FQXi Essay Contest - Is Reality Digital or Analog?
Continuous Spacetime From Discrete Holographic Models by Moshe Rozali
We argue that discrete models of quantum gravity can be reconciled with continuous spacetime, and in particular with local Lorentz invariance. The fundamental discreteness cannot be manifested merely as a short distance cutoff. Rather it is expressed in subtle and non-local correlations, of the sort which are most familiar in the context of the quantum mechanics of black holes. Hence, holography is essential in reconciling any fundamentally discrete view of the universe with our continuous and local description thereof.Author Bio
I am an associate professor in the department of Physics and Astronomy at the University of British Columbia, Vancouver, Canada. I was educated in Tel-Aviv University (degrees in Mathematics and in Physics) and in the University of Texas at Austin (PhD in Physics). Before joining the faculty at UBC, I was a postdoc in the University of Illinois at Urbana-Champaign, and in Rutgers University.Download Essay PDF File
You wrote: "Nevertheless, to the author it seems clear that the discreteness
associated with quantum gravitational effects has little to do with any truncation of spacetime at short distances. Rather it is manifested in global correlations between the degrees of freedom making up spacetime."
I still don't see any proof of the above statement in the essay. Is this a speculation?
"The difference between space and time manifests itself in the difference between boosts and rotations"
How about when this is a screw motion?
Your statement below describes a model I have been thinking about for some years - a face-centered-cubic universal lattice:
You wrote: "To see the tension between this requirement [Lorentz invariant at the Planck scale] and fundamental discreteness, imagine at first the most naive possible discrete model, the assumption that our space is arranged into a cubic lattice of sides whose length is precisely lp. It is immediately clear that this statement depends on your reference frame, your friend in possession of the latest spaceship model will zoom by this lattice and will see it Lorentz contracted. The principle of relativity is clearly at odds with this very naive picture."
My short response is that the cubic lattice will not be seen from the spaceship, let alone contract because it is not itself made of matter nor does the lattice constitute a frame of reference! The lattice has no background in which it is embedded - it itself creates space, matter and everything else. My other observation about this very basic and important picture you conjured is that in such a lattice the spaceship will not move as if it were in an inertial frame, but will self-convolute as a soliton pattern leap-frogging across the field of lattice nodes, as I have outlined in my original Beautiful Universe
paper on which my present fqxi paper is based.
I hope this makes sense and that it helps. Vladimir
Thanks for a first rate essay. Your classification of operators is a super great way to put things in perspective. And your expert explanation of Lorentz Invariance hopefully will correct a lot of misconceptions among relativity doubters (who seem to be more abundant in these forums than elsewhere)about the role of "empty space."
If you don't mind a little side observation, I like the way your philosophy and results smoothly descend from Leiniz down to contemporary researchers such as Hermann Weyl and Gregory Chaitin. Appealing to Leibniz's continuity principle, Weyl wrote in 1949*, "Only in the infinitely small may we expect to encounter the elementary and uniform laws, hence the world must be comprehended through its behavior in the infinitely small."
* Philosophy of Mathematics and Natural Science, Princeton.
I the conclusion of your paper you state: "Nevertheless, to the author it seems clear that the discreteness associated with quantum gravitational effects has little to do with any truncation of spacetime at short distances. Rather it is manifested in global correlations between the degrees of freedom making up spacetime. This is the best way to avoid the tension between
Lorentz invariance and the underlying discreteness of quantum gravity."
A much more serious problem is the experimentally demonstrated non-locality in quantum mechanics. Even if you can find a mathematical way to reconcile quantum gravity with Lorentz invariance, something that it is not demonstrated in the paper, the issue still remains: relativity obeys locality and in quantum mechanics you have nonlocality confirmed by the existence of Greenberger-Horne-Zeilinger relations. These two theories cannot be reconciled physically. One must be fundamentally wrong in the sense that it is not a true theory of nature but only an instrument for making predictions. I don't know which but in my opinion, trying to invent some way to keep both is like having the cake and eating it too.
All the best.
you said "These two theories cannot be reconciled physically. One must be fundamentally wrong in the sense that it is not a true theory of nature but only an instrument for making predictions. I don't know which but in my opinion, trying to invent some way to keep both is like having the cake and eating it too."
Yum indeed. I propose a solution in my essay. Your opinion is most welcome.
Thanks everyone for the encouragement and the interesting comments, lots of food for thought.
Holographic type of discreteness can prove with Holometer
http://holometer.fnal.gov/ and my non-standart idea
Efthimios you said " These two theories cannot be reconciled physically. One must be fundamentally wrong in the sense that it is not a true theory of nature but only an instrument for making predictions. I don't know which but in my opinion, trying to invent some way to keep both is like having the cake and eating it too. "
That is exactly right. GR and QM both 'work' but both are based on irreconcilable assumptions. In my paper here I have traced these assumptions to two of Einstein's arbitrary and experimentally unfounded ideas: 1- the constancy of the speed of light. 2- a photon being a point object . The first worked brilliantly in SR but vastly complicated GR. The second led to the unnecessary confusions of explaining QM in terms of probability. Unless physics is restructured without these two concepts it will never be unified in a simple and direct way close to nature.
You might read what I replied to Georgina today in my thread 833.
Your essay was a good reading, and I gave it a good score. My essay discusses the issue of conformal completion and quotient group with AdS.
Further down the discussion list is a reply of mine to Tejindar Singh, where I indicate something about noncommutative geometry in quantum gravity.
Thanks for your interest and for your comments Lawrence, I'll take a look.
Do you think that black holes destroy information? Some speak of holograms to explain that BHs don't.
Do you have details on this?
Thanks for your interest. I think that information is not likely to be lost, and the point of my last section is trying to learn from possible ways information can be preserved in black hole physics. For quantum mechanical black holes you'd expect discrete spectrum and finite entropy, and one might instinctively attempt to explain this fact by assuming that the horizon is made of tiny discrete bits at short distances. In this example at least, this is unlikely that things work this way, because any such discretization would be at odds with the principle of equivalence and the experience of the infalling observer. We need the holographic principle to make things work while avoiding the problems that would follow from making spacetime discrete. Maybe there is a lesson there for more general situations.
If you want to read more on the holographic principle and black hole physics, I'd recommend Lenny Susskind's books: "The Black Hole War" for non-technical exposition, and "An Introduction To Black Holes, Information And The String Theory Revolution: The Holographic Universe" for semi-technical exposition.
Fascinating essay! Well-written and researched. Interesting speculations about the black hole energy spectrum.
Paul Halpern, The Discreet Charm of the Discrete
Thanks very much Paul.
Can this essay help to you for final answer?
Thanks for your interest Yuri, I will take a look.
Congratulations on your dedication to the competition and your much deserved top 35 placing. I have a bugging question for you, which I've also posed to all the potential prize winners btw:
Q: Coulomb's Law of electrostatics was modelled by Maxwell by mechanical means after his mathematical deductions as an added verification (thanks for that bit of info Edwin), which I highly admire. To me, this gives his equation some substance. I have a problem with the laws of gravity though, especially the mathematical representation that "every object attracts every other object equally in all directions." The 'fabric' of spacetime model of gravity doesn't lend itself to explain the law of electrostatics. Coulomb's law denotes two types of matter, one 'charged' positive and the opposite type 'charged' negative. An Archimedes screw model for the graviton can explain -both- the gravity law and the electrostatic law, whilst the 'fabric' of spacetime can't. Doesn't this by definition make the helical screw model better than than anything else that has been suggested for the mechanism of the gravity force?? Otherwise the unification of all the forces is an impossiblity imo. Do you have an opinion on my analysis at all?