam tells the amplitude of a certain transition, say from initial to final state.

    Crane understands the structure I just described in terms of category theory and group representation theory---the catch is the labels, or how you understand the Hilbertspace of states associated with a given graph. He understands it in some way that allows him to talk about a "tetron" field defined on the spinfoam, i.e. on the spacetime: the evolving geometry which the spinfoam describes in a discrete fashion.

    This sounds reasonable to me---like some math we could have expected to eventually be done---even though I don't yet understand it in detail. So the paper seems quite interesting.

    Crane does not always show the arxiv number of his references, so to save time I will post them now and then.
    Reference [10] refers to http://arxiv.org/abs/hep-ph/0211393
    Reference [23] (Baez Huerta, note change of title) http://arxiv.org/abs/0904.1556

    Some Crane-related FQXi pages: http://fqxi.org/grants/large/awardees/view/__details/2006/crane
    http://fqxi.org/community/articles/display/127
     
    Last edited: Jan 4, 2012
  5. Re: String field from Loop/SpinfoamQG:how to build SM matter on discrete quantum geom

    I think the link between A4 and E6 is potentially of interest beyond the EPRL model.

    A4 is the symmetry group of a tetrahedron when you don't allow reflections. Tetrahedra show up in EPRL as boundaries between 4-simplexes. That is, the space-time is represented as a complex of 4-dimensional hypertetrahedra which are joined to each other by 3-dimensional tetrahedral "hyperfaces", and Crane talks about the line from the center of one 4-simplex to the center of its neighbor, which will pass through one of these tetrahedral hyperfaces. Being a tetrahedron, the hyperface has an A4 symmetry, and the A4/E6 quantum duality applies to families of representations which are associated with the line passing through it.

    But tetrahedra show up in lots of places in physics.

    Crane also mentions that A4 symmetry might explain tribimaximal mixing for neutrinos. But the recent results from T2K experiment (nonzero theta13 mixing angle) indicate a strong deviation from tribimaximal mixing. There was a recent paper on the nuMSM which tried to explain the mixing angles (and lepton mass hierarchies) using a Q6 symmetry. That's the "dicyclic group" or "binary dihedral group" of order 12.
     
  6. marcus

    marcus 24,508
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    Re: String field from Loop/SpinfoamQG:how to build SM matter on discrete quantum geom

    I believe you are right. Both you and Crane give reasons why it is that go beyond the nice fit with this one QG model.

    ==quote page 3==
    A large body of recent work on grand unified models [17] inspired by the neutrino oscillations has focussed on symmetry groups which are a product of a finite discrete group with a Lie group; and A(4) × E6 seems to be the most successful of the possibilities...

    Discrete symmetry as a fundamental symmetry of nature is quite a puzzle. In a continuum theory it would be hard to avoid domain walls. The conceptual foundation of the BC and EPRL models, in which spacetime is modelled by a simplicial complex and NOT by a manifold, is necessary to include discrete symmetry as in this paper...
    ==endquote==

    I think that part of what we see in Crane's work and that of a number of quantum relativists more generally is a mathematical instinct to explore NON-MANIFOLD representations of spacetime geometry.

    The urge to go beyond the smooth manifold picture of continuum---perhaps one could date it 1850, with Riemann---has already been simmering in the math community for many decades, surely going back to before 1970s. The sticking point has perhaps been with the conservatism of physicists.

    But simply to have a quantum theory of spacetime geometry seems almost to necessitate a discrete representation. And now Crane is pointing to an additional justification, the broader interest of discrete symmetry in matter.
     
  7. ohwilleke

    ohwilleke 702
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    Re: String field from Loop/SpinfoamQG:how to build SM matter on discrete quantum geom

    "The sticking point has perhaps been with the conservatism of physicists."

    I think that the sticking point until scientists started to consider them was that we didn't have the computational power to implement and apply discrete models, which couldn't be analyzed with the tools of mathematical analysis available in continuum models. Euler, Lagrange and Laplace were doing very fancy real and complex analysis in physics applications before George Washington became President of the United States in 1789. The Univac, the first mainframe computer, came on line in 1951, four years before Einstein died. Stanford physicists and engineers through the late 1960s were limited to slide rules, punch cards, French curves and graph paper. There still wasn't enough computing power available even in supercomputers to do things like interesting 4D Lattice QCD during Feynmann's lifetime. LQG is only possible in any even conceivably applicable version because of advances associated with massive computing power.
     
    Last edited: Jan 5, 2012
  8. marcus

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    Re: String field from Loop/SpinfoamQG:how to build SM matter on discrete quantum geom

    I see your point. Continuum math affords beautiful and elegant means of calculation. This could explain the enduring preference for that kind of model.
    Now it seems to me that (rightly or wrongly) Crane is arguing that continuum geometry is unphysical on theoretical grounds. Here he has just been talking in the preceding section about the state sum models of EPRL and BC quantum gravity.

    ==quote page 4 http://arxiv.org/abs/1201.0525 ==
    2. THE MATHEMATICAL STRUCTURE OF SPACETIME
    The state sum models for quantum gravity should not be thought of as approximations corresponding to triangulations of an underlying smooth manifold [11]. In quantum Physics coupled to relativity, such a point of view is unphysical, because no information can be communicated from a sub-Planck scale region. The attempt to recover a continuum limit is the one unsuccessful part of the interpretation of the model, and is naive.
    Although the replacement of manifolds by simplicial sets is a mathematically well understood point of view for geometry and topology [19] [20], it seems very foreign to physicists.
    Let us take some simple region of spacetime and compare two descriptions of it, one as a subset of a smooth manifold, the other as a simplicial complex, thought of as a discrete combinatorial object, not a point set....
    ==endquote==
     
    Last edited: Jan 5, 2012
  9. Re: String field from Loop/SpinfoamQG:how to build SM matter on discrete quantum geom

    What does a discrete spacetime model do to the differential equations of quantum mechanics that differentiate with respect to continuous time and space?
     
  10. atyy

    atyy 10,325
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    Re: String field from Loop/SpinfoamQG:how to build SM matter on discrete quantum geom

    I think the reason for wanting a continuum limit is that without it the model may not be unique. This is the reason for Rovelli's summing = refining.

    Does Crane do EPRL with or without the continuum limit that Rovelli and Smerlak discuss?
     
  11. tom.stoer

    tom.stoer 5,489
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    Re: String field from Loop/SpinfoamQG:how to build SM matter on discrete quantum geom

    First it should be clear that "differentiate with respect to continuous time and space" refers to coordinates only and is therefore not as physical as it may seem.

    Second we know that topologically and even w.r.t. to differential topology there is (nearly) identity between topological manifolds, differential manifolds and piecewise linear manifolds. But piecewise linear manifolds and "triangulations" are again nearly identical (I don't want to be precise regarding the meaning of "nearly identical"). It may be the case that we don't need continuous differential structures at all.

    Third: a discrete spacetime model will turn differential equations into difference equations (LQG and LQC do exactly this) and it may very well be that these are "physically equivalent".

    Even the following may be true, namely that we are allowed to use continuous models which give us identical results. Let's make a very simple example: Assume we are small quanta living in a space of weakly coupled harmonic oscillators. Usually we observe discrete energy levels |n> and we are jumping from n to n+1 etc. we can calculate jumping rates, etc. using creation and annihilitation operators. Now we organize a conference and invite two quanta called deBroglie and Schrödinger and they present us their theory of waves ψ(x), wave equations etc. Of course we are rather shocked b/c nobody has ever observed continuous space x, all we know about are quantum jumps and probabilities. But then Schrödinger proves the equivalence of his equations with our description of the world in terms of creation and annihilation operators. We would still feel that his equations are strange, but we accept the math. We will even understand that in a so-called macrsoscopic world we are not aware of but to which Schrödinger is always referring to his strange description may be helpful.

    That means that both continuous strings and discrete spinfoams can be correct descripton of nature (or let's be slightly more careful: may point towards reasonable pathways towards the construction of candidates for unified model of 'quantum matter and quantum geometry' ;-)
     
    Last edited: Jan 6, 2012
  12. marcus

    marcus 24,508
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    Re: String field from Loop/SpinfoamQG:how to build SM matter on discrete quantum geom

    I like this story a lot!
     
    Last edited: Jan 6, 2012
  13. MTd2

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    Re: String field from Loop/SpinfoamQG:how to build SM matter on discrete quantum geom

    If he allowed the volume of the Tetrahedra to have reflections, since we are talking about volume as side of a 4-simplex, wouldn't we have half the symmetries that he wants to get the SM? This also would be a way of implement CPT symmetry.
     
  14. marcus

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  15. Re: String field from Loop/SpinfoamQG:how to build SM matter on discrete quantum geom

    I have to wonder what a discrete spacetime would do to wave propagation. If there is an effect, then I suppose it would become obvious at cosmological scales.
     
  16. marcus

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    Re: String field from Loop/SpinfoamQG:how to build SM matter on discrete quantum geom

    Tom already addressed the general issue. I assume the answer would depend on what kind of discreteness you are imagining and what mathematical models you are using.

    Good things to keep in mind would, I suppose, be what Bohr said (quantum physics focuses not on what nature IS but on what we can say about it) and Rovelli's clarification (it's not what nature IS but how it responds to measurement).

    A quantum model of geometry should embody the limitations of geometrical measurement. If we can only make a finite number of measurements and the readings we take of length angle area etc can only take on a discrete spectra of values then we expect the model to keep track of these limitations for us.

    If you like you can picture a continuous continuum equipped a discrete geometry.

    But that is not really on the main topic which is Crane's paper.
     
  17. marcus

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    Re: String field from Loop/SpinfoamQG:how to build SM matter on discrete quantum geom

    I don't know the answer to that, Atyy :biggrin:

    He refers to some kind of continuum limit concern as naive. I don't know if he is specifically talking about their work or not.
     
  18. marcus

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    Re: String field from Loop/SpinfoamQG:how to build SM matter on discrete quantum geom

    Basically what we're focusing on in this thread is the program sketched in the first few paragraphs of Louis Crane's 3 January 2012 paper http://arxiv.org/abs/1201.0525

    These first few paragraphs lay out the basic idea, which is then elaborated in more detail in the next dozen or so pages.

    ==quote Crane 1201.0525==
    In the last few years, a new development [1] [2] [3] [4] has largely resolved the problems of the old BC model [5] for quantum gravity. It is now a natural task to study extensions of the EPRL model which would include realistic matter fields. It would be extremely desirable to find an algebraic extension of the EPRL model which was essentially unique or at least had a small number of possibilities and which gave us the standard model, rather than some random collection of particles and fields.

    During the same time frame, a series of delicate experiments [6] has given us a detailed picture of the oscillations of the neutrinos of the three generations (electron muon and tau neutrinos) into one another. The oscillations turned out to be much larger than the ones for quarks, and seem to be well approximated by a form called the tribimaximal matrix [7].

    The neutrino masses indicate that the neutrino has a right handed component, called the sterile neutrino because it does not interact under any force. This changes the form of the three generations of fermions; each now has 16 Weyl fermions.

    The new neutrino physics is crucial to the search for a unified theory; it is as if we have been working a jigsaw puzzle with a missing piece. If the proposal in this paper is correct, the missing piece was the most important one.

    In explaining the tribimaximal matrix, a number of researchers have proposed that the particles appear in representations of the discrete group A(4) [8], the alternating group on 4 letters, and that the interactions with the Higgs particle be invariant under this symmetry. This reproduces the tribimaximal form very nicely.

    The fact that A(4) has three one dimensional representations and one irreducible three dimensional representation allows the quark and lepton mixing matrices to be very different. Quarks and neutrinos get different A(4) representational labels in this approach.

    The A(4) symmetry cannot be explained as a broken or residual symmetry from a Lie group [9], it can only be understood as a fundamental symmetry of nature. This lends itself much more naturally to coupling to a discrete model such as EPRL rather than to a continuum Yang Mills theory.

    The EPRL model, like the BC model which preceeded it, is constructed from the representation categories of Lie groups, which are used to assign quantum geometrical variables to the faces of a four dimensional simplicial complex. The tensor structure of the representation categories is used in a natural way to construct the model. The EPRL model, in particular, uses the representation categories of SO(3) and SO(3,1), together with a functor connecting them [11].

    The group A(4) which was interesting to the neutrino physicists is also the group of symmetries of the tetrahedron...

    ==endquote==

    Here are some of Crane's previous papers that might help us understand this one:
    http://arxiv.org/abs/1006.1248
    Holography in the EPRL Model (this earlier one seems to me the more helpful of the two)
    http://arxiv.org/abs/1105.6036
    Discrete symmetry in the EPRL model and neutrino physics
     
    Last edited: Jan 9, 2012
  19. atyy

    atyy 10,325
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    Re: String field from Loop/SpinfoamQG:how to build SM matter on discrete quantum geom

    Thanks, it's very helpful to have the earlier papers! He answers my question in the first of them. He rejects Rovelli and Smerlak's proposal: "Summing over complexes ala GFT returns us to a formal divergent expression, which is tragic after finiteness has come to us in such a magical way."
     
  20. marcus

    marcus 24,508
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    Re: String field from Loop/SpinfoamQG:how to build SM matter on discrete quantum geom

    From your quote one sees that back in June 2010 Crane was not especially clairvoyant as to the future course of other people's research. But that's OK, nobody should be expected to be so. Of primary interest here are his own ideas, which are imaginative and in some cases I think original.

    http://arxiv.org/abs/1112.2511
    q-Deformation of Lorentzian spin foam models
    Winston J. Fairbairn, Catherine Meusburger
    (Submitted on 12 Dec 2011)
    We construct and analyse a quantum deformation of the Lorentzian EPRL model. The model is based on the representation theory of the quantum Lorentz group with real deformation parameter. We give a definition of the quantum EPRL intertwiner, study its convergence and braiding properties and construct an amplitude for the four-simplexes. We find that the resulting model is finite.
    12 pages, 2 figures, Proceedings of the 3rd Quantum Gravity and Quantum Geometry School (Zakopane, 2011), to appear in PoS
     
    Last edited: Jan 9, 2012
  21. marcus

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    Re: String field from Loop/SpinfoamQG:how to build SM matter on discrete quantum geom

    The effort to show a refinement limit for EPRL may actually NOT be misguided and may in fact be successful. I would not take Crane's word as gospel about the world of other people's research. But his own ideas about the structure of spacetime (e.g. page 4 section 2 of the current paper) are interesting and I would like to understand them better. Partially recapitulating, I'll quote the whole of section 2.

    ==quote 1201.0525 page 4==
    2. THE MATHEMATICAL STRUCTURE OF SPACETIME
    The state sum models for quantum gravity should not be thought of as approximations corresponding to triangulations of an underlying smooth manifold [11]. In quantum Physics coupled to relativity, such a point of view is unphysical, because no information can be communicated from a sub-Planck scale region. The attempt to recover a continuum limit is the one unsuccessful part of the interpretation of the model, and is naive.

    Although the replacement of manifolds by simplicial sets is a mathematically well understood point of view for geometry and topology [19] [20], it seems very foreign to physicists. Let us take some simple region of spacetime and compare two descriptions of it, one as a subset of a smooth manifold, the other as a simplicial complex, thought of as a discrete combinatorial object, not a point set. Both of these have associated to them a differential graded complex, the differential forms for the manifold, and the cocycles for the complex.

    If we compute the cohomology of the region using both complexes, we find there is an important difference: the differential forms can only give vector spaces, while the cocycles can be calculated using integer coefficients, and give finite group components to the cohomology as well. This is known as the torsion. Thus finite group invariants appear more naturally in the complex description.

    Similarly, it is very easy to construct topological quantum field theories from the representations of finite groups on a simplicial complex, while a continuum Lagrangian is hard to imagine.

    We interpret the discovery of discrete symmetry as a fundamental symmetry in neutrino Physics as an indication that the simplicial complex description is favored by nature.
    ==endquote==

    It seems to me (at my current stage of appreciating his work) that Crane may be RIGHT about simplicial complex being a good math description of spacetime geometry to be using. I don't think it follows that one needs to scoff at physicists for wanting a unique limit of the transition amplitude under refinement. Of course the physicists want this and they are right to devote effort to attain it.

    If I am not mistaken, in pragmatic terms, operationally, this is all "continuum limit" means in the simplicial state sum context. "Continuum limit" does not necessarily mean that Nature IS a smooth manifold or that the manifold structure is always the best for putting matter and geometry together! We are always concerned with how she responds to measurement, not with what she is. But in any case, whatever she is, there has to be a unique limit of the transition amplitude under refinement. So it is not naive to look for it and try to prove it exists.
     
    Last edited: Jan 9, 2012
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