Structure and Interpretation of Classical Mechanics (2014)

(Sorry, it's been a while, but iirc the code comments call it the amazing bug, with credit to Radul)

The main thing that is at the root of all of them is the word "things". In QM, the ground truth of the world is the wavefunction of the system. The wavefunction assigns some amplitude (potentially 0) to any possible state of the system that it describes. It then evolves purely deterministically from past to future, according to Schrodinger's equation (or Dirac's equation, if you want to discuss speeds close to that of light). The only kink is interaction with a measurement device (what constitutes a measurement device is one of the big mysteries that we don't yet have an answer for). After a measurement, the wavefunction collapses non-deterministically to one of the states that the measurement device was set up to detect, with a probability that is proportional to the amplitude of the wave function of that state.

Now, this is the "ground truth" of QM. Everything else, such as particles and space-time and so on are just stories we tell to make sense of the wavefunction and its behavior. Sometimes those descriptions break down, and they start assigning weird fanciful ideas, such as retrocausality etc - but these just prove that the stories are wrong, that they are misinterpreting the math of the wavefunction.

I'd also note that the main "time is weird" factoid you encounter related to QM experiments, the delayed-choice quantum eraser, is mostly a misunderstanding / sensationalization of the actual physics and the experiment. It again only proves that certain interpretations of what the wavefunction and its collapse represent are not temporally consistent, but the direct conclusion from this should be that the interpretations are wrong, not that "cause and effect goes for a toss, as behavior of time is different".

* PNC: at most one is true; both can be false

* PEM, at least one is true; both can be true

* PNC + PEM, exactly one is true, exactly one is false

If you stick to the familiar computational complexity classes P=co-P, but NP != co-NP, and those both relate to the accessibility of T/F, where both P and NP are by definition decision problems, specifically the ones that can be verified in poly time.

If you ignore the Heisenberg uncertainty principle to avoid that complexity, the standard model of QM is just mapping continuous functions to discrete space, and will have problems with the above.

This happens in math too, where we use the rationals over the reals or Cauchy sequences to construct the reals, because almost all reals are normal and non-computable, and even equality between to "real" real numbers is undecidable.

This is also related to why after Weierstrass show that almost all continuous functions are nowhere smooth, we moved to the Epsilon-Delta definition of a limits etc...

We have the lier's paradox, which is easier to understand than Berry's paradox, which relates to Chaitin proof that there is an upper limit to what any algorithm can prove.

Things at Quantum scale do act very different than our typical intuition, but lots of maps from a continuous space to discrete categories can exhibit the same behavior even at macro scale, we just can often use a model that lets us ignore that to accomplish useful work.

Often we can even use repeated approximation or other methods to reduce those problems to something that is practical, but that is still the map and not the territory.

Superposition is just that:

    f(a + b) = f(a) + f(b)

    f(sa) = s*f(a)

If you think about Heisenberg uncertainty as indeterminacy instead, Independence in math (Like ZFC + CH) which are neither provable as false or true, Chatlin's complexity limits, the breakdown of Laplacian determinism, or even modal logic all apply at multiple levels.

The amazing thing in my mind is that we found useful models despite these limits and using methods which are effective.

But IMHO it is best to think that in the quantum world, we aren't so lucky rather than those limits aren't still lurking under the surface at the macro scale, which they very much are.

The Hamiltoninan (and Lagrangian) are much more amenable to actual physical calculations, at least on a computer, than the Newtonian formulations of classical mechanics - but otherwise they are perfectly equivalent mathematically. I'm not sure where you'd need any kind of dynamical laws in the building of a cabinet, on the other hand. Are you trying to arrange for a system of inclined planes and pullies to slot the pieces into place?

> Perhaps more accurately, our senses mostly believe pre-Newtonion approximations, which is why it took until Newton to realize how inaccurate they were.

This is a bit of misnomer. Our senses and intuitions are in fact remarkably accurate for a certain range of values, and quite equivalent to what Newton's laws of motion say about these. To some extent, Newton "only" found a simple formalism to represent our existing intuitions. Our intuitions of course break down in other places, such as at very high speeds, or very high altitudes , where relativistic corrections start to become significant.

QM however is a paradigm shift in how the world is described, and it is completely non-intuitive, even in regimes where its predictions are fully aligned with our intuitions and senses. You can use QM to compute the collision of two ideal balls on an ideal plane, and the results will exactly match your intuitions. But the computation and even the representation of the system will not, in any way.

We do that every day using things like finite elements. It’s just a different Hamiltonian that accounts for the fact that we simplify bunches of atoms into a continuum.

> Again, 99.999% of all functional mechanics don't involve epistemology.

Discussing the nature of scientific theories is epistemology. The parent’s point is epistemological in nature.

> ! You're so close to seeing the point... There are multiple levels of approximation (at least two), and the one we all experience is Newtonian.

You are off base. There are many, many approximations that may or may not overlap. It’s not just onion layers.

> You're pretty radically moving the goalposts here. GP was talking about Newtonian mechanics, not Hamiltonian.

GP was talking about different theories applying at different scales. Sorry, it may not have been clear in context but Hamiltonian here is not just the generalisation of Newtonian mechanics we learn about in 2nd year Physics. these theories (quantum mechanics, Newtonian mechanics, and relativity) can be written using the Hamiltonian formalism.

The one we all experience is Aristotelian.

That is only one approach to QM. As you rightly point out, Hamiltonian and Lagrangian approaches are always two sides of the same coin: one is only the Legendre transformation of the other and so they describe the same physics.

So to that end there is a neat QM Lagrangian representation: the path integral formulation. You can apply it to basic QM as well as to QFT or even QFT in curved spacetimes and string theory.

So if your goal is to "trace your way from simple postulates", that is a good way: assume your system can be described by an action, and go from there. It works in pretty much every scenario. In most research I've been involved, you always end up constructing generic actions whose coefficients eventually determine the behaviour of the theory.

And to get back to what I assume is your real conundrum (why do we extremize something to begin with), I just don't think there's any true answer as to why nature behaves this way.

What we can answer is: what is the action, what does it represent, what does it mean to extremize it? The short answer to this (provided by the path integral formulation I mentioned earlier) is that the action is essentially controlling a probability distribution of paths in a given geometry. In quantum mechanics, we interpret this distribution with that of actual particles. When you extremize the distribution, you essentially find the most likely trajectory, and if your distribution is peaked enough around that trajectory, then you can take this path as representative of your system when fluctuations are ignored.

So in the classical limit of QM, that trajectory is all that's left (and that would be the classical mechanics trajectory).

Interestingly a similar interpretation exists in statistical physics. If you "complexify" your time dimension, your action is again on a Euclidean (instead of lorentzian) spacetime and the time direction behaves like a circle whose radius sets a scale akin to a temperature. This might sound a bit complex but where I'm going with this is that once again, you can think of this euclidean path integral as a distribution of paths over fluctuations (this time thermal), and the extremum is this time the system's behaviour when at equilibrium.

There is a very interesting approach in the quantum physics book by Eisberg and Resnick, section 5.2

To arrive at the Schrödinger equation Eisberg and Resnick construct what they refer to as a plausibility argument.

The goal: to arrive at a wave equation that when solved for the Hydrogen atom will have the electron orbitals as set of solutions.

Eisberg and Resnick state 4 demands:

-1. Must be consistent with the de Broglie/Einstein postulates. wavelength=h/p, frequency=E/h

-2. Must be such that for a quantum entity followed over time the sum of potential energy and kinetic energy is a conserved quantity.

-3. Must be such that the equation is linear in \Psi(x,t): any linear combination of two solutions \Psi_1 and \Psi_2 must also be a solution of the equation. (Motivation: in experiments electron diffraction effects are observed. Interference effects can occur only if wave functions can be _added_.)

-4. In the absence of a potential gradient the equation must have as a solution a propagating sinusoidal wave of constant wavelength and frequency.

Eisberg and Resnick proceed to show that the above 4 demands narrow down the possibilities such that arriving at the Schrödinger equation is made inevitable.

To me the second demand is particularly interesting. The second demand is equivalent to demanding that the work-energy theorem holds good. The recurring theme: the work-energy theorem.

I have a (html)-transcript of the Eisberg & Resnick treatment that I can make available to you.

There is a youtube video with a presentation that is based on the Eisberg & Resnick plausibility argument.

https://youtu.be/2WPA1L9uJqo

In that video the presentation of the plausibility argument is in the first 18 minutes, the rest of the video is about application of the Schrödinger equation.

Physics Explained Ultraviolet Catastrophe: https://youtu.be/rCfPQLVzus4

This Veritassium Video goes back further in time and talks about Action, and the Genesis of this idea. https://youtu.be/qJZ1Ez28C-A

Chemistorian, The history of Atomic Theory. https://youtu.be/SqYPrA7upiE

Repeating the links: Page dedicated to the case of a potential proportional to the cube of the displacement: http://cleonis.nl/physics/phys256/stationary_action.php

From F=ma to Hamilton's stationary action: http://cleonis.nl/physics/phys256/energy_position_equation.p...

There are other demonstrations available that go from the newtonian formulation to Hamilton's stationary action. I believe the one in my resource is the most direct demonstration. (As in: a more direct path doesn't exist, I believe.)

(If you are interested, I can give links to the other demonstrations that I know about.)

One section of that will be replaced in a day or two: the last part of section 2. I completed a new diagram, that diagram will allow me to cut a lot of text. I believe the change will be a significant improvement.