[alancalverd]

Measurement implies interaction, transfer of energy, transfer of momentum....so it's entirely different from propagation.

I don't think we need to look for "explanations" beyond the statement that physics is about creating predictive mathematical models consistent with observation.

Do you think it's possible that photons can be observed as both, or as neither?

I think this is badly expressed. You can make an experiment whose results are predicted by consecutive models, and AFAIK these still fall into the categories of propagation (wave model) and detection (particle model).

Generation may be a quantum (particulate) phenomenon or a continuum (wave) phenomenon, the former predominating at high photon energies and where the propagated particle has a rest mass, the latter predominating for low energy e.m. radiation.

I know this approach is the inverse of school textbook physics, but as with relativity, I think the time has come when we should teach the general case and point out that where v or E is fairly low, there is a convenient classical/continuum approximation. 

[varsigma (OP)]

I know this approach is the inverse of school textbook physics, but as with relativity, I think the time has come when we should teach the general case and point out that where v or E is fairly low, there is a convenient classical/continuum approximation. 

When I studied electronics, mostly it was about designing different types of filters, mostly with op-amps. It was all radio-frequency stuff, so directly connected to communications. Everything is continuous.

But if you want to transmit quantum information, things are a bit different. Continuous at a boundary becomes important; unitarity too. At a classical scale there isn't really any way to see any of it--there is no "quantized information", there is a heat bath instead.

[varsigma (OP)]

One approach to not getting into the weeds, when the subject of information is the game, is to start with sets and categories of sets.

Category theory lets you literally define stuff out of thin air. But, well, that's what logic and numbers are about.
Duality, or, pairs of "adjoint" algebraic structures, are extremely simple things from this viewpoint. The idea, once you have some algebra, is to refine it, and to exhaust the number of representations.

In some sense, all physics experiments represent some symmetry of nature, we are in the process of exhausting some algebra(s) as we refine them, and (categorists say) these have maps between them.

So how do we start with sets, so we end up with equations for SHM?

. . . ahem, we define sets of functions; functions of rotating vectors are useful here. I could conjecture that once a vector space is defined, then rotating vectors about an origin is like extending something I can call a module. I start with a vector algebra and add an algebra of rotations. But then I show that this is a frequency domain, and dually there is a time domain. There are transforms or algebraic maps between these. We can thank Fourier, Gauss, Laplace, Lagrange and other earlier mathematicians for the details.

 But that isn't enough, we should understand what the mathematics of frequencies in some domain, says about dualities in physics. Maybe

[varsigma (OP)]

I'll see if this makes any sense:

My thinking was that equations are the "compressed" version of a theory. Of course they only mean something to someone who recognises what they are (the information content).

If you've studied relativity and someone texts you Enstein's field equations, the text has low entropy. You get high entropy when someone sees the same message, but doesn't know what the symbols mean, they haven't seen them before.

If you write the equations out in full, that text has higher entropy even if you still know what it all represents, it's "encoded in a bigger language" there is a lot more "grammar".

Yeah? nah

[varsigma (OP)]

Another observation.

Equations in differential form that define oscillations in a LC circuit have the same form as equations in kinematic oscillations. So you could posit a "theory of oscillatory motion". Say this theory has a basis of ODE "polynomials" whose form is an invariant of the theory.

In Maxwellian dynamics, the symmetries of an obviously more complex system than a simple string or spring pendulum, have a problem with algebras of rotations, or with the quaternions. Although there is a quaternionic representation of Maxwell's "theory of electromagnetism", it doesn't have an obvious map to any Hopf algebra structure.

Hopf algebras are ubiquitous in physics, so this makes the obstruction an interesting kind of algebraic problem. Sometimes knowing why the math won't work can be as informative as knowing why it does. A theory that does or doesn't preserve structure of an algebra is the vehicle needed.

And no, I don't really have much idea why there's a problem, or even if it matters.

[varsigma (OP)]

Refinement is what gives an algebra a way to generalise, and to unify concepts. Maxwell eventually unified all forms of electricity with all forms of magnetism, his theory proposed that it was all because electric and magnetic fields behave the same way, in free space as in conductors, or when interacting with other forms of matter.

The forms  of matter that lie in the domain of kinematics are often pendulums with oscillatory motion. This motion is a common feature of all kinds of physical pendulums i.e. plug in some real physical stuff and the equations are the same.
So oscillatory motion is a physical thing which itself unifies the different kinematic systems, and this has some dual kind of unification (replace L and C with other kinds of reactive devices).

Or at least, that's the theory.

[alancalverd]

Say this theory has a basis of ODE "polynomials" whose form is an invariant of the theory.

But it doesn't have anything to do with polynomials! The underlying mathematics of harmonic motion is precisely continuous and analytic. sinωt is an exact statement, even if any calculated value (other than at t = 0) is necessarily an approximation.

Maybe an apology is needed here - this discussion is running rather close to another thread on the nature of mathematics!

[varsigma (OP)]

But it doesn't have anything to do with polynomials!

That depends on how you define polynomials. The rule seems to be you start with a field.
If you can show there is a way to grade your expressions over this field, then you have polynomials.
Derivatives can be first, second, etc order so there's the graded algebra. ODE and PDE expressions are polynomials.

And, in physics it's usually possible to expand any expression into a Taylor polynomial.

[varsigma (OP)]

Here's the story, or at least a version of.

Suppose a program is written which does everything that Newton's Laws of motion say objects with mass can do, when forces act on them. This program though, has no specification that it outputs anything, but it works and it can be proven that it does.

An output is then a kind of postcondition, a requirement that is imposed--the program has to be observed to be working, not just proven. Where does this go? Algorithms have no "duty to report", unless we add that to them.

Just saying. That this seems to connect observation of motion to whether a program outputs anything. As long as the program faithfully represents the same motion.

[alancalverd]

And, in physics it's usually possible to expand any expression into a Taylor polynomial.

...which is still an approximation to the analytic statement.

[varsigma (OP)]

...which is still an approximation to the analytic statement.

There is something a bit weird about analytic continuation. If you expand a unitary "space" by functionalising it, that is, you map a pair of complex matrix exponentials, along with their inverses such that their product is 1, but you still have a Taylor series.

Because of continuity you can show that you still get Euler's formula, even with just the first three terms of the Taylor polynomial.  Unitarity is a real thing though. It also has something to do with normalisation and measurement.

One way I think about the Taylor expansion is, how many dots do I need to see a pattern that isn't random? The dots, their positions, are the "polynomial expression", it has to be unitary in that, I need a large enough volume for all the individual probabilities to add up to "a pattern".

If that's no help at all, yeah, I guess. I've been watching Sir Roger Penrose on Youtube, talk about Schrodinger and how that man was sure he had it wrong, even though it works.

That stuff about expanding matrix exponentials was part of an online MIT course by Seth Lloyd, the goal was to show that for electrons, or charged particles generally, an external magnetic field can rotate some abstract angle. I recall the three normalised real unit vectors were in there, this was meant to be something that fermions do in real 3-space, or as physicists like to call it, a lab.