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

Dear Anton,

thank you for your comment. I will look at your essay.

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.

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.

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

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.

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.

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...

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

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...

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

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 Tobias,

I'm adding here some figures that I did to better understand your point.Attachment #1: triangular4.pdfAttachment #2: triangular6.pdf

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 :)