Tong’s Integers

David Tong Physics and the Integers Foundational Questions Institute (FQXi) 2011

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.

This seems an accessible account as far as it goes, but as a mathematician I would like to say more.

A History Lesson
“God made the integers, the rest is the work of man” — Leopold Kronecker [1]

I have never really understood this quote. It may be fine for mathematicians, but it doesn’t seem to gel with how I understand the laws of physics. In part, the purpose of this essay is to explain why.

Tong gives a useful discussion. But it is perhaps worth noting that in Kronecker’s time the relationship between mathematics and physics was largely misunderstood, as perhaps it still is by some. For example, it was assumed that the points and lines of geometry were necessarily adequate to model ‘real’ space, whereas since Einstein and Eddington we recognize that such questions are the province of physics, and open to dispute and investigation. Thus the questions of mathematical ‘correctness’ and physical ‘correctness’ are independent. Thus it is perfectly possible to suppose that God made both the (mathematical) integers and ‘real’ space-time, without necessarily supposing any particular correspondence between the two. As with classical geometry, it may simply be that mathematical ‘models’ (discrete or otherwise) are useful, perhaps even valid models of some concept of reality, as distinct from being models of reality ‘as such’. As in economics or epidemiology it is sometimes (perhaps often) ‘pragmatic’ to gloss over this distinction, but sometimes not. Perhaps Tong’s essay can help us appreciate the difference?

The Integers in Nature
It is not, a priori, obvious that the integers have any role to play in physics.

Indeed. Others have noted more generally that it is not obvious why any mathematics should be so effective. Possibly mathematics is effective only in so far as it is the language of contemporary concepts about reality, and has (so far) proven itself capable of sufficient extension to cover experimental findings.

The Integers are Emergent
The examples above show that discrete objects undoubtedly appear in Nature. Yet this discreteness does not form the building blocks for our best theories.

One could easily interpret geometry as being about real-world space, arithmetic as being about real-world counting, and mathematical probability as being about real-world uncertainty. But in so far as these are mathematical concepts they have nothing to say about reality: that is the province of the physicist. But when a physicist says that ‘integers are emergent’, what does this say about mathematics?

Searching for Integers in the Laws of Physics
While the quanta of quantum mechanics are emergent, we could also play a more fundamental version of the “count-the-planets” game, this time trying to find integers within the laws of physics themselves. Are there inputs in the fundamental equations that are truly discrete? At first glance, it is obvious that there are; at closer inspection, it is more murky. I should warn the reader that this small section is substantially more technical than the rest of the paper and little of the general argument will be lost if it is skipped

Here, I think, is the ‘meat’ of the paper, from the point of view of physics. Tong notes:

The dimensions of space are not the robust objects that they appear. At heart, the issue is exactly the same as that described above: the dimensions of space count the number of degrees of freedom of a particle that lives in it. And, like ripples on the pond, the number of degrees of freedom is not so easy to count in interacting theories.

This seems to me to mirror the observation of the mathematician C.L. Dodgson: that a finite number observations can only reduce the dimensionality of possible reality by a finite amount, and so no finitistic model can ever be justified except by assuming some a priori limitation on the dimensionality of reality.

Could the Known Laws of Physics be Fundamentally Discrete?
Above I have argued that the presence of discrete structures in Nature is either emergent or illusory. Either way, there is no evidence for an underlying digital reality. But absence of evidence is not evidence of absence. Is it possible, nonetheless, that Nature is at heart discrete?

If I had been asked, I would have suggested clarifying the heading. My understanding is that to be acceptable as ‘laws of physics’, theories must be finitistic, and yet (as above) one can never be sure that reality is finitistic. Indeed, there seems a good argument that given any definite finitistic reality there ‘must’ be the potential for the emergence of additional reality. At least, any contrary view would seem to lead down a metaphysical rabbit-hole.

But still, maybe reality is not constrained by logic?

… no one knows how to write down a discrete version of the current laws of physics.

An important point. But the broader question of whether, at some point in the future, the ‘known laws of physics’ could be amended to be discrete remains.

Could the Unknown Laws of Physics be Fundamentally Discrete?
Until now, I have tried to resist speculation, preferring to mostly restrict attention to a study of the known, confirmed laws of physics. But what does this tell us about the underlying reality of the world? Are the true laws of physics discrete? Is spacetime discrete? Should we all be searching for the green digital rain seen by Neo at the end of the The Matrix?

To clarify (I think) the ‘known, confirmed laws of physics’ means ‘known and confirmed to be the accepted laws, as distinct from necessarily being ‘true to reality’. Tong concludes:

I find it striking that the discreteness we see in the world is not sewn into the equations of physics, but arises only when we solve them.

… 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.

Quite so. If by ‘computer’ one means a classical, classically programmed Turing-Machine type, then we could potentially break out being smarter than the programmer. If one considers a ‘machine-learning’ system then we might ‘break out’ of any current constraints by teaching the computer. If there are some aliens who intervene in the computer, we could aspire to outsmart or teach them. In any case, we would want to understand the nature of our constraints, if any.

It may also be worth considering the possibility that the universe is not finitistic and/or not compact.

Dave Marsay

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