A Quantum-Digital Universe by Giacomo Mauro D\'Ariano

What I forgot to say yesterday is that the essay nevertheless is one of the most fascinating ones and contains some ingenious insights :)

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

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

[math]B_{kn}=k\cdot B_n[/math]

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

[math]E(p)=M_{ij}p^i p^j + O(p^3)[/math]

where M_ij is something like an "inverse mass tensor" and summation is implied. When

[math]M_{ij}=\delta_{ij}[/math]

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.

The previous post is mine, my login had expired...

Also, the energy-momentum relation is missing a square root.

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

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

[math]

\lim_{n\rightarrow\infty}\frac{1}{n}B_n

[/math]

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

[math]

P = \mathrm{conv}(B_1)

[/math]

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,

[math]

B_n = \{x_1+\ldots+x_n\:|\:x_1,\ldots,x_n\in B_1\}

[/math]

Similarly,

[math]

\frac{1}{n}B_n=\left\{\frac{1}{n}x_1+\ldots+\frac{1}{n}x_n\:|\:x_1,\ldots,x_n\in B_1\right\}

[/math]

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

[math]

\vec{v}=\frac{\vec{s}}{t}

[/math]

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

[math]

Q=\lim_{n\rightarrow\infty}\frac{1}{n}B_n

[/math]

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.

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.

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.

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.

I think reality how it is in itself is analog, while appearances are digital.

https://www.academia.edu/7347240/Our_Cognitive_Framework_as_Quantum_Computer_Leibnizs_Theory_of_Monads_under_Kants_Epistemology_and_Hegelian_Dialectic