Physics and the Integers by David Tong

David Tong

Essay Abstract

I review how discrete structures, embodied in the integers, appear in the laws of physics, from quantum mechanics to statistical mechanics to the Standard Model. I argue that the integers are emergent. If we are looking to build the future laws of physics, discrete mathematics is no better a starting point than the rules of scrabble.

Author Bio

David Tong is a theoretical physicist at the University of Cambridge and adjunct professor at the Tata Institute of Fundamental Research, Mumbai.

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Marcel-Marie LeBel

David,

A very interesting essay! Philosophy is just that. Stepping back, get a general picture and then zoom on one general "given" assumption and question it.

For Nature, mathematics is also geometry. To me, $10 in my pocket and $10 at the bank and I have $20! But in Nature, a planet adds mass or get more mass only by getting that mass to come close to its surface. The heuristic method used with kids in bringing beads together to "add them" is the way Nature actually works! The addition in the mind and the one Nature performs are different. Natural addition is bringing things closer together.

Marcel,

[deleted]

Dear David,

You wrote:"Discreteness in the world is simply the Fourier transform of compactness." May I ask you to elaborate?

If you will read my [hint:http://www.fqxi.org/community/forum/topic/833] essay [/link] you will understand why I would like to know whether or not the cosine transform instead of Fourier transform might be sufficient. Admittedly, I do not share your opinion that "God did not make the integers. He made the complex numbers and the rest is the work of the Schrödinger equation." My favorites are Euclid and to some extent Brouwer.

Regards,

Eckard

David Tong

Hi Eckard,

If you put a quantum particle in a compact box then its momentum and energy is necessarily discrete. If you put the same particle in an infinite line, then its momentum can be anything at all.

The same argument works in reverse. If the momentum of a particle is restricted in its range (i.e. a Brillouin zone) then the space it lives on is dicrete (i.e. a lattice).

The maths that underlies this is simply the Fourier transform. Your cosine transform is just the real part of the Fourier transform.

Best, David

Eckard Blumschein

Dear David,

I hope you will be ready to read and understand my essay. You wrote: "The maths that underlies this is simply the Fourier transform. Your cosine transform is just the real part of the Fourier transform." Do you consider me stupid? Why do you believe that cosine transform is not sufficient?

Regards,

Eckard

[deleted]

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

[deleted]

'Allo, monopole!

[deleted]

Some might say you've got it backward:

"Clearly, a number of important questions remain open. Of these, we mention here two. The first refers to continuous variables. The problem there is that with continuous variables, one has in principle an infinite number of complementary observables. One might tackle this question by generalizing the definition of (3.4) to infinite sets. This, while mathematically possible, leads to conceptually difficult situations. The conceptual problem is in our view related to the fact that we wish to define all notions on operationally verifiable bases or foundations, that is, on foundations which can be verified directly in experiment. In our opinion, it is therefore suggestive that the concept of an infinite number of complementary observables and therefore, indirectly, the assumption of continuous variables, are just mathematical constructions which might not have a place in a final formulation of quantum mechanics.

"This leads to the second question, namely, how to derive the Schrödinger equation. ..."

-- Caslav Brukner and Anton Zeilinger: "Quantum Physics as a Science of Information" (2005)

[deleted]

"It is not entirely clear what to make of the second piece of evidence presented in this essay: our inability to simulate the Standard Model on a computer. It is difficult to draw strong conclusions from a failure to solve a problem and the huge progress made in this area a decade ago suggests that it is probably just a very difficult issue waiting to be solved with fairly conventional techniques. I included it in this essay mostly as a warning shot to those who would insist that it is obvious that Nature is digital. But it may be worth considering the possibility that the difficulty in placing chiral fermions on the lattice is telling us something important: the laws of physics are not, at heart, discrete. We are not living inside a computer simulation."

Why would a discrete universe necessarily imply a computational one? You can't meaningfully say that something's a computation unless you can produce the algorithm. There's a whole domain of problems that are algorithmically intractable (as someone else has already pointed out in other threads, protein folding appears to be one such problem). There exist NP-Complete problems, and not even a quantum computer (as soon as one comes into being, if it does) will be able solve those in polynomial time.

How would that be proof of continua, infinities and infinetesimals?

David Tong

Hi Rick,

There may well be distinctions between discreteness and computability but the main point remains:

No one knows how to write down a discrete version of the laws of physics.

No one knows how to simulate the laws of physics on a computer.

Where by "laws of physics" I mean those that have already been established, in particular the Standard Model. I could well imagine that the first problem above is solved but not the second (there is still something called the Fermi sign problem to overcome). I could also imagine that both problems are solved. But, at the present time, both of the above statements are true.

[deleted]

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

[deleted]

Dear David,

I enjoyed your essay. I think you're right that unless discrete models can capture all the key symmetries of nature, including the properties of the chiral fermions you mentioned, we can't expect them to be useful.

I build discrete models of physical phenomena and like a good challenge. Replicating chiral fermions sound like a fine one.

Alex

David Tong

Hi Alex,

Good luck! It's a worthy problem!

Best, David

[deleted]

Hello,

You wrote: "The examples above show that discrete objects undoubtedly appear in Nature".

Undoubtedly is the word. I thought math and specifically algebra and numbers is our tool for modeling nature. I never thought that tool is nature itself.

Do I sense correctly that discrete is equated to integer in this essay? Reality may be discrete but laws may not be. Maybe I am missing something.

Lawrence Crowell

Your essay is interesting, though I tend to think the discrete and continuous aspects of physics are a type of complementarity. I don't see one as superior to the other. It is not possible to formulate a theory which conserves Noetherian currents in a discrete setting. However, in what I have done discrete structure (quotients) do have content with charges. These two perspectives are complements.

Cheers LC

Edwin Klingman

David,

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.

Edwin Eugene Klingman

Peter Jackson

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

James Hoover

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

John Benavides

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

[deleted]

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

[deleted]

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

[deleted]

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.