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

    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.

      • [deleted]

      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

      • [deleted]

      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

      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.

      • [deleted]

      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

      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

      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.

      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

      • [deleted]

      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...

        • [deleted]

        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

        • [deleted]

        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...

        • [deleted]

        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

        • [deleted]

        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

        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...

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

        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

        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

          • [deleted]

          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.