I What has changed since the Copenhagen interpretation?

[Demystifier]

With you being the resident expert on BM, I'm very curious to hear your opinion about this paper

I believe that wave functions which are nowhere differentiable do not appear in nature.

 

[martinbn]

I believe that wave functions which are nowhere differentiable do not appear in nature.

Interesting! I thought that physicists felt differently. What about all the elements in a typical $L^2$ space? I thought those were essential for QM.

 

[Demystifier]

Interesting! I thought that physicists felt differently. What about all the elements in a typical $L^2$ space? I thought those were essential for QM.

They exist mathematically, but I don't see how they are essential. Perhaps they are needed for a rigorous mathematical formulation of QM, but it doesn't mean that they appear in nature.

 

[martinbn]

They exist mathematically, but I don't see how they are essential. Perhaps they are needed for a rigorous mathematical formulation of QM, but it doesn't mean that they appear in nature.

None of the functions appear in nature, they appear in the mathematical description of nature. What I find interesting is that you have a preference on which functions should be used in the models. My impression was physicists are not that committed. After all, you all like things as the Dirac delta function.

 

[Demystifier]

After all, you all like things as the Dirac delta function.

I like it only as an idealization with which it is easy to make explicit computations.

 

[martinbn]

I like it only as an idealization with which it is easy to make explicit computations.

Hm, some would say that for the differentiable functions.

 

[Demystifier]

Hm, some would say that for the differentiable functions.

But if the wave function is not differentiable, then it cannot satisfy the Schrodinger equation (which is a partial differential equation).

 

[Auto-Didact]

I believe that wave functions which are nowhere differentiable do not appear in nature.

Why exactly? I'm assuming you are arguing based on the adherence of some physical principle (or else based on some aesthetic criteria as Sabine might put it).

How would their existence be precluded in terms of physics?

They exist mathematically, but I don't see how they are essential. Perhaps they are needed for a rigorous mathematical formulation of QM, but it doesn't mean that they appear in nature.

By this argument no actual fractals (of an infinite path length or equivalent criteria) exist in nature. I would assume you mean then that all actually occurring fractals in nature actually only scale up/down up to some limit.

Speaking not only as a physicist, but from the perspective of canonical classical physics, how would you explain the occurrence of strange attractors in phase space then? Are these not physical objects?

None of the functions appear in nature, they appear in the mathematical description of nature. What I find interesting is that you have a preference on which functions should be used in the models. My impression was physicists are not that committed. After all, you all like things as the Dirac delta function.

Yes, this also surprises me somewhat, maybe even very much. I was under the suspicion that other physicists today, especially after the Dirac delta function issue and the subsequent discovery in later decades of hyperfunction/distribution theory, more openly embraced what were once, for very good reasons, seen as mathematical pathological functions.

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.

But if the wave function is not differentiable, then it cannot satisfy the Schrodinger equation (which is a partial differential equation).

As the article puts it, non-differentiability doesn't hold for $$i\hbar \partial_t\Psi_t(x)=\hat H \Psi_t (x)$$but it does hold for $$[i\hbar \partial_t - \hat H] \Psi_t (x)=0$$The above is called a weak solution in PDE and has been studied extensively.

 

[martinbn]

But if the wave function is not differentiable, then it cannot satisfy the Schrodinger equation (which is a partial differential equation).

It can in a weak sense.

 

[atyy]

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.

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

 

[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.

 

[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]

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 a lot 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?

 

[Demystifier]

Can you give a reference for the latter?

https://www.ncbi.nlm.nih.gov/pubmed/10894533
Do you think it's a challenge for your thermal interpretation of QM?

 

[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?

 

[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.