["Quantum Theory from Quantum Gravity" (Markopoulou and Smolin, 17 Nov 2003)] - A New Kind of Science: The NKS ForumA New Kind of Science: The NKS Forum
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"Quantum Theory from Quantum Gravity" (Markopoulou and Smolin, 17 Nov 2003)
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Posted by: Chris Weed
See http://arxiv.org/PS_cache/gr-qc/pdf/0311/0311059.pdf
RE: NKS - Ch. 9, Sec. 16
ABSTRACT -- "We provide a mechanism by which, from a background independent model with no quantum mechanics, quantum theory arises in the same limit in which spatial properties appear. Starting with an arbitrary abstract graph as the microscopic model of spacetime, our ansatz is that the microscopic dynamics can be chosen so that 1) the model has a low energy limit which reproduces the non-relativistic classical dynamics of a system of N particles in flat spacetime, 2) there is a minimum length, and 3) some of the particles are in a thermal bath or otherwise evolve stochastically. We then construct simple functions of the degrees of freedom of the theory and show that their probability distributions evolve according to the Schrodinger equation. The non-local hidden variables required to satisfy the conditions of Bell’s theorem are the links in the fundamental graph that connect nodes adjacent in the graph but distant in the approximate metric of the low energy limit. In the presence of these links, distant stochastic fluctuations are transferred into universal quantum fluctuations."
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For more background, see:
http://www.math.princeton.edu/~nelson/books/bmotion.pdf
"A dialog on quantum gravity" (Carlo Rovelli, 12 Oct 2003)
http://arxiv.org/PS_cache/hep-th/pdf/0310/0310077.pdf
Also see Lee Smolin's two popular books and his article "Loop Quantum Gravity" in the January 2004 issue of Scientific American (featured on the cover).
On causal sets:
http://www.maths.qmul.ac.uk/~pjc/csgnotes/causetbib.pdf
http://arxiv.org/PS_cache/gr-qc/pdf/9909/9909075.pdf
Posted by: Tony Smith
This is looking more and more like a rerun of the bridge building that went on half a generation ago to link black holes and big bangs. Those of us who were happy to leave the mathematical rigours to others more competent were able to race ahead and see just where that bridge might lead us in our quest for a viable history of the world we find ourselves in.
Now we can see the ghostly outline of an equally profound bridge that will eventually be fleshed out between Wolfram's work and Smolin et al's work. The result will surely be that the fundamental building blocks of physicality will be seen to be the simple nodes and edges of an effectively limitless graph.
Mark Suppes recently started what became a rather active thread here tackling a Mechanical Gravity Theory which hints where our wilder speculations might lead us. At this stage Markopoulou and Smolin have constrained themselves to surer theoretical ground, relying on some unlikely simplifying assumptions to demonstrate that both quantum theory and gravitational theory could conceivably be derived from graph theory.
In other places, Smolin has emphasised the importance of a "background independent model" in contrast to the implicit background assumptions that bedevil string theory. (This is basically the issue of whether the model produces space and maybe time, or spacetime is just assumed.) In the paper Chris Weed brings to our attention here, Markopoulou and Smolin "assume that (...) some of the nodes of the graph correspond to the positions of particles in three dimensional space". They talk of the nodes being embedded in 3D space, an idea which might be better understood by contrasting it with the section of NKS on "Space as Network" (pp. 475-480) where the network is seen as producing space rather than being embedded in it. With Smolin's emphasis on background independence I am confident that is the kind of place he too wants to go.
Markopoulou and Smolin also assume "that the edges of the graph (...) do not evolve in time." This is in many ways the antithesis of what I would really expect the fundamental graph to do. It certainly flies in the face of the hard evidence of an expanding universe.
Posted by: Chris Weed
The LQG literature will make it perfectly clear that these assumptions were only made for the purposes of this paper, to show concretely and precisely how non-relativisitic quantum mechanics can emerge from a discrete substructure underlying ordinary three-dimensional space. Obviously this result needs to be refined and generalized in way that is directly connected with the much more fully developed notions of spin networks and spin foams that are the putative substructures of space and spacetime in LQG (and not simply embedded in them!). I'm sure the authors' hope is that their paper will inspire others to reconsider the relationship of the foundations of quantum theory to more complete formulations of what is currently known as quantum gravity. It is clear that Stephen Wolfram has already done this qualitatively, in a way that shows striking similarities with the notions presented by Markopoulou and Smolin.
I'm inclined to offer for consideration an appellation derived from John Wheeler's term "pregeometry" [Gravitation (MTW), Chapter 44] -- pregeometric discrete stochastic dynamics. By the way, I think that Wheeler's vision of "pregeometry as the calculus of propositions," although vague, was prescient. Any deterministic dynamics on a finite space of states can in theory be represented as a set of Boolean transition rules specifying how the bits or cells of a binary encoding of those states changes in successive "time steps". A particular significance of this, I would argue, is that one must consider such arbitrary Boolean networks in order to avoid assuming a priori fixed structure in the space of states or in the dynamics. Whatever structure ultimately appears must be a consequence of the dynamics, which must have the background independence characteristic of general relativity and loop quantum gravity.
Having said the above, I would like to offer here a concrete suggestion of how an initial correspondence between spin networks and Boolean networks might be established. The suggestion is simply this: The transition rules of the network *are* the network; they dictate its topology and geometry. The rules are linked together to the extent that any given pair of rules (Boolean functions) share at an input, ie, a cell state in the network. Furthermore, one must realize that although the transition rules might be stated in terms of a labeling scheme, distinctions between rule sets cannot depend on the choice of labeling or on any preexisting relations those labels might appear to imply.
The ideas presented in NKS clearly anticipate this "labeling invariance", although the notion of background independence needs to be more fully developed in this context. This is absolutely crucial if one takes the central ideas of general relativity seriously. The developers of LQG have argued forcefully and persuasively that most string theorists have failed to do so until recently.
Posted by: Jason Cawley
I'm not a physicist, but I am highly interested in non-local deterministic or discrete underlying models in physics, for philosophical reasons. I thought people here might find the following pointers to aspects of the literature on this useful.
The base method paper, reference 5 in the posted paper, is E. Nelson, Derivation of the Schrodinger equation from Newtonian mechanics, Phys. Rev. 150, pp. 1079-1085, 1966.
http://fangio.magnet.fsu.edu/~vlad/...p/fs2_14021.htm
(Unfortunately that link is now dead, the site having shut down).
A full length book developing the math etc, not limited to the QM use of it, came in 1967 -
E. Nelson, Dynamical Theories of Brownian Motion, Princeton Univ. Press, 1967
http://www.math.princeton.edu/~nelson/books/bmotion.pdf
(That one is still available and what I would now recommend)
Non-locality is still required; any simple classical version of this scheme will fail.
A useful collection of links to older papers for background (where the first Nelson paper above comes from, for example) is here -
http://fangio.magnet.fsu.edu/~vlad/.../chap14_toc.htm
(This is also a dead link, being from the same site as the above. Sorry, but that's the way of the web...)
Smolin made use of this before, in an eigenvalues of random matrices approach, in 1985 -
L. Smolin, "Derivation of quantum mechanics from a deterministic non-local hidden variable theory", in Quantum Concepts in Space and Time, edited by C. J. Isham and R. Penrose, Oxford University Press, 1985.
http://www.amazon.com/exec/obidos/t...777445?v=glance
A brief net review describes that paper thus -
He shows that, under certain conditions, the eigenvalues of ensembles of large random matrices are described by Nelson's stochastic processes and thus by the Schroedinger equation. He envisions such matrices as describing non-local interactions in the universe and concludes that given such, quantum mechanics follows from statistical non-local physics.
A more recent paper on the same (previous) line of development is his
"Matrix models as non-local hidden variables theories"
A link for all of Smolin's stuff is -
http://www.qgravity.org/writings/
What seems to be new here compared to the previous use of the Nelson criteria is getting non-locality just from embedding a discrete graph.
Meanwhile Hooft has also been investigating possible underlying deterministic generators for QM. Local information loss gives the required non-locality. Here are some relevant links for what he's been doing.
http://arxiv.org/abs/quant-ph/0212095
The abstract reads in relevant part -
Contrary to common belief, it is not difficult to construct deterministic models where stochastic behavior is correctly described by quantum mechanical amplitudes, in precise accordance with the Copenhagen-Bohr-Bohm doctrine. What is difficult however is to obtain a Hamiltonian that is bounded from below, and whose ground state is a vacuum that exhibits complicated vacuum fluctuations, as in the real world.
Beneath Quantum Mechanics, there may be a deterministic theory with (local) information loss.
Here is a condensed, recent overview of his work along these lines -
http://www.phys.uu.nl/~thooft/quantloss/index.htm
A full list of Hooft's publications can be found here -
http://www.phys.uu.nl/~thooft/gthpub.html
From the NKS book, many of the notes to chapter 9 are also relevant to all this. Particularly the history notes, from 1041 (on relativity) right through to the end of the section on 1065 (on Bell's inequalities).
http://www.wolframscience.com/nksonline/page-1041c-text
There are relevant notes on pages 1043, 1047, 1053, 1054, 1056, 1059, 1061, 1062, 1064, and 1065. So it is best just to read through that whole section of the notes.
I hope this is interesting.
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