I What has changed since the Copenhagen interpretation?

[Demystifier]

It can in a weak sense.

So suppose that we have a weak non-differentiable solution of the Schrodinger equation. Is your point that the Bohmian trajectories are not defined then? Or perhaps they could still be defined in some weak sense?

Suppose that the Bohmian trajectories are not defined for such solutions. What does it mean physically? For an analogy, consider the Hamilton-Jacobi (HJ) equation of classical mechanics. The solution S(x,t) of the HJ equation defines classical particle trajectories, in very much the same was as the solution of the Schrodinger equation defines Bohmian trajectories. But now someone may object that HJ equation has weak non-differentiable solutions for which classical particle trajectories are not defined. What does it mean physically? Does it mean that classical particle trajectories do not exist? Or that such non-differentiable solutions are just not physical? For me, it seems obvious that the second answer is the right one. And by analogy, it seems reasonable to extrapolate the same answer to QM and Bohmian trajectories as well.

 

[martinbn]

I don't know what the situations is in these two cases, but you are missing another possibility. That the model is not fundamental, in the sense that it is only an approximation, and a better more accurate one may describe nature (in two words it may be that BM is no good). In fact that seems the only possibility if physically meaningful initial/boundary conditions lead to weak solutions, which are not regular. You cannot just say those function/distributions are not physical. If they are not physical then the theory is simply inadequate (it predicts unphysical things). I don't know if that is possible in the case of Schrodinger, HJ, BM. On the other hand I don't see what is so unphysical about a continuous function, which is not differentiable. In any case I was just surprised that you (a physicists) holds such a strong view which functions are relevant to physics.

 

[martinbn]

I don't understand. How is this

http://blog.jessriedel.com/2017/07/...cs-part-7-quantum-chaos-and-linear-evolution/

a response to this

If physics can not allow non-differentiable things, then all of nonlinear dynamics (chaos, turbulence, catastrophe, Feigenbaum universality, etc) directly goes out of the window. To me, because of experimental facts, this position is clearly untenable.

 

[Auto-Didact]

http://blog.jessriedel.com/2017/07/...cs-part-7-quantum-chaos-and-linear-evolution/

I will read this later and get back to you.

So suppose that we have a weak non-differentiable solution of the Schrodinger equation. Is your point that the Bohmian trajectories are not defined then? Or perhaps they could still be defined in some weak sense?

My point is exactly the opposite: being 'well defined' mathematically at a certain point in history - such as the very concept of differentiability discovered a few centuries ago by the analysts - need not be a proper criteria for judging whether or not a phenomenon exists in nature; what is considered 'not well defined' today might turn out to be 'well defined' tomorrow depending on what mathematicians will discover in the theory of mathematics.

In other words, being well defined or not is at best a pragmatic rule of thumb which can tell a physicist about what mathematicians have discovered about the tools which physicists use, not an error-free scientific methodology to talk about the properties of nature or decide what is 'good physics' or 'bad physics'. As @martinbn points out, it simply means that what you believe to be fundamental (differentiability and the thereby resulting fundamental physics principles) simply might actually be an approximation.

To make clear how subjective differentiability is as a criterion of 'good physics', why not go for smoothness? Or analyticity? Or holomorphicity? The fact that physicists go for analyticity is a sociological effect: all the 'fundamental theories' studied so far haven't seemed to require more than analyticity; because of this the aesthetic sense in the physicist community became attuned to this mathematical property for physical tools and models.

To me, such a standpoint as 'physics needs to be differentiable' is clearly, purely pragmatic - even semi-fallacious - reasoning, because it has turned out in the past more than once that such mathematical criteria often end up getting amended once new mathematical facts are discovered, i.e. such as in the case when distribution theory was discovered and the Dirac delta suddenly became a proper mathematical object.

 

[Auto-Didact]

To make my point even stronger, my main day job is as a physician in intensive care medicine: there are tonnes of medical entities which have resisted technical exact description due to their overwhelming complexity; frankly speaking, they can be considered neither mathematically, nor scientifically well-defined, yet it is an unquestionable experimental fact that these entities do exist.

Can I therefore, based upon physics knowledge, conclude that these things simply don't exist? Clearly not, it is instead the knowledge in contemporary physics which is inadequate to explain some medical entities at this point in time. (NB: having always been a theorist at heart, I actually did exactly think that the opposite was true i.e. that many medical entities did not actually exist; this is until I had to actually start doing rounds in practice and so was forced to learn to think in a completely different manner than I do in mathematics, physics or in science more generally).

This just shows that being well defined is not a scientific problem, but an aesthetic criteria: nice to have, but given some phenomenon, even if you don't have a way to define it properly yet, you still have to somehow make do; this of course, applies to all scientific discoveries before they were discovered. To paraphrase Feynman, nature exists in the exact way that she does whether we are capable of discovering or describing her or her properties or not.

 

[A. Neumaier]

But now someone may object that HJ equation has weak non-differentiable solutions for which classical particle trajectories are not defined. What does it mean physically? Does it mean that classical particle trajectories do not exist? Or that such non-differentiable solutions are just not physical? For me, it seems obvious that the second answer is the right one.

The problem is that most partial differential equations, and in particular generic Schroedinger equations or the Navier-Stokes equations, have only a weakly differentiable solutions at times $t>0$ even when the initial condition at time $t=0$ is very smooth. There are good reasons why mathematicians work with weeak differentiability...

 

[martinbn]

How sure are you about Navier Stokes?

 

[Auto-Didact]

http://blog.jessriedel.com/2017/07/...cs-part-7-quantum-chaos-and-linear-evolution/

This entire piece isn't directed at my point; chaos literally can not occur or be explained mathematically by linear mathematics, which is why I don't believe that the concept often called 'quantum chaos' can have any mathematical existence at all, let alone physical existence; if chaos actually arises in QM at all, it is due to currently badly understood aspects of QM, buried in the formalism by simplification, and therefore will constitute a clue that orthodox QM based on unitarity is mathematically incomplete.

It is in this same way that I believe that hydrodynamics, a continuum theory, can in principle not ever be explained by quantum theory, a form of linear(ized) mathematics; in other words, 'quantum hydrodynamics' if it is meant to be quantized hydrodynamics is mathematical nonsense, either a non-starter or some kind of application of (experimental) physics bringing the two fields together.

Here is where things get confusing, there is actually a field of research called 'quantum hydrodynamics'; this however is a field in mathematical physics which has nothing whatsoever to do with quantizing hydrodynamics; perhaps the name is a misnomer and it should be called 'hydrodynamic quantum mechanics' instead, but I digress.

What researchers in this field try to do is study the mathematical objects in quantum theory i.e. the Schrödinger equation (SE), Dirac equation, Klein-Gordon equation etc as mathematical entities, i.e. PDEs, and then try to generalize these equations to equations from hydrodynamics, i.e. Euler (type) equations and Navier-Stokes (type) equations, but then properly respecting the tantalizing presence of $i$ in the SE.

I will clarify this by example: mathematically, the SE is a linear PDE, namely a (complex) diffusion equation of the form (with $k$ a constant and $i$ omitted for simplicity): $$\partial_t u = k \nabla^2 u$$All such diffusion equations are 'internal parts' of a nonlinear PDE, namely the (incompressible, one dimensional) Navier-Stokes (type) equations with the form: $$\partial_t u + (u \cdot \nabla)u = k \nabla^2 u - \nabla l + m$$ As is immediately clear from inspection, the earlier linear PDE is literally part of this nonlinear PDE; this means it can be obtained through linearization, simplification and a careful choosing of constants.

I'm not sure if other physicists realize this, but this means that a linear PDE can always be generalized i.e. non-linearized into such a non-linear PDE; this non-linearization process tends to be non-unique i.e. there are multiple ways to end up with a nonlinear PDE and the road ahead is not exactly clear; incidentally, this is also exactly why it is so immensely difficult to solve nonlinear (P)DEs, explaining why practitioners of mathematics often like to avoid them.

In any case, there is no guarantee that such a generalized nonlinear PDE might even exist mathematically, let alone in terms of physics; however, experience teaches us otherwise: when you automatically get more out of a derivation than what you put in, then this is a clue you might be onto something. A good strategy is to try to generalize your equation towards a known PDE, such as Korteweg-de Vries, Born-Infeld or even the Einstein field equations; the field of research called 'quantum hydrodynamics' tries to do precisely this with the Navier-Stokes equation.

 

[Demystifier]

The problem is that most partial differential equations, and in particular generic Schroedinger equations or the Navier-Stokes equations, have only a weakly differentiable solutions at times $t>0$ even when the initial condition at time $t=0$ is very smooth. There are good reasons why mathematicians work with weeak differentiability...

How about the Hamilton-Jacobi equation?

 

[martinbn]

How about the Hamilton-Jacobi equation?

Not absolutely certain, but I believe so.

 

[Demystifier]

Not absolutely certain, but I believe so.

Believe what?

 

[martinbn]

That it has weak solutions to regular initial data.

 

[Demystifier]

That it has weak solutions to regular initial data.

So what are the physical consequences of this on the existence of classical particle trajectories?

 

[A. Neumaier]

How sure are you about Navier Stokes?

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.

 

[A. Neumaier]

How about the Hamilton-Jacobi equation?

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.

 

[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]

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 a lot 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.

 

[A. Neumaier]

Y
Edit: this has a lot 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]

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.