Against "interpretation" - Comments

[Demystifier]

Greg Bernhardt submitted a new blog post

Against "interpretation"

Continue reading the Original Blog Post.

 

[PeterDonis]

Interesting viewpoint! I tend to dislike talk about "interpretations" as well, but I'm not sure if it's for the same reason, so let me briefly explain my reason.

In the article you give the example of two theories, T1 and T2, that both make the same experimental predictions. That means that it is not possible to decide by experiment which of the two theories is correct (or "more correct", or "better", or whatever comparative term you want to use). So any preference that someone has for one theory over the other can only be based on subjective judgments of one sort or another. And disagreements about such preferences can never be resolved (which is why PF threads on such topics tend to go on and on until one of the Mentors gets fed up enough to close the thread).

To me, the word "interpretation" (particularly when talking about QM) is a signal that the above situation exists, i.e., that whatever disagreement is being discussed can't be resolved, because the two theories (or "interpretations") being discussed both make the same experimental predictions. And since such a disagreement can't be resolved, I see very limited value, at best, in discussing it.

In the particular case of QM, at least, there is another aspect as well. Interpretations of QM not only all make the same experimental predictions, they do so via provably mathematically equivalent machinery. So the rationale for calling them different theories, as opposed to different interpretations of the same theory, is weaker than it would be if the different interpretations used mathematically distinct machinery, that just happened to result in the same predictions for experiments.

 

[fresh_42]

I follow your argumentation and appreciate its clarity. But don't you just merge the two terms and therewith shift all problems the term interpretation is charged with into the now new definition of theory? My impression is, that you merely shifted the problem of inconsistency into the unregarded territory of theory and model. The term theory is to me a very unfortunate one, not the least as it is regularly in conflict with what commonly is meant by it, namely a hypothesis. I admit that the term model may partly be in conflict with the specific meaning it has in logic, but that shouldn't be a problem here.

Thus my suggestion to solve the discrepancies wouldn't be the merger of theory and interpretation, but the merger of theory and model. I think this is what physicists usually mean: a theory - or model - is a mathematical framework to describe the experimental results, including possible predictions. On the other hand, an interpretation is merely an informal description of named theory / model to describe the mathematical framework in common language and by the frequent use of aphorisms and metaphors. In this sense, T1 and T2 are different models, which either coincide, i.e. are different but equivalent sets of mathematical language, or they make distinguishable predictions. Interpretations of T1 and T2, however, may differ a lot, as they were only descriptions a lecturer uses to explain the models behind them.

So my understanding is, that T1 (Heisenberg) and T2 (Schrödinger) are two different models of the same physics, and as such equivalent theories or models, whereas Kopenhagen and MWI are actually interpretations, namely metaphors to visualize the equations. This is in my opinion closer to the actual use of the terminology and doesn't need the merger theory with interpretation, the more as theory and model has already been merged. Furthermore, it doesn't create the next problem to explain why interpretation and model will be the same, nor does it leave a vacancy, namely: what are verbal descriptions of mathematical frameworks be called?

 

[A. Neumaier]

So you think that Copenhagen and MWI are two different theories that happen to share 99.9% of the theoretical part and disagree only about 0.1% of interpretation? Then quantum mechanics would be a collection of 20 or so theories that differ only marginally?

 

[Klystron]

Borrowing from military jargon, the professionals -- scientists, engineers, mathematicians, and their students -- work to understand and utilize theory and models. Civilians; depending on their training, ability, and access to correct reliable information; are often satisfied with various interpretations.

As an example consider the science writer James Gleick and his still popular interpretation of chaos theory Chaos: Making a New Science. I first read his book back in the 1970's in order to discuss it with my father, a trained chemist and electrical engineer but also a fan of popular science. Gleick's writing led to productive conversations. We even drove to Half-Moon Bay to observe the "fractal coastline" (and a neat sunset). Despite using aspects of chaos theory in my work, I never referenced Gleick's books, instead using math and science textbooks for actual applications.

[Edit: Change "electrical" to electronics engineer at ESL Sylvania as a project manager for the USAF.]

Accurate factual interpretations perhaps with the math explained rather than read as-is, may be a fair method for civilians to understand science. Students and teachers should be held to the more rigorous standard of understanding theory, the underlying models, and the limits of corroborating data.

 

[bhobba]

I think one the issues regarding interpretations is to consider them complete in some sense. Copenhagen fell into that trap eg the state is a complete description of the quantum system. I have posted it before, but I consider the following debate between Dirac and Heisenberg instructive:
http://philsci-archive.pitt.edu/1614/1/Open_or_Closed-preprint.pdf

Personally I am with Dirac on this one. I do not think QM is complete - it will slowly and gradually be superseded and in that vein you can look on interpretations as attempts to understand what direction the next step may take us.

Already we have no-go theorems putting constraints on acceptable interpretations that help us see in what direction research may lead us.

Thanks
Bill

 

[hagopbul]

T1 &T2 are approximations just like classical mechanics to special relativity

 

[ade]

Demystifier,

The admirable deed is that you walked all the way to the border!
When you manage to cross it, you will see wonderful things that reasoning cannot even imagine!

Keep it on!

 

[Peter Morgan]

Insofar as a given theory is some specific mathematical construction from some set of axioms and all its mathematical consequences, and all equivalent constructions from different axioms, different interpretations of that one theory (1) put different emphasis on one axiomatic construction or another, all isomorphic to each other, and/or (2) give a different relationship between elements of the theory and whatever experimental raw data there may be. I like Lakatos's term for (2), bridge principles, which, being between things mathematical and things in the world are neither of mathematics nor of the world. I think it's the bridge principal concept that your polemic misses. Mathematics is unworldly and not everything is mathematics: we have to learn and teach and experience and come to an intuition of the connection between mathematics and the world.
I suppose I'm saying that theory/interpretation/world is just a variant of the map/how to grok the map/territory problem, over which much ink has been spilt, so battle lines will have to be drawn.

 

[atyy]

Should Wilson's interpretation of renormalization be considered theory or interpretation?

 

[bhobba]

Should Wilson's interpretation of renormalization be considered theory or interpretation?

I would call it an approach. My understanding is its equivalent to say dimensional regularization - but others that know more about it than me may correct me on that.

Thanks
Bill

 

[martinbn]

What is an example of T1 and T2?

 

[Demystifier]

So my understanding is, that T1 (Heisenberg) and T2 (Schrödinger) are two different models of the same physics, and as such equivalent theories or models, whereas Kopenhagen and MWI are actually interpretations, namely metaphors to visualize the equations.

Then what about Bohmian mechanics? Is it a T3, or is it just a metaphor?

 

[Demystifier]

What is an example of T1 and T2?

There are many examples, but my favored one is this: T1 = standard textbook QM, T2 = Bohmian mechanics.

 

[atyy]

Traditionally, an interpretation is a solution of the measurement problem:

However, another meaning of interpretation is simply the meaning physical or operational meaning assigned to mathematical operations.

A third meaning of interpretation is the one that is disfavoured in the Insight of the OP.

Thus the Copenhagen Interpretation is the only valid interpretation (since BM is not an interpretation, but a different theory) of QM. However, the Copenhagen Interpretation is not a valid interpretation in the sense of being a possible solution of the measurement problem (it is basically the poster boy for having a measurement problem). Bohmian Mechanics is not an interpretation, but a different theory. However, since it is the only currently known possible solution of the measurement problem for a sufficiently broad domain of QM, BM is the only valid interpretation of QM even though it is a different theory.

 

[martinbn]

There are many examples, but my favored one is this: T1 = standard textbook QM, T2 = Bohmian mechanics.

I thought that BM gives different predictions in some cases?

 

[Demystifier]

I thought that BM gives different predictions in some cases?

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

 

[Demystifier]

Should Wilson's interpretation of renormalization be considered theory or interpretation?

If only more Bohmians understood the depth and importance of this question ...

 

[anorlunda]

And disagreements about such preferences can never be resolved (which is why PF threads on such topics tend to go on and on until one of the Mentors gets fed up enough to close the thread).

That raises the question of why we allow public debate of interpretations on PF. Or why respected textbooks include chapters on interpretations.

If it was just a case of competing equivalent models, as @fresh_42 pointed out, then there is nothing to debate. The wise analyst keeps all possible models in his toolkit ready to use when advantageous. The value of a model is its utility.

Personally I am with Dirac on this one. I do not think QM is complete - it will slowly and gradually be superseded and in that vein you can look on interpretations as attempts to understand what direction the next step may take us.

That may be the answer to why allow it and why we support continued research. An interpretation might be described as a theory extension wannabe. Researchers hope to find a case where T₁ and T₂ predict different outcomes, so that experiment may prove one of them useful and the other not useful. But such hope should not be eternal. At some point, the rational thing to do is to declare the search futile, and deploy our resources elsewhere.

 

[Dr. Courtney]

Well articulated Insight. Thanks!

 

[martinbn]

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

Can you give another example?

 

[Demystifier]

Can you give another example?

T1 = "Copenhagen" with collapse induced by measurement, T2 = many worlds

 

[martinbn]

T1 = "Copenhagen" with collapse induced by measurement, T2 = many worlds

I expected a non quantum example. So, you have in mind only QM interpretations, and you think they should be called theories. My opinion is that they are correctly called interpretations. The all start with QM or at least the core of QM, then add a bit more, yet don't get new predictions. To me that is not a different theory. To be a different theory it should build on something else, it should be possible to get to that theory even if you have never seen QM. And that is not the case of the interpretations.

 

[Auto-Didact]

There is another complication not yet explicitly mentioned, namely the role of the underlying mathematics from the viewpoint of mathematical physics: as an object in the theory or as a tool used to calculate objects in the theory. Key mathematical distinctions, unconventional among physicists, may then be trivialized prematurely. Here is a common example in physical theory.

A theory of some phenomenon most naturally formulated explicitly in terms of complex analysis may have very non-trivial conceptual differences (e.g. geometrically, topologically, algebraically, etc) from a less complete theory of the same phenomenon which was tentatively first formulated purely in terms of the way more familiar tools of real analysis.

For example, most theories start off describing the phenomenon in question in some kind of simplified and often linearized setting for simplicity; however when unsimplified to more general, complicated settings and/or fully formulated in terms of some general curved manifold, the theory then actually turns out to be mathematically very distinct from the simplified real analytic theory in some extremely non-trivial way.

This non-trivial difference may however not be directly visible from the surface, i.e. neither theoretically by textbook presentations and analytic calculations (especially when ad hoc linearization tools have become available widespread and conventional), nor experimentally from direct verification of the most straightforward extremely trivial laboratory setting in comparison with the natural settings of the phenomenon.

The problem of simplification and convention then multiplies enormously once the theoreticians - having become accustomed to the simplistic formulation of the theory together with their arbitrary ad hoc toolkit for handling difficulties - then end up not taking the underlying complex analytic structure seriously, which directly predicts the complications of the phenomenon.

Hereby a theory can become stuck in being viewed by both theoretician and experimentalists in a highly simplified fashion, with the conventional ad hoc simplification tools - both theoretically and experimentally - actually carefully masking the very departure of the simplified theory to the more general setting.

 

[atyy]

How about
T1 Weierstrassian analysis
T2 Non-standard analysis
?

 

[fresh_42]

How about
T1 Weierstrassian analysis
T2 Non-standard analysis
?

Two (mostly equivalent) theories models, no interpretation anywhere.

 

[Auto-Didact]

How about
T1 Frequentist statistics
T2 Bayesian statistics
?

To be a different theory it should build on something else, it should be possible to get to that theory even if you have never seen QM. And that is not the case of the interpretations.

To be fair, mathematically there actually are several ways to derive BM without ever having seen or even taken QM (of course, it helps to be able to recognize the correct derivation if one has ever seen e.g. the SE).

 

[atyy]

Two (mostly equivalent) theories models, no interpretation anywhere.

Are you using "model" in the sense of "standard and non standard models of arithmetic"?

 

[fresh_42]

Are you using "model" in the sense of "standard and non standard models of arithmetic"?

I use model as synonym for a mathematical calculus, in the sense of framework. I do not like theory, as it has far too many connotations and in the end it's only the calculations we're interested in, hence the term calculus. Standard and non standard are no mathematical categories, they already include an evaluation. Either a calculus is without contradictions, or at least those can be resolved as we are forced to do since we know that Hilbert's second is undecidable, or a calculus is none because it's useless.

 

[atyy]

I use model as synonym for a mathematical calculus, in the sense of framework. I do not like theory, as it has far too many connotations and in the end it's only the calculations we're interested in, hence the term calculus. Standard and non standard are no mathematical categories, they already include an evaluation. Either a calculus is without contradictions, or at least those can be resolved as we are forced to do since we know that Hilbert's second is undecidable, or a calculus is none because it's useless.

I guess the normal terminology in English is that the theory is Peano Arithmetic and the models are either standard or nonstandard.

 

[PAllen]

@Demystifier , where do you see distinctions like Hamiltonian and variational methds, versus forces and Diff.. Eq. ?

 

[Orodruin]

How about
T1 Frequentist statistics
T2 Bayesian statistics

They are not physical theories. They are mathematical tools that share some similarities but make fundamentally different statements.

 

[Demystifier]

@Demystifier , where do you see distinctions like Hamiltonian and variational methds, versus forces and Diff.. Eq. ?

Yes, those could also be thought of as different theories with identical measurable predictions.

 

[Demystifier]

I expected a non quantum example. So, you have in mind only QM interpretations, and you think they should be called theories. My opinion is that they are correctly called interpretations. The all start with QM or at least the core of QM, then add a bit more, yet don't get new predictions. To me that is not a different theory. To be a different theory it should build on something else, it should be possible to get to that theory even if you have never seen QM. And that is not the case of the interpretations.

For a non-quantum example see my post above. In your terminology, they would not be different theories too.

The problem with your terminology is that it is ambiguous. For instance, Schrodinger developed his wave quantum theory without referring to the Heisenberg matrix quantum theory. It was only later realized (by Dirac) that they were equivalent. Would you say that Schrodinger QM and Heisenberg QM are different theories?

 

[Lord Jestocost]

a theory - or model - is a mathematical framework to describe the experimental results, including possible predictions. On the other hand, an interpretation is merely an informal description of named theory / model to describe the mathematical framework in common language and by the frequent use of aphorisms and metaphors.

I think @fresh_42's comment #3 puts it in a nutshell. Regarding quantum mechanics and the interplay between theory and interpretation, the following is said by Hans Primas in his book “Chemistry, Quantum Mechanics and Reductionism, Perspectives in Theoretical Chemistry”:

“It is tempting, but mistaken, to believe that we can use the formalism of quantum mechanics without entering into a discussion of its interpretation. Only interpreted mathematical formulas express laws of nature, the interpretation gives a physical meaning to the mathematical symbols used. Interpretations never are inherent in the formalism but must be assigned to it. Mathematical construction is such a marvelously effective creative principle that the formal structure of a new theory may be arrived at prior to the establishment of a consistent interpretation. That is, we may be confronted with the amazing situation that we know and can use the mathematical equations of a new theory without exactly knowing what they mean. Pioneer quantum mechanics is an example: it is the most controversial theory we have and even today no general agreement exists on its interpretation [compare for example the review given by Jammer, 1974).”

 

[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