Against "interpretation" - Comments

[atyy]

Some versions of BM do, but the standard "minimal" version doesn't.

But isn't one difference that the minimal version allows a measurement to be reversed in principle but not FAPP, whereas Copenhagen does not allow a measurement to be reversed in principle and FAPP since its principle is FAPP.

Also, is the equivalence of minimal BM and Copenhagen exact, or does minimal BM have small but FAPP unmeasurable differences from Copenhagen?

Thinking about the Wilson case, many expositions say that Wilsonian renormalization does produce terms different from old fashioned renormalization, but the differences are too small to be measured. So if one uses theory to mean exact equivalence, then Wilsonian renormalization is a different theory. However, since interpretation is also supposed to have the meaning of "solving a common sense problem with the theory (measurement for Copenhagen QM or nonsensical subtraction of infinities for old-fashioned renormalization", I would like to say Wilsonian renormalization is both an interpretation (solves the conceptual problem) and a different theory.

 

[Demystifier]

But isn't one difference that the minimal version allows a measurement to be reversed in principle but not FAPP, whereas Copenhagen does not allow a measurement to be reversed in principle and FAPP since its principle is FAPP.

Also, is the equivalence of minimal BM and Copenhagen exact, or does minimal BM have small but FAPP unmeasurable differences from Copenhagen?

Thinking about the Wilson case, many expositions say that Wilsonian renormalization does produce terms different from old fashioned renormalization, but the differences are too small to be measured. So if one uses theory to mean exact equivalence, then Wilsonian renormalization is a different theory. However, since interpretation is also supposed to have the meaning of "solving a common sense problem with the theory (measurement for Copenhagen QM or nonsensical subtraction of infinities for old-fashioned renormalization", I would like to say Wilsonian renormalization is both an interpretation (solves the conceptual problem) and a different theory.

BM is observationally equivalent to Copenhagen only in the FAPP sense.

Can you give a reference for an exposition saying that Wilsonian renormalization produces terms different from old fashioned renormalization?

 

[atyy]

BM is observationally equivalent to Copenhagen only in the FAPP sense.

For that reason, I think even if one is not against "interpretation", minimal BM can be considered a different theory. In contrast, the Newtonian, Lagrangian, and Hamiltonian formulations would be identical even in principle so they are the same theory. Similarly, the Schroedinger and Heisenberg pictures are the same theory.

Can you give a reference for an exposition saying that Wilsonian renormalization produces terms different from old fashioned renormalization?

http://www.solvayinstitutes.be/pdf/doctoral/Adel_Bilal2014.pdf (p69)
Theories like QED are presently thought to be only effective theories ... Such an effective theory then has an effective Lagrangian obtained by “integrating out” the very heavy additional fields that are present in such theories. This necessarily results in the generation of (infinitely) many non-renormalizable interactions ... From the previous argument it is then clear that at energies well below this scale these additional non-renormalizable interactions are completely irrelevant, and this is why we only “see” the renormalizable interactions. Our “low-energy” world is described by renormalizable theories like QED not because such theories are somehow better behaved, but because these are the only relevant ones at low energies: Renormalizable interactions are those that are relevant at low energies, while non-renormalizable interactions are irrelevant at low energies.

 

[Demystifier]

For that reason, I think even if one is not against "interpretation", minimal BM can be considered a different theory. In contrast, the Newtonian, Lagrangian, and Hamiltonian formulations would be identical even in principle so they are the same theory. Similarly, the Schroedinger and Heisenberg pictures are the same theory.

http://www.solvayinstitutes.be/pdf/doctoral/Adel_Bilal2014.pdf (p69)
Theories like QED are presently thought to be only effective theories ... Such an effective theory then has an effective Lagrangian obtained by “integrating out” the very heavy additional fields that are present in such theories. This necessarily results in the generation of (infinitely) many non-renormalizable interactions ... From the previous argument it is then clear that at energies well below this scale these additional non-renormalizable interactions are completely irrelevant, and this is why we only “see” the renormalizable interactions. Our “low-energy” world is described by renormalizable theories like QED not because such theories are somehow better behaved, but because these are the only relevant ones at low energies: Renormalizable interactions are those that are relevant at low energies, while non-renormalizable interactions are irrelevant at low energies.

Brilliant insight atyy!

 

[PeterDonis]

BM is observationally equivalent to Copenhagen only in the FAPP sense.

What in principle predictions does BM make that are not equivalent to predictions of Copenhagen?

 

[Demystifier]

What in principle predictions does BM make that are not equivalent to predictions of Copenhagen?

In principle, Bohmian particles may be far from the quantum equilibrium, in which case the probabilities of measurement outcomes can be totally different.

 

[bolbteppa]

This reads as an attempt to legitimize alternative 'interpretations' to Copenhagen such as dBB, referred to as nonsense by some of the founders of QM, e.g.

The term 'Copenhagen interpretation' suggests something more than just a spirit, such as some definite set of rules for interpreting the mathematical formalism of quantum mechanics, presumably dating back to the 1920s. However, no such text exists, apart from some informal popular lectures by Bohr and Heisenberg, which contradict each other on several important issues[citation needed]. It appears that the particular term, with its more definite sense, was coined by Heisenberg in the 1950s,[4] while criticizing alternate "interpretations" (e.g., David Bohm's[5]) that had been developed.[6] Lectures with the titles 'The Copenhagen Interpretation of Quantum Theory' and 'Criticisms and Counterproposals to the Copenhagen Interpretation', that Heisenberg delivered in 1955, are reprinted in the collection Physics and Philosophy.[7] Before the book was released for sale, Heisenberg privately expressed regret for having used the term, due to its suggestion of the existence of other interpretations, that he considered to be "nonsense".

https://en.wikipedia.org/wiki/Copenhagen_interpretation#Origin_of_the_term

Calling alternatives such as dBB which steal equations from QM (valid only in the non-relativistic limit of all things, a galling thing to do) and then postulate those stolen equations as axioms only to then (gallingly) use them to contradict the most fundamental claims about QM (paths not existing), I mean it is correct to say this not an 'interpretation' but rather it's own separate theory analogous to the theory of angels causing quantum weirdness who give us Schrodinger equations that only work non-relativistically... The fact that one is led to do things like deny things like relativity as fundamental https://www.physicsforums.com/insights/stopped-worrying-learned-love-orthodox-quantum-mechanics/ and rationalize away such basic, basic, concepts of physics should prove to most people why words like "nonsense" for these alternatives are appropriate.

 

[Auto-Didact]

the founders of QM, e.g.

If you want to take that road: Einstein, Schrodinger, Born and Dirac all spoke out against QM as being incomplete, tentative or in need of revision; moreover, Dirac also rightfully criticized QFT.

such as dBB which steal equations from QM

In fact, Madelung published his equations (Nov 1926) a month before Schrodinger (Dec 1926). But arguing about priority is childish nonsense.

The fact that one is led to do things like deny things like relativity as fundamental https://www.physicsforums.com/insights/stopped-worrying-learned-love-orthodox-quantum-mechanics/ and rationalize away such basic, basic, concepts of physics should prove to most people why words like "nonsense" for these alternatives are appropriate.

Yeah, just like how Newton being 'obviously correct' for two centuries invalidated Einstein's attempt to rewrite the canon of physics... oh wait.

Actually, being unable to make theoretical leaps in order to progress may only be the sign of a cautious and preservative mind; however prohibiting others from doing so and even going so far as to ridicule those who boldly do is what distinguishes the real ignoramuses from the mere cowardly.

 

[bhobba]

Theories like QED are presently thought to be only effective theories

Indeed. QED is even thought to be trivial, but I do not think anyone has proven it rigorously. If so that is strong evidence it could only be an effective theory - and of course we now know it is since its part of the electro-weak theory at high enough energies.

Is the elctro-weak theory trivial - that is something I have not seen anything written about - but my guess is probably.

Thanks
Bill

 

[*now*]

(Snip) Schrodinger (Dec 1926) (snip).

Do you mean the paper published in January 1926? “Quantisierung als Eigenwertproblem. Erste Mitteilung” (Quantization as a problem of proper values, part one), which he sent to the Annalen der Physik on 26 January 1926. In this paper, he first formulated his famous wave equation...” https://www.uzh.ch/en/about/portrait/nobelprize/schroedinger.html

 

[Auto-Didact]

Do you mean the paper published in January 1926? “Quantisierung als Eigenwertproblem. Erste Mitteilung” (Quantization as a problem of proper values, part one), which he sent to the Annalen der Physik on 26 January 1926. In this paper, he first formulated his famous wave equation...” https://www.uzh.ch/en/about/portrait/nobelprize/schroedinger.html

He indeed derived it back in 1925 and published it quite early on. But as I said, arguing about priorities is childish especially seeing the extensive influencing the early founders had on each other (including Planck, Einstein, de Broglie et al.).

Claiming that the import of a mathematical equation from one theory into another - upon which the equation is then naturally expanded upon by using a more general form of the same type of mathematics (differential equations) - is theft is pure ludicrous, especially seeing BM was first formulated by de Broglie.

 

[Auto-Didact]

To get back on topic: orthodox QM has several problems; however, from both a pure mathematical point of view as well as the mathematical physics point of view, the most important problem is the ad hoc nature of the Born rule as an axiom; no other canonical physical theory formulated as a set of differential equations has such problems. Even worse; in fact all ad hoc procedures (e.g. stationary principles) in classical canonical physics can be shown to be a necessary consequence i.e. derived purely mathematically from first principles from a more complete formulation of the differential equations.

Somehow most physicists - not being sufficiently trained to recognize proper canonical forms of differential equations based on their symplectic geometric formulation - don't take this mathematical issue seriously, instead ranting on about the fundamental importance of symmetries. For mathematicians and mathematical physicists however - especially those well versed in algebraic geometry, algebraic topology, differential geometry and complex manifolds - this is as clear as daylight.

The inability to derive the Born rule from the theory, i.e. directly from the differential equations, actually makes orthodox QM a disjointed mathematical framework. This remains a fact regardless of any reformulations using or invocations of matrices, operator algebras and/or Hilbert space; none of these help in this matter, they only make matters worse by enabling physicists to confuse the map for the territory. Moreover, going to many-body QM or QFT does absolutely nothing to solve this mathematical problem.

If a differential equation cannot be understood and solved immediately, this implies that it has somehow been stated in an implicit form which hides essential properties of the equations through unwanted algebraic simplification; this means that the equation is actually incomplete, which is no shame in itself. There are in fact many incomplete differential equations which took several decades if not centuries to be properly understood and many more which have remained so to this day.

In contrast to what physicists often claim, a differential equation not being complete is not merely a mathematical problem, but also distinctly a fundamental physics one. The way these differential equations are completed is by algebraically rewriting the incomplete implicit form into a more complete explicit form; this for example is also how Dirac derived his equation from first principles. It is not an exaggeration to say that rewriting differential equations into more explicit forms is almost the entire history of theoretical physics in a nutshell.

What is important to note is that each rewriting of an implicit differential equation can have different subsequent completions and extensions which need not be immediately consistent with each other; differential equations are like different species, they often need to be analyzed and dissected before their relation to one another becomes more clear. This is why understanding differential equations is so hard, i.e. why theoretical physics is so difficult: there is no clear foolproof route forward, only careful reasoning, mathematical experimentation and intuition.

If physicists on the other hand however choose not only to forego completing the equation, but instead go on to claim that the equation actually requires additional axioms in order to be able to interpret it, it is a signal that something has gone terribly wrong. Even worse, if they then proceed to claim that invoking new axioms in order to be able to interpret equations is proper practice, it means that they either lack mathematical insight, theoretical standards or both.

Those who today still continue to say and think that the Schrodinger equation, stated in the implicit standard form, actually precludes the existence of the Bohm quantum potential, which was always already there in the more explicit Madelung form, are as misguided as those who say that the electromagnetic potential in electrodynamics is just a mathematical trick with no actual physical existence. This is why BM is actually practically a different theory from orthodox QM: it is the most natural direct mathematical completion of orthodox QM - similar to adding the cosmological constant to the Einstein field equations - instead of an extension such as Dirac theory.

In contrast to orthodox QM, BM derives Born's equation from first principles i.e. from within the theory as part of the differential equations without any special pleadings or unwarranted axioms. The derivation is very similar to reformulating the Maxwell equations into a more brief and more symmetric form by restating everything in terms of the electromagnetic potential; this automatically makes BM a fundamentally more coherent mathematical framework than the disjointed mess that is orthodox QM.

 

[bolbteppa]

The fact that you think "BM derives Born's equation from first principles i.e. from within the theory as part of the differential equations without any special pleadings or unwarranted axioms" completely explains why you think the Schrodinger equation in BM is not simply stolen from QM.

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, $p^{\mu} = \hbar k^{\mu}$ 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...

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...

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... 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, the idea we need to derive the premise on which the whole theory is built is simply shocking, if properly understood it's like saying we need to derive F = ma or the principle of least action from nothing... 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.

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, 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 - just because Newton and Einstein got the force laws (i.e. part of the ode's) allowing us to predict the motion wrong, 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, 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...

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...

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", 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"?

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).

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...

 

[bhobba]

and ironically in QM you can't derive the non-relativistic Schrodinger equation

Have you seen chapter 3 of Ballentine where its derived from probabilities are frame independent? If so can you elaborate on how it fits in with the above?

Thanks
Bill

 

[bhobba]

To get back on topic: orthodox QM has several problems; however, from both a pure mathematical point of view as well as the mathematical physics point of view, the most important problem is the ad hoc nature of the Born rule as an axiom

Gleason's Theorem? It of course depends on non-contextuality. But that is hardly ad-hoc - its very natural.

than the disjointed mess that is orthodox QM.

IMHO QM is very very elegant, not a disjointed mess. What it means is another matter, but the theory itself is very elegant. If you accept Gleason (ie non-contextuality) it all follows from just one axiom as shown in Ballentine.

Thanks
Bill

 

[bolbteppa]

Have you seen chapter 3 of Ballentine where its derived from probabilities are frame independent? If so can you elaborate on how it fits in with the above?

Thanks
Bill

My post said "without Galilean symmetry... you can't derive the non-relativistic Schrodinger equation BM'ers seem to think is all of reality", chapter 3 of Ballentine is mostly about applying Galilean symmetry in the edition I have seen anyway.

 

[bhobba]

My post said "without Galilean symmetry... you can't derive the non-relativistic Schrodinger equation BM'ers seem to think is all of reality", chapter 3 of Ballentine is mostly about applying Galilean symmetry in the edition I have seen anyway.

Yes - without something to be symmetrical in you can't apply symmetry. That's is the paradox about physics relation to symmetry despite its fundamental importance.

Thanks
Bill

 

[bolbteppa]

Yes - without something to be symmetrical in you can't apply symmetry. That's is the paradox about physics relation to symmetry despite is fundamental importance.

Thanks
Bill

I would love to see that chapter re-written as though it were directly a non-relativistic version of Weinberg's QFT chapter 2

 

[atyy]

The fact that one is led to do things like deny things like relativity as fundamental https://www.physicsforums.com/insights/stopped-worrying-learned-love-orthodox-quantum-mechanics/ and rationalize away such basic, basic, concepts of physics should prove to most people why words like "nonsense" for these alternatives are appropriate.

But what about comments like:

http://www.staff.science.uu.nl/~hooft101/lectures/basisqft.pdf
"Often, authors forget to mention the first, very important, step in this logical procedure: replace the classical field theory one wishes to quantize by a strictly finite theory. Assuming that physical structures smaller than a certain size will not be important for our considerations, we replace the continuum of three-dimensional space by a discrete but dense lattice of points ... If this lattice is sufficiently dense, the solutions we are interested in will hardly depend on the details of this lattice, and so, the classical system will resume Lorentz invariance and the speed of light will be the practical limit for the velocity of perturbances."

https://arxiv.org/abs/1106.4501
"Since, emergent rotational and translational symmetry and locality of couplings is both common and familiar, let us start by assuming we have a continuum quantum field theory with these properties, but without insisting on Lorentz invariance ... This problem is quite general in weakly coupled field theories for multiple particle species, but at strong coupling the flow to Lorentz invariance can be robust."

 

[atyy]

In principle, Bohmian particles may be far from the quantum equilibrium, in which case the probabilities of measurement outcomes can be totally different.

Can one also get small deviations under the assumption of quantum equilibrium? In QM, when one shifts the cut to include the measurement apparatus, for example in decoherence or in the Hay and Peres paper https://arxiv.org/abs/quant-ph/9712044, one seems to find small deviations from QM with a smaller quantum system (ie. when the quantum side excludes the measurement apparatus). So in a sense, it seems that QM with different cuts is also in principle different theories. Since BM uses decoherence, could one argue that BM like QM with a bigger quantum system also has small deviations, even if quantum equilibrium is not violated? Thus if QM is not the same theory as QM, then BM is not the same theory as QM.

 

[martinbn]

@Auto-Didact I know you that you've already written a long post, but can you be a bit more specific. You are using a lot of phrases that I personally find hard to guess what they mean. For example what is a canonical form based on symplectic geometric formulation, say what is that for the heat or the Laplace equations? What is the problem with having axioms such as the Born rule? What is an implicit form of a differential equation and why is the Schrodinger equation implicit? Generally what makes the Schrodinger equation so different from any other to say that the theory has a problem? What does it mean for a differential equation to be incomplete/complete? And what does it mean to complete it? Say why is the Schrodinger equation incomplete and why are the equations from classical physics complete?

 

[DarMM]

Indeed. QED is even thought to be trivial, but I do not think anyone has proven it rigorously. If so that is strong evidence it could only be an effective theory - and of course we now know it is since its part of the electro-weak theory at high enough energies.

Is the elctro-weak theory trivial - that is something I have not seen anything written about - but my guess is probably.

Thanks
Bill

It's a genuinely uncertain issue. There are known cases where adding an $SU(2)$ gauge field to otherwise trivial theories renders them non-trivial and there are numerical simulations and simplified or limiting theories suggesting this might be what occurs in the Electroweak theory. So we currently don't actually know if the standard model is trivial.

A good intro to this stuff is still:
D.J.E. Callaway, Triviality pursuit: can elementary scalar particles exist? Phys. Rep. 167(5), 241–320 (1988)

 

[Auto-Didact]

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

 

[Demystifier]

Thus if QM is not the same theory as QM, then BM is not the same theory as QM.

I certainly agree with that.

 

[*now*]

He indeed derived it back in 1925 and published it quite early on.

Yes, the first communication article instead seems to be the article in question. Responding to a request in another thread here I’d linked a paper recently that discusses the first two of a number of his articles published in 1926, including the first communication article, hence some interest in the facts. The paper I recently linked also describes aspects of notions involved in the wave equation as misleading, hence less interest in primacies here : https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.2017.0312.

 

[Auto-Didact]

@Auto-Didact I know you that you've already written a long post, but can you be a bit more specific. You are using a lot of phrases that I personally find hard to guess what they mean.

Good questions, I will try to answer each of them in a manner understandable to as wide an audience as possible; this means that I will use the most elementary notations that everyone who has taken calculus should be able to recognize.

For example what is a canonical form based on symplectic geometric formulation

This is the key question, so I will spend most time on this one: stated simply in words first, it means that one can do geometry and calculus in an extended phase space, which topologically is a very special kind of manifold. I will illustrate this by utilizing analytical mechanics with phase space trajectory $\Gamma(t)$, $H(q,p,t)$ and $\delta S[\Gamma] = \delta \int_{t_0}^{t_1} L(q,\dot q, t) dt = 0$ as a case study:

Let $$H=p\dot q- L \Leftrightarrow L=p\dot q- H$$It then immediately follows that $$\begin{align}
\delta \int_{t_0}^{t_1} L dt & = \delta \int_{t_0}^{t_1} (p \frac {dq}{dt}-H) dt \nonumber \\
& = \delta \int_{t_0}^{t_1} (p \frac {dq}{dt}-H\frac {dt}{dt}) dt \nonumber \\
& = \delta \int_{t_0}^{t_1} (p \frac {dq}{dt}+0\frac {dp}{dt}-H\frac {dt}{dt}) dt \nonumber \\
\end{align}$$where obviously $L = p \frac {dq}{dt}+0\frac {dp}{dt}-H\frac {dt}{dt}$.

Now group the terms on the RHS of $L$ as two vectors, namely: $$\vec X = (p, 0, -H) \text { & } \vec Y = (\frac {dq}{dt},\frac {dp}{dt},\frac {dt}{dt})$$It then immediately follows that $$L = \vec X \cdot \vec Y$$Now we can continue our earlier train of thought: $$\begin{align}
\delta \int_{t_0}^{t_1} L dt & = \delta \int_{t_0}^{t_1} (p \frac {dq}{dt}+0\frac {dp}{dt}-H\frac {dt}{dt}) dt \nonumber \\
& = \delta \int_{t_0}^{t_1} \vec X \cdot \vec Y dt \nonumber \\
& = \delta \int_{\Gamma} \vec X \cdot d \vec {\Gamma} = 0 \nonumber \\
\end{align}$$ where the last equation is a line integral along the phase space trajectory $\Gamma$.

Now one may say I just did a bit of algebra and rewrote things and yes, that actually is trivially true. However the more important question naturally arises: are the vector fields $\vec X$ and $\vec Y$ simply mathematics or are they physics? The answer: they are physics, more specifically they are properties of analytical mechanics in phase space, with $\vec Y$ being the displacement vector field in phase space.

More specifically, what is $\vec X$? Remember that these are vectors in a 3 dimensional space $(q,p,t)$. So let's just do some vector calculus on it, specifically, take the curl of $-\vec X$: $$ \nabla \times (-\vec X) =
\begin{vmatrix}
\hat {\mathbf q} & \hat {\mathbf p} & \hat {\mathbf t} \\
\frac {\partial}{\partial q} & \frac {\partial}{\partial p} & \frac {\partial}{\partial t} \\
-\vec X_q & -\vec X_p & -\vec X_t
\end{vmatrix} = (\frac {\partial H}{\partial t},- \frac {\partial H}{\partial t}, 1)$$ Now if you haven't seen the miracle occur yet, squint your eyes and look at the last part $\nabla \times (-\vec X) = (\frac {\partial H}{\partial t},- \frac {\partial H}{\partial t}, 1)$. More explicitly, recall $\vec Y = (\frac {dq}{dt},\frac {dp}{dt},\frac {dt}{dt})$. It then immediately follows that $$(\frac {dq}{dt},\frac {dp}{dt},\frac {dt}{dt}) = (\frac {\partial H}{\partial t},- \frac {\partial H}{\partial t}, 1)$$ or more explicitly that $\vec Y = - \nabla \times (\vec X)$. These are Hamilton's equations! In other words, $- \vec X$ is the vector potential of $\vec Y$. There is even a gauge choice here making $X_q = p$ and $X_p = 0$, but I will not go into that.

Now Hamilton's principle - i.e. that $\delta \int_{t_0}^{t_1} Ldt = 0$ - follows naturally from Stokes' theorem: $$ \begin{align} \delta \int_{\Gamma} \vec X \cdot d \vec {\Gamma} & = \int_{\Gamma} \vec X \cdot d \vec {\Gamma} - \int_{\Gamma} \vec X \cdot d \vec {\Gamma'} \nonumber \\
& = \oint_{\Sigma} \nabla \times \vec X \cdot d \vec {\Sigma} \nonumber \\
\end{align}$$ where $\Sigma$ is a closed surface between the two trajectories $\Gamma$ and $\Gamma'$. From vector calculus we know that it is necessary that $\oint_{\Sigma} \nabla \times \vec X \cdot d \vec {\Sigma} = 0$ because $\vec Y = - \nabla \times \vec X$ and this means that $\nabla \times \vec X$ is tangent to $\Sigma$ verifying our proof that the integral vanishes.

In other words, Hamilton's principle is just an implicit consequence of $\vec Y = - \nabla \times \vec X$, i.e. of Hamilton's equations in phase space. If $\vec Y$ has a vector potential $- \vec X$, then $\vec Y$ is automatically a solenoidal vector field, i.e. $$\vec Y = - \nabla \times \vec X \Leftrightarrow \nabla \cdot (\nabla \times \vec X) = 0 \Leftrightarrow \nabla \cdot \vec Y = 0$$demonstrating that variational principles w.r.t. action - indeed even the very existence of calculus of variations as a mathematical theory - is purely a side effect of Liouville's theorem applying to Hamiltonian evolution in phase space; that in a nutshell, is symplectic geometry in it's most simple formulation.

Lastly, if you remove the requirement of Liouville's theorem, you leave the domain of Hamiltonian mechanics and automatically arrive at nonlinear dynamical systems theory, practically in its full glory; this is theoretical mechanics at its very finest.

say what is that for the heat or the Laplace equations?

As DEs, the heat equation $\nabla^2 u = \alpha \frac {\partial u} {\partial t}$, the Laplace equation $\nabla^2 u = 0$ and the Poisson equation $\nabla^2 u = w$ are all elliptical PDEs, meaning all influences are instantaneous (cf. action at a distance in Newtonian gravity). Carrying out the phase space analysis goes a bit too far for now.

However it is immediately clear upon inspection of the equations that the heat equation is a more general Poisson equation, which is itself a more general Laplace equation; as stated above this is what is meant by saying the one is an implicit form of a more explicit form. More, generally all of them are all special instances of the more general Helmholtz equation which has as its most explicit form $$\nabla^2 u + k^2 u = w$$It is the set of all solutions of an implicit DE, i.e. the set $U$ containing all possible functions $u$, which decides what the explicit form of the DE is. Unfortunately, this set is typically for quite obvious reasons unknown.

These relationships between DEs becomes even more obvious once one applies these same mathematical techniques to the study of sciences other than physics, where these differential equations naturally tend to reappear in their more explicit forms. If one steps back - instead of mindlessly trying to solve the equation - an entire taxonomy with families of differential equations slowly becomes apparent. Actually we don't even have to leave physics for this, since the Navier-Stokes equation from hydrodynamics and geometrodynamic equations tend to rear their heads in lots of places in physics.

What is the problem with having axioms such as the Born rule?

The answer should become obvious once rephrased in the following manner: What is the problem with having the Born rule - a distinctly non-holomorphic statement - for needing to be able to understand what an analytic differential equation is describing?

What is an implicit form of a differential equation and why is the Schrodinger equation implicit?

An implicit form is as I stated above: the Laplace equation is an implicit form of the Poisson equation with $w = 0$ implicitly.

Generally what makes the Schrodinger equation so different from any other to say that the theory has a problem?

There is nothing special about the Schrodinger equation, that is my point. What has a problem is orthodox QM, which consists of a mishmash of SE (DE) + Born rule (ad hoc, non-analytic)+ measurement problem + etc. No other canonical physical theory has the mathematical structure where all consequences of the theory aren't directly derivable from the DE and the mathematics (i.e. analysis, vector calculus, differential geometry, etc).

What does it mean for a differential equation to be incomplete/complete?

It means that stated in it's implicit form there are terms missing, i.e. implicitly made to be equal to zero and therefore seemingly not present, while when rewritten into the most explicit form the terms suddenly appear as out of thin air: the terms were there all along, they were just hidden through simplification by having written the equation in its implicit form.

And what does it mean to complete it?

It means to identify the missing terms which are implicitly made to equal zero; this is done purely mathematically through trial and error algebraic reformulation, by discovering the explicit form of the equation. There is no straightforward routine way of doing it, it cannot be done by pure deduction; it is instead an art form, just like knowing to be able to handle nonlinear differential equations.

Completing an equation is a similar but not identical methodology to extending an equation; extending is often mentally more taxing since it tends to involve completely rethinking what 'known' operators actually are conceptually, i.e. just knowing basic algebra alone isn't sufficient. Extending is how Dirac was able to derive his equation purely by guesswork; he wasn't just blindly guessing, he was instead carefully intuiting the underlying hidden structure Lorentzian structure in the d'Alembertian's implicit form and then boldly marching forward using nothing but analysis and algebra.

Say why is the Schrodinger equation incomplete and why are the equations from classical physics complete?

The Schrodinger equation once completed in the manner described above actually has the Madelung form with an extra term, namely the quantum potential. This is purely an effect of studying the equation as an object in the theory of differential equations and writing it in the most general form without simplifying. I'll give a derivation some other time, if deemed necessary.

 

[bolbteppa]

But what about comments like:

http://www.staff.science.uu.nl/~hooft101/lectures/basisqft.pdf
"Often, authors forget to mention the first, very important, step in this logical procedure: replace the classical field theory one wishes to quantize by a strictly finite theory. Assuming that physical structures smaller than a certain size will not be important for our considerations, we replace the continuum of three-dimensional space by a discrete but dense lattice of points ... If this lattice is sufficiently dense, the solutions we are interested in will hardly depend on the details of this lattice, and so, the classical system will resume Lorentz invariance and the speed of light will be the practical limit for the velocity of perturbances."

https://arxiv.org/abs/1106.4501
"Since, emergent rotational and translational symmetry and locality of couplings is both common and familiar, let us start by assuming we have a continuum quantum field theory with these properties, but without insisting on Lorentz invariance ... This problem is quite general in weakly coupled field theories for multiple particle species, but at strong coupling the flow to Lorentz invariance can be robust."

If you read the whole page and the top of the next of t'Hooft he points out the whole goal in quantization is to make sure we obtain a Lorentz invariant theory in the continuum limit, I'm not sure how one particular human-made method of promoting a classical theory to a quantum theory (which is not the only way) requires some step whose effects have to not matter in the final result (continuum limit) does in any way "deny things like relativity as fundamental", just as say the use of ghosts in path integral quantization don't deny spin-statistics as being fundamental.

 

[bolbteppa]

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.

That is crystal clear in light of unbelievable statements such as

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.

I've never heard someone go this far outside the bounds and try to pretend we can derive things like the POLA from anything other than something equivalent to Newton's laws, let alone Stokes' theorem, this is an even worse misunderstanding than thinking we should be able to derive the Born rule (unless you unquestioningly assume either the insanely complicated and specialized non-relativistic Schrodinger equation or these weird completely unjustified concepts like energy and momentum to then get the Schrodinger equation which normal QM actually defines as charges from those symmetry conservation laws you don't like, then you can at least pretend you are deriving things, but to pretend we can derive the POLA is as out there as pretending we can derive Newton's first law...). Statements such as

What has a problem is orthodox QM, which consists of a mishmash of SE (DE) + Born rule (ad hoc, non-analytic)+ measurement problem + etc. No other canonical physical theory has the mathematical structure where all consequences of the theory aren't directly derivable from the DE and the mathematics (i.e. analysis, vector calculus, differential geometry, etc).

illustrate a deep misunderstanding of the most elementary claims in physics, no amount of mathematics is going to alleviate the fact that we need to assume some primitive notions in physical theories, it's simply a shocking misunderstanding to claim things like the POLA can be derived from mathematics...

 

[atyy]

If you read the whole page and the top of the next of t'Hooft he points out the whole goal in quantization is to make sure we obtain a Lorentz invariant theory in the continuum limit, I'm not sure how one particular human-made method of promoting a classical theory to a quantum theory (which is not the only way) requires some step whose effects have to not matter in the final result (continuum limit) does in any way "deny things like relativity as fundamental", just as say the use of ghosts in path integral quantization don't deny spin-statistics as being fundamental.

It means that a theory without fundamental Lorentz invariance may be indistinguishable in the experimentally relevant regime from a theory with fundamental Lorentz invariance. That is what the quote from Raman Sundrum also means.

Further on, 't Hooft also points out that some of our current theories may not have a continuum limit. At the physics level of rigour, QCD is thought to have a continuum limit, but QED is thought to have a Landau pole.
p50: "If a theory is not asymptotically free, but has only small coupling parameters, the perturbation expansion formally diverges, and the continuum limit formally does not exist."
p58: "Landau concluded that quantum field theories such as QED have no true continuum limit because of this pole. Gell-Mann and Low suspected, however, ... it is not even known whether Quantum Field Theory can be reformulated accurately enough to decide"

 

[Auto-Didact]

I've never heard someone go this far outside the bounds and try to pretend we can derive things like the POLA from anything other than something equivalent to Newton's laws, let alone Stokes' theorem, this is an even worse misunderstanding than thinking we should be able to derive the Born rule (unless you unquestioningly assume either the insanely complicated and specialized non-relativistic Schrodinger equation or these weird completely unjustified concepts like energy and momentum to then get the Schrodinger equation which normal QM actually defines as charges from those symmetry conservation laws you don't like, then you can at least pretend you are deriving things, but to pretend we can derive the POLA is as out there as pretending we can derive Newton's first law...).

Demonstrate to me purely mathematically how the calculus of variations is not a direct consequence of Stokes' theorem.

Statements such as ... illustrate a deep misunderstanding of the most elementary claims in physics, no amount of mathematics is going to alleviate the fact that we need to assume some primitive notions in physical theories, it's simply a shocking misunderstanding to claim things like the POLA can be derived from mathematics...

There is no misunderstanding here, I perfectly understand the conventional way of understanding these matters; I just consciously choose to reject it for different - mathematical, theoretical and methodological - reasons as well as based on my knowledge of the history of physics, where I see the same type of mistakes keep getting made again and again.

I believe that the contemporary conventions in theoretical physics are possibly mistaken; this seems most obvious to me because, many theoreticians, when pushed, do not seem to really know anything in depth about how the unconventional theories of pure higher mathematics feature in the foundations of physics, except for trivial procedural knowledge i.e. how to mindlessly carry out some calculations.

When asked legitimate questions about mathematical issues w.r.t. physics they tend to either go off on irrelevant tangents, blatantly avoid the question, just assume that the problem is not a real problem, or worse, assume without any verification that it is already solved. Even worse, there are some who really only truly seem to be worried about having an academic job and the social status gained from their career; this while theoretical physics as a discipline continues on in its current period of stagnation.

 

[bolbteppa]

It means that a theory without fundamental Lorentz invariance may be indistinguishable in the experimentally relevant regime from a theory with fundamental Lorentz invariance. That is what the quote from Raman Sundrum also means.

Further on, 't Hooft also points out that some of our current theories may not have a continuum limit. At the physics level of rigour, QCD is thought to have a continuum limit, but QED is thought to have a Landau pole.
p50: "If a theory is not asymptotically free, but has only small coupling parameters, the perturbation expansion formally diverges, and the continuum limit formally does not exist."
p58: "Landau concluded that quantum field theories such as QED have no true continuum limit because of this pole. Gell-Mann and Low suspected, however, ... it is not even known whether Quantum Field Theory can be reformulated accurately enough to decide"

My bad/developing understanding is the latter points are stem from the necessity of renormalisation, a necessity also in classical electromagnetism (originally motivating renormalisation in qed), the necessity of which is due to the fact that we unavoidably (due to relativity) work with point particles, until string theory came along as the first (and only ) legitimate way to potentially bypass the point particle model which is still being discovered, with these lattice models being nothing but approximation methods, and none of this in any sense questioning relativity.

 

[bolbteppa]

I just consciously choose to reject it

That says it all.

 

[Auto-Didact]

That says it all.

Congratulations, you know how to generate soundbites! Say, do you actually have any original thoughts at all or have you just mastered the art of repeatedly parroting consensus opinions?

There are good mathematical reasons for questioning the conclusions of the picture bestowed upon us by Wilsonian EFT, namely the identification of a deeper mathematical theory of renormalization instead of the conventional version Wilsonians cling to; this however goes way off-topic from this thread.

In any case, you haven't answered my mathematical challenge, so if that's all you have to say, then I accept your concession.

 

[PeterDonis]

Thread closed for moderation.

 

[PeterDonis]

Demonstrate to me purely mathematically how the calculus of variations is not a direct consequence of Stokes' theorem.

That's not how it works. You made the positive claim, so it's up to you to prove it. If you have such a proof, or a reference to one, feel free to PM it to me.

My bad/developing understanding

That says it all.

Given that you admit your understanding is bad/developing, you should not be so quickly dismissive of what other people post.

Both of you are now banned from further posting in this thread.