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List of new quantum spacetime/matter initiatives

  1. Nov 30, 2005 #1

    marcus

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    we are fortunate to be introduced to several approaches for combining matter with a quantum spacetime---enough so there is actually a profusion of "quantum spacetime/matter" alternatives

    so we had better list the ones that have been introduced to us recently

    1. Garrett Lisi

    2. Torsten & Helge

    3. Sundance Bilson-Thompson preon model put on spin networks

    As time permits, I will add brief descriptions and gather some links to threads discussing these. If anyone else want to add things, or take over the job, please do! You are very welcome to take charge. I only think we need to list these new approaches so we do not accidentally forget any.
     
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  3. Nov 30, 2005 #2

    marcus

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    I will try to make a very brief list with links to other threads

    1. Garrett Lisi type stuff
    https://www.physicsforums.com/showthread.php?t=100984
    keywords:
    "Garrett Lisi bid to join Std. Model w/ gravity"
    "Clifford bundle formulation of BF gravity generalized to the standard model"
    CarlB has been discussing his ideas, along similar lines, with Garrett

    2. Torsten and Helge stuff
    https://www.physicsforums.com/showthread.php?t=101540
    keywords:
    "Differential Structures - the Geometrization of Quantum Mechanics"


    3. Preon Spin Networks....Preon Spin Foams----Bilson-Thompson/Smolin
    https://www.physicsforums.com/showthread.php?t=100183
    work just getting started on this
    see links in that thread
    also the slides and recording of Smolin's talk at the Loops '05 conference
    http://loops05.aei.mpg.de/index_files/abstract_smolin.html

    4. pure Quantum Spacetime Dynamics
    by pure I mean not linked to the Standard Model of matter, instead simply having to do with quantum geometry of spacetime itself
    4A Renate Loll
    4B Martin Reuter
    both these lines of research find dimensional collapse at microscopic scale.

    5. there was something new here, have to find it later when I get back:
    https://www.physicsforums.com/showthread.php?p=842964#post842964

    I have the feeling there are other things that should be on the list.
    At least these things have been recently discussed at PF.
    ferment is a good word for what is going on. there is a lot of ferment these days
     
    Last edited: Dec 1, 2005
  4. Dec 10, 2005 #3

    marcus

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    Yes CHARLES WANG should be on this list. That link is to some Charles Wang papers. I temporarily forgot I was trying to gather a list of REBEL APPROACHES.

    and here is another I just noticed:

    http://arxiv.org/abs/gr-qc/0508104
    Towards a fully consistent relativistic quantum mechanics and a change of perspective on quantum gravity
    Johan Noldus
    17 pages, submitted to CQG

    "This paper can be seen as an exercise in how to adapt quantum mechanics from a strict relativistic perspective while being respectful and critical towards the experimental achievements of the contemporary theory. The result is a fully observer independent relativistic quantum mechanics for N particle systems without tachyonic solutions. A remaining worry for the moment is Bell's theorem."

    I just had what felt like a moment of enlightenment, so watch out here will probably be coming a little sermon.

    These people are, several of them, OUTLAWS BUT NOT CRACKPOTS. We really need QG outriders at this time with the courage to go off on their own. How else do you explore what the other people haven't looked at?

    It is actually harmful to the scientific enterprise (when many people agree that radically new ideas are needed in QG) to have a system where these people CANNOT GET JOBS.
    The job market (in the US) is tied up by string and EVEN IF THEY ARE QUALIFIED COMPETENT academic people they can barely even get a postdoc---or hardly that in the US.

    It may not be this parochial in europe, but here in US I get the impression that if someone will not go with the herd, and if he will not do string, this is seen as a sign that he must be mentally or emotionally unstable. But in fact that might not be true! He might simply have a backbone of intellectual independence but be otherwise quite sensible.

    I remember a quote from Richard Feynmann where he actually urged young researchers to take this hard life path of developing their own ideas, essentially for the good of physics----because it can often lead the person to an obscure second-rate career. does anyone know where to find this quote from Feynmann?
     
  5. Dec 10, 2005 #4

    marcus

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    Now I see that this PF forum can actually sometimes serve as an OUTLAW CAFE in some of its threads. We can help compensate for deficiencies in the system.

    One way to do this is simply to LIST the divergent QG approaches and to try our best to shoot them down.

    I guess I should make the point that these approaches are not enemies or rivals to each other! The deviant QG are all natural ALLIES because they share the main objective of getting attention focused out in the bushes off the beaten track---they all need to coherently send the message that alternative ideas are possible and even may be more interesting!

    BTW I class Renate Loll as an outlaw. she just has had a little bit of success and now has gathered some funds and a group. this is what all outlaws should do.

    If these novel approaches are natural allies, not rivals, then why should we concentrate on shooting them down? BECAUSE THAT IS WHAT YOU ALWAYS TRY TO DO WITH PHYSICS IDEAS, and it is GOOD FOR THEM.

    I am pleading with someone to shoot down Charles Wang. I would love to have an excuse to drop him from the list. Charles Wang approach looks really interesting to me. What is wrong with it?

    Also NOLDUS, whom I just noticed. What is wrong with Noldus ideas. He has a way to reform quantum mechanics from what he calls a "diehard" gen rel perspective. Well at least on the surface that sounds great. The possibility really should be seriously considered that whatever is keeping QM from merging with GR is basically QM's fault. People are reluctant to look at it this way, but Noldus attempts to bend QM into shape
    ========================
    OK so let's maintain the list:

    1. Garrett Lisi
    https://www.physicsforums.com/showthread.php?t=100984
    keywords:
    "Garrett Lisi bid to join Std. Model w/ gravity"
    "Clifford bundle formulation of BF gravity generalized to the standard model"


    2. Torsten and Helge
    https://www.physicsforums.com/showthread.php?t=101540
    keywords:
    "Differential Structures - the Geometrization of Quantum Mechanics"

    3. Preon Spin Networks....Preon Spin Foams----Bilson-Thompson/Smolin
    https://www.physicsforums.com/showthread.php?t=100183

    http://loops05.aei.mpg.de/index_files/abstract_smolin.html

    4. Loll triangulations (with Monte Carlo simulation of quantum spacetime)

    5. Reuter (assymptotically safe QG, nonper. renormalization, oddly similar results to Loll)

    5. Charles Wang
    https://www.physicsforums.com/showthread.php?p=842964#post842964

    6. Johan Noldus (just noticed)

    http://arxiv.org/abs/gr-qc/0508104
    Towards a fully consistent relativistic quantum mechanics and a change of perspective on quantum gravity
    17 pages, submitted to CQG

    "This paper can be seen as an exercise in how to adapt quantum mechanics from a strict relativistic perspective while being respectful and critical towards the experimental achievements of the contemporary theory. The result is a fully observer independent relativistic quantum mechanics for N particle systems without tachyonic solutions. A remaining worry for the moment is Bell's theorem."

    ==================

    we have already had considerable discussion of some of these here at PF (Loll, Reuter, Garrett Lisi, Torsten and Helge) and some also of Sundance Bilson-Thompson's new preon model. what about these others?
    can anyone express why we should dismiss the ideas of Wang and Noldus? Or, conversely, does anyone LIKE what they have to say and find it interesting?
     
  6. Dec 10, 2005 #5
     
  7. Dec 10, 2005 #6

    marcus

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    impressive and concise comment, thanks!

    I am very much looking forward to seeing whatever (in addition) you wish to share with us of your reaction to the Wang paper.
     
  8. Dec 11, 2005 #7

    garrett

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    Hey Marcus,

    I finally got around to spending a couple of hours looking at CDT. I skimmed through the papers linked off your sig. I want to make sure I have the basics right:

    Regge calculus is an old and well established method for discretizing GR using simplexes -- basically using finite elements to carve up spacetime in a dynamical triangulation. For a 4D pseudo-Riemannian manifold these simplexes are the analogs of 3D tetrahedra. The GR action, usually an integral over spacetime curvature, becomes a sum over these configurations of simplexes and their properties: angles and leg-lengths as well as how they're glued together. This model approaches continuous GR in the limit that the simplexes get small. Now, if you were to try to do quantum gravity numerically by brute force, the thing to do is sum over all possible configurations of simplexes weighted by the appropriate quantum amplitude obtained from the action. Since there are an infinite number of configurations, you sum as many as you can with appropriate weights. Apparently though, this doesn't work -- it blows up and gives you nonsense. (is that right?) Presumably because (A) you are including sums over configurations that represent different topologies, and/or (B) you are including simplexes that can be arbitrarily stretched out.

    The CDT idea, as far as I can tell, is to restrict the dynamical triangulation to fix (A) and (B). To fix (B) the simplexes are restricted to have space-like legs with all the same lengths and time-like legs with all the same lengths. To fix (A) the topology implied by the configuration of simplexes is restricted to some arbitrary fixed topology, like S1xS3. Your dynamically triangulated spacetime can't branch, and this is what puts the "causal" in CDT. In practice, this means a possible configuration representing a spacetime is built from another one by a collection of moves that add or remove simplexes in a set of special ways.

    (Does that sound right so far?)

    The dimension of a spacetime represented by a dynamical triangulation is determined by hopping a test particle from vertex to nearby vertex until it comes back to the spatial place of origin. If it has a low probability of returning -- that's a large dimension. And you can adjust the scale by allowing more or fewer jumps for the test particle.

    Given this setup, it makes sense that the dimension, measured this way, would give 4D for many jumps (large scale) since you are using a collection of 4D simplexes; and it makes sense that it gives 2D for small scales since you're then only hopping between a few vertices.

    So... it looks like CDT can be summarized as a restricted quantum Regge calculus? They took the evil behaving QRC and constrained it until it played nice. But I'm not sure I like it. Are there justifications for the restrictions? And I kind of like the idea of there being topology change down at the Planck scale.

    Of course, I'm sure there's stuff I'm missing.
     
  9. Dec 12, 2005 #8

    marcus

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    It certainly does! Thanks for looking at CDT, Garrett. the next Loll paper that comes out, we can have a thread. we will review it and see whats new.

    I like the visual way you quickly summed up the essentials.

    maybe we shouldn't even wait for a new CDT paper, but start one now.
    BTW your intuition about it makes sense that, in this setup... outraces mine at one point. which is fine , it's to be expected, and people's intuition ought to be different anyway
     
    Last edited: Dec 12, 2005
  10. Dec 12, 2005 #9

    garrett

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    Here's another TOE I just stumbled on that might be worth adding to the list: Thomas Larsson, who makes regular net appearances, builds something close to the standard model by using one of the few exceptional Lie Superalgebras, as espoused by V. Kac and recently worked on by very few others. This is a fascinating idea, since it appears to give all the funky standard model pieces from a fundamental bit of mathematics. I'm just now reading and learning about this, but these papers seems a good place to start:
    http://arxiv.org/abs/math-ph/0103013
    http://arxiv.org/abs/math-ph/0202025
    http://arxiv.org/abs/math-ph/0202023
    Does anyone here on PF understand Lie Superalgebras well enough to explain them to physicists -- preferably by describing an illustrative example? They are related to, but not dependent on, supersymmetry.
     
  11. Dec 12, 2005 #10

    garrett

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    Hmm, now I'm replying to myself...
    I have to downgrade my initial excitement about the exceptional Lie superalgebra TOE. It seems the quark hypercharges don't work perfectly. (I applaud the author for pointing this out directly, rather than obfuscating it.) It's still a neat idea -- and I'll keep Lie superalgebras on my list of things to learn about.
     
  12. Jan 4, 2006 #11
    Could you please give the references to discussion of these differences?
     
  13. Jan 4, 2006 #12
    Concerning the self field approach itself you might have noticed that I gave some references in the ``why quantize gravity ?´´ thread, post 28. Now, you should be able to find such discussions in a good book on quantum chemistry (I was informed about the procentual differences by a good quantum chemist during a conversation), probably Barut himself has written on that too but I do not know that for sure - perhaps the webpage of Dowling is a good place to start with for some material on this. If you want to, I will look it up myself, but perhaps Vanesch or ZapperZ can alleviate my task (they are more knowledgable about quantum chemistry than I am) and give a reference to a good book which discusses this.

    Cheers,

    Careful
     
  14. Jan 4, 2006 #13
    Thank you, I have references to Barut's theory.
    Thank you again, actually you've essentially answered my question (I wanted to hear about your source). Somehow I would be very much surprised to find any :-) references to Barut's theory in books on quantum chemistry (I know, though, that Deumens wrote both on quantum chemistry and SFED, but I don't remember any numerical comparisons). Furthermore, it is my understanding that Barut e.a. claim coincidence with QED to at least first order in \alpha, so differences may occur in the second order in \alpha, that would be much less than 5%? On the other hand, one may ask: 5 percent of what?
    As for Dowling, he wrote (Found. Phys., 28, 855 (1990) (http://baton.phys.lsu.edu/~jdowling/Dowling98c.pdf): "there is some debate on whether the quantum version of the Barut self-field approach to QED is equivalent to the traditional second-quantized formalism." So at least in 1998 he was not even sure there were any differences at all.
     
  15. Jan 5, 2006 #14
    ** Somehow I would be very much surprised to find any :-) references to Barut's theory in books on quantum chemistry (I know, though, that Deumens wrote both on quantum chemistry and SFED, but I don't remember any numerical comparisons). **

    Ah, so thank you.

    **
    Furthermore, it is my understanding that Barut e.a. claim coincidence with QED to at least first order in \alpha, so differences may occur in the second order in \alpha, that would be much less than 5%?
    On the other hand, one may ask: 5 percent of what? **

    I mean : five percent of known extensive quantities (such as energies of bound states and so on ). **

    As for Dowling, he wrote (Found. Phys., 28, 855 (1990) (http://baton.phys.lsu.edu/~jdowling/Dowling98c.pdf): "there is some debate on whether the quantum version of the Barut self-field approach to QED is equivalent to the traditional second-quantized formalism." So at least in 1998 he was not even sure there were any differences at all**

    I am not interested in the quantum version, but in the classical one.
     
  16. Jan 5, 2006 #15
    Then this seems to contradict the claim of Barut e.a. I would think that first order in \alpha should give pretty good figures for energies of bound states.
    Sorry for this misunderstanding: I believe that what you call "classical", he calls "quantum" in his context. You mean "classical" because it's not second-quantised, and he calls it "quantum" because matter is first-quantised (as, say, the Dirac equation is used, rather than, say, relativistic Newton equations).
     
  17. Jan 5, 2006 #16
    ** Then this seems to contradict the claim of Barut e.a. I would think that first order in \alpha should give pretty good figures for energies of bound states.**

    Look, I know of course why you think the approximation might be much better than 5 percent, since 1/274 is much smaller than 1/20. However, I did not want to make any further comments on this since I do not know by heart how fast the expansion series is converging. :smile: And if it is not known yet whether you might dispose all along of second quantization or not (using the Barut self field approach), is fantastic news.

    **Sorry for this misunderstanding: I believe that what you call "classical", he calls "quantum" in his context. You mean "classical" because it's not second-quantised, and he calls it "quantum" because matter is first-quantised (as, say, the Dirac equation is used, rather than, say, relativistic Newton equations). **

    Yeh, I was confused about what you meant by this :wink:

    But anyway, I regard this approach as a very useful intermediate step in deepening our understanding of QM (I would be glad to hear some thougts of you about it).


    Cheers,

    Careful
     
    Last edited: Jan 5, 2006
  18. Jan 5, 2006 #17
    It is worse than that (or better than that, depending on the point of view :-) ) If SFED emulates QED up to the first power of \alpha (inclusive), then the discrepancy should be of the order of \alpha to the second power or less. SFED seems to describe the Lamb shift correctly, among other things, and some 60 years ago measurement of the Lamb shift became a landmark experiment, so it is much less than 5% of the energy of the bound state. Anyway, I did not intend any criticism, I am just interested in SFED and trying to understand its status - in comparison with experimental data and QED, and in perception of the physics community.
    I would also welcome a success of a theory of this kind, but so far I haven't had enough time to understand SFED better.
    That was not me, that was Dowling :-)
    I tend to agree, but I do feel I should reserve my opinion until I understand SFED better. I started to look for something like SFED just half a year ago when I tried to understand why Coulomb interaction of electron with itself is not included in, say, Shroedinger equation, although the relevant term is definitely present in the QED Lagrangian and is not small, to put it mildly. So we first delete this term in quantum mechanics for some unfathomable reason, and then reintroduce it in QED. So the final result is correct, but the procedure is less than straightforward :-) Finally I found references to SFED and was enthusiastic about it. However, today I tend to be more cautious, as I don't understand some things about SFED. I have recently better understood criticism of SFED by Bialynicki-Birula, Phys. Rev. A 34, 3500 (1986), and two things bother me most: first, using Feynman propagator seems to mean complex electromagnetic field, and second, I cannot understand to what power in \alpha SFED and QED results coincide. nightlight wrote that at least to the fourth or fifth power (in this case they may be experimentally undistinguishable at the moment - could it be too good to be true?), but so far I have not found clear confirmation in Barut's articles. There is another thing. Both Barut and nightlight seem to lay their hopes on monopole solutions to describe free particles, and I am less than enthusiastic about that. On the other hand, in my article http://www.arxiv.org/abs/quant-ph/0509044 I consider a possibility of doing without monopoles in SFED.
     
  19. Jan 8, 2006 #18
    Hm. This seems like I have to abandon my safe lurker status. With Careful on board, it might be worth the effort.
    A Lie superalgebra is like a Lie algebra except that some commutators have been replaced by anti-commutators. This means that you insert minus signs at the right places (keeping track of the right places is a nuisance) and add the prefix "super" to every word in sight. A supersymmetry is a superalgebra with extra structure: a Poincare subalgebra which is identified physically with spacetime transformations. However, the concepts are blurred, e.g. worldsheet SUSY, but to qualify as a SUSY at least the Hamiltonian should have a fermionic square root, H = {Q,Q}. This condition leads to positive energy (since H = Q^2) and to superpartners (since the eigenstates |psi> and Q|psi> have the same energy).
    I admit that I was guilty of excessive enthusiasm. In recent years I have sobered considerably.

    E(3|8) was never meant as a ToE, since it does not contain gravity. There is a 1-1 correspondence between E(3|8) and SU(3)+SU(2)+U(1) irreps, so the idea was to find some fancy generalization of the standard model, without gravity. However, I lost interest for several reasons. First, I never managed to transform this insight into physics. Second, I suspect that if an E(3|8) theory exists, it probably predicts new physics of a kind which has already been ruled out by experiments - the standard model always wins! Third, a 1-1 correspondance with the SM irreps is not so special - this property is shared by any finite-dimensional Lie algebra whose Dynkin diagram has four nodes in a row, i.e. SU(5), Sp(8), SO(9) and F_4 (in two ways).

    As an excuse, let me point out that the idea really wasn't mine but Victor Kac', who is an eminent mathematician but not a physicist. In contrast, the discovery of projective representations of the diffeomorphism algebra in more than 1D is directly related to quantum gravity. The reason is very simple. To understand quantum systems with rotation symmetry, you should know about SO(3), including its projective spinor representations. To understand systems with diffeomorphism symmetry, like gravity, you should understand the representations of the diffeomorphism group, including the projective ones. This leads immediately to the multi-dimensional generalization of the Virasoro algebra, which was discovered by Rao, Moody and myself in the early 1990s.

    The standard objection is that diffeomorphisms in GR generate a gauge symmetry, i.e. a mere redundancy of the description, and hence we only need to know about the trivial rep. However, it is simply not true that all gauge symmetries remain redundancies after quantization (counterexample: free subcritical string). Moreover, even if we insist that every gauge anomaly should vanish (which I think is wrong), we must still know about them. Otherwise, we would know e.g. that there is something special about 26D in the bosonic string.
     
  20. Jan 8, 2006 #19
    **Hm. This seems like I have to abandon my safe lurker status. With Careful on board, it might be worth the effort. **

    I do not quite know how to read that, but fine.

    **The standard objection is that diffeomorphisms in GR generate a gauge symmetry, i.e. a mere redundancy of the description, and hence we only need to know about the trivial rep. However, it is simply not true that all gauge symmetries remain redundancies after quantization (counterexample: free subcritical string). Moreover, even if we insist that every gauge anomaly should vanish (which I think is wrong), we must still know about them. Otherwise, we would know e.g. that there is something special about 26D in the bosonic string. **

    Fine. I would like to hear more about your viewpoint on the problems of QG (which undoubtedly differ from mine) before I read your papers (which seem to be very interesting). As you might have read, IMO there are still lots of crucial insights to be gained from GR and QM before we even start such an enterprise (cat problem, vacuum state, rigorous QFT, ...). To open perhaps the discussion, let me give some of the alternatives I see for adressing the problem of QG (and more in particular the ``problem of time´´ in classical GR):

    (a) Add a covariant prescription for a time function to the classical Lagrangian. Now, there seem at least two ways to proceed:
    (i) find a classical dynamics for the microworld (of course guided by QM) and you are done.
    (ii) Keep the notion of time classical and quantize the ``spatial´´ degrees of freedom (which is what CDT does in a sense for T = Gaussian time)

    (b) Quantize the constraint algebra (well this could be the classical Dirac algebra (which is of course not a Lie algebra), the Lie Algebra in the Kuchar Isham framework of dynamical embeddings, or using the constructions in the Histories framework - which I am not too familiar with):
    (i) *hope* that the constraint algebra contains anomalies which would dynamically single out a notion of time
    (ii) Pick out your favorite observable T, call it time and write out the evolution equations wrt to T (which is what LQG proposes). It is here still possible to ``quantize´´ the dynamical time prescription of (a) and proceed from that (but this has proven to be difficult).

    Apart from (a) (i) which deals with the cat (and other) problem(s) directly, it seems that all other approaches face the equally difficult tasks of finding realistic vacuum states as well as a prescription which singles out the vacuum state of our universe. However, I am not so worried about the meaning or implementation of quantum covariance one choses to adopt; much more troublesome in my view is an adequate resolution of the problem of macroscopic reality (a solution to the latter might crucially determine what view one wishes to entertain about quantum covariance). The orthodox, statistical and MWI views on this matter are clearly inadequate (the orthodox one not even being logically consistent). To my taste and knowledge, only three reasonable mechanisms have been proposed so far:
    (a) Penrose gravitational reduction
    (b) Ghirardi, Rimini and Weber stochastic reduction theories
    (c) local realism


    As you know very well, the views on what (even classical) covariance really means are very diverse (of course I am aware of the matematical viewpoints of gauge transformations and reduced symplectic manifolds - I am mainly talking about the *physical* meaning here), and certainly not as ``rectilinear´´ as some people claim (for instance Smolin); the debate being initiated by Kretschmann some 90 years ago. Also, the views on what it means on the quantum level are of course even more divergent (as I illustrated above). This is probably the best social indicator that it is *not* the main problem to worry about. :smile:
    Anyway, I guess this is more than enough for one post.

    Cheers,

    Careful
     
    Last edited: Jan 8, 2006
  21. Jan 8, 2006 #20
    All the conceptual difficulties in quantum gravity have a counterpart in the representation theory of the diffeomorphism group. This is not surprising, because the difficulties have to do with general covariance rather than with the specifics of the Einstein action, and the diffeomorphism group is the mathematics of general covariance.

    The diff algebra in 1D is well understood - after quantization, we get the Virasoro algebra of string theory. The construction of Fock representations proceeds in three steps:
    1. Start with a classical rep, i.e. a primary field = scalar density.
    2. Introduce canonical momenta.
    3. Normal order.
    This recipe gives rise to a well-define rep of the Virasoro algebra.

    However, it does not work in higher dimensions, because
    1. In order to define normal ordering, we must single out a privileged energy or time direction. This gives us the usual problems with a foliation, which is at best ugly, and probably worse than that.
    2. Normal ordering of a bilinear expression always gives rise to a central extension (central = commutes with everything). However, the Virasoro extension is not central, except in 1D.
    3. It is ill defined. Normal ordering gives rise to an unrestricted sum over transverse degrees of freedom, which leads to an *infinite* central extension, i.e. nonsense. This does not happen in 1D, where there are no tranverse directions.
    This problem is a manifestation of the usual infinities in QFT. In 1D, normal ordering is enough to remove them, but not so in higher dimension. I have tried to invent some kind of renormalization to remove these infinities, unsuccessfully. This is not so surprising, since GR is not renormalizable.

    Instead, one must proceed along the following path:
    1. Start with a classical rep, i.e. a tensor density.
    2. Expand all fields in a Taylor series around a 1D curve, which I call "the observer's trajectory".
    3. Truncate the Taylor series at order p.
    4. Introduce canonical momenta, both for the Taylor coefficients and for the observer's trajectory.
    5. Normal order wrt frequency.
    This gives us a well-defined representation of an extension of the diffeomorphism algebra on a linear space. The reason why this prescription works is that in step 3 we have a classical realization on *finitely* many functions of a *single* variable (the parameter along the trajectory), which is precisely the situation where normal ordering works. The realization is non-linear, which leads to a non-central extension.

    Let me write down the formula for the Taylor expansion, because it is the crucial new ingredient which resolves all problems. For every field f(x) (x is a point in spacetime), we write

    f(x) = sum_m 1/m! f_m(t) (x-q(t))^m

    where t is a single parameter, living on a circle, say. With multi-index notation, this formula makes sense in any number of dimensions. Instead of formulating physics in terms of the fields f(x), I use the Taylor coefficients f_m(t) and the observer's trajectory q(t). Classically, this is completely straightforward - the Euler-Lagrange equation for f(x) turns into a hierarchy of algebraic equations for f_m(t).

    But we now have a privileged time variable t, and we can define the Fock vacuum by demanding that all negative-frequency components of f_m(t) and q(t) (and their canonical momenta) annihilate it. This gives us a notion of a privileged time variable without having to break covariance. Of course, we have added an extra dimension, and the RHS really defines a field f(x,t), so one must eventually impose some constraints on the f_m(t) to eliminate the t dependence. Nevertheless, f_m(t) still depends on t, even though f(x) does not, so the notion of lowest-energy representation still makes sense.

    After this lengthy prelude, let us return to physics. The basic observation is that in any general-covariant theory, the Hilbert space carries a rep of the diffeomorphism group. Some clarifications:
    1. For definiteness, consider the kinematical Hilbert space, on which the representation certainly is non-trivial. In the presence of anomalies, the rep must act non-trivially, and in addition unitary, on the physical Hilbert space as well.
    2. In quantum theory, representations are always of lowest-weight type (the polymer reps of LQG apparently lack a lowest weight). This is true even for the kinematical Hilbert space, whose rep is not unitary, but it does have a lowest weight.
    3. By diffeomorphisms I always mean space-time diffeomorphisms, which is the constraint algebra of GR in covariant formulations. Classically, the Dirac algebra is physically equivalent to the 4-diffeomorphism algebra, although they are mathematically distinct. However, apart from many other problems with the introduction of a foliation, I don't know how to construct reps of the Dirac algebra, so I have nothing to say about that.

    In recent years I have tried to apply this math to the quantization of gravity. Unfortunately, none of the standard quantization schemes seems appropriate. The concepts from representation theory (action of a group on a Hilbert space) do not fit naturally into the path-integral formalism, whereas the Hamiltonian formalism breaks covariance which is precisely what the diffeomorphism group is all about. Moreover, there are big problems as soon as one deals with non-local objects, e.g. the Hamiltonian or the action which are integrals over space and spacetime. The Taylor expansion above is a manifestation of strong locality; the field f(x), which lives throughout spacetime, is replaced by data which live on the observer's trajectory only.

    However, there is one formulation of dynamics which is both local and covariant, namely the Euler-Lagrange equation. Therefore, I have tried to invent a canonical quantization scheme based on this formulation of dynamics: manifestly covariant canonical quantization. It is in fact well known, from the days of Lagrange, that phase space is a covariant concept - it is the space of solutions to the Euler-Lagrange equations, modulo gauges (caveat about anomalies). My idea is to regard dynamics as a constraint on the space of virtual histories, and to quantize first and to constrain afterwards. In this way I can capitalize on the construction of diffeomorphism algebra modules, which describe virtual histories, without dynamics.

    One should note that diffeomorphisms act in a well-defined way at every stage. This is because I primarily view the Hilbert space as a projective diffeomorphism algebra module.

    By truncating the Taylor expansion at some finite order p, the normal-ordering infinites were removed. However, this is a regularization, and at the end one should take the limit p -> infinity; only infinite Taylor series can be identified with the original fields. It comes as no surprise that the infinities resurface in this limit. However, by considering more general representations starting from several fields, both bosonic and fermionic, one can cancel the infinite parts, but not the finite parts, of the anomalies. It turns out that this condition naturally singles out four spacetime dimensions. Although it would be a lie to claim that I predict 4D, this is still a promising hint.

    There are of course also problems: I don't have an invariant inner product and thus no notion of unitarity, and one must probably develop perturbation theory within this formalism to actually compute something. There is also one rather serious flaw: classically, my construction gives the space of differential operators over phase space, rather than the desired space of functions over phase space equipped with the Poisson bracket. I think I know how to tackle these problems, but I haven't done so yet.

    The most striking thing is the presence of the observer's trajectory. This leads to several conceptual observations:

    1. Time is defined locally by the observer's trajectory, as the direction parallel to the vector q(t).

    2. There is no global foliation, and no global notion of time. Everything is formulated in terms of the Taylor coefficients f_m(t), which live on the observer's trajectory.

    3. The theory does hence not explicitly deal with objects away from the observer's worldline, except subtly in convergence issues, which I don't have anything to say about anyway.

    4. The awkward notion of dynamical spacelikeness is also absent, since all data on the trajectory are causally related.

    5. One can define a genuine local Hamiltonian, rather than a Hamiltonian constraint, as the generator which translates the fields relative to the observer, i.e. which acts on f_m(t) but not on q(t). There might be some problems here, though.

    6. We can combine diff invariance with locality, in the sense of correlation functions depending on separation, but only in the presence of an anomaly. This is in fact well known in conformal field theory, which is covariant under analytic diffeomorphisms in one complex dimension. This is a important point, because the scaling operator does not depend on the metric, and anomalous dimensions are thus background independent.

    It is getting late and this post is already too long. I am not sure how many of your points I have addressed, but this has to do for now.
     
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