One thing needs to be made crystal clear: the argument I'm making is as I said from a pure mathematics or mathematical physics viewpoint. All physical theories can and have been judged from this viewpoint, and its criteria are different from those of regular and theoretical physics because their respective direct goals and intentions are truly different.
This is why I made the distinction 'as a mathematical framework' instead of saying 'as a physical theory'; this distinction is not vacuous, instead it is where the very notion of a theory being fundamental or not comes from in the practice of physics. Notice that many theoreticians today have turned this notion on its head and instead just subjectively reserve the right to claim fundamentality for their particular viewpoint; such anarchy is exactly what happens once physicists decide to forego the prime role of bestowing upon mathematical physics the task of identifying what is fundamental.
bolbteppa said:
The fact that you think we need to formulate theories in terms of differential equations is to unavoidably assume that classical paths must exist, and so to literally deny/misunderstand the most basic claim of QM that paths don't exist - if paths don't exist, all of classical physics is wrong and we have absolutely nothing...
It is not my viewpoint, it is a standard one in mathematical physics, because historiclly almost all more fundamental mathematical reformulations were discovered from within this very viewpoint of practicing mathematical physics.
bolbteppa said:
Please go and read Bohm's original papers (
http://cqi.inf.usi.ch/qic/bohm1.pdf) and show me where he derived the Schrodinger equation - you won't find it because he didn't, he assumed it out of thin air, which is what many BM sources do. The ones that try harder try and derive it from something along the lines of these here
https://en.wikipedia.org/wiki/De_Broglie–Bohm_theory#Derivations which are either complete nonsense (to be explained in a moment) or are using concepts that assume standard Copenhagen QM (I mean really, pμ=ℏkμpμ=ℏkμp^{\mu} = \hbar k^{\mu} as your starting point, where do these strange concepts of energy or momentum even come from? and we are talking about a theory that is not a 'disjointed mess') and so defeat the whole purpose of BM, i.e. to save classical physics and deny what science actually tells us...
##p=hk## is just another way of stating the de Broglie wavelength, which was invented in 1924 before Schrodinger came up with his equation. Again, it is irrelevant what happened first: the mathematical veracity of equations do not depend on when someone first writes them down or for whatever reasons they were written down.
bolbteppa said:
All we can say before the Born rule is that paths don't exist because that's what experiments tell us, and therefore that classical theories (non-relativistic and relativistic) are wrong, and so we literally have nothing...
Experiments say nothing of the sort, it is an interpretation of the theory and experiment together which talks about the non-existence of paths.
bolbteppa said:
The very fact you think we should be able to derive the Born rule illustrates an extremely fundamental misunderstanding of what QM says - if the very first thing it says is that path's don't exist, and so without paths we have nothing,
Again it's not my own viewpoint, it is a legitimate standard viewpoint in the practice of mathematical physics.
The SE is just a (complex) differential equation, like all other differential equations; this means that it can be studied purely mathematically from within the theory of differential equations just like any other differential equation. The fact that the Born rule is mathematically derivable in this manner makes your point moot.
bolbteppa said:
the idea we need to derive the premise on which the whole theory is built is simply shocking,
It is not shocking because all canonical physical theories, except for QM, were eventually able to be derived in such a manner, once reformulated into the specific mathematical framework in which the physical theory best fits (i.e. into vector calculus, or exterior calculus, or differential forms, or differential geometry, or bispinor calculus, or complex manifolds, etc) by the mathematicians and mathematical physicists.
bolbteppa said:
if properly understood it's like saying we need to derive F = ma or the principle of least action from nothing...
The principle of least action is directly derivable from Stokes theorem; calculus of variations is not an independent framework but a direct consequence of not taking exterior calculus and differential forms to heart.
On the other hand ##F=ma## can actually be derived from experiment directly, just like ##E=h\nu## as Planck and Einstein did.
bolbteppa said:
In order to state something to build a theory we need to admit that we have the existence of classical mechanics in 'some sense', i.e. the to-be-defined quasi-classical limit, and so try to merge the fact that paths don't exist in experiments with paths existing in some approximate sense which leads to needing what we call the Born rule, which is why QM is so nuts - we unavoidably need classical mechanics to formulate it.
This is just one way of seeing it, i.e. an interpretation. It is however not merely an interpretation of an equation but an interpretation of methodology as well; i.e. it is a purely pragmatic FAPP philosophy. In terms of mathematical physics, such FAPP philosophies are unnecessary assumptions since the question to be answered in mathematical physics isn't a question to be answered FAPP, but instead a question to be answered in principle.
bolbteppa said:
Without standard QM you are literally banned from using concepts like wave functions as if they were fundamental, it is simply madness to even think of something like a wave function if the notion of a path exists in any sense, nothing but a decision to ignore inherently obtainable information for no reason,
This is just pure hogwash. Wave functions are just mathematical objects, taken literally functions describing waves. They arise naturally not only in physics, but in all different kinds of manners in empirical and phenomenological science studied by applied mathematicians and/or non-physicist scientists.
bolbteppa said:
and the ironic reason for this is differential equations, which tell us that if particles follow any kind of path at all in any sense, we should be able to predict the path no matter what the equations which control it's motion are because it's just basic mathematics
The theory of differential equations absolutely says no such thing; what can and cannot be done depends on the class of the differential equation. It is a severe misapprehension of mathematics to think otherwise. It is not a non-trivial issue because the theory of differential equations is still a work in progress, meaning many physicists, focussed solely on applications, remain unaware of such issues.
bolbteppa said:
just because Newton and Einstein got the force laws (i.e. part of the ode's) allowing us to predict the motion wrong
Again what I stated applies to all of canonical physics, up to and including statistical mechanics, critical phenomenon, geometrodynamics, etc.
bolbteppa said:
if the paths exist in any sense, you'd have to deny differential equations if you want to pretend we can never know what the path was for some given special example, which is why said 'derivations' of the Schrodinger equation are complete nonsense
Opinion, not fact. Whether this opinion is popular among physicists says absolutely nothing about the veracity of the claim.
bolbteppa said:
the idea that these random concepts like wave functions should mean anything if paths exist is simply human bias, of course it's a bias motivated by BM'ers trying to copy orthodox QM because they have to for unexplained reasons despite the fact that they should be able to do way more fundamental things like actually predict paths if what they claimed made any sense... In other words, there are good reasons why the founders made such bold claims about complementarity e.g. paths not existing and why this is all they could come up with without committing basic logical errors...
The reasons Bohr et al. made such strong claims were due to reasons of practicality and ignorance of more advanced mathematics; they were at the cutting edge in their time. Realizing that much more was left to be understood experimentally, physicists generally just ignored the problem in the foundations of QM for almost a century, merely pretending that these were resolved, which is why the foundational problems still haunts the theory until this very day.
However, someone today seriously making the exact same argument as Bohr et al. did a century ago just means that this person is just hopelessly out of touch with the progression of science and mathematics since then: it was justified then because there was more to discover and there was the hope the issue would resolve itself; more was indeed discovered but the issues have not resolved themselves. Playing make belief in the name of FAPP philosophy is fine until one hits a wall where experiment gets stuck; suffice to say, physics seems to have hit that wall since.
bolbteppa said:
Landau's QM spends a good few pages stressing the technical points here, I don't know how anybody could try and imply that orthodox QM is flawed because the Born rule can't be found via differential equations if they understood the very first claim of QM is that paths simply don't exist so that no differential equation could ever dictate it's most fundamental claim...
Easy, two different ways:
- by challenging the viewpoint made by Landau and Lifshitz; the books are good, stellar even, but not holy.
- by approaching the question from the point of view of mathematical physics instead of based on FAPP philosophy; this issue would necessarily arise sooner or later due to the
mathematical problem of merging GR and QT.
bolbteppa said:
Even more laughable is the idea that a quantum theory which fails so spectacularly at dealing with relativity is "a fundamentally more coherent mathematical framework than the disjointed mess that is orthodox QM",
That depends on the intent of the formulation. The intent in mathematical physics is to give QM a solid mathematical foundation instead of parroting FAPP philosophy; no one seems to question the non-FAPP intent of mathematical physics when Wightman et al. attempted to give a rigorous foundation of QFT. I'm guessing you would say that Newton-Cartan theory has absolutely no scientific merit either and studying it was a complete waste of time for physicists.
Making any theory consistent with relativity is another step in the process of building foundations; for BM this step is still a work in progress. You are pretending for some strange reason that all steps have to be taken at once, else thrown out immediately. This is a strange and overambitious methodology which can not possibly be rigorous enough to be seen as legitimate practice in fundamental physics.
bolbteppa said:
as I've already pointed out one of the ways people claim to be able to do this is to literally deny that special/general relativity is more fundamental non-relativistic classical mechanics, this should be beyond shocking, yet in here we are implying this is "more coherent"?
That is a way, not the only way. And yes, it is still mathematically more coherent independent of whether it is the correct theory of nature. That is another question entirely! Fact: BM as well as Newton-Cartan theory are more coherent mathematical frameworks than orthodox QM. This just implies that mathematical coherence alone is not sufficient nor the best guide for judging the utility or veracity of a theory for physics; this is obvious, that role belongs to experiment.
Fact: all experiments done so far cannot distinguish between the outcomes of orthodox QM and BM.
bolbteppa said:
Finally, the reason physicists are "ranting on about the fundamental importance of symmetries" is because without symmetries we can do almost nothing, e.g. without Galilean symmetry we can't go far beyond the statement of the principle of Least action in non-relativistic mechanics, and ironically in QM you can't derive the non-relativistic Schrodinger equation BM'ers seem to think is all of reality, and it's merely the failure of Galilean symmetry that leads to special relativity, with both Galilean and Einsteinian relativity based on the primitive notion of a path existing, unlike QM... (Again, all in Landau).
Without symplectic geometry there is no principle of stationary anything. Working on BM as a project in terms of mathematical physics does not in any way imply that those who work on it believe it to be all of reality; that is just pure projection, which in fact sounds very much like a soundbite that a politician would make to smear his opponents.
bolbteppa said:
So yes, BM is "actually practically a different theory from orthodox QM" because it begins by contradicting the most basic claim QM makes and then tries to still get the results of the theory it denies by assuming it's equations out of thin air, it's no wonder people like Heisenberg used words like "nonsense" for alternatives this logically flawed, with the relativity denial issues taking this over the top. These are the kinds of serious flaws that an essay like this is trying to legitimize...
I see no issue whatsoever with constructing intermediate mathematical frameworks in order to arrive at a new physical theory or in trying to formulate rigorous foundations where they are sorely lacking in an existing physical theory. The progression in the foundations of physics is not helped at all by physicists who believe that appealing to FAPP philosophy actually solves foundational problems, thereby giving them a license to bark at those actually attempting to solve such foundational problems.
