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Is Reality Digital or Analog? Essay Contest (2010-2011)
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A Quantum-Digital Universe by Giacomo Mauro D'Ariano
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Author Giacomo Mauro D'Ariano wrote on Feb. 14, 2011 @ 13:07 GMT
Essay AbstractCan Reality be simulated by a huge Quantum Computer? Do we believe that Reality is made of something more than interacting quantum systems? The idea that the whole Physics is ultimately a quantum computation---a strong quantum version of the Church-Turing hypothesis well synthesized by the Wheeler's coinage "It from bit"---is very appealing. It is theoretically very parsimonious: an Occam razor's quality-guaranteed description of the world. But, if this is the case, then we need to understand the entire Physics as emergent from the quantum computation. Here I will make a short exploration on how this may come about.
Author BioI am professor at the University of Pavia, where I teach "Physical Theory of Information" and "Foundations of Quantum Mechanics", and enjoy research with a marvelous group of much younger collaborators.
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nikman wrote on Feb. 14, 2011 @ 19:54 GMT
I like the Zeilinger-Brukner, or IQOQI Interpretation of Wheeler (out of Bohr). "It from Bit" is about what we can KNOW, not about "what is". One fundamental quantum system yields one classical bit. A qubit is a fundamental quantum system and you know what you get when you measure it. This is fascinating because our minds are binary too. We construct a world from binary propositions.
Brukner suggests in another couple of papers that we're just crap (coarse-grained) measuring devices. This seems more than plausible. Interestingly, neither of those guys lets the concept of universe-as-quantum-computer pass through his fingertips.
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Giacomo Mauro D'Ariano replied on Feb. 14, 2011 @ 20:57 GMT
Well said. One thing which has not been appreciated yet is the crucial role of the quantumness of Information in a digital universe. A classical computer would not work, not just in obviously reproducing efficiently reality (that's would be a tautology, being the world quantum), but in being efficiently reversible, efficiently addressable in different directions in a network, and more ...
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nikman wrote on Feb. 15, 2011 @ 00:19 GMT
I didn't express myself as well as I should have in reference to the "quantum computer".
How can we assume the universe is a computer unless we can verify its computations? Why does it need to be a computer? Isn't it possible that the universe isn't even computable in terms of computation as we understand computation?
Aren't you assuming there's stuff going on inside of qubits that can't be measured?
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Giacomo Mauro D'Ariano wrote on Feb. 15, 2011 @ 13:25 GMT
Dear Nikman,
there is no stuff hoing inside of the qubits, there are only qubits from which stuff is emerging!
It is a theoretical description of reality. The reality being perfectly simulated by a quantum computer (David Deutsch physical Church Turing principle), is everything you need for physics. The rest is for metaphysics.
I understand that is an hard to swallow ontology, but, this is the theme of the context. Otherwise, what else does iit mean a "digital Reality"?
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nikman wrote on Feb. 15, 2011 @ 14:39 GMT
I love ontology! I love "It from Bit" (at least as interpreted by Brukner, Zeilinger et alia). What I have a problem with is the computer analogy, which B & Z do not adduce. You mention Deutsch, which brings up another issue: many worlds. That seems to go hand-in-hand with the universe-as-computer approach, and which Zeilinger is on record as having no use for.
Anyway, it needs to be noted that any such quantum computer universe would NOT be a quantum computer of the type that may someday be realized here in the human realm. They're not, as Scott Aaronson regularly notes, "known to be able to solve NP-complete problems in polynomial time." People get really confused about this and expect almost supernatural stuff to begin happening once qcomputers get on line, if they do. It's a public service to disabuse them.
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egal wrote on Feb. 15, 2011 @ 16:27 GMT
How is your view any different from Seth Lloyd's view? I think Lloyd as you may be giving the quantum a too prominent place when in fact is not needed at all. Of course one can think that quantum mechanics is at the basis of everything else, but you are then already making a choice assuming such a prior statement and then jumping to say that the universe is a quantum computer. In that case it is not longer what a computer means. The standard quantum computer is Turing computable, but quantum mechanics actually allows infinite number of states, so whether the universe is a computer in one or another way is actually as open as the original digital vs. analog question…
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Mauro replied on Feb. 16, 2011 @ 13:53 GMT
Dear Egal,
there are two different problems related to your point.
First: if you believe in quantum field theory, then the problem is to see if a discrete field theory (= quantum computation) has some strengths compared to the continuum. And, as I wrote in my essay, it has many, e.g. covariance is a free bonus, all problem of the continuum disappear (especially localization), no need of quantization rules, emergence of Hamiltonian, Dirac as free flow without covariance, and more ...
The second problem is if a classical computer would do the same job. Answer: It will doit only at the expense of a very complicate computational network! And there are things that will never work, in a classical computer, e.g. simply directing the information by state-preparation (you need a kind of telephone system in the network!). There is more than that. Quantum Mechanicsis the only possibility to have a digital nature: the quantumness is part of the fabric of space-time.
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Anton W.M. Biermans wrote on Feb. 18, 2011 @ 04:00 GMT
Dear Giacomo,
You write
---"The big question is now where gravity comes from."---
The answer is surprisingly straightforward. If a universe is to create itself, then its particles must create themselves, each other. If (the properties of) particles then are as much the product as the source of their interactions, of their energy exchange, then their mass also must be as much the product of their exchange, of gravity between them as its source. As the force between particles then also is a much the product as the source of their interactions, a force obviously cannot be either attractive or repulsive (that is, at least at quantum level). A universe which discovers how to create itself, can hardly stop doing so. It is this continuing creation process which powers, or is powered by gravity, the force we associate with the contraction of masses and the expansion of spacetime between the mass concentrations. For details see my essay.
Best regards, Anton
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Giacomo Mauro replied on Feb. 21, 2011 @ 06:30 GMT
Dear Anton,
thank you for your comment. I will look at your essay.
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Thomas J. McFarlane wrote on Feb. 20, 2011 @ 08:45 GMT
Giacomo,
Thank you for the engaging essay. I have a question about this passage:
"The real entities are the events, facts of the world describable by the basic language obeying the rules of predicate logic (the 'facts' of Wittgenstein's Tractatus). ...The notion of 'event' must be regarded as truly primordial: events do not happen in space- time, they build-up space-time. Stated in other words: space-time is our way of organizing events."
If events are fundamental and space-time derivative, as you propose, then to avoid circularity events can not be specified or described in any way that makes reference to space or to time. Yet you define events as "facts of the world". Could you give an example of such 'facts of the world' that can be specified without reference to time or space?
Thanks,
Tom
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Giacomo Mauro replied on Feb. 21, 2011 @ 09:26 GMT
Dear Thomas,
please, call me Mauro: it is my middle name, but this is the one that everybody uses.
Thank you very much indeed for your comment, which brings up a relevant and often raised point. Due to length limitations, I didn't have the space to write about this in the essay, and this is a good occasion for doing it. This is also why I like the idea of this Forum: it gives a...
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Dear Thomas,
please, call me Mauro: it is my middle name, but this is the one that everybody uses.
Thank you very much indeed for your comment, which brings up a relevant and often raised point. Due to length limitations, I didn't have the space to write about this in the essay, and this is a good occasion for doing it. This is also why I like the idea of this Forum: it gives a unique opportunity for receiving genuine feedback, the one that you don't usually get from colleagues e.g. at conferences.
Essentially your question is: how can you state an "event" in the basic language without reference to space? The answer looks hard, but it is indeed surprisingly simple. And this is the case for every event, if you ask yourself "where" the information about space-time comes from. Whenever you specify the location in space and time of an event you use local information At Your Place and Time (AYPT). In a everyday description of an event you get a rough idea of the distance from your place from the size of the image on your retina compared to your previous experience, and/or from your previous knowledge about measurements that you did in the past. The specification of time is a NOW, or it is the recollection-memory of a reading of a clock made AYPT in the past. In a very accurate specification of space-time using the best state-of-the-art technology you use a very precise clock under your control AYPT, send an em beam toward the place of the event (using previous information AYPT: you have "seen" the place toward which you direct the beam), use signals AYPT from satellites to locate the event with precision in space-time relative to YPT. Every complete and precise description of the event ultimately is given in terms of observations made AYPT, and/or based on previous knowledge and observations made AYPT, including the calibration of the apparatuses, the sending of the satellites, the made of the clock, etc. Therefore, ultimately, the event is specified in terms of events happening AYPT (i.e. at the observer), using information AYPT and relations with previous events.
In short, the whole space-time is made of events connected to the AYPT word-line.
Now, it doesn't matter if is you or another observer, it is again a relation with your AYPT + storing of information AYPT.
Another of way of saying: ultimately space-time is made of positions and times of events, it is made of events, each of which is specified with information AYPT. Ultimately space-time is reduced to a single point: AYPT.
And, this is not my idea: it is the lesson of Albert Einstein, who thought us to synchronize clocks, and to use clocks and the speed of light to measure distances. The building up of space-time that I gave you above gives the equation "reference-frame" = "observer". Einstein did "believe" in an objective reality, but he needed the observer to define his relativity operationally.
It will be nice to hear from you again, Thomas
Thank you.
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Thomas J. McFarlane replied on Feb. 25, 2011 @ 04:10 GMT
Dear Mauro,
Thank you for the helpful and detailed response to my question. I understand you as saying an event is simply defined as information in the spaceless here and timeless now. Space and time then arise as ways to organize events in such a way that they are given a spatio-temporal structure. Is this accurate?
If so, it is still not clear to me how information for an event is measured without any reference to space and time. I can imagine a reference frame where local space and time are defined, and then using that to develop a global spacetime structure as Einstein did. But how are the measurements made in the observer's reference frame without even time and space defined for it?
To illustrate, you used the example of space arising from comparisons of the sizes of an image on the retina. But the measurement of size on the retina seems to presuppose space. And the example of time arising from comparing clock measurement now with clock measurement of a memory also seems to presuppose a temporal distinction between the now and the past memory. It also seems to presuppose space to measure the movement of the clock's dials in space.
Your further clarification would be appreciated.
Best regards,
Tom
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Jacek Safuta wrote on Feb. 21, 2011 @ 21:43 GMT
Dear Mauro,
Thank you for the beautiful essay. It seems to have something in common with Computational LQG? And maybe I do not get it. But this is not my point.
You write: what else is out there more than interacting quantum systems? Is it space? No, space is a “nothingness”.
In my essay I propose a very simple “thought experiment”: we observe a small region in spacetime (the size of an elementary particle radius) deformed in the way that the wave we actually detect is not emitted or reflected by the observed object but it comes back to us along the geodesic (as the notion of a "straight line" in general relativity). In fact we observe only a strongly deformed spacetime region, “empty” inside and redirecting our wave but apparently… we perceive a particle. Our measuring instruments and our language out of the force of habit say so.
You also write: gravity must be a quantum effect.
In general I fully agree. But I propose to look at the gravitation not as a fundamental but emergent interaction. Details in my essay if you are interested. However it is highly speculative.
Best regards,
Jacek
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Mauro replied on Feb. 24, 2011 @ 06:49 GMT
Dear Jacek,
thank you for your post! I'm not sure that I have something in common with COmputational LQG, of which, however, I'd like to know more. I think that both we agree that gravity is emergent as a quantum effect. The point on which apparently we don't agree is that also space-time is emergent.
I'll read your essay.
Best regards
Mauro
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Jacek Safuta replied on Feb. 24, 2011 @ 18:34 GMT
Dear Mauro,
I hope that you will read my essay and you will find that in my view the space-time is not emergent.
Jacek
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Tobias Fritz wrote on Feb. 22, 2011 @ 17:55 GMT
Dear Mauro,
there is a lot I would like to comment on, mostly very positive things, but there is one point which seems important enough to merit its own post. My question is about Figure 4, where you state that "In a computational network made with tetrahedra [...] the maximal information speed is the same in all directions". Can you provide a proof or a reference for this?
I have thought about exactly this issue of information speed in a lattice, albeit in the two-dimensional case, and come to the conclusion that isotropic information speed is impossible. You can see this by marking all the vertices which can be reached by traversing 2 edges, then all those which can be reached by traversing 4 edges, 6 edges, and so on. If you interpret, in each case, these points as vertices of a polytope, you will notice that these polytopes are identical (up to scaling): the shape of the wave fronts does not change! This is easy to understand in terms of Minkowski sums: the 4-edge polytope is the Minkoswki sum of the 2-edge polytope with itself, and similarly for all the other ones.
Now it seems to me that the same reasoning applies in the 3-dimensional case to show that the wave fronts are polytopes of a fixed shape. In particular, if this is correct, then the speed of information is not isotropic.
In any case, can you explain in a little more detail how the tetrahedra in figure 4 are arranged? It's not quite obvious to extract this information from the picture.
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Mauro replied on Feb. 24, 2011 @ 06:45 GMT
Dear Tobias,
thank you for raising your point. The problem of digitalization of field theory in 3D is much more interesting of what I imagined at the beginning. Its is really fascinating. But it is also not easy to formalize mathematically.
I don't have an analytic proof of the isotropy of information speed in the Regge-like causal network. For the moment, the proof is just the visual one of Fig. 4. Let me try to explain the figure. The figure is a representation of the lattice seen from the top in the direction of time axis. Unfortunately, all planes are merged in the same 2D figure. I think it is very important to understand the way in which the tetrahedra are arranged, and that's probably the reason of the disagreement with your results. I tried to make a 3D, but it comes out not easy to understand either (I need to write a code for Mathematica…)
Build the lattice in this way. Put 6 tetrahedra with a face on a common horizontal plane to make an hexagon. Take a plane passing through the top vertices of the tetrahedra, and use it to mirror another 6 tetrahedra on the top, having each the verities in common on the mirroring plane. You have now an hexagonal cylinder. Use the cylinder as a tile to span an infinite slab. Stack slabs one over the other, sharing vertexes on the planes. Done!
On Fig. 4 you see paths that belongs to different planes: each path is raising at each step!
You can cheek yourself that for increasingly large circle, the number of paths reaching the border are increasing, with increasing number of directions, and the shortest paths all have the same length also in terms of steps!
I hope that it is now clear.
I'll try to make a 3D figure as soon as I'll have the time, and try to post it.
Or else, please, send me an email, and I'll send to you when available!
Cheers
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Tobias Fritz replied on Feb. 24, 2011 @ 12:10 GMT
Dear Mauro,
thank you for the vivid explanation, it is very clear now what you mean!
I may be wrong, but I still think your claim is incorrect. If I connect all the points reachable in 2 steps, I get a hexagon. If I connect all those reachable in 4 steps, I get a hexagon. Likewise for 6 steps, 8 steps, ... My (maybe not so clear) argument above in terms of Minkowski sums shows my these are all hexagons of the same shape.
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Mauro replied on Feb. 24, 2011 @ 16:55 GMT
Dear Tobias,
with 6 steps in my Fig 4. on the red circle, for 6 steps you get a 12-side polygon, whereas for 2 steps you get an hexagon! However, I like the way in which you are addressing the issue. Thank you for your feedback!
Mauro
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Tobias Fritz replied on Feb. 24, 2011 @ 21:04 GMT
Dear Mauro,
I feel embarrassed for still not being able to agree, but at least we have pinned the issue down to the question of which points can be reached in 6 steps.
Basically, your figure consists of a big hexagon, with a little triangle attached at each side. Right? Then which points can be reached in 6 'zig-zag' steps from the center? Well, all the vertices on the big hexagon--without the little triangles--can be reached in 6 steps: for some of these vertices, you have drawn the 6-step paths into the figure. For some of the blue paths, in particular for the ones heading to the lower left, one can simply change the direction of the very last step and one ends up at a corner of the big hexagon, outside of the red circle, in 6 steps just the like. This demonstrates how all vertices of the big hexagon can be reached in 6 steps.
However, none of these outer vertices can be reached in less than 6 steps; this one can see by simple examination. Hence, the farthest one can go in 6 steps is precisely this hexagon.
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Mauro replied on Mar. 6, 2011 @ 15:45 GMT
Dear Tobias,
sorry for not having replied to you soon, but I didn't see your further post. I did some experiments for larger circles, and I noticed that I cannot have more than 12 sides. This maybe connected to your point. In such case, this idea doesn't work and one needs other ways, such as using the depth of events due to clock imprecision, or some other ideas. In the meanwhile I noticed your wonderful paper, and I'm going to leave my feedback on your blog.
Cheers
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Anonymous replied on Mar. 6, 2011 @ 16:37 GMT
Dear Tobias,
I'm adding here some figures that I did to better understand your point.
attachments:
triangular4.pdf,
triangular6.pdf
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Tobias Fritz replied on Mar. 8, 2011 @ 13:55 GMT
Dear Mauro,
thanks for getting back to it! It's good to sort this out, luckily it's a mere mathematical point and therefore has a clear and unequivocal answer.
In fact, using the idea of Minkowski sum it is relatively simple to prove that a regular lattice will never give an isotropic propagation speed. Let me know in case I should explain more details.
The figures are amazing!! How did you make these?
I think I understand the first one, but the second one I unfortunately could not make sense of...
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Tobias Fritz replied on Mar. 9, 2011 @ 17:15 GMT
What I forgot to say yesterday is that the essay nevertheless is one of the most fascinating ones and contains some ingenious insights :)
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Author Giacomo Mauro D'Ariano replied on Mar. 10, 2011 @ 10:34 GMT
Dear Tobias,
thank you for your nice words. Regarding the figures, I've even better ones: the one that I posted were done only for the sake of this discussion with you, and I spent no more than 30 minutes to do them just using Xfig (available for unix-linux, or Mac Fink).
Coming back to Physics: it seems to me that our two works are much more connected than what may appear at first sight.
I'm very interested in the mathematical proof that you are mentioning that a regular lattice will never give an isotropic propagation speed (this clearly refers only to space dimensions d>1). Indeed, the only notion of "Minkowski sum" that I know is an operation between subsets of an affine space. Can you give me more information, e.g. a place where to look for your mentioned proof, or can you please give me more details?
What you say is very interesting. However, at first sight it seems to contradict the possibility of simulating the Dirac equation (which is covariant!) by a quantum computer with a a periodic network of gates. This is the case also of your graphene simulator. I believe that your proposal of the grapheme simulator is a great idea, and I want to prove it correct. But, how we reconcile a quantum computer simulation of Dirac with an anisotropic maximum speed of propagation of information?
I will post also a reply in your blog, continuing our two parallel discussions.
Let me say that from my positive experience about these blogs, the idea of FQXi of this contest is starting to pay real dividends to research in terms of interesting discussions.
Cheers
Mauro
Anonymous replied on Mar. 14, 2011 @ 18:31 GMT
Dear Mauro,
sorry for the delay, sometimes it's difficult when one has a day job, but I suppose you know that ;)
So about proving anisotropy of propagation speed in a regular lattice: The notion of Minkowski sum I mentioned is indeed the one you are familiar with. The anisotropy proof goes as follows: think of the lattice as projected onto space, ignoring the time dimension. Designate a certain starting point as the origin. Then define the "ball" B_n to be the set of points which can be reached in n steps from the origin. Clearly, every B_n is a polytope, i.e. is the convex hull of finitely many points. For a certain n (n=2 in your case), the extreme points of B_n are all translates of the origin. Then how far can we get in n+n=2n steps? From the origin, we can get to all the outer points of B_n; from each outer point, we can then traverse another n steps. And then the distance traversable in these n steps is precisely given by a translated copy of B_n! Therefore, B_2n is the Minkowski sum of B_n with itself. Hence B_2n coincides with B_n scaled by a factor of 2. The same argument applies inductively to show that

In particular, the shape of the balls B_kn is independent of k.
Concerning the comparison to graphene, yes, that's an excellent question! One difference is that we are now looking at wave functions instead of classical point particles. Then the characteristic quantity of the system is the energy-momentum relation E(p) of the (quasi-)particles. A Taylor expansion of this quantity yields precisely something of the form
where M_ij is something like an "inverse mass tensor" and summation is implied. When
holds, then the low-energy excitations have isotropic propagation speed! And as I mentioned in my essay, it is in fact only the low-energy excitations for which the whole emergence of the massless Dirac equation holds. (In light of this discussion, this is a point which I should have emphasized more...) For higher-energy excitations, isotropy does not hold. In the graphene case, anisotropies occur which are known as "trigonal warping"; I haven't been able to find a good reference for this, but google turns up a whole lot of papers on that.
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Tobias Fritz replied on Mar. 14, 2011 @ 18:33 GMT
The previous post is mine, my login had expired...
Also, the energy-momentum relation is missing a square root.
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Author Giacomo Mauro D'Ariano replied on Mar. 15, 2011 @ 11:16 GMT
Dear Tobias,
thank you very much for your simple and clever mathematical proof!
I think that, however, there must be a way to improve the isotropy especially for the massless case, which you comment in your blog in replying to my last post. Here a mechanism that I recently devised. As you may know from my paper in 1plus1 dimensions there is a renormalization of the speed c coming from the mass coupling between field operators left and right due to unitariety. In short, one has that the sum of the square-modulus of the matrix elements of the local U (in 2 steps) must be one, and this turns to be the sum of mass and speed squared. I thought that there maybe a way that in d>2 space dimensions the coupling with a larger number of modes provides more non-vanishing matrix elements (you must have a larger matrix in larger dimensions, with dimension 4 in 3plus1, with the gate shaped as two pentachorons (5-simplex), connecting 4 wires with 4 wires in space). One now has the chance of an anisotropic refraction index coming from unitariety, curing the problem. And, this maybe the way to cure also the massless case, which still needs more matrix elements in U, even without the mass coupling. Even without the mass coupling there is the need of coupling the four field modes exiting from the vertices of the pentachoron in order to recover the three 2nd order partial derivatives from a 4x4 matrix (H=U-U^\dag).
I cannot believe that the massless field has no digital analog, there must be a way! Otherwise we are proving that the world is not digital!!
Author Giacomo Mauro D'Ariano replied on Mar. 17, 2011 @ 17:12 GMT
Tobias,
there is something odd, which I cannot understand in your beautiful proof (which I'd like to be correct, since it would be a simple argiment). Apparently your assertion that the "ball" B_n (the set of points which can be reached in n steps from the origin) is a polytope is not true. See e.g. the figure here attached. Where am I wrong?
attachments:
Minkowski_small.pdf
Author Giacomo Mauro D'Ariano replied on Mar. 17, 2011 @ 19:51 GMT
I made another two drawings which describe your theorem. It seems that for some lattice (e.g. fixed coordination number) and for sufficiently large n your theorem works, but it seems that B_kn=kB_n is not generally true.
attachments:
Minkowski2.pdf,
Minkowski3.pdf
Anonymous replied on Mar. 18, 2011 @ 17:44 GMT
OK, this may become a very long post... maybe we should switch to email? Or is anyone else following this discussion here on the forum?
First of all, I think the statement is true in both your Minkowski2.pdf (as I interpret it) and also in your Minkowski3.pdf. Note that I was not claiming B_kn = k*B_n to be true for all n. Rather, I said that there exists a certain n such that this holds...
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OK, this may become a very long post... maybe we should switch to email? Or is anyone else following this discussion here on the forum?
First of all, I think the statement is true in both your Minkowski2.pdf (as I interpret it) and also in your Minkowski3.pdf. Note that I was not claiming B_kn = k*B_n to be true for all n. Rather, I said that there exists a certain n such that this holds for all k. In Minkowski3.pdf, you have drawn all B_n from B_1 to B_5. The relevant value for n here is n=2. And we have indeed B_4 = 2*B_2, as claimed.
However, your previous post did point out a problem in my proof. So I have been going back to the drawing board and thought about it all again. By now, I think I have a mathematically precise formulation of the statement as well as a rigorous proof. Here it comes.
The setting is the following: let us consider any
periodic graph which is an infinite, connected and locally finite graph G=(V,E) together with an embedding of G into R^d, where d is arbitrary. The main assumption is that this embedding is periodic: there is a group Z^d acting as translations on R^d which maps the embedded graph to itself. Hence R^d decomposes into isomorphic unit cells of finite size, which are all translates of each other. In your example, the unit cell can be taken to be a hexagon made up out of 6 equilateral triangles.
Now fix any point of the graph as origin and take B_n to be the set of all vertices of the graph which can be reached from the origin by traversing at most n edges. So in contrast to my previous terminology, B_n is only a set of vertices, and not a polytope anymore; in particular, talking about "convexity" of B_n is meaningless. B_n is the set of points which can be reached in n time steps.
We are interested in how the shape of B_n / n depends on n. In particular, whether it is possible that this "velocity set" tends to a Euclidean ball as n --> oo.
*Claim:* The set
is a polytope. (More accurately: there is a polytope P such that the
Hausdorff distance between P and B_n / n converges to 0 as n--> oo.)
*Proof:* For simplicity, let us consider first the case where every unit cell contains only one vertex of G. Then any vertex can be mapped into any other by a translation preserving the graph. In this case, I will now prove that the velocity polytope is precisely the convex hull
To see this, note that, as in the previous "proof", we get B_2 by translating a copy of B_1 to all the vertices of B_1. We obtain B_3 by translating copies of B_1 to all vertices of B_2. And so on. Hence,
Similarly,
This clearly lies in the convex hull of B_1; morevoer, as n grows, we can approximate any point in the convex hull of B_1 by a point of this form. (Such a point is a convex combination of elements of B_1. Approximate the coefficients of this convex combination by rational numbers with denominator n.)
This proves the claim in the case that each unit cell contains exactly one vertex of the graph. The argument for the general case follows now. Define a an "admissible velocity" to be a vector
where \vec{s} and t are given as follows: there needs to exist a path in the graph which begins at the origin x, ends at a vertex which is a translate Tx of x, and does not traverse any other translate of x, such that t is the number of edges in the path, and \vec{s} is the direction vector from x to Tx.
Then it is clear that there is only a finite number of admissible velocities. The convex hull of these velocities is a polytope Q. The claim now is that
To see this, we prove the two inclusion separately. So why is the left-hand side contained in the right-hand side? The reason is that admissible velocities can be concatenated with each other any number of times, and this gives convex combinations of velocity vectors as above. Why is the right-hand side contained in the left-hand side? For any very long path which begins at the origin x, adding or removing a few edges does not change much, so we can assume that it ends at some translate of the origin Tx. But then we are back to a convex combination of velocity vectors.
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Author Giacomo Mauro D'Ariano replied on Mar. 18, 2011 @ 20:31 GMT
Dear Tobias,
I'm sure that the statement of the theorem is correct. I will reread the proof more closely. We'll discuss this in person (e.g. where you prove that the number of extremal points of the polytope Q is bounded, or, whatever, why is not a ball). I know that the statement is true for some lattices, but I want to exclude the existence of lattices where the limiting polytope approaches a circular ball. By the way, star-shaped sets are even farer from the circular ball!
The theorem and the proof are interesting, and you should publish them somewhere!
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Anonymous wrote on Feb. 27, 2011 @ 16:18 GMT
Very nice presentation (clear and well written), congratulations!
However, I'm having some problems understanding how space-time (time in particular) can emerge from the causal structure of the network: I really don't see how you can define "cause" and "effect" without the notion of "time", because a cause must by definition precede IN TIME an effect. (Of course there are also other further requirements for such definition.)
In other words, it seems that the clock time tau of the computation of the quantum computer is "sneaked in" to constitute the elementary building block of time, which then does not really emerge, but it is "sneaked in". [By "clock" here, I mean the clock in a computer, namely the time it takes for a single elementary operation to complete, which in normal computers gives the processor speed.]
I have similar concerns also for the emergence of space.
Probably I'm missing something! Thanks in advance for the clarifications...
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Mauro replied on Feb. 28, 2011 @ 20:16 GMT
Dear Anonymous Reader,
thank you very much for your kind appreciation!
Here's the point that is needed to understand the emergence of space-time from the causal network.
First, causality must be defined in a way that is independent on the arrow of time: otherwise, you cannot consider even the mere possibility that information could be sent from the future. Or, equivalently: you cannot even imagine the possibility of time-travel!
Causality is defined in my Ref. [8] (and also in [10]) without reference to time. There you have events in input-output connection: causality is the assumption that the marginal probability of any event does not depend on the set of events connected at the at its output. We then assume the time-arrow coincide with the causality arrow, i.e. with the in-out direction. In short: cause and effect are defined simply through an asymmetric dependence of marginal probability.
The emergence of time (as well as space) should now be regarded as the emergence of the Minkowsky "metric" from pure "topology" through event-counting. And this can be done via building-up of foliations over the quantum circuit. The time tau and distance a are just the digital-analog conversion from pure dimensional numbers (event counting) to the usual seconds and meters.
I hope that I answered your question!
You can convince yourself that space-time is always referred to events (not that events happen within space-time), by taking the lesson from Einstein literally: time and space must be defined operationally through measurements. Then, ultimately, each measurement is referred to a single observer, the AYPT (at your place and time) through an history of previous observations (please, read my answer to Thomas). Thus whatever happens in the four-dimensional Minkowsky space-time is precisely contained in a zero space-dimensional local memory. It is like the stream of bits of a 3D movie.
Let me know your opinion now!
Cheers
Mauro
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Anonymous replied on Mar. 2, 2011 @ 09:51 GMT
Thanks for the clarification! I'm still a little confused though. If events are "facts of the world describable by the basic language obeying the rules of predicate logic", I'm not sure how I can assign a probability to an event (and hence calculate a marginal). The event either happens, or it doesn't happen. Probabilities pertain to our predictions only (namely to our ignorance of some fact). What exactly do you mean by "probability of an event"?
Also, when you speak of the events connected to the input and to the output, you are implying that the input happens before (IN TIME) than the output. I would say that is implicit in the notion of input-output. Can you instead define input and output without resorting to time?
In other words, I'm sorry, but still don't see how you can relate events without assuming time...
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Mauro replied on Mar. 4, 2011 @ 06:07 GMT
Dear Anonymous,
From your answer I infer that for you the impossibility of time-travel is a tautology, still many authors believe that time-travels are possible!
In a time-travel the input is in the future and the output is in the past...
Cheers
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Guilford Robinson wrote on Mar. 11, 2011 @ 17:51 GMT
Giacomo
I enjoyed reading your article. I have been interested in how the quantum computer would combine the digital and the analog properties.
Relative to the mass-dependent refraction index of the vacuum, what would the effect be if there is a different Planck mass employed? One that is on the order of the mass of the electron and the mass of the proton--at the same time, keeping the Planck length. See my article, and review the connection of the Planck length realm and the election-proton realm.
Guilford Robinson
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Author Giacomo Mauro D'Ariano replied on Mar. 15, 2011 @ 23:48 GMT
Dear Guilford,
thank you for your interest and your compliments! I just downloaded your paper: I'll take a closer look at it (it doesn't look easy to follow at first sight).
The refraction index of vacuum is a function of the ratio between the Compton wavelength and the distance 2l between two next neighbor in-out independent gates, and the same ratio expressed in terms of mass ratio gives the Planck Mass if you take 2l equal to the Planck length. One would need indeed another good reason to chose 2l as the Plank length: the only thing that I can say is that it is the minimum distance in principle between causally independent events. Clearly, if you take 2l larger than the Planck length, you may incur in imaginary refraction indexes (corresponding to absorption?) which is odd. Whence the Planck mass must be the largest possible mass of the field, and information halts at such mass value!
I hope that this is what you were looking for. Please let me know.
Peter Jackson wrote on Mar. 16, 2011 @ 00:17 GMT
Dear Guacamo
In a last minute 'trawl' of essays I hadn't read I was pleased to come across yours. Your clear and lucid description of a quantum computer was very interesting and refreshing, and a new angle on my own model.
I hope you'll read my rather analogue version of what seems to be QC=SR, entirely equivalent, explaining special relativity logically with a quantum mechanism and deriving Equivalence with a = g.
Probably too late for you to vote now, but I'd like your take on it anyway. The lower string gives some good analogies. http://fqxi.org/community/forum/topic/803
Best wishes
Peter
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Author Giacomo Mauro D'Ariano replied on Mar. 17, 2011 @ 20:54 GMT
Dear Peter,
thank you for your appreciation.
I downloaded your paper. It looks very nice, but with a lot of physics that I cannot check myself.
Best regards,
Mauro
basudeba wrote on Mar. 20, 2011 @ 06:23 GMT
Sub: Possibility of manipulation in judging criteria – suggestions for improvement.
Sir,
We had filed a complaint to FQXi and Scienticfic American regarding Possibility of manipulation in judging criteria and giving some suggestions for improvement. Acopy of our letter is enclosed for your kind information.
“We are a non-professional and non-academic entrant to the Essay...
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Sub: Possibility of manipulation in judging criteria – suggestions for improvement.
Sir,
We had filed a complaint to FQXi and Scienticfic American regarding Possibility of manipulation in judging criteria and giving some suggestions for improvement. Acopy of our letter is enclosed for your kind information.
“We are a non-professional and non-academic entrant to the Essay contest “Is Reality Digital or Analog”. Our Essay under the same name was published on 29-12-2010. We were associated with Academic Administration as a part of our profession before retirement. From our experience, we were concerned about the problems and directions of current science. One example is the extended run and up-gradation given to LHC, (which was set up to finally prove that Standard Model and SUSY were wrong), even when Tevatron is closing down. Thus, after retirement, we were more focused on foundational works addressing, in one of its many facets, our understanding of the deep or “ultimate” nature of reality.
Specifically we were concerned about the blind acceptance of the so-called “established theories” due to the rush for immediate and easy recognition even on the face of contradictions raising questions on the very theories. One example is the questions being raised on the current theories of gravitation after the discovery of Pioneer anomaly. While most students know about MOND, they are not aware of the Pioneer anomaly. Most of the finalists of this contest have either not addressed or insufficiently addressed this question. We hold that gravity is a composite force that stabilizes. This way we can not only explain the Pioneer anomaly and the deflection of the Voyager space-craft, but also the Fly-by anomalies.
Similarly, we were concerned about the blind acceptance of some concepts, such as inertial mass increase, gravitational waves, Higg’s boson, strings, extra-dimensions, etc. Some of these are either non-existent or wrongly explained. For example, we have given a different explanation for ten spatial dimensions. Similarly, we have explained the charge interactions differently from the Coulomb’s law. We have defined time, space, number and infinity etc., differently and derived all out formulae from fundamental principles. There are much more, which we had discussed under various threads under different Essays. We are the only entrant who defined “reality” and all other technical terms precisely and strictly used this definition throughout our discussion.
Though our essay was on foundational concepts and we derived everything from fundamental principles, it was basically alternative physics. Moreover, we are not known in scientific circles because we did not publish our work earlier. Hence it is surprising that even we got a community rating of 3.0 and (12 ratings) and Public Rating of 2.5 (2 ratings). We have no complaints in this regard. However, we have serious reservations about the manner in which the finalists were chosen.
A set of thirty-five finalists (the “Finalists”) have been chosen based on the essays with the top Community ratings that have each received at least ten ratings. The FQXi Members and approved Contest entrants rate the essays as “Community evaluators”. Since many of the FQXi Members are also approved Contest entrants, this effectively makes the contestant as the judge for selection of the finalists. This process not only goes against the foundational goals of the Contest, but also leaves itself open for manipulation.
Most contestants are followers of what they call as “mainstream physics”. Thus, they will not be open to encourage revolutionary new ideas because it goes against their personal beliefs either fully (like our essay) or partially (like many other essays that did not find place in the final list. One example is Ms Georgina Parry. There are many more.) The prime reason for such behavior is cultural bias and basic selfish instinct of human beings. Thus, truly foundational essays will be left out of the final list.
In support of the above, we give a few examples. While there are some really deserving contestants like Mr. Julian Barbour, who really deserve placement in the final listing, the same cannot be said for many others. Mr. Daniele Oriti, who tops the list of finalists, says that whether reality is digital or analog “refers, at least implicitly, to the ‘ultimate’ nature of reality, the fundamental layer.” He admits that “I do not know what this could mean, nor I am at ease with thinking in these terms.” Then how could he discuss the issue scientifically? Science is not about beliefs or suppositions. His entire essay exhibits his beliefs and suppositions that are far from scientific descriptions. He admits it when he talks about “speculative scenario”. Yet, his essay has been rated as number one by the Community.
The correspondence between us and Mr. Efthimios Harokopos under his Essay and our comments under the various top ranking finalists show the same pattern. One example is Mr. Paul Halpern. We have raised some fundamental questions under the essay of Mr. Hector Zenil. If the answers to these questions are given, most of the finalists will be rejected. If the idea is to find out the answers to these questions, then also most of the finalists will be rejected.
The public that read and rated the essays are not just laymen, but intelligent persons following the developments of science. Their views cannot be ignored lightly. Mr. Daniele Oriti, who tops the list of finalists as per community rating, occupies 35th place in public rating. Mr, Tejinder Singth, who is 7th among the list of finalists as per community rating, occupies 25th place in public rating. If public rating is so erroneous, it should be abolished.
Secondly, the author and interested readers (including FQXi Members, other contest entrants, and the general public) are invited to discuss and comment on the essay. Here personal relationship and lobbying plays an important role. An analysis of the correspondence between various contestants will show that there was hectic lobbying for mutual rating. For example: Eckard Blumschein (Finalist Sl. No. 15) had written on Mar. 15, 2011 to Mr. Ian Durham (Finalist Sl. No. 3) “Since you did not yet answered my question you give me an excuse for not yet voting for you.” There are many such examples of open lobbying. One of the first entrants visited most contestants and lobbied for reading his essay. Thus, not only he has received the highest number of posts under his Essay, but has emerged as one of top contenders.
The above statement gets further strengthened if we look at the voting pattern. More than 100 essays were submitted between Feb.1-15. Of these 21 out of 35 are the finalists. Of these the essays of 14 contestants were published in 5 days between Feb. 14-18. Is it a mere coincidence? For some contestants, maximum rating took place on the last day. For example, on the last date alone, Mr. Paul Halpern rose from 14th place to 5th place, Mr. Donatello Dolce rose from 35th place to 14th place, and Mr. Christian Stoica came into the top 35. All these cannot be coincidental.
Thirdly, no person is allowed to submit more than one essay to the Contest, regardless if he or she is entering individually or as part of a collaborative essay. Yet, we suspect that some have indulged in such activities. For example, we commented below the essay of one contestant on March 4. We got a reply from the next contestant the same day. The correspondence continued. The original contender has not replied to us. In fact he has only replied twice in 20 posts. This is surprising.
In view of the above, we request you to kindly review your judging process and forward all essays to an independent screening committee (to which no contestant or their relatives will be empanelled), who will reject the essays that are not up to the mark and select the other essays without any strict restriction on numbers to the final judges panel. This will eliminate the problems and possibilities discussed by us. This will also have the benefit of a two tier independent evaluation.
Our sole motive for writing this letter is to improve the quality of competition. Hence it should be viewed from the same light”.
Regards,
Basudeba.
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Sridattadev wrote on Jun. 9, 2011 @ 19:15 GMT
Dear Mauro,
Congratulations. Are'nt we the quantum computers our selves hooked in the network of the universe?
who am I? I am virtual reality, I is absolute truth.
I am a quantum computer, I is the network.
Love,
Sridattadev.
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William Amos Carine wrote on Jul. 29, 2013 @ 20:07 GMT
Hello Mauro,
Does the Dirac equation always deal with electron? Where it does, is there an upper limit of how much energy or information it can have, a point that to go above or further would be a violation of some principle or mathematical sense?
Thanks for directing me back to your other two essays.
Best,
Amos.
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Darius M wrote on Jun. 22, 2014 @ 09:48 GMT
I think reality how it is in itself is analog, while appearances are digital.
https://www.academia.edu/7347240/Our_Cognitive_Frame
work_as_Quantum_Computer_Leibnizs_Theory_of_Monads_under_Kan
ts_Epistemology_and_Hegelian_Dialectic
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