The high-level description of classical mechanics was formulated by Hamilton, who was starting from optics. He saw a mathematical analogy between the equations for light and the equations for mechanics. The principle of least time (Fermat's principle) for light became the principle of least action for mechanics.
But the principle of least time does not predict diffraction, just the geometric path of a light ray. It fails when the wavelength of the light is large compared to whatever it's interacting with.
At the time, the equations for mechanics were clearly failing for small systems. Here's where Schrodinger had his incredible insight: what if mechanics broke in the same way as optics? Could matter itself display a kind of "diffraction" when its "wavelength" was similar in size to the objects it was interacting with? Could this explain the success of de Broglie's work, which treated small particles like waves?
Guided by that, he was able to add "diffraction" to the equations of matter and come up with the Schrodinger equation.
It's worth reading the original paper if you have a physics background -- probably grad-level (just being realistic.) I've been wanting to write a blog post about this because the physics lore is something like "Schrodinger just made a really good guess" but that totally undersells the depth of his reasoning.
> The high-level description of classical mechanics was formulated by Hamilton, who was starting from optics. He saw a mathematical analogy between the equations for light and the equations for mechanics. The principle of least time (Fermat's principle) for light became the principle of least action for mechanics.
I think you are attributing to Hamilton what was developed by others some time before. The jump from optics to mechanics was about 100 years before Hamilton published his principle. Maupertuis was the one who said that physical objects should follow shortest paths in their phase space in a way analogous to light according to Fermat’s principle. This was developed and generalised by Euler and Lagrange (with the Euler-Lagrange equations to derive equations of motions), and then Hamilton’s principle is another generalisation.
It does not affect the point you are making about Schrödinger, though.
Before Hamilton, "action" meant "accumulated living force", i.e. the integral of the kinetic energy.
The principle of the minimum action of Maupertuis was true only in some restricted cases, and it was false in most mechanics problems.
Hamilton has introduced a new physical quantity, for which he has used only the name "function S". He formulated a variational principle for the "function S", analogous to the principle of the minimum action, but from which it is possible to deduce the system of equations of Lagrange, so it has general applicability.
Hamilton's "function S" is relativistically invariant and nowadays it is usually called Hamilton's action. It is the integral of the Lagrangian, not of the kinetic energy, like the traditional action. It is proportional with the phase of the wave function in quantum mechanics. In relativistic mechanics, the Lagrangian is the component of the momentum-energy that is tangent to the trajectory in space-time, so "function S" is the line integral of the momentum-energy over the trajectory in space-time.
So Hamilton's variational principle is quite different in meaning and applicability from the principle of the minimum action that existed before him. It remains true in all forms of physics that have been discovered after Hamilton.
For those of us who didn't major in physics... where did the whole "action" thing (let alone the thesis that it's minimized) itself even come from? The whole notion of "action" feels entirely foreign and unintuitive for someone who's just studied Newtonian mechanics. At least I've never managed to find a real world feel for what it is, unlike with force or energy.
I think of the Lagrangian, what we integate to get the "action", as some sort of energy related function. I don't really attribute much meaning to it other than the fact that minimizing it implies the equations of motion, which are something we can phyiscally grasp.
For a particle in one dimension,
L = L(x(t),v(t))
The solution to the minima is where the "gradient" of L with respect to x and v is zero. However, position x
and velocity v are not independent, so that "gradient = 0" equation implies:
dL/dx = d/dt(dL/dv)
- You can define dL/dv is the generalized momentum.
- You can think of dL/dx as a force.
This gives you newtons equation, but you can say you derived it.
F = d/dt(p)
Granted, we didn't really start from a more fundamental place. But then this starts to make more sense when you realize the world is governed by quantum mechanics. And this least action principal results from the fact that, in the classical physics regime, the only part of the "trajectory" (wave function) that gives a meaningful contribution is the part along with minima of the lagrangian.
Honestly it's a great question. Even the typical physics major isn't going to be able to give you a great answer because learning how this was derived isn't a part of any curriculum that I know of.
But if you can accept the principle of least time (that a light ray will travel along the path that takes the shortest time) then you kind of already accept that (ordinary, classical) light somehow knows the time it takes to go along all possible paths, then chooses the minimum.
The action is a kind of thing, discovered by Hamilton, by analogy, that plays the same role in mechanics as time does in optics. I image he just stared at the equations of optics for a while and had an "ah-ha" moment. He had been working on this stuff for decades, on top of being a pretty smart guy already.
It's extremely unintuitive that classical systems should minimize anything like a time or an action. Think about it: they travel along the minimum path, but how do they know that that path is the minimum? Do they sample the other paths to know?
Well interestingly, Feynman's thesis was about exactly this idea. What happens if you start from the assumption that particles just sample all possible paths (weighted by something having to do with the action/time)? It turns out you can get the Schrodinger equation (and optics equations) from that too. It partially explains how paths "find the minimum." It turns out they don't, but a nice cancellation happens that makes the minimizing path the most probable one.
> For those of us who didn't major in physics... where did the whole "action" thing (let alone the thesis that it's minimized) itself even come from?
It comes from Maupertuis’ work in the 18th century (about a century before Hamilton). The initial insight is that a physical object follows the “shortest” possible “path”, in the same way as light follows the quickest path as in Descartes’ law. The difficulty is that the path is in a phase space with more than our usual 3 dimensions, so the whole thing is a bit abstract and calculations are a bit counter-intuitive at first. The approach is still useful because it helps solving problems that are very difficult to solve using Newton’s equations, like systems with constraints or couplings between objects.
> The whole notion of "action" feels entirely foreign and unintuitive for someone who's just studied Newtonian mechanics.
Action is a tool to calculate these shortest paths, and because the actual trajectory corresponds to an extremum of the action, and most of the time to a minimum, the principle is sometimes called the “least action principle”. Fundamentally, that’s almost all there is to it. The rest is defining the action, and processing it to get equations of motion. Action kind of looks like a weird energy in classical mechanics,
It is foreign from Newtonian mechanics. If you want to understand how it works you need to consider Lagrangian mechanics, which was a generalisation of Maupertuis’ principle and paved the way for Hamiltonian mechanics (which are another step in abstraction). Newtonian mechanics are built on calculus and the concept of derivative; Lagrangian mechanics are built on variational calculus and the concept of functionals.
> At least I've never managed to find a real world feel for what it is, unlike with force or energy.
Action is actually quite similar to energy, conceptually. Energy is whatever gets minimised in a Newtonian system at equilibrium. Energy changes are governed by differential equations that we can solve to calculate simple trajectories. Action is a function that is minimised along the trajectory of a physical system.
This approach is extremely powerful. The same principle can be used to derive the equation of motion for systems following classical or quantum mechanics, or general relativity by “simply” considering different definitions for the action (or, equivalently, different definitions of what we call the Lagrangian function, which is more common). It’s a bit difficult to explain more in this format; if you want to dig deeper you should start by looking into Lagrangian mechanics.
It's a great question that, as far as I can reason, has no answer. Newtonian vibes that are familiar to us will only take you so far, and intuitive interpretations of physical quantities often break down when you try to relate them to the scale, experiences, and stimuli of humans.
Let's take momentum, energy, and charge, things that you probably have a strong "real world feel" for. It's worth noting that our intuition for these quantities is actually pretty far-removed from their mathematical origin. Maybe you consider these as different loosely related quantities that pop up in different loosely related calculations, which is a useful and powerful mental model. Momentum is a thing that..."gives velocity to inertial bodies". Energy is a thing...that "does work". Charge is a thing that..."causes forces in the presence of an electric field". If you try to define the terms within each definition, you'll find yourself in some circular definitions, and it'll become unclear which definition, if any, is "most fundamental".
But these quantities are actually quite similar in the sense that they can all be defined in terms of action! Specifically, these are quantities that are conserved because there exists some nice symmetries in the Lagrangian (roughly speaking, a derivative of action). So our intuitive definitions of these things are really just less generalized/more specific understanding of structure that is emergent from action.
Can we look at a physical system and say "oh this one's got a lot of action" or "nature's doing a great job of minimizing the action over here"? No, but we can look at a physical system and say "wow, everything that's happening in here lines up with what I'd observe if this little quantity I defined just so happened to be minimized"
I think no matter how many Lagrangians we integrate or variational calculations we perform, we'll probably never gain a better intuition for action beyond "The Thing That Explains A Lot Of Seemingly Unrelated Physics When It's Minimized." To me, it's both deeply unsatisfying for its abstract and unintuitive nature, but also deeply profound for its universal explanatory power.
tldr; when it comes to action, reject real world feels and embrace mathematical structure.
I have created a demonstration of Hamilton's stationary action with interactive diagrams, (supported with discussion of the mathematics that is involved).
Interestingly: it is possible to go in all forward steps from Newtonian mechanics to Hamilton's stationary action. That is the approach of this demonstration. (How Hamilton's stationary action came into the physics community is quite a convoluted story. With benefit of hindsight: a transparent exposition is possible.)
The path from F=ma to Hamilton's stationary action goes in two stages:
1) Derivation of the work-energy theorem from F=ma
2) Demonstration that in cases where the work-energy theorem holds good Hamilton's stationary action will hold good also
Also interesting:
Within the scope of Hamilton's stationary action there are also classes of cases such that the true trajectory corresponds to a maximum of Hamilton's action.
In the demonstration it is shown for which classes of cases the stationary point corresponds to a minimum of Hamilton's action, and for which classes to a maximum.
The point is: it is not about minimization.
The actual criterion is that which both have in common:
As you sweep out variation: in the variation space the true trajectory is the one with the property that the derivative of Hamilton's action is zero. The interactive diagrams illustrate why that property holds good (it follows from the work-energy theorem).
Hamilton's stationary action is a mathematical property. When the derivative of the kinetic energy matches the derivative of the potential energy: then the derivative of Hamilton's action is zero.
(Ycombinator does not give control over the layout of the text I submit. I insert end-of-line, to structure the text, but they are eaten.)
The good thing is that the least action principle is fundamental and very flexible. All the physics are encapsulated in the Lagrangian. So you can come up with any crazy Lagrangian you want, plug it into Hamilton’s principle or Euler-Lagrange equations and see what you get. That way, you can build a whole theory from an insight somewhat easily as the fundamental framework is already in place.
The Schrödinger equation emerges from classical mechanics most closely (well ok that's a bit subjective) from the Hamilton Jacobi frame work, and it was indeed here that Schrödinger saw, in hindsight, because in the beginning he pretty much guessed it, the biggest connection to classical dynamics. This is also related to the optic-mechanical relation that abstracts mechanics to the point it becomes comparable to optics.
Ah, you've given me a thought I'm grateful for. Thanks!
I'm someone who's had a gut feeling about something in some random niche of science for several years. I've spent that time slowly gathering evidence from the literature to validate my hunch. It feels less like a "guess" and more like a high dimensional observation (of a form that's hard to cite or trace origins for) that first needs to be re-grounded in "real research".
Though maybe it DID feel like a guess to Schrödinger...! but if he didn't say it that way, I'd assume it's not quite so accurate a framing :) though it is an entertaining way to communicate it, and I appreciate that it lends a sense of serendipity and happenstance and luck, which is perhaps the most important thing to telegraph about how science happens... to take a swing at the false inevitability and certainty that has its hooks in our histories!
I've spent that time slowly gathering evidence from the literature to validate my hunch.
That is most likely the wrong way to go about this, you should probably look for evidence that your hunch is wrong, that it is in conflict with established physics.
It was a guess in a sense, but a very educated guess. Schrödinger didn’t get lucky, he was hard working, talented and very educated in his field. He was already one of the most revered physicist at the time he came up with the Schrödinger equation.
I don't know if there is a rule about science papers links, but I think using the journal paper link [1] is more suitable. The paper is open access, so no need for research gate.
Reminds me of a paper by by Hardy[1] where he introduces five reasonable axioms (his words). Classical and quantum probability theory obeys the first four. However the fifth, which states that there exists continuous transformations between pure states, is only obeyed by the quantum theory.
In that sense he argues that quantum theory is in a sense more reasonable than classical theory.
There's also an interesting link between this and entanglement[2] which seems to rule out other probability theories, leaving only quantum theory able to exhibit entanglement.
Not my field at all though, just find these foundational things interesting to ponder.
If I wanted to know what the community thought of a particular paper, is there a place where I can find a discussion of it? I thought maybe researchgate was the place, but I usually don't see discussion on the paper submission there. I know sometimes you can find the peer reviewer comments before the paper got published, but what I mean is comments from other scientists.
Scientists comment on papers by writing papers. For a paper that just appeared, wait a year or so, and check Google Scholar for papers that cite this paper. Check again every few months.
If you know physicists with an interest in this field, you can ask them if they’ve seen the paper and what they think of it. If they have an opinion they’ll probably share it with you freely, but they won’t write it down anywhere.
Maybe math overflow or physics overflow might work in rare cases... For most papers, I don't think there's really much a layperson can actively do to find out what experts think.
I have not yet read the linked paper, but seismologists have used the Schroedinger wave equation in seismic imaging since at least the 1970s [1], certainly a "classical" system.
This is not guesswork, if one evening you lie in the garden feeling bad because of a breakup or other reasons and watch the shadow of the lights, you can get similar results. Introducing Fourier transform into optics can indeed explain some phenomena, I can't recall the specifics, but it is related to the shape formed between the light and the fence.
This paper smells like crack pot stuff. That is probably why it collected only two citations in more than 10 years. It also mentions the experiment from Couder et. al. in the summary, which has been debunked several years ago: https://www.quantamagazine.org/famous-experiment-dooms-pilot...
Behind sophisticated math it hides a beginners understanding of physics. Classical mechanics emerges from Quantum Mechanics in the same way as wave optics emerges from ray optics.
If it would be otherwise, you would also argue, that wave optics emerges from ray optic. The experimental evidence is clear against such an interpretation.
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The high-level description of classical mechanics was formulated by Hamilton, who was starting from optics. He saw a mathematical analogy between the equations for light and the equations for mechanics. The principle of least time (Fermat's principle) for light became the principle of least action for mechanics.
But the principle of least time does not predict diffraction, just the geometric path of a light ray. It fails when the wavelength of the light is large compared to whatever it's interacting with.
At the time, the equations for mechanics were clearly failing for small systems. Here's where Schrodinger had his incredible insight: what if mechanics broke in the same way as optics? Could matter itself display a kind of "diffraction" when its "wavelength" was similar in size to the objects it was interacting with? Could this explain the success of de Broglie's work, which treated small particles like waves?
Guided by that, he was able to add "diffraction" to the equations of matter and come up with the Schrodinger equation.
It's worth reading the original paper if you have a physics background -- probably grad-level (just being realistic.) I've been wanting to write a blog post about this because the physics lore is something like "Schrodinger just made a really good guess" but that totally undersells the depth of his reasoning.
I think you are attributing to Hamilton what was developed by others some time before. The jump from optics to mechanics was about 100 years before Hamilton published his principle. Maupertuis was the one who said that physical objects should follow shortest paths in their phase space in a way analogous to light according to Fermat’s principle. This was developed and generalised by Euler and Lagrange (with the Euler-Lagrange equations to derive equations of motions), and then Hamilton’s principle is another generalisation.
It does not affect the point you are making about Schrödinger, though.
The principle of the minimum action of Maupertuis was true only in some restricted cases, and it was false in most mechanics problems.
Hamilton has introduced a new physical quantity, for which he has used only the name "function S". He formulated a variational principle for the "function S", analogous to the principle of the minimum action, but from which it is possible to deduce the system of equations of Lagrange, so it has general applicability.
Hamilton's "function S" is relativistically invariant and nowadays it is usually called Hamilton's action. It is the integral of the Lagrangian, not of the kinetic energy, like the traditional action. It is proportional with the phase of the wave function in quantum mechanics. In relativistic mechanics, the Lagrangian is the component of the momentum-energy that is tangent to the trajectory in space-time, so "function S" is the line integral of the momentum-energy over the trajectory in space-time.
So Hamilton's variational principle is quite different in meaning and applicability from the principle of the minimum action that existed before him. It remains true in all forms of physics that have been discovered after Hamilton.
For a particle in one dimension,
L = L(x(t),v(t))
The solution to the minima is where the "gradient" of L with respect to x and v is zero. However, position x and velocity v are not independent, so that "gradient = 0" equation implies:
dL/dx = d/dt(dL/dv)
- You can define dL/dv is the generalized momentum. - You can think of dL/dx as a force.
This gives you newtons equation, but you can say you derived it.
F = d/dt(p)
Granted, we didn't really start from a more fundamental place. But then this starts to make more sense when you realize the world is governed by quantum mechanics. And this least action principal results from the fact that, in the classical physics regime, the only part of the "trajectory" (wave function) that gives a meaningful contribution is the part along with minima of the lagrangian.
https://groups.csail.mit.edu/mac/users/gjs/6946/sicm-html/bo...
for an approach that might be especially appealing to hackers.
But if you can accept the principle of least time (that a light ray will travel along the path that takes the shortest time) then you kind of already accept that (ordinary, classical) light somehow knows the time it takes to go along all possible paths, then chooses the minimum.
The action is a kind of thing, discovered by Hamilton, by analogy, that plays the same role in mechanics as time does in optics. I image he just stared at the equations of optics for a while and had an "ah-ha" moment. He had been working on this stuff for decades, on top of being a pretty smart guy already.
It's extremely unintuitive that classical systems should minimize anything like a time or an action. Think about it: they travel along the minimum path, but how do they know that that path is the minimum? Do they sample the other paths to know?
Well interestingly, Feynman's thesis was about exactly this idea. What happens if you start from the assumption that particles just sample all possible paths (weighted by something having to do with the action/time)? It turns out you can get the Schrodinger equation (and optics equations) from that too. It partially explains how paths "find the minimum." It turns out they don't, but a nice cancellation happens that makes the minimizing path the most probable one.
It comes from Maupertuis’ work in the 18th century (about a century before Hamilton). The initial insight is that a physical object follows the “shortest” possible “path”, in the same way as light follows the quickest path as in Descartes’ law. The difficulty is that the path is in a phase space with more than our usual 3 dimensions, so the whole thing is a bit abstract and calculations are a bit counter-intuitive at first. The approach is still useful because it helps solving problems that are very difficult to solve using Newton’s equations, like systems with constraints or couplings between objects.
> The whole notion of "action" feels entirely foreign and unintuitive for someone who's just studied Newtonian mechanics.
Action is a tool to calculate these shortest paths, and because the actual trajectory corresponds to an extremum of the action, and most of the time to a minimum, the principle is sometimes called the “least action principle”. Fundamentally, that’s almost all there is to it. The rest is defining the action, and processing it to get equations of motion. Action kind of looks like a weird energy in classical mechanics,
It is foreign from Newtonian mechanics. If you want to understand how it works you need to consider Lagrangian mechanics, which was a generalisation of Maupertuis’ principle and paved the way for Hamiltonian mechanics (which are another step in abstraction). Newtonian mechanics are built on calculus and the concept of derivative; Lagrangian mechanics are built on variational calculus and the concept of functionals.
> At least I've never managed to find a real world feel for what it is, unlike with force or energy.
Action is actually quite similar to energy, conceptually. Energy is whatever gets minimised in a Newtonian system at equilibrium. Energy changes are governed by differential equations that we can solve to calculate simple trajectories. Action is a function that is minimised along the trajectory of a physical system.
This approach is extremely powerful. The same principle can be used to derive the equation of motion for systems following classical or quantum mechanics, or general relativity by “simply” considering different definitions for the action (or, equivalently, different definitions of what we call the Lagrangian function, which is more common). It’s a bit difficult to explain more in this format; if you want to dig deeper you should start by looking into Lagrangian mechanics.
Let's take momentum, energy, and charge, things that you probably have a strong "real world feel" for. It's worth noting that our intuition for these quantities is actually pretty far-removed from their mathematical origin. Maybe you consider these as different loosely related quantities that pop up in different loosely related calculations, which is a useful and powerful mental model. Momentum is a thing that..."gives velocity to inertial bodies". Energy is a thing...that "does work". Charge is a thing that..."causes forces in the presence of an electric field". If you try to define the terms within each definition, you'll find yourself in some circular definitions, and it'll become unclear which definition, if any, is "most fundamental".
But these quantities are actually quite similar in the sense that they can all be defined in terms of action! Specifically, these are quantities that are conserved because there exists some nice symmetries in the Lagrangian (roughly speaking, a derivative of action). So our intuitive definitions of these things are really just less generalized/more specific understanding of structure that is emergent from action.
Can we look at a physical system and say "oh this one's got a lot of action" or "nature's doing a great job of minimizing the action over here"? No, but we can look at a physical system and say "wow, everything that's happening in here lines up with what I'd observe if this little quantity I defined just so happened to be minimized"
I think no matter how many Lagrangians we integrate or variational calculations we perform, we'll probably never gain a better intuition for action beyond "The Thing That Explains A Lot Of Seemingly Unrelated Physics When It's Minimized." To me, it's both deeply unsatisfying for its abstract and unintuitive nature, but also deeply profound for its universal explanatory power.
tldr; when it comes to action, reject real world feels and embrace mathematical structure.
Interestingly: it is possible to go in all forward steps from Newtonian mechanics to Hamilton's stationary action. That is the approach of this demonstration. (How Hamilton's stationary action came into the physics community is quite a convoluted story. With benefit of hindsight: a transparent exposition is possible.)
Recommended: read the following two articles in this order: Introduction to calculus of variations: http://cleonis.nl/physics/phys256/calculus_variations.php
Hamilton's stationary action: http://cleonis.nl/physics/phys256/energy_position_equation.p...
The path from F=ma to Hamilton's stationary action goes in two stages: 1) Derivation of the work-energy theorem from F=ma 2) Demonstration that in cases where the work-energy theorem holds good Hamilton's stationary action will hold good also
Also interesting: Within the scope of Hamilton's stationary action there are also classes of cases such that the true trajectory corresponds to a maximum of Hamilton's action.
In the demonstration it is shown for which classes of cases the stationary point corresponds to a minimum of Hamilton's action, and for which classes to a maximum.
The point is: it is not about minimization. The actual criterion is that which both have in common: As you sweep out variation: in the variation space the true trajectory is the one with the property that the derivative of Hamilton's action is zero. The interactive diagrams illustrate why that property holds good (it follows from the work-energy theorem).
Hamilton's stationary action is a mathematical property. When the derivative of the kinetic energy matches the derivative of the potential energy: then the derivative of Hamilton's action is zero.
(Ycombinator does not give control over the layout of the text I submit. I insert end-of-line, to structure the text, but they are eaten.)
https://en.m.wikipedia.org/wiki/Action_(physics)
I bet a lot of movie fans regretted stumbling onto this page. No, I wanted action!
Hamilton Jacobi theory: https://en.m.wikipedia.org/wiki/Hamilton%E2%80%93Jacobi_equa...
Optic-mechanical analogy: https://en.m.wikipedia.org/wiki/Hamilton%27s_optico-mechanic...
Schrödinger equation from HJ theory: https://www.reed.edu/physics/faculty/wheeler/documents/Quant...
Ah, you've given me a thought I'm grateful for. Thanks!
I'm someone who's had a gut feeling about something in some random niche of science for several years. I've spent that time slowly gathering evidence from the literature to validate my hunch. It feels less like a "guess" and more like a high dimensional observation (of a form that's hard to cite or trace origins for) that first needs to be re-grounded in "real research".
Though maybe it DID feel like a guess to Schrödinger...! but if he didn't say it that way, I'd assume it's not quite so accurate a framing :) though it is an entertaining way to communicate it, and I appreciate that it lends a sense of serendipity and happenstance and luck, which is perhaps the most important thing to telegraph about how science happens... to take a swing at the false inevitability and certainty that has its hooks in our histories!
That is most likely the wrong way to go about this, you should probably look for evidence that your hunch is wrong, that it is in conflict with established physics.
[https://iopscience.iop.org/article/10.1088/1742-6596/361/1/0...]
https://doi.org/10.1088/1742-6596/361/1/012015
Deleted Comment
In that sense he argues that quantum theory is in a sense more reasonable than classical theory.
There's also an interesting link between this and entanglement[2] which seems to rule out other probability theories, leaving only quantum theory able to exhibit entanglement.
Not my field at all though, just find these foundational things interesting to ponder.
[1]: https://arxiv.org/abs/quant-ph/0101012
[2]: https://arxiv.org/abs/0911.0695v1
If you know physicists with an interest in this field, you can ask them if they’ve seen the paper and what they think of it. If they have an opinion they’ll probably share it with you freely, but they won’t write it down anywhere.
I made a link cause I didn't see one via search
Feedback: Searching by Topic, Title or arxiv ID would be well augmented by a DOI search, especially for the current post.
Best you can do is look for papers that cite this paper.
[1] https://pubs.geoscienceworld.org/geophysics/article-abstract...
Behind sophisticated math it hides a beginners understanding of physics. Classical mechanics emerges from Quantum Mechanics in the same way as wave optics emerges from ray optics.
https://physics.stackexchange.com/questions/397694/what-make...
If it would be otherwise, you would also argue, that wave optics emerges from ray optic. The experimental evidence is clear against such an interpretation.