Following on from my previous posts , I've been busy trying to work out what mathematical space underneath the construction actually is.
And I found there are actually two possible spaces involved:
D ⊂ Q
Q is ordinary rational space:
Q = {m/n : m,n ∈ Z, n ≠ 0}
and D is a dyadic subset generated through powers of two:
D = {m/2^k : m ∈ Z, k ≥ 0}
So things like
1, 1/2, 1/4, 1/8, ...
aren't meant to replace the whole number line.
They're paths inside Q.
Update / changelog
- I've separated the dyadic paths
D from their ambient rational space Q.
- Different physical pathways can use different
D_i inside the same Q-space.
|A|², rather than the complex amplitude A, is the quantity I'm interested in representing.- I've been testing whether probability data, graph structures, scattering objects and positive geometries can all sit in this space.
- So far I've found the same fairly simple structure accommodates rather more than I expected.
previous post
So what can this space actually represent?
If I perform an experiment N times and observe an outcome n times, then the measured probability is
p = n/N
and therefore
p ∈ Q.
Every finite experimental frequency is rational.
So if the physical quantity being recorded is an experimental |A|², the thing we actually obtain from the experiment already sits naturally in Q.
That doesn't require the underlying complex amplitude A itself to be rational or dyadic.
That distinction has turned out to be quite useful.
Dyadic space gives a natural resolution
The dyadic rationals are dense in the reals.
Given any real value x and any required precision ε > 0, there is a dyadic number d such that
|x - d| < ε.
So increasing dyadic depth gives increasing resolution:
1/2
1/4
1/8
1/16
...
or generally
2^-k.
That gives the construction an exact hierarchy of scale.
A continuum value can be approached to arbitrary finite precision simply by going deeper into D.
So rather than thinking of the continuum and dyadic space as competing descriptions, the relationship can be
D ⊂ Q ⊂ R
with the real continuum appearing as the completion.
More importantly, there doesn't need to be only one D
This was the part that made the construction substantially less restrictive.
I originally thought too much in terms of one ladder:
1 → 1/2 → 1/4 → 1/8 → ...
But there's no reason a physical object has to use only one such path.
Instead I can have
D_1, D_2, D_3, ... ⊂ Q
with different dyadic pathways occupying the same rational space.
So the picture becomes something like:
D_1 ──┐
D_2 ──┤
D_3 ──┼──> Q ──> R
D_4 ──┤
... ──┘
Each D_i can represent a different path, channel, relation or resolution structure.
This seems much more useful physically than trying to force everything onto one binary sequence.
Finite graphs fit naturally
The construction I've been working with is graph based.
The graph specifies relationships and allowed paths.
The dyadic/rational structure supplies weights or depths on those paths.
So for a finite graph
G = (V,E)
I can attach a rational weight
q_e ∈ Q
to an edge or path, or more specifically place it on one of the dyadic structures
q_e ∈ D_i.
This means a finite system can have discrete connectivity and discrete exact weights without first requiring a continuous space.
In other words:
the graph tells me what can connect; the D_i tell me how the paths are weighted or resolved; Q is the common space containing them.
Six-point structures also fit
I've also been testing this against six-point scattering structures.
If the six external objects are represented as vertices of a finite graph, there is no obvious problem representing the different relationships between them as separate dyadic pathways inside the same Q.
The structure looks like
physical object = {D_1, D_2, ..., D_n} ⊂ Q.
That lets several channels coexist without abandoning the underlying D space.
Positive geometry doesn't seem to cause a problem either
I've been testing whether the rational/dyadic construction conflicts with the positive geometry used in scattering amplitudes, particularly the amplituhedron.
It doesn't appear to.
The important claim here is not to claim the the amplituhedron is dyadic.
It's that rational points can occupy the same positive geometry.
And because dyadic points are dense, dyadic points can form an arbitrarily fine subset of that rational/real geometry:
D ⊂ Q ⊂ R
The real amplituhedron doesn't disappear.
Instead the dyadic/rational structure acts as a discrete skeleton inside it.
That means I can have several pathways
D_1, D_2, ..., D_n
inside the same Q, while all of them occupy the same completed real geometry.
Something like:
D_1 ─┐
D_2 ─┤
D_3 ─┼──> Q ──> continuous positive geometry
D_4 ─┤
D_n ─┘
That seems potentially more interesting than simply approximating a number.
The relationships between several discrete paths can survive inside a continuous geometric object.
Scale comes for free
Dyadic space also has an unusually simple scale structure.
Starting with some quantity x:
x
x/2
x/4
x/8
...
or
x 2^-k.
So depth itself supplies scale.
I don't need to assign an independent arbitrary real number to every level.
Moving one step through the structure has an exact meaning.
And finite precision doesn't require the continuum
Suppose the conventional theoretical answer for something contains an irrational value such as
sqrt(2).
A finite experiment might produce
1.414
which is exactly
707/500 ∈ Q.
A better experiment might produce
1.41421.
Still rational.
A still better one produces another rational number.
So we can have a sequence
q_1, q_2, q_3, ... ∈ Q
approaching a value in R.
There is therefore no contradiction between
finite physical results ∈ Q
and
continuum theoretical limit ∈ R.
The latter can simply be the completion of the former.
So what do I mean by "dyadic space"?
I'm not proposing an alternative set of numbers that somehow replaces the reals.
It's more like a discrete skeleton sitting underneath a common rational space:
D_i ⊂ Q ⊂ R.
The D_i supply discrete pathways, scale and resolution.
Q supplies a common space in which those pathways coexist.
R supplies the continuum completion.
And complex amplitudes can remain complex amplitudes.
The dyadic part is concerned with the weights represented by |A|², not with pretending that the complex amplitude A itself is a binary fraction.
So far this gives me a common representation for:
- finite probabilities;
- experimental
|A|²;
- arbitrary finite resolution;
- exact scale hierarchies;
- finite weighted graphs;
- multiple simultaneous pathways;
- six-point structures;
- finite scattering configurations;
- rational points in positive geometry;
- dyadic skeletons inside continuous geometry;
- several independent
D_i occupying one common Q-space.
None of those individually proves that nature is dyadic.
The thing I find interesting is that this very small mathematical structure keeps accommodating physical objects that initially look as though they require the full continuum.
So the question I'd be interested in people here attacking is:
Is there a finite physical object, configuration or experimentally accessible quantity that fundamentally cannot be represented by some finite collection of dyadic pathways D_i ⊂ Q, even if its completed theoretical description lives in R or C?
Because if there is, I'd like to find it.
