# I What has changed since the Copenhagen interpretation?

#### martinbn

Almost sure. Navier-Stokes equations generate turbulent flow, which is unlikely to have a smooth description. To be 100% sure someone would need to solve one of the Clay Millennium problems.
Exactly, and the answer might turn out to be that there are global regular solutions.

#### Demystifier

2018 Award
Except in the completely integrable case, these probably have only weak solutions, too. Breakdown of smoothness is probably due to the caustics already visible in the WKB approximations. In tractable cases (e.g., Burgers equation) these lead to discontinuous shock waves.
So can you answer my question in #223?

#### A. Neumaier

Exactly, and the answer might turn out to be that there are global regular solutions.
might, but very unlikely. Also it could depend on the initial conditions - small initial conditions may well behave differently from large ones.

#### martinbn

So what are the physical consequences of this on the existence of classical particle trajectories?
I am not sure what bothers you here. If you have a weak solution that is say an $L^2$ function, then it has derivatives in the weak sense, which can be also $L^2$. The fact that it may not have pointwsie derivatives, shouldn't change the physical meaning.

#### A. Neumaier

So what are the physical consequences of this on the existence of classical particle trajectories?
In classical mechanics, we have ordinary differential equations with locally Lipschitz continuous right hand sides, and existence of smooth point particle trajectories is no problem unless particles collide exactly, which happens with probability zero. Problems with smoothness usually appear in field theories. From the mathematical point of view, the Schroedinger equation is a field theory. It would be up to you to check whether Bohmian trajectories inherit from the Schroedinger equation their nonsmoothness.

#### Auto-Didact

existence of smooth point particle trajectories is no problem unless particles collide exactly, which happens with probability zero.
Is this point related to "runaway" solutions which were problematic in Farnes' model with positive/negative mass collisions as described here?

#### A. Neumaier

Is this point related to "runaway" solutions which were problematic in Farnes' model with positive/negative mass collisions as described here?
I assumed pure gravitational interactions with of course positive masses. There motion is smooth until a collision occurs. Negative masses are unphysical.

#### Auto-Didact

I assumed pure gravitational interactions with of course positive masses. There motion is smooth until a collision occurs. Negative masses are unphysical.
Of course they aren't physical, but that's not what I'm asking. I mean, from a purely mathematical standpoint as an aspect of PDE theory, is the non-smoothness of point particle trajectories due to exact collisions in the HJ equations equivalent to the "runaway" solution problem in Farnes' model?

#### A. Neumaier

Of course they aren't physical, but that's not what I'm asking. I mean, from a purely mathematical standpoint as an aspect of PDE theory, is the non-smoothness of point particle trajectories due to exact collisions in the HJ equations equivalent to the "runaway" solution problem in Farnes' model?
We are talking about ODEs not PDEs.

From a mathematical point of view, as long as the right hand side is Lipschitz continuous (i.e., no collision), the solution can be shown to be continuous differentiable until it reaches either the boundary or infinity. Thus the solution exists either for all times, or there is a collision, or there must be a finite time for escape to infinity.

#### Auto-Didact

We are talking about ODEs not PDEs.
Ah yes, of course. Forgive my sloppiness, I tend to just regard ODEs as special cases of PDEs without giving any proper formal mathematical justification.

#### A. Neumaier

Ah yes, of course. Forgive my sloppiness, I tend to just regard ODEs as special cases of PDEs without giving any proper formal mathematical justification.
They behave very differently in practice, hence it is rarely appropriate to treat ODEs as special PDEs. One rather does the opposite, and treats time-dependent PDEs as ODEs in appropriate Banach spaces!

#### Auto-Didact

They behave very differently in practice, hence it is rarely appropriate to treat ODEs as special PDEs. One rather does the opposite, and treats time-dependent PDEs as ODEs in appropriate Banach spaces!
Yes, I know. My 'classification' isn't based on how to solve equations, but instead more of an attempt to more easily taxonomize DEs as mathematical entities based purely on the visual form of the equation.

Edit: this has alot to do with my biased view of always viewing DEs as dynamical systems (and sometimes geometrically) and applying qualitative methods like drawing phase diagrams, etc.; in my research I usually only tackle nonlinear dynamical systems.

Now that I recall, this was in fact also the way that I was first introduced to HJ theory, namely by drawing vector fields, analyzing orbits in phase space and classifying their stability, before learning Hamiltonian/Lagrangian mechanics and QM.

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#### A. Neumaier

Y
Edit: this has alot to do with my biased view of always viewing DEs as dynamical systems (and sometimes geometrically) and applying qualitative methods like drawing phase diagrams, etc.; in my research I usually only tackle nonlinear dynamical systems.
But this would mean that you view everything as an ODE, not as a PDE!?

#### Auto-Didact

But this would mean that you view everything as an ODE, not as a PDE!?
I'm not sure, I suspect I have internalized it so much that I don't consciously make the distinctions anymore; in practice, the situation almost never arises that I actually need to manually solve a DE anymore: instead I just feed it into Mathematica, occasionally only needing to rewrite things a bit using Fourier or Laplace transforms before Mathematica is able to spit out an answer.

I think the rise of computer algebra systems, such as Mathematica, have in a sense made me somewhat lazy/sloppy and simultaneously increased productivity enormously. As a result, I (as well as others) just tend to forego formal labeling altogether and just refer to all differential equations and difference equations (iterative maps) collectively as dynamical system.

#### Demystifier

2018 Award
But this would mean that you view everything as an ODE, not as a PDE!?
I like to view a PDE as an uncountably infinite set of coupled ODE's.

#### Auto-Didact

I suspect an analogy can be made with the following:
if the problem is a broken bone, depending mainly on the size of the fractured bone, the frequency of this kind of fracture and the simplicity of the surgery, it is either carried out by a traumatologist (a surgeon) or a orthopaedist (a bone specialist); the orthopaedist classifies fractures carefully (using mechanics and geometry) and reasons logically towards a plan of attack, while the average traumatologist just repairs routinely using the tools at hand: if the problem is too difficult, he will refer it to the orthopaedist.

Replace 'fracture' with '(difficult) equation', 'traumatologist' with 'physicist/engineer/etc' and 'orthopaedist' with 'mathematician'. In fact the only reason I suspect I think ODEs can be seen as special cases of PDEs is because both the textbook said so and my mathematician friend, who I tend to consult when specialist care is really required for some problem, also says so.

#### A. Neumaier

As a result, I (as well as others) just tend to forego formal labeling altogether and just refer to all differential equations and difference equations (iterative maps) collectively as dynamical system.
But this can be quite misleading. There is nothing dynamical in an elliptic pde.
because both the textbook said so and my mathematician friend, who I tend to consult when specialist care is really required for some problem, also says so.
I would have strong reservations towards such a textbook or mathematician.

#### Demystifier

2018 Award
I suspect an analogy can be made with the following:
if the problem is a broken bone, depending mainly on the size of the fractured bone, the frequency of this kind of fracture and the simplicity of the surgery, it is either carried out by a traumatologist (a surgeon) or a orthopaedist (a bone specialist); the orthopaedist classifies fractures carefully (using mechanics and geometry) and reasons logically towards a plan of attack, while the average traumatologist just repairs routinely using the tools at hand: if the problem is too difficult, he will refer it to the orthopaedist.

Replace 'fracture' with '(difficult) equation', 'traumatologist' with 'physicist/engineer/etc' and 'orthopaedist' with 'mathematician'. In fact the only reason I suspect I think ODEs can be seen as special cases of PDEs is because both the textbook said so and my mathematician friend, who I tend to consult when specialist care is really required for some problem, also says so.
I thought it was me who likes to use funny metaphors.
By the way, do you like Dr. House, who also likes funny metaphors?

#### Auto-Didact

But this can be quite misleading. There is nothing dynamical in an elliptic pde.
You are correct of course, but as I'm sure you are aware old habits die hard. In my defense, static in physics or science often is just a case of dynamic equilibrium, just like zero velocity also is a perfectly reasonable velocity.
I would have strong reservations towards such a textbook or mathematician.
The textbook was pretty good though (Kreyszig), but I know what having beef with a textbook (Ballentine) means. In any case, that's why you are the mathematician and I am not.
I thought it was me who likes to use funny metaphors.
By the way, do you like Dr. House, who also likes funny metaphors?
Yeah for sure, learned alot like personally avoid the patient at all costs, it's never lupus and everybody lies

Incidentally, I have also, like him, on a number of occasions consider(ed) to leave medicine and go study dark matter

#### A. Neumaier

In my defense, static in physics or science often is just a case of dynamic equilibrium, just like zero velocity also is a perfectly reasonable velocity.
Thus you view algebraic equations as a particular case of ordinary differential equations???

#### A. Neumaier

The statement that macroscopic world obeys classical laws is quite obsolete, because there are many counterexamples. For instance, superconductor in a superposition of macroscopic currents in the opposite directions.
Can you give a reference for the latter?

2018 Award

#### A. Neumaier

https://www.ncbi.nlm.nih.gov/pubmed/10894533
Do you think it's a challenge for your thermal interpretation of QM?
Thanks for the reference from 2000. A more recent (2018) review of macroscopic quantum state preparation is here:

Fröwis, Florian, et al. "Macroscopic quantum states: Measures, fragility, and implementations." Reviews of Modern Physics 90.2 (2018): 025004.

It is primarily an experimental challenge. But there are no associated foundational problems as quantum mechanics is not violated in the experiments.

Why should it be a challenge for the thermal interpretation? Is it a challenge for Bohmian mechanics?

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#### Auto-Didact

Thus you view algebraic equations as a particular case of ordinary differential equations???
You misunderstand me: when doing research into some phenomenon characterized by an equation, my null hypothesis is usually that all equations (algebraic or transcendental) and (partial or ordinary, discrete or continuous) differential equations are actually specific parts, properties or aspects of dynamical systems until demonstrated otherwise; this does not mean that the equations are so in and of themselves, but that they are so when looked at from the right perspective in the correct context, i.e. when the right scientific question is asked. For me, the right question is almost always the interesting question in a scientific context and specifically not any questions in the context of mathematical formalism.

It goes without saying that I'm biased and focus on some equations more than others, e.g. equations with a deep established model behind them, often directly from the context of physics, than just random equations or obviously trivial equations. This quickly gets complicated because many empirical equations of phenomenon that are encountered are simplifications, truncations, linearizations, regressions and so on and they require care to reveal their deeper nature. Looking at an equation naively such as a beginner would is, I think, a mistake of premature closure of classification, because doing that too strictly makes one incapable of correctly generalizing with as a result that the person is only able to see the equations (the trees), not the larger classes they belongs to (the forest); many of these classes are essentially uncharacteristed by mathematicians so far, or still the subject of ongoing research.

There tends to be a stark difference between how physicists and mathematicians approach the subject of mathematics as a theory; moreover, it seems as if most practitioners say one thing (e.g. believe in formalism) while do something else (e.g. practice Platonism). In either case, the view I'm arguing for is aligned with how most classical physicists (from Newton up to Fourier, Laplace, Lagrange et al. up to Poincaré and some dynamicists today) viewed the relationship between mathematics and physics. I suspect that not just physics, but all advanced applied mathematics (mathematical biology, economics and so on) has this same form; this would in some sense be the answer to Wigner's observation regarding the unreasonable effectiveness of mathematics in the natural sciences.

From my experience in doing research it turns out more often than not, that my null hypothesis is true, with the caveat that what exactly the original equation is w.r.t. the dynamical system requires a very careful characterization: they don't all share the same relationship to some dynamical system, but so far they all fall into a set of specific themes. In my idiosyncratic view, this is the correct theoretical methodology of how to practice theoretical physics based on advanced pure and applied mathematics; I think many physicists and mathematicians actually mean this when they refer to 'being guided by mathematical beauty' with beauty being specifically the experience of recognizing a relation to the same kind of equations they were exposed to (i.e. the canonical equations of physics) during training.

"What has changed since the Copenhagen interpretation?"

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