Physics and the Integers by David Tong

David,

A fine, well argued essay.

I agree with you on the above, second case, to the extent that current computational technology must necessarily convert the continuous function differential equations of classical physics to difference equations, and settle for approximations.

In the first case, the statement is true as you phrase it. However, writing down the laws is not necessarily equivalent to computing results. We've always known that the theory of quantum mechanics is mathematically incomplete. That does not obviate the possibility of quantum computing accurately modeling a physical quantum process.

All best, and good luck in the contest. I also have an entry ("Can we see reality from here?")

Tom

David

An enjoyable and not too testing essay. I've been testing models using something similar to the laws of Scrabble (and a few decades of research) which seem to have thrown up a possible fundamental solution which is highly falsifiable empirically, and which I nor anyone else so far can find the scientific problem with. I'd be eternally grateful if you'd have a look over my essay and tell me if you can see it.

Very many thanks

Peter

David,

the laws of physics are not, at heart, discrete. We are not living inside a computer simulation.

Nice job, but your ending leaves us hanging. I like analogue though I stick my neck out.

Jim Hoover

Dear David

You have written an extraordinary essay, I enjoyed very much. On my essay I try to explain something similar within other context. What you are exposing is just the fact that the properties we see about reality are strongly related to the tools we use to model and understand it. To put it in a few words what I try to explain on my essay, is that the duality between discrete and continuum is just a consequence of use classical logic for understand our continuum rulers and pointers. In this context the apparent discreteness of quantum reality is strongly related to the measurement, which is the tool that allow to use classical logic in our partial understanding of the quantum world. I also try to explain why we should go beyond this classical logic scheme if we want to understand completely quantum reality. I would like to hear you opinion about it.

Regards,

J. Benavides

Dear David

Although you do not share my discrete religion :), I can say that your essay is very nicely and clearly written.

I had not yet heard for the Kronecker's sentence, but it is very deep.

However, I will still analyse your thoughts.

I do not understand enough, how undiscrete quantum field theory agree with Planckian discretness of space-time? Do you have any hints on this question?

I was too late for this contest. So my discrete theory is here:

http://vixra.org/pdf/1103.0025v1.pdf

Guessed formulae gave me, that measured particle masses are really time average of integer values...

Do you have any comment or counter-arguments.

Regards Janko Kokosar

Dear David,

My theory Quantum statistical automata shows how laws of nature can arise from Integer numbers only. As a matter of fact you cannot design a dynamic universe from a simple fundamental entity(just a number)in any way other than the one shown.In another word, the correct way to look at how the universe works is to design it(or see how its been designed) using fundamental entities which have to be numbers i.e. have no sub-structure.

Humans are right about their astonishment of reality, while it is easy to understand that numbers and some relationship between them do have a reality but that a particular and unique relationship has given rise to our reality is the ultimate astonishment.

httt://www.qsa.netne.net

David,

Just to be pedantic: upthread you probably should've said "fermion" sign problem. Hardly crucial. It's also known more generically as the "numerical sign problem." It's officially NP-Hard. In theory the physics should be computable or simulable on a Turing Machine, but as there might not be enough time left in the universe there's no way of proving that the theory actually equals reality.

So consider the simulation of all related physics as unlikely ever to happen unless it's suddenly proven that P=NP, which isn't generally regarded as likely either.

David,

Intriguing and well argued. I agree with your premise that the integers are emergent. In fact, I described a method in my ICCS 2006 paper by which (with no appeal to Zorn's lemma) well ordering proceeds from inherent properties of the complex plane and Euler's geometric interpretation thereof. This is purely mathematical, of course; I find it awfully interesting that you get the result from a physical construction.

My essay in this competition, which I hope you get a chance to read before tomorrow, takes a non-technical tack, but I think it does expose, as your essay also does, the great subtlety of the continuous vs. discrete question.

Nice job. Thanks.

Tom

LOL. I must be getting essay fatigue. Scanning replies, I noticed that I had already made a remark early on, where I commented on the commentary, rather than on the piece itself -- which surely deserves a careful reading, which I am happy to say that I accomplished, along with awarding a deservedly high score.

Tom

Hey David, good to "see" you here. Just dropping by to say hi and let you know that enjoyed your essay. Hope to run into you soon...

Cheers,

Moshe

Excellent essay, a breath of fresh air. Hope you will win a prize.

Hello dear David, dear Eckard,

It's relevant indeed dear Eckard, the complexs ah these complexs....after all they dance inside the sphere....conclusion viva el Reals fortunally.....

ps good luck dear finalists.....

Steve

Dear David,

Congratulations on your dedication to the competition and your much deserved top 35 placing. I have a bugging question for you, which I've also posed to all the potential prize winners btw:

Q: Coulomb's Law of electrostatics was modelled by Maxwell by mechanical means after his mathematical deductions as an added verification (thanks for that bit of info Edwin), which I highly admire. To me, this gives his equation some substance. I have a problem with the laws of gravity though, especially the mathematical representation that "every object attracts every other object equally in all directions." The 'fabric' of spacetime model of gravity doesn't lend itself to explain the law of electrostatics. Coulomb's law denotes two types of matter, one 'charged' positive and the opposite type 'charged' negative. An Archimedes screw model for the graviton can explain -both- the gravity law and the electrostatic law, whilst the 'fabric' of spacetime can't. Doesn't this by definition make the helical screw model better than than anything else that has been suggested for the mechanism of the gravity force?? Otherwise the unification of all the forces is an impossiblity imo. Do you have an opinion on my analysis at all?

Best wishes,

Alan

David,

I enjoyed reading your essay very much. I quite agree with you about the emergence of the integers as something special. A discreteness emerges from continuous semigroups. Take spheric to toric as an example of where the path integral proves useful. We are able to constrain divergence in order to commute a system in a computer simulation, and this indeed is the correct approach. I believe that the integers themselves are the culprit of breaking symmetry. Your essay was very clear, and next year I will endeavor to make my essay more like yours, You hit the soul of philosophical thinking that I like very much. Emergence allows us to do all sorts of things that we have in math that physics may have to wait a long time to realize.

Dear Alan

This is my guess:

There are Base Fermion and Base Boson of the Universe.

Both have radius size 10^-13sm

Base Fermion is proton(neutron) Mpr=10^-24 g

Base Boson is Hawking black hole Mhbl=10^16 g

Mplank; Mpl=10^-4g

Mpl=sqrt(Mpr x Mhbl)=10^-4g

10^16g/10^-24g=10^40

Fgr/Fem =1/10^40

Rounding values.

Regards

Yuri

Hi Yuri,

Thanks for the reply. I only understand pictures at the moment, my mind has become exclusive to numerical arguments I'm affraid! You may well be onto something though.

P.S I feel guilty about not reading David's essay and giving him a critique. The replies he got give me a good idea that he's 'on our side'.

Cheers,

Alan

David,

I can only re-iterate Edwin's remarks:

[quote]You have truly written a masterful essay. Simple enough to be read and enjoyed by all visitors to fqxi and yet insightful enough to teach experts a few things. Thank you for your excellent contribution.[end quote]

I wish I'd have read this during the early stages of the competition. I would have scored this a 10 with ease. The best essay here by far. I really hope that you win something.

Alan

Right from the start arithmetic was connected to music, so this is the most obvious place to look for integers. Standing waves are the paradigm: anything non integer interferes with itself and gets destroyed. So, they provide their own scale and select only multiples. Most of discrete physics boils down to this. (And if you need some extra integers, take the values of Euler's gamma function.)

Dear David,

A nice read in which one feels you know whereof you speak because of the elegance and ease in the flow of ideas! You single out the fact that no one can can write a discrete version of the Standard Model because of the need to express its chirality. Since fqxi is devoted to asking fundamental questions that remain unsolved in physics, can't we simply say that the Standard Model is far from being the last word - a simpler and more powerful theory may entirely supplant it one day.

Something similar may have already happened to the the "no-go" Nielsen and Ninomiya theorem proving that chirality cannot be represented on a lattice. I refer to this admitting the technical aspects are beyond my understanding, but the author claims he has done so: Weyl Neutrinos on a Lattice: An Explicit Construction.

I would be honored if you look at my qualitative and incomplete suggestions for a lattice theory of everything in which chirality is built in at the most fundamental 'building block' level: a universal lattice of identical dielectric nodes spinning mostly in one direction transferring angular momentum in units of (h). These ideas are in my present fqxi contest contribution, and also in the earlier 2005 Beautiful Universe paper on which it is based.

With best wishes for your success, Vladimir