Is Reality Digital or Analog? by Jarmo Matti Mäkelä

The partition of integers is important in counting the number of states on a black hole horizon. The area of a black hole is composed of little quanta of areas given by a sum of integers n_i >= 0,

A = 4 π a(n_1 n_2 ... n_m)

where this total number N = n_1 n_2 ... n_m can be written according to the integer partition. Another way of thinking about this is that the string modes can exist in a distribution which is an integer parition. This is the holographic principle in action, where the event horizon or stretched horizon is composed of a "gas" of strings.

The density of states for a string is tr(w^N) , which for N = \sum_nα_{-n}α_n the string number operator. Given there are 24 string operator the computation of this generating function is tr(w^N) = f(w)^{-24} for

F(w) = prod_{n=1}^∞(1 - w^n)

This is a form of the Dedekind η-function and the remaining calculation leads to a form of the Hardy-Ramanujan approximation for the integer partitions.

The black hole in the holographic setting has a stretched horizon which is a gas of strings. If we consider the string to be the bosonic string in d = 26 then 24 correspond to the SO(24) group for the graviton plus dilaton and a gauge field. So the Newtonian insight here seems to be pointing in this direction.

Cheers LC

Hi,

thank you for your answer. From the Newton character I understand that:

(1) Unruh effect => "a sort of atomic structure of spacetime"

But how do we know about the Unruh effect? It is not an experimental fact in search of an explanation. We know about it because it follows from the principles of QFT. So we have:

(2) QFT => Unruh effect

From (1) and (2) we have

(3) QFT => "a sort of atomic structure of spacetime"

But QFT is based on continuous spacetime. So, we have something like

continuous spacetime (combined with other principles) => "a sort of atomic structure of spacetime"

This is what I don't understand.

Best regards,

Cristi

Hi Jarmo,

Actually you agree with me that you take as a hypothesis that the simplexes you use for discretization live in the "ordinary, classical spacetime", so you build upon a continuous background. I was under the impression that your Newton character claims that reality is discrete.

Just in case you think that "simplex in a riemannian space" is sufficiently flexible, let me give you a distance distribution with the property: there is no embedding of this finite metric space into any riemannian manifold. Take the simplex vertices to be O, A, B, C, D and distances between them as follows: OA=OB=OC=OD=1, AB=AC=AD=BC=BD=CD=2.

The Question wether Reality is digital or analogue struck me as extremely unrealistic: those are just different ways humans can handle what they believe to be reality. Of course reality is neither and is beyond any human approach except by intuition. Therefore I think this question can only be put by extremely short-sighted people. It is not really aserious problem at all.

I know this is not the subject here, but I saw a possibility to make my point.

Jarmo Matti Mäkelä,

I just printed off your essay. I will be looking for why you asked Newton about the 2nd law of thermodynamics instead of asking Clausius:

"I am not quite sure what Newton really had in mind. I think that he simply meant that at appropriate length scales the metric tensor, which depends on the quantum states of the constituents of spacetime, has a signature (-,+,+,+). The process, which brings this result may or may not have something to do with decoherence, but I am not sure about that."

I don't believe that either of them, even today, would state the quote above. I think that both of them would have first determined what thermodynamic entropy is before skipping past it to what I consider to be sidestepping the question. In other words, giving it later assigned meaning that I do not see applying to Clausius' definition.

James

An excellent and entertaining entrance to your essay Jarmo, congratulations on your imagination and ingeniuty. I have a burning question which I've always wanted to ask Newton though, which is this:

Q: Since he equated the ancient greek philosophy of the smallest irreducible particle, called an atom, with the motions of the planets as observed by Galileo Galilei, does he want to know what his very large unspoken logical assumption was, which has now meant that humanity has been led down the wrong scientific path?

Ans: He assumed that the cores of the planets and sun are composed of the same everyday matter which is found on the external crust. (It's not necessarily the case and so invalidates the whole of Einstein's space-time concept imo and also invalidates the results of the Cavendish experiment to 'weigh' the Earth).

Eckard, An excellent comment. I found Valdes-Marin's essay on 'Structure and Force' to define causality rather well.

Jarmo, An interesting and enjoyable format.

Actually, the Hawking effect (black hole radiation) also follows from ordinary QFT, although applied in curved spacetime. Also, the derivations of the Hawking and the Unruh effects are petty similar. The point is that in both effects one may (at least in principle) measure some temperature for matter coming out of the vacuum. Since spacetime interacts with matter, one may consider the temperature of the matter as the temperature of spacetime from the point of view of an appropriate observer. If spacetime has temperature, it presumably has some internal structure, which produces that temperature. For more dwetails you may have a look at Refs. [13] and [14] in my essay.

Best,

Jarmo

Enormously clever! And I don't just mean the conceit of talking to Isaac Newton (with a nice paradox-saving ending). I mean the journey through algebra, combinatorics, geometry, to the 4-simplex whose edges correspond to the 10 non-redundant points of the Riemann metric tensor. That is literally where "the rubber meets the road," as they say -- where the discrete comes smack up against the continuous, where " ... the causal properties of spacetime ..." imply the existence of " ... a still unknown law of nature ..." that we hope quantizes spacetime.

I'm betting that you're a very popular lecturer. Bravo.

Tom

A temperature for spacetime is an interesting possibility, but one which might have some problems. To assign spacetime an internal structure with a metric means the entropy of spacetime can be very large. This is an enormous number of degrees of freedom. However, the temperature of spacetime might just be confined to horizons, which reduces the numbers of degrees of freedom, and fluctuations or dynamics outside an event horizon has a map or holographic realization on an event horizon.

Cheers LC

I said that no one knows what thermodynamic entropy is. The point I am making is that Clausius' definition of entropy is the real definition of entropy. The transferrence of Boltzmann's constant onto Boltzmann's entropy was the beginning of losing sight, if we ever had it in sight, of the meaning of Boltzmann's constant. Boltmann's interpretation of entropy was the first step in ignoring Clausius' definition and finding some more simplistic substitute. I think that Newton would have first said: What is thermodynamic entropy? Perhaps there are others here who will clarify the meaning of thermodynamic entropy for me?

James

Hi Mäkelä,

Your article is ingenious because you put in Newton autority mouth, your ideas.

Guessing, I supose that Newton will agree more with an universal gravitation law modified than with Einstein TGR.

If you replace Beta in the partition formula by one divided Boltzmann constant times Temperature, and as Hawking temperature in a Black Hole is proportional to the inverse of Energy the formula doesn´t diverge.

Bests,

JC

Hi,

When you calculate the partition function, you must assume that the inverse temperature Beta is a constant, and therefore independent of the energy. Once after you have calculated the partition function, you will find the relationship between energy and temperature using the formula of Note 6.

Best,

jarmo

Your metric tensor, and your Newton analogy are very clever. I would like to understand how that contributes to the quantum limit as you propose.

Could you elucidate this point more clearly ?

Andy

Dear Jarmo

I am embarrassed to admit I seem to have a confession to make. Some time ago I heard someone fall down some stairs and went to help, he was in a bad way, mumbling is some alien language and delirious. I picked him up with his papers and bits and carried him off to hospital.

I found afterwards I still had his papers and a strange little machine with HG Wells stamped on it.. but the poor fellow had mysteriously disappeared. I read the papers but they had no name or address. They were entitled "The Complete Quantum Theory of Gravitation". I really didn't know what to do with them. If I'd known they were yours I'd have put them in a time capsule...

As it was I thought I must try to put them and the machine to good use. They both rather scared me! I was worried about causality. What I decided to do was pop forward the 300 years the machine was set to and check if all was ok and we'd managed to find the theory.

Imagine my horror when I discovered it was my fault that we hadn't, ..and that science seemed to have gone backwards in time! I didn't have a clue if you were around or where to find you. The world was strange to me but I managed to survive by popping back and making the odd bet and investment, and have had fun directing some great blockbuster films about a land I visited using your... well never mind about that. I spent a few years checking the theory and it seemed spot on so I eventually decided it best to write it up in my own way and give it to the world. - the complete atomic structure of space time, unifying SR, GR and QM.

But apparently the world doesn't understand it! As only you will I hope you'll read my essay, gently slipping in that the sub space-time medium condenses into an atomic structure to produce space-time itself. The quantum states are ions, re-polarised by black holes, and distributed as plasma in such a way that the every day notion of time and causality are recovered. Actually I think that wording is yours, I mean Sir Isaac's, not mine!

It's been a difficult time, but a massive relief to have found it's owner. I did tentatively refer to it in last years contest, and touch on it quite lightly in my current essay languishing some way below yours (2020 Vision..) But it's better covered in a number of preprints I can pass to you. Here are some recent ones; http://vixra.org/abs/1001.0010 and http://vixra.org/abs/1102.0016

I hope you're able to finally confirm I'm not totally insane. The only way I've managed to hide from the men in white coats so far is to be secretly the 3rd richest man in the world.

I pass it all to you, in Isaac's name. - Do you think I should withdraw or just give you both credits and citations in the paper currently accepted for review!

With massive relief

Peter Jackson

PS. You have a well earned 10 from me. But please do reply asap.

Dear Jarmo,

I must agree with many of your other readers, in that you have given a very creative presentation. The introduction and conclusion are very well and clearly written and are enjoyable. Unfortunately, not being a physics specialist, I found the middle of the essay too technical to follow. Also as I have limited mathematical skill I can not judge if what you have said is technically correct and rigorously argued. I will have to leave that judgment to others.I can see some lively debate and much interest in your comments thread. It is certainly topical and foundational.I suspect it is far more interesting and goes far deeper than I can appreciate and understand.I wish you good luck, Georgina.

Hi,

Thanks for your interesting post. It would really be wonderful, if the papers written by Sir Isaac still existed! I have been reading your pre-prints. They are interesting, but I am not quite sure, whether they are quite similar to the ones, which I was reading just before that unfortunate fall in Newton's stairs. Maybe you have misunderstood some points in Newton's draft? Do you still have the originals? If you do, please take photocopies of them, and send them to the address mentioned in my essay. Many thanks for taking me to the hospital after my fall in Newton's stairs, although I have no recollection of that.

Best,

Jarmo

Dear FQXi community,

There are three kinds of essays in our contest: 1) the essays with original physics research where all physics' information was created by their authors. Often such papers contain some errors or unclear information because it is very difficult to create a NEW physical theory (information). 2) There are essays-stories about physics which contain physics' information copied from physics textbooks or other published papers (for example authors Jarmo Makela, Singh, Durham, Funakoshi and so on). The author's commentaries like 'this theory is good, or not are neither original physics research nor new physics information. These essays-stories cannot have any errors by definition because all physics' information was copied from the textbooks and other published papers. (However, if these authors have added original physics research information in their articles, I'm sure we'll find physical errors in their papers).3) There are essays of mixed type containing mixed information (original research plus physics' information copied from the textbooks and papers). It is clear that the authors of the essays-stories have advantages because their essays never contain errors since all Physics' information was copied from the textbooks. However, it does not mean these essays-stories are better than essays with original research.

What kind of the essays must FQXi community support? If we support the essays-stories then we'll transform FQXi community into the entertainment community. For example, instead of my ''interpretation of quantum mechanics'' I could send the anecdotes about Bohr, Einstein or stories like Gamov's Mr. Tompkins in paperback. It would be very interesting and fun. Another option is to create essays-discussions with Einstein, Bohr, or Aristotle following the example of Jarmo Makela. In this context, the next logical step is to organize a banquet for the authors of essays where we tell jokes and funny stories about physics. What is Our Purpose?

However, since the goals of the FQXi (the "Contest") are to: ''Encourage and support rigorous, innovative, and influential thinking about foundational questions in physics and cosmology; Identify and reward top thinkers in foundational questions'' therefore I ask readers to vote for essays with original physics research rather than for essays-stories about physics. In this way we'll encourage the fundamental physics research but not entertainment essays.

Sincerely,

Constantin

Hi,

Thanks for your message. I you want a really technical account on how metric tensor arises from the quantum states of the constituents of spacetime, (I suppose that was your question?) I advice you to have a look on my paper in arXiv:0805.3952 (Ch. 5). When expressed in very broad, non-technical terms the basic idea is that when a sufficiently large aggregate of the constituents of spacetime is considered, it becomes possible to talk meaningfully about four-simplices. The areas of the triangles of those four-simplices depend on the quantum states of their constituents, and we may express the components of the metric tensor in terms of those triangle areas.

Best,

Jarmo