Against “interpretation”
I am against “interpretations” of Quantum Mechanics (QM) in a sense in which John Bell [1] was against measurement in QM and Travis Norsen [2] is against realism in QM. Bell was not against doing measurements, he was against using the concept of measurement as a central concept in quantum foundations. Norsen does not think that realism does not exist, he thinks that the existence of realism is so obvious and basic that one should not even talk about it. In a similar spirit, I do not think that physicists should not study interpretations, I think that it is misleading to talk about interpretations as something different from theories. The titles “Against measurement” [1] and “Against realism” [2] were chosen by Bell and Norsen to provoke, by imitating the provocative style of Paul Feyerabend – the famous philosopher of science who was “Against method” [3]. My intentions here are provocative too.
Physicists often say that in physics we need theories that make new measurable predictions and that we don’t need interpretations that make the same measurable predictions as old theories. I think it’s nonsense. It’s nonsense to say that theories are one thing and interpretations are another. The interpretations are theories. Making a distinction between them only raises confusion. So we should ban the word “interpretation” and talk only about the theories.
Let me explain. Suppose that someone develops a theory called T1 that makes measurable predictions. And suppose that those predictions were not made by any previous theory. Then all physicists would agree that T1 is a legitimate theory. (Whether the predictions agree with experiments is not important here.)
Now suppose that someone else develops another theory T2 that makes the same measurable predictions as T1. So if T1 was a legitimate theory, then, by the same criteria, T2 is also a legitimate theory. Yet, for some reason, physicists like to say that T2 is not a theory, but only an interpretation. But how can it be that T1 is a theory and T2 is only an interpretation? It simply doesn’t make sense.
To resolve that issue, one might say that both T1 and T2 are interpretations. Fine, but then what is the theory? T1 was a legitimate theory before someone developed T2, but now T1 ceased to be a theory just because someone developed T2? It doesn’t make sense either.
Or perhaps the theory is just the set of final measurable predictions of T1 and T2, while all the other “auxiliary” elements of T1 and T2 are the “interpretation”? It doesn’t make sense either, because no theory in physics deals only with measurable predictions. All physics theories have some “auxiliary” elements that are an integral part of the theory.
Or perhaps an interpretation is a theory that emphasizes philosophical aspects? I think this is what most physicists mean by interpretation, even if they don’t want to say it explicitly. The problem with this definition is that it cannot be put into a precise form. All theories have some philosophical aspects, some theories more, some less. So exactly how much of philosophy does a theory have to have to call it an interpretation? It’s simply impossible to tell. And where exactly is the borderline between philosophy and non-philosophy? There is no such borderline.
To conclude, we can talk about a theory, we can distinguish the measurable predictions of the theory from other elements of the theory that cannot be directly measured, but it doesn’t make sense to distinguish an interpretation from a theory. There are no interpretations of QM, there are only theories.
References:
[1] J. Bell, Against measurement, https://m.tau.ac.il/~quantum/Vaidman/IQM/BellAM.pdf
[2] T. Norsen, Against “realism”, http://de.arxiv.org/abs/quant-ph/0607057
[3] P. Feyerabend, Against method, https://en.wikipedia.org/wiki/Against_Method
Read my next article on Anyons
Theoretical physicist from Croatia
I believe you intend to include both the mapping to experiment and also metaphysical claims about reality.Good point I hadn't thought of before.
John Baez's writings has often influenced me in my views on interpretations:
http://math.ucr.edu/home/baez/bayes.html
In particular:
'It turns out that a lot of arguments about the interpretation of quantum theory are at least partially arguments about the meaning of the probability!'
You have to have an interpretation of probability to do the mapping. Interpretations like the ensemble do only that. I would call them minimal.
An interesting observation is that in math we generally do not worry about interpretations of probability – we either apply it as most books like Feller's classic do or we simply look at the consequences of the Kolmogorov axioms as books on rigorous probability theory do. People generally do not get caught up much in the interpretation issue – but in Quantum Theory we have all sorts of, how to put it, 'vigorous' discussions about it. That always has struck me as, well interesting.
But others go further – even Copenhagen goes further (at least in some versions – there seems no standard version). But it generally seems to be something like (from a blog discussion on it):
1. A system is completely described by a wave function ψ, representing an observer's subjective knowledge of the system. (Heisenberg)
2. The description of nature is essentially probabilistic, with the probability of an event related to the square of the amplitude of the wave function related to it. (The Born rule, after Max Born)
3. It is not possible to know the value of all the properties of the system at the same time; those properties that are not known with precision must be described by probabilities. (Heisenberg's uncertainty principle)
4. Matter exhibits a wave–particle duality. An experiment can show the particle-like properties of matter, or the wave-like properties; in some experiments both of these complementary viewpoints must be invoked to explain the results, according to the complementarity principle of Niels Bohr.
5. Measuring devices are essentially classical devices, and measure only classical properties such as position and momentum.
6. The quantum mechanical description of large systems will closely approximate the classical description. (The correspondence principle of Bohr and Heisenberg)
The above contains quite few debatable points:
1. Is a quantum system completely described by the wave function?
2. Wave particle duality – its really neither wave or particle – it's quantum stuff.
3. There are in a sense no classical systems – its all really quantum stuff. If you do not view it as all quantum stuff you face a problem – exactly where is the dividing line?
Every one of those really requires a thread of their own, so I will not discuss them here except to say modern interpretations like decoherent histories realize they are issues and try to correct them – which was the view of the blog I got it from. But we should not be too harsh, Copenhagen was formulated in the early days of QM – things have moved on a lot since then.
On thing that always brings a bit of a smile to my face is Einstein was the original champion of the Ensemble interpretation. It seems to have come through mostly unchanged to modern times. But Copenhagen, championed his old sparring partner, and good friend, Bohr, didn't. Could it be Einstein, after his debates with Bohr saw to the heart of it better? Einstein was wrong to object to QM so strongly at it's birth, but eventually he came to accept it as correct. To be fair though his objections did strengthen the theory. But to his dying day thought it incomplete – which due to various unresolved issues like quantum gravity is of course true – but may change in the future – or actually be shown as incomplete.
Thanks
Bill
This says exactly what I claimed.I am not convinced that this is exactly the same as what you were claiming. First, this is the definition of interpretation, not the definition of theory. The definition of theory is not consistent with your definition of theory. The theory itself includes the mathematical framework as well as the mapping to experiment. It specifically rejects your definition of theory as being only the math.
Now, as to whether this section on interpretation is consistent with your view of interpretation depends a little on what is meant by “corresponds with reality”.
I believe you intend to include both the mapping to experiment and also metaphysical claims about reality. In that case there is some overlap between the definition of theory and interpretation since they both include the mapping to experiment. This usage would be consistent with the term “minimal interpretation” to describe that mapping.
However, the phrase “corresponds to reality” could be taken to refer exclusively to the metaphysical statements only. After all, it is possible to assert a relation to measurement while not asserting whether or not the results of measurements are “real”.
I don’t think that definition is as strong a support for your position as you think. At best it gives a kind of messy overlap between theory and interpretation where the useful part (link to measurement) is part of both.
In either case, the theory consists of the portion that is experimentally testable, the mathematical framework and the mapping to experiment. If you like the overlapping concept then you could talk about the objective interpretation and the subjective interpretation to distinguish between the scientific and philosophical portions of the interpretation.
I would only agree that the objective interpretation is what makes a theory useful, and that is already part of the theory itself.
This is just arguing over definitions, right?Yes, because the Insight article defining this thread tries to change definitions:
It doesn’t make sense to distinguish an interpretation from a theory. There are no interpretations of QM, there are only theories.Without good reasons, I think; see my post #4.
This is just arguing over definitions, right? Maybe the only thing that can be said is there is enough disagreement that when these words are important, they should be defined in each discussion. Even if one definition is 90% popular, that's still pretty ambiguous. If PF had a mentor-editable glossary that might cut down on the convergence times.
Theory is the formal, purely mathematical part, and interpretation tells how this formal part relates to reality / observation. In simple cases, the interpretation is simply done by choosing the right words for the formal concepts, but in relativity, more is needed since it is no longer intuitive, and in quantum mechanics, much more is needed since the meaning is – a mess.Another independent wikipedia source also follows my notion of interpretation:
Attempts to formalize the principles of the empirical sciences use an interpretation to model reality, in the same way logicians axiomatize the principles of logic. The aim of these attempts is to construct a formal system that will not produce theoretical consequences that are contrary to what is found in reality. Predictions or other statements drawn from such a formal system mirror or map the real world only insofar as these scientific models are true.
In my opinion, there is a problem with interpretations in that when you have a different one, you cannot know in advance that there are no circumstances where you will not get to either different outputs, or easier ways of going about something. I know here you are not supposed to mention your own work, but with QM there is a small group of interpretations where it is assumed there is a physical wave (De Broglie, Bohm). Now, if you assume the wave is the cause of diffraction in the two slit experiment, then you might consider the wave has to travel with the particle. This gives a physical relationship not present in standard QM, and when coupled with Euler's complex number theory (from which the antinode is not complex) you get a much simpler means of calculating properties of the chemical bond. (You also get a relationship that has not been noted in standard theory.) Now, whether simplified means of calculating is worth bothering about is a matter of opinion, but for me it is.
Wikipedia says "A scientific theory is an explanation of an aspect of the natural world that can be repeatedly tested and verified in accordance with the scientific method" https://en.wikipedia.org/wiki/Scientific_theory where clearly a theory must be testable. The purely mathematical concept of theory that you propose is not testable, so it does not fit the Wikipedia definition.Since you quote wikipedia, let me also quote it:
An interpretation of quantum mechanics is an attempt to explain how the mathematical theory of quantum mechanics "corresponds" to reality. […] An interpretation (i.e. a semantic explanation of the formal mathematics of quantum mechanics) […]This says exactly what I claimed. The same meaning is also echoed in another wikipedia page not related to quantum mechanics:
The mathematics of probability can be developed on an entirely axiomatic basis that is independent of any interpretation: see the articles on probability theory and probability axioms for a detailed treatment.From another well-known common source on quantum mechanics:
Mathematically, the theory is well understood […] The problems with giving an interpretation […] are dealt with in other sections of this encyclopedia. Here, we are concerned only with the mathematical heart of the theory, the theory in its capacity as a mathematical machine, and — whatever is true of the rest of it — this part of the theory makes exquisitely good sense…. and by implication, everything else is interpretation, about which ''there is very little agreement''. Very little is said in the cited article about how an observable or a state is related to reality, no operational definition is given how to measure a state or an observable. Loose connections are given in Section 3.4 (Born's rule) and statement (4.2) (special case of eigenstates). The second connection is too special to be representative of the meaning of QM; the first connection is already interpretation dependent (the formulation assumes collapse, a controversial feature) and nevertheless fraught with problems, as is said explicitly on the same page:
But without contexts of type 2, nothing at all follows about the relation between the formalism and measurable cross sections or detection events. Thus the uninterpreted theory must be silent about the latter.
No, it is the operative definitions of how to relate mathematical concepts of the theory to measurable quantities that make a theory useful. This is not interpretation in the common nomenclature typically used here, regardless of what Born and Schrödinger thought about the issue.So please spell out the operative definitions that relate the mathematical concepts of quantum theory to measurable quantities. You'll find that this is impossible to do independent of any of the interpretations of quantum mechanics that can be found in the literature. (Shut-up-and-calculate works only because it leaves the interpretation to the community without spelling out precisely what it consists of.)
Thus interpretation is a prerequisite for making quantum theory useful.
But to give precise references – if you still want them – I need to do some research.So I found a few references that clearly disagree with your definition of "theory" at least. Wikipedia says "A scientific theory is an explanation of an aspect of the natural world that can be repeatedly tested and verified in accordance with the scientific method" https://en.wikipedia.org/wiki/Scientific_theory where clearly a theory must be testable. The purely mathematical concept of theory that you propose is not testable, so it does not fit the Wikipedia definition.
I also found a paper entitled "What is a scientific theory?" by Patrick Suppes from 1967 (Philosophy of Science Today) who says "The standard sketch of scientific theories-and I emphasize e word “sketch”-runs something like the following. A scientific theory consists of two parts. One part is an abstract logical calculus … The second part of the theory is a set of rules that assign an empirical content to the logical calculus. It is always emphasized that the first part alone is not sufficient to define a scientific theory".
As he describes this as the "standard sketch" and as this also agrees with the Wikipedia reference and my previous understanding, then I take it that your definition of theory is not that which is commonly used. I have not found a similar clear definition of "interpretation", but clearly the term theory includes the mapping to experimental outcome that is necessary to make it useful on its own for designing and analyzing experiments. Thus, by the standard usage it is also not the interpretation which makes a theory useful, the theory is already useful without an interpretation.
It is the interpretation that makes a theory useful.No, it is the operative definitions of how to relate mathematical concepts of the theory to measurable quantities that make a theory useful. This is not interpretation in the common nomenclature typically used here, regardless of what Born and Schrödinger thought about the issue.
It is the interpretation that makes a theory useful.Even using your definitions I would disagree with this claim. With your definition it is only the so-called “minimal interpretation” that makes the theory useful. All other interpretations are subjective per your terms.
I disagree. The theory must include the relation to observation. Otherwise it is useless.It is the interpretation that makes a theory useful.
Sorry, but in my mind this is severely twisting the meaning of the word "interpretation" in this discussion.I just observe that Schrödinger and Born thought differently about the issue. At the time where the interpretation problems in quantum mechanics became relevant, there was good theory, and there was disagreement about the relation to observation in general – just pieces that were undisputable but others that were hoghly contentuous. Indeed, the meaning of the relation to observation changed during the first few years.
To me what is scientifically important is the distinction between the portions of a model which can be experimentally tested using the scientific method and the portions that cannot. I don’t care too much about the terminology, but that distinction is important so it should have some corresponding terminology. In my usage that would be “theory” vs “interpretation”.
What words would you personally use to make that distinction?objective = testable and subjective = untestable.
If theory = testable and interpretation = untestable there would have not been nearly 100 years of dispute about the interpretation issues.
As I said, in simple cases, the interpretation is simply calling the concepts by certain names. In the case of classical Hamiltonian mechanics, ##p## is called momentum, ##q## is called position, ##t## is called time, and everyone is supposed to know what this means, i.e., to have an associated interpretation in terms of reality.Sorry, but in my mind this is severely twisting the meaning of the word "interpretation" in this discussion.
But to give precise references – if you still want them – I need to do some research.I would appreciate that and I will look for similar explicit definitions as well. My “implicit” definitions are quite opposed to yours.
To me what is scientifically important is the distinction between the portions of a model which can be experimentally tested using the scientific method and the portions that cannot. I don’t care too much about the terminology, but that distinction is important so it should have some corresponding terminology. In my usage that would be “theory” vs “interpretation”.
What words would you personally use to make that distinction?
But this is not standard terminology. Theory is the formal, purely mathematical part, and interpretation tells how this formal part relates to reality / observation. In simple cases, the interpretation is simply done by choosing the right words for the formal concepts
That is not how I have seen the distinction. Do you have an authoritative reference for this usage? (What you are calling “theory” I have seen called “mathematical framework”)It is surely implicit in the discussions of 1926-1928 about the interpretation of quantum mechanics by their originators. Schrödinger's and Heisenberg's theories were proved to be equivalent (i.e., the mathematical frameworks were interconvertible), but views about the interpretation differed widely. Moreover, different interpretations even made different predictions, and the analysis turned out to give a harmonizing Copenhagen interpretation, both relaxing the incomatible hardliner positions that Born and Schrödinger originally had.
But to give precise references – if you still want them – I need to do some research.
It is the interpretation that makes a theory useful.
Classical Hamiltonian mechanics is surely a theory. But it needs interpretation to be used: How to interpret energy in terms of reality/observation is clearly not part of the theory.
I disagree again. It is the prediction of measurable quantities that makes a theory useful.
[Classical Hamiltonian mechanics] certainly does not need interpretation to be used and tested. You do not need to give a "deeper meaning" to the Hamiltonian to test Hamiltonian mechanics or to give a meaning to why the Poisson brackets with the Hamiltonian give the time evolution of a system. You need a description of phase space, an expression for the Hamiltonian, and the measurable predictions resulting from it.As I said, in simple cases, the interpretation is simply calling the concepts by certain names. In the case of classical Hamiltonian mechanics, ##p## is called momentum, ##q## is called position, ##t## is called time, and everyone is supposed to know what this means, i.e., to have an associated interpretation in terms of reality.
Of course, to be useful, a theory must not only have an interpretation but also give valid predictions of measurable results.
It is the interpretation that makes a theory useful.I disagree again. It is the prediction of measurable quantities that makes a theory useful.
Classical Hamiltonian mechanics is surely a theory. But it needs interpretation to be used: How to interpret energy in terms of reality/observation is clearly not part of the theory.It certainly does not need interpretation to be used and tested. You do not need to give a "deeper meaning" to the Hamiltonian to test Hamiltonian mechanics or to give a meaning to why the Poisson brackets with the Hamiltonian give the time evolution of a system. You need a description of phase space, an expression for the Hamiltonian, and the measurable predictions resulting from it.
But this is not standard terminology. Theory is the formal, purely mathematical part, and interpretation tells how this formal part relates to reality / observation.That is not how I have seen the distinction. Do you have an authoritative reference for this usage? (What you are calling “theory” I have seen called “mathematical framework”)
Sure I agree. That has never been the question. The question is whether a lattice model (at finite spacing) could provide a non-perturbative definition for the currently successful experimental predictions of QED, QCD and the standard model. If that is a reasonable research programme (at least as reasonable as looking for a continuum 4D QED theory), then one can say that the standard model may be consistent with non-relativistic QM. It is not an "either-or" question. One could believe that both research programmes are reasonable.I agree, when I said numerical results in my initial post I was referring to Lattice theories and you'll find plenty of discussions about Lattice versions of the Standard Model suggesting non-triviality in Callaway's paper that I referenced. I also consider both programs reasonable.
Then just to be clear I was stating something else, that for theories involving ##SU(2)## gauge fields there are strong arguments that they are not trivial, so I was rather referencing some evidence against triviality for the Standard Model.
However I share your doubts about typical arguments against ##phi^{4}_{4}## and ##QED_4## as I don't think the Landau pole is a particularly strong argument. It's just a perturbative suggestion that a particular approach to the continuum limit is blocked. Alan Sokal's PHD thesis "An Alternate Constructive Approach to the ##phi^{4}_{3}## Quantum Field Theory, and a Possible Destructive Approach to ##phi^{4}_{4}##" has some interesting material on this. He uses the sum of bubble graphs to argue for triviality of the continuum.
For anybody reading there is the possibility that there are non-trivial continuum ##QED_4## and ##phi^{4}_{4}## theories. It's simply that they aren't the ##a rightarrow 0## limit of a lattice theory and so the triviality of the lattice theories when taking the continuum limit isn't a definitive proof of triviality.
My personal gut intuition is that is that ##phi^{4}_{4}## is trivial on its own, but not when embedded in the electroweak theory. I suspect ##QED_4## is not trivial as you do.
In general I strongly suspect that properly controlled non-perturbative quantum field theory will show that plenty of folk wisdom about QFT is just wrong. For example it might emerge that having a simple Higgs is the only way of having massive gauge bosons that has a nonperturbative definition and alternates like technicolor aren't defined. Similarly many parameters that look like they can take any value perturbatively and non-rigorously might be restricted to certain ranges non-perturbatively. Also the Standard Model might be much more natural and less adhoc seeming, perhaps only theories of its form exist non-perturbatively in 4D.
Basically we're currently operating under the assumption that the space of QFTs in 4D is identical to to the space of field theories that are perturbatively renormalizable. However this is incorrect as ##Gross-Neveu_3## is pertrubatively non-renormalizable and yet non-perturbatively exists.Sure I agree. That has never been the question. The question is whether a lattice model (at finite spacing) could provide a non-perturbative definition for the currently successful experimental predictions of QED, QCD and the standard model. If that is a reasonable research programme (at least as reasonable as looking for a continuum 4D QED theory), then one can say that the standard model may be consistent with non-relativistic QM. It is not an "either-or" question. One could believe that both research programmes are reasonable.
Example of papers within a research programme for a lattice standard model are:
https://arxiv.org/abs/0912.2560
https://arxiv.org/abs/1809.11171
atyy's statement sounded like it. He thinks that Landau poles are the death blow to a continuum theory and wants to substitute finite lattices for the true, covariant theories. But in fact the Landau pole of QED just says that the lattice approximation of QED is always poor, so it is actually the death blow to his lattice philosophy. We had discussed this in several threads:
https://www.physicsforums.com/threads/lattice-qed.943462/
https://www.physicsforums.com/threads/does-qft-have-problems.912943/
Yes. I am convinced that ##phi_4^4## and ##QED_4## exist, though I don't know how to prove it. But I have been collecting ideas and techniques for a long time, and one day I might be prepared to try….
Klauder has some nonrigorous ideas how to do perturbation theory from a different starting theory: https://arxiv.org/abs/1811.05328 and many earlier papers propagating the same idea. Nobody seems to take up Klauder's challenge and tries; hence I don't know whether it has merit. Do you see any obvious faults in his proposal?Then just to be clear I was stating something else, that for theories involving ##SU(2)## gauge fields there are strong arguments that they are not trivial, so I was rather referencing some evidence against triviality for the Standard Model.
However I share your doubts about typical arguments against ##phi^{4}_{4}## and ##QED_4## as I don't think the Landau pole is a particularly strong argument. It's just a perturbative suggestion that a particular approach to the continuum limit is blocked. Alan Sokal's PHD thesis "An Alternate Constructive Approach to the ##phi^{4}_{3}## Quantum Field Theory, and a Possible Destructive Approach to ##phi^{4}_{4}##" has some interesting material on this. He uses the sum of bubble graphs to argue for triviality of the continuum.
For anybody reading there is the possibility that there are non-trivial continuum ##QED_4## and ##phi^{4}_{4}## theories. It's simply that they aren't the ##a rightarrow 0## limit of a lattice theory and so the triviality of the lattice theories when taking the continuum limit isn't a definitive proof of triviality.
My personal gut intuition is that is that ##phi^{4}_{4}## is trivial on its own, but not when embedded in the electroweak theory. I suspect ##QED_4## is not trivial as you do.
In general I strongly suspect that properly controlled non-perturbative quantum field theory will show that plenty of folk wisdom about QFT is just wrong. For example it might emerge that having a simple Higgs is the only way of having massive gauge bosons that has a nonperturbative definition and alternates like technicolor aren't defined. Similarly many parameters that look like they can take any value perturbatively and non-rigorously might be restricted to certain ranges non-perturbatively. Also the Standard Model might be much more natural and less adhoc seeming, perhaps only theories of its form exist non-perturbatively in 4D.
Basically we're currently operating under the assumption that the space of QFTs in 4D is identical to to the space of field theories that are perturbatively renormalizable. However this is incorrect as ##Gross-Neveu_3## is pertrubatively non-renormalizable and yet non-perturbatively exists.
I disagree. The theory must include the relation to observation. Otherwise it is useless.It is the interpretation that makes a theory useful.
Classical Hamiltonian mechanics is surely a theory. But it needs interpretation to be used: How to interpret energy in terms of reality/observation is clearly not part of the theory.
Theory is the formal, purely mathematical part, and interpretation tells how this formal part relates to reality / observation.I disagree. The theory must include the relation to observation. Otherwise it is useless. The problem comes with interpretations imposing some sort of unneseccary ”reality” on top of this, which unless you can provide observational differences will always remain purely philosophical.
I think the whole ordeal started from QM being a theory based on experiments and model fitting more or less. The interpretation is needed because the theory does not show the origin of mass or charge …etc. They must be emergent from a more fundamental concept.
If we don’t use the word “theory” for the parts which can be scientifically tested and “interpretation” for the parts which cannot be testedBut this is not standard terminology. Theory is the formal, purely mathematical part, and interpretation tells how this formal part relates to reality / observation. In simple cases, the interpretation is simply done by choosing the right words for the formal concepts, but in relativity, more is needed since it is no longer intuitive, and in quantum mechanics, much more is needed since the meaning is – a mess.
Or perhaps the theory is just the set of final measurable predictions of T1 and T2, while all the other “auxiliary” elements of T1 and T2 are the “interpretation”? It doesn’t make sense either, because there is no theory in physics that deals only with measurable predictions. All physics theories have some “auxiliary” elements that are an integral part of the theory.I disagree with the T1 and T2 analogy entirely, but particularly with this paragraph here.
Regardless of the existence of an interpretation-free theory, it is useful to distinguish the parts of a model which can be tested with the scientific method from the parts that cannot. If we don’t use the word “theory” for the parts which can be scientifically tested and “interpretation” for the parts which cannot be tested with the scientific method, then what terms should we use to distinguish them? We would need to coin some new terms for the same concepts.
No, the standard terminology is fine. It is pointless to change the names since the distinction between testable and untestable is scientifically important and captured in the current terminology.
That's certainly true, were we saying otherwise?atyy's statement sounded like it. He thinks that Landau poles are the death blow to a continuum theory and wants to substitute finite lattices for the true, covariant theories. But in fact the Landau pole of QED just says that the lattice approximation of QED is always poor, so it is actually the death blow to his lattice philosophy. We had discussed this in several threads:
https://www.physicsforums.com/threads/lattice-qed.943462/
https://www.physicsforums.com/threads/does-qft-have-problems.912943/
Do you mean there might be a non-trivial continuum theory that is not the limit of lattice approximations?
The kind of thing suggested in Gallavotti and Rivasseau's review paper from 1984 for example.I am convinced that ##phi_4^4## and ##QED_4## exist, though I don't know how to prove it. But I have been collecting ideas and techniques for a long time, and one day I might be prepared to try….
Klauder has some nonrigorous ideas how to do perturbation theory from a different starting theory: https://arxiv.org/abs/1811.05328 and many earlier papers propagating the same idea. Nobody seems to take up Klauder's challenge and tries; hence I don't know whether it has merit. Do you see any obvious faults in his proposal?
Actual triviality just means that the continuum limit of the lattice theory is trivial. It says nothing about nonexistence of the continuum theory. See the discussions here and here.That's certainly true, were we saying otherwise?
Do you mean there might be a non-trivial continuum theory that is not the limit of lattice approximations?
The kind of thing suggested in Gallavottiv and Rivasseau's review paper from 1984 for example.
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.
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.Actual triviality just means that the continuum limit of the lattice theory is trivial. It says nothing about nonexistence of the continuum theory. See the discussions here and here.
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"Note that the Landau pole of QED is at physically irrelevant energies, while QCD has (due to infrared issues) a Landau pole at experimentally accessible energies! Thus a Landau pole says nothing about existence or nonexistence, only about troubles in certain renormalization schemes.
Thread reopened.
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.
Thread closed for moderation.
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.
I just consciously choose to reject itThat says it all.
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 :DD) 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.
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.
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"
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…
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.
@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 formulationThis 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 $$H=pdot q- L Leftrightarrow L=pdot q- H$$It then immediately follows that $$begin{align}
delta int_{t_0}^{t_1} L dt & = delta int_{t_0}^{t_1} (p frac {dq}{dt}-H) dt nonumber \
& = delta int_{t_0}^{t_1} (p frac {dq}{dt}-Hfrac {dt}{dt}) dt nonumber \
& = delta int_{t_0}^{t_1} (p frac {dq}{dt}+0frac {dp}{dt}-Hfrac {dt}{dt}) dt nonumber \
end{align}$$where obviously ##L = p frac {dq}{dt}+0frac {dp}{dt}-Hfrac {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: $$begin{align}
delta int_{t_0}^{t_1} L dt & = delta int_{t_0}^{t_1} (p frac {dq}{dt}+0frac {dp}{dt}-Hfrac {dt}{dt}) dt nonumber \
& = delta int_{t_0}^{t_1} vec X cdot vec Y dt nonumber \
& = delta int_{Gamma} vec X cdot d vec {Gamma} = 0 nonumber \
end{align}$$ 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.
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.
Thus if QM is not the same theory as QM, then BM is not the same theory as QM.I certainly agree with that. :biggrin:
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 wont 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=hnu## 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 mathematicsThe 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 wrongAgain 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 nonsenseOpinion, 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.
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
BillIt'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 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?
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.
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."
Yes – without something to be symmetrical in you cant apply symmetry. That's is the paradox about physics relation to symmetry despite is fundamental importance.
Thanks
BillI would love to see that chapter re-written as though it were directly a non-relativistic version of Weinberg's QFT chapter 2 :DD
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 cant apply symmetry. That's is the paradox about physics relation to symmetry despite is fundamental importance.
Thanks
Bill
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
BillMy 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.
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 axiomGleason'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