I Does the Principle of Least Action Have a Physical Meaning?

[martinbn]

The PLA does not have physical meaning in the sense that it is not always unique. for example, the (classical) Einstein field equations can be derived from several different actions.

I have always wondered why physicists prefer actions/Lagrangians over the equations! At the end of the day arn't the equations that matter?

 

[Dr. Courtney]

The PLA does not have physical meaning in the sense that it is not always unique. for example, the (classical) Einstein field equations can be derived from several different actions.

Potentials are not unique either. Does this mean that they do not have a physical meaning?

Physical meaning comes from the predictions made by a model. Being the only model to make those predictions is not a requirement for physical meaning.

 

[sophiecentaur]

Could one likewise argue, "If you can construct realistic theories of mechanics without Newtonian forces, one might argue that Newtonian forces are not "fundamental" and thus lack physical meaning?

If so, how are we going to teach intro physics?

But, just imagine: someone might invent an entirely different way of describing 'concrete' events in a non Newtonian way which would make sense (perhaps even better sense) to a class of eleven year olds. Bigger paradigm changes have taken place in the past - what a thought!

 

[Dr. Courtney]

But, just imagine: someone might invent an entirely different way of describing 'concrete' events in a non Newtonian way which would make sense (perhaps even better sense) to a class of eleven year olds. Bigger paradigm changes have taken place in the past - what a thought!

You missed the point. I am disproving the notion that alternate models demonstrate that a given model is not physical. If one accepts the notion that an accurate model needs to be unique to have a physical meaning, then as soon as a phenomenon has two equally accurate models, then neither one of them has a physical meaning. (Or picking which one has the physical meaning is a subjective beauty contest rather than an objective scientific endeavor.) "Making sense" to eleven year olds is a subjective criteria. Paradigm shifts in science come from making better predictions, not by making the same predictions in a way that is more aesthetically pleasing or by making more sense to eleven year olds.

 

[anorlunda]

I am disproving the notion that alternate models demonstrate that a given model is not physical. If one accepts the notion that an accurate model needs to be unique to have a physical meaning, then as soon as a phenomenon has two equally accurate models, then neither one of them has a physical meaning.

That can be problematical. Are Newton/Lagrangian/Hamiltonian approaches different models or different ways to express the same model? Can we make a precise definition of "model?"

 

[Dr. Courtney]

That can be problematical. Are Newton/Lagrangian/Hamiltonian approaches different models or different ways to express the same model? Can we make a precise definition of "model?"

I view them as different models which happen to be completely equivalent in terms of testable predictions. (As are matrix and wave quantum mechanics). Newtonian mechanics knows nothing of kinetic energy, potential energy, or action. Does the introduction of kinetic energy, potential energy, and action in later formulations mean forces have no physical meaning? Or do the forces in Newtonian mechanics mean the kinetic energy, potential energy, and action have no meaning in later formulations?

A scientific model is a set of principles that can be used to make testable predictions.

Whether we realize that two models are equivalent does not change their validity or physical meaning. Their physical meaning and validity rest in their ability to make accurate predictions about experiments, not in their uniqueness from other models (either generally, or for specific experiments in question).

 

[Auto-Didact]

A model is any mathematical description of some phenomenon, usually capturable by some set of differential equations or something mathematically equivalent to that.

The mathematics of Newtonian, Lagrangian and Hamiltonian mechanics are different physical theories but deeply interrelated mathematical procedures, wherein which calculations of a model of the same phenomenon based on these procedures will always lead to the same answers, i.e. the physical theories are in some sense mathematically equivalent. This would be mysterious were it not that they are essentially different aspects of the same theory, namely mechanics, and that the mathematics of mechanics has all these different properties which can each be seperately exploited and further understood independently on their own terms.

The physical content however, i.e. what is intrinsically psychologically implied to the physicist by the mathematics regarding the phenomenon and other analogous phenomenon, of the three approaches are quite distinct and these distinctions can themselves suggest different ways of modifying concepts of the theory in order to generalize to some grander theory; in theoretical science, these modifications and generalizations can lead to very different kinds of theories, which aren't necessarily equivalent anymore to some other theories after such a modification is made compared to the situation before the modification was made where the theories were equivalent.

Feynman, having as usual already thought about these issues extensively, explains this marvelously in a matter of minutes here:

 

[atyy]

Potentials are not unique either. Does this mean that they do not have a physical meaning?

Physical meaning comes from the predictions made by a model. Being the only model to make those predictions is not a requirement for physical meaning.

Yes, that's exactly what I was thinking of. It is quite conventional to say within classsical electrostatics that potentials themselves are not physically meaningful, because they are not unique. Rather it is the potential difference that is physically meaningful.

One can choose to use other language, but this choice is very reasonable.

 

[sophiecentaur]

I am disproving the notion that alternate models demonstrate that a given model is not physical.

Paradigm shifts in science come from making better predictions, not by making the same predictions in a way that is more aesthetically pleasing or by making more sense to eleven year olds.

I was rather assuming that the necessity of Physicality would apply to individuals with only Concrete Cognitive skills. From this

If so, how are we going to teach intro physics?

I thought you were thinking along the same lines.
My view of the term 'physical' is that an idea is readily accepted in terms of already familiar and very concrete terms. By definition, a Physical explanation excludes factors that are new to the learner. One has only to examine one's own appreciation of Physics (and a lot of other stuff, of course) to realize that 'Physical' is mostly synonymous with 'familiar'.
Physical descriptions include "Nature abhors a vacuum" and 'the music of the spheres' and we have advanced beyond them by using notions of pressure and gravitational laws which would have been very non-physical at one time.

 

[Dr. Courtney]

Another example: the choice of a coordinate system is part of a physical model from the earliest physics courses. Dropping a rock from tower. Is y = 0 m the starting position or the ending position? The choice impacts the model, and while the two models are equivalent (make the same predictions), they are different models.

At some point (early, I hope) those who practice physics realize that there are no right or wrong choices of coordinate systems, though the math is sometimes easier or harder as a result of our choices. But the fact that the choice of coordinate system is not _unique_ does not mean that the choice of coordinate system (and the resulting mathematical models) do not have physical meaning. Of course, the choice of coordinate system has a physical meaning.

 

[atyy]

Another example: the choice of a coordinate system is part of a physical model from the earliest physics courses. Dropping a rock from tower. Is y = 0 m the starting position or the ending position? The choice impacts the model, and while the two models are equivalent (make the same predictions), they are different models.

At some point (early, I hope) those who practice physics realize that there are no right or wrong choices of coordinate systems, though the math is sometimes easier or harder as a result of our choices. But the fact that the choice of coordinate system is not _unique_ does not mean that the choice of coordinate system (and the resulting mathematical models) do not have physical meaning. Of course, the choice of coordinate system has a physical meaning.

Yes, that too is another example where it is quite conventional to say that coordinates are physically meaningless. So in the same sense that potentials and coordinates are meaningless, a non-unique action is physically meaningless. It is the underlying invariant geometric object that is physically meaningful.

As before, one can choose different language, but this is quite conventional.

 

[Orodruin]

Yes, that too is another example where it is quite conventional to say that coordinates are physically meaningless. So in the same sense that potentials and coordinates are meaningless, a non-unique action is physically meaningless. It is the underlying invariant geometric object that is physically meaningful.

As before, one can choose different language, but this is quite conventional.

You can get rid of the "physically meaningless" part of a potential by specifying equivalence classes of potentials. Same thing goes for gauge choices and actions. I would not consider an action meaningless, but rather say that it contains meaningful (and some meaningless) stuff.

 

[gmax137]

I'm still stuck on what "more fundamental" and "physically meaningful" mean to you guys. Are these the same as, "what is really happening"? As in, "what is going on inside the watch to make it work"?

I have pretty much resigned myself to the idea that that question is outside the purview. If you have, as Dr. Courtney describes, two models that give the same predictions, we might expect the 11th grader to ask, "yes so which model is true?" We don't expect the physicist to ask that question, because s/he knows there is no way to answer.

 

[Orodruin]

I'm still stuck on what "more fundamental" and "physically meaningful" mean to you guys. Are these the same as, "what is really happening"?

To me, definitely not. To me something "physically meaningful" is something that tells you something (a priori) measurable about a system.

 

[sophiecentaur]

Of course, the choice of coordinate system has a physical meaning.

To you, possibly but that may not apply to everyone. It's not an "of course" matter. The direction of +y is a small problem for someone new to the SUVAT equations but is the "physical" appreciation of Up, when a ball is thrown up or Down when it is released from a tower, really Physical? But, once you have started using SUVAT, you have left the physical behind and you just accept the outcome and then believe the answer to the equations will tell you where you are and how fast you're going. What happens then is that you turn the suvat results into a 'physical' idea in your head. You made a mental jump to increase your set of Physical things.
At the other end, if you use a Hamiltonian approach to a problem, it will deliver an up / down / how far answer too, which is also Physical It will tell you whether or not you will land safely on the Moon, which is also Physical. The Maths is just a black box which is divorced from the Physical 'till the other end of the process.

 

[Orodruin]

The choice impacts the model, and while the two models are equivalent (make the same predictions), they are different models.

I disagree. If two models make the same predictions, then they are by definition the same model. A model in physics is not the mathematics behind the predictions, it is the sum of its testable predictions. If two mathematical frameworks lead to the same testable predictions, then they are for all purposes (apart from computational) to be regarded as the same model. You cannot tell them apart by empirical observation.

 

[Auto-Didact]

I disagree. If two models make the same predictions, then they are by definition the same model. A model in physics is not the mathematics behind the predictions, it is the sum of its testable predictions. If two mathematical frameworks lead to the same testable predictions, then they are for all purposes (apart from computational) to be regarded as the same model. You cannot tell them apart by empirical observation.

This is simply incorrect, for it completely ignores:
a) that physics is an incomplete science, i.e. all statements made in physics are uncertain too some extent and in order to progress in theoretical physics we must actively remain uncertain by keeping open the possibility that any experimentally verified statement may merely be an approximation which can be corrected or improved by further theoretical guesswork and inductive reasoning,
b) that we are humans that think about theories in our minds and our minds simply are not perfect logical deductive machines, i.e. intuition is an actual part of physics reasoning and it is not reducible to deductive logic; this fact alone already ensures that theoretical physics clearly is not completely reducible to formal mathematics and in contrary to the opinion of many scientists and mathematicians, this is not a bug, it is a feature.

The point is that mathematical equivalence does not imply psychological equivalence. Feynman adresses this exact point in the video I posted earlier, but here it is again in text:

Suppose you have two theories, A and B, which look completely different psychologically, with different ideas in them and so on, but that all the consequences that are computed from each are exactly the same, and both agree with experiment. The two theories, although they sound different at the beginning, have all consequences the same, which is usually easy to prove mathematically by showing that the logic from A and B will always give corresponding consequences. Suppose we have two such theories, how are we going to decide which one is right? There is no way by science, because they both agree with experiment to the same extent. So two theories, although they may have deeply different ideas behind them, may be mathematically identical, and then there is no scientific way to distinguish them.

However, for psychological reasons, in order to guess new theories, these two things may be very far from equivalent, because one gives a man different ideas from the other. By putting the theory in a certain kind of framework you get an idea of what to change. There will be something, for instance, in theory A that talks about something, and you will say, ‘I’ll change that idea in here’. But to find out what the corresponding thing is that you are going to change in B may be very complicated it may not be a simple idea at all. In other words, although they are identical before they are changed, there are certain ways of changing one which looks natural which will not look natural in the other. Therefore psychologically we must keep all the theories in our heads, and every theoretical physicist who is any good knows six or seven different theoretical representations for exactly the same physics. He knows that they are all equivalent, and that nobody is ever going to be able to decide which one is right at that level, but he keeps them in his head, hoping that they will give him different ideas for guessing.

 

[cosmik debris]

I disagree. If two models make the same predictions, then they are by definition the same model. A model in physics is not the mathematics behind the predictions, it is the sum of its testable predictions. If two mathematical frameworks lead to the same testable predictions, then they are for all purposes (apart from computational) to be regarded as the same model. You cannot tell them apart by empirical observation.

I'm not sure I agree with this. A model is a story with its associated mathematical description. I might model relativity using the usual Einstein method or use the LET (Lorentz Ether Theory) model. The story and the mathematics may be different different but the predictions of each model are the same.

Cheers

 

[quickAndLucky]

Could one likewise argue, "If you can construct realistic theories of mechanics without Newtonian forces, one might argue that Newtonian forces are not "fundamental" and thus lack physical meaning?

If so, how are we going to teach intro physics?

If there were some realistic theories of mechanics that could not be represented using forces then I would argue that forces might not be fundamental. But even if forces weren't fundamental they could still be practical to use in in most cases and definitely worth teaching.

 

[Dr. Courtney]

Of course, the choice of coordinate system has a physical meaning.

To you, possibly but that may not apply to everyone. It's not an "of course" matter.

Perhaps some may disagree, but I think most would agree that a heliocentric model of the solar system has a different physical meaning than a geocentric model.

Further a choice of a coordinate system in which a system is separable gives much more insight that one in which the motion cannot be separated. Take the hydrogen atom in an electric field. Sure, at the end of long and difficult calculations, solving the problem in spherical coordinates may give the same answer as solving it in a coordinate system where the system separates. But a coordinate system in which the system separates also gives the very physical insight that the system is separable. Separation of variable and the associated constants of motion do have physical meanings.

As a teacher, I've been very successful convincing students that models in different coordinate systems are equivalent in that they make the same testable predictions. But it's a hard sell that a model that requires calculus is the same as a model that can be solved with only algebra or that a model that requires numerical integration on a computer is the same as a model that can be solved in a few lines of pencil and paper. A clever choice of coordinate system is part of the model insofar as it is often the difference between whether or not a problem is solvable with the available tools. (Likewise, a clever choice of a basis set in atomic and molecular physics is often the difference between whether a problem is solvable with the available computing power.)

I disagree. If two models make the same predictions, then they are by definition the same model. A model in physics is not the mathematics behind the predictions, it is the sum of its testable predictions. If two mathematical frameworks lead to the same testable predictions, then they are for all purposes (apart from computational) to be regarded as the same model. You cannot tell them apart by empirical observation.

That's certainly a possible definition of model, but I'd prefer to say that the two models were equivalent, but not the same. Consider the hydrogen atom. It is separable in several coordinate systems, including spherical, parabolic, and semi-parabolic. But add a uniform electric field (Stark effect), and not it is only separable in parabolic and semi-parabolic coordinates. Are two models really the same if one handles a specific perturbation much easier than another?

 

[mpresic3]

What is the criteria used to judge if something has “physical meaning”? The lack of consensus on the answer is probably more due to a lack of consensus about “physical meaning” than about the principle of least action.

I had a professor who rephrased the question: Does a photon have a physical reality?
His rephrasing was practical and a little bit facetious. He said by this question, one means "Can the photon(s) be bought and sold".

But in answer to the original post, I offer this thought.

The Feynman path integral formulation in non-relativistic quantum mechanics suggests it is not just the least action path that contributes to the motion. All paths contribute, but the action of interference cancels all but the paths near the least action. This idea suggests the PLA has a physical meaning and not just mathematical

 

[Orodruin]

This is simply incorrect, for it completely ignores:
a) that physics is an incomplete science, i.e. all statements made in physics are uncertain too some extent and in order to progress in theoretical physics we must actively remain uncertain by keeping open the possibility that any experimentally verified statement may merely be an approximation which can be corrected or improved by further theoretical guesswork and inductive reasoning,
b) that we are humans that think about theories in our minds and our minds simply are not perfect logical deductive machines, i.e. intuition is an actual part of physics reasoning and it is not reducible to deductive logic; this fact alone already ensures that theoretical physics clearly is not completely reducible to formal mathematics and in contrary to the opinion of many scientists and mathematicians, this is not a bug, it is a feature.

The point is that mathematical equivalence does not imply psychological equivalence. Feynman adresses this exact point in the video I posted earlier, but here it is again in text:

No, you are simply wrong here. I never said there is no point in thinking differently about a theory, but as a theory, two theories that make exactly the same predictions are equivalent and should be considered the same theory. This has absolutely nothing to do with uncertainties in measurements. It is just about whether two theories are, even in principle, distinguishable.

Your Feynman quote touches on something different, namely if there is a point to having several ways of thinking about the same model - there is, because there may be a deeper model that makes different predictions that more closely conforms to one or the other.

I'm not sure I agree with this. A model is a story with its associated mathematical description. I might model relativity using the usual Einstein method or use the LET (Lorentz Ether Theory) model. The story and the mathematics may be different different but the predictions of each model are the same.

Cheers

No, this is simply not the case. Distinguishing between LET and standard SR is physically irrelevant. It is a purely philosophical question that has nothing to do with physics, which is an empirical science. In an empirical science, a model is nothing more than the sum of its predictions.

Then there is another question about whether it is most helpful to think about LET or standard SR.

Perhaps some may disagree, but I think most would agree that a heliocentric model of the solar system has a different physical meaning than a geocentric model.

This depends on the meaning you put into these concepts. In the strictest interpretation (an object is the centre of the universe), both are in conflict with classical mechanics.

Further a choice of a coordinate system in which a system is separable gives much more insight that one in which the motion cannot be separated. Take the hydrogen atom in an electric field. Sure, at the end of long and difficult calculations, solving the problem in spherical coordinates may give the same answer as solving it in a coordinate system where the system separates. But a coordinate system in which the system separates also gives the very physical insight that the system is separable. Separation of variable and the associated constants of motion do have physical meanings.

Again, this is a completely different question. Insights about how a model works is different from the predictions of the model. Separation of variables is a computational aid, not a model in itself. Ask yourself the question: ”how can I experimentally distinguish separation of variables from other means of computation?” The answer clearly is that you cannot. The constants of motion will be constants of motion regardless of whether you used separation of variables or not.

A clever choice of coordinate system is part of the model insofar as it is often the difference between whether or not a problem is solvable with the available tools. (Likewise, a clever choice of a basis set in atomic and molecular physics is often the difference between whether a problem is solvable with the available computing power.)

Again, this has nothing to do with the model itself and everything to do with our difficulty in extracting the useful predictions from the model.

Are two models really the same if one handles a specific perturbation much easier than another?

Yes. Computational difficulties are irrelevant. Regardless of what coordinates you use, you are solving the same problem in the same model.

 

[Dr. Courtney]

I disagree. If two models make the same predictions, then they are by definition the same model. A model in physics is not the mathematics behind the predictions, it is the sum of its testable predictions. If two mathematical frameworks lead to the same testable predictions, then they are for all purposes (apart from computational) to be regarded as the same model. You cannot tell them apart by empirical observation.

No, you are simply wrong here. I never said there is no point in thinking differently about a theory, but as a theory, two theories that make exactly the same predictions are equivalent and should be considered the same theory. This has absolutely nothing to do with uncertainties in measurements. It is just about whether two theories are, even in principle, distinguishable.

I see now, you are using the word model to have the identical meaning in science as the word theory. I think that is certainly one possible meaning of the word model in science, and there certainly are overlaps in the Venn diagram of how the words model and theory are used in science, but there are certainly plenty of counterexamples to an insistence that the words are identical in all uses. A few examples:

1. A model need not have the same confidence level from experimental validation as a theory. A hypothesis which has not yet been experimentally tested can give rise to models (indeed it needs to) which provide predictions which are then subject to experimental testing. Likewise, at some point, scientists can have so much confidence in a theory, scientists consider it to be a fact or a law.

2. The word model can be used to apply a specific theory to a specific problem (or system) and thus have a narrower meaning. Thus the model of the hydrogen atom in non-relativistic quantum mechanics is different from the model of the helium atom, and so on.

3. The word model can be used to apply the same theory to the same problem, but make different approximations. Neglecting air resistance in a specific projectile motion problem is different from using a linear model of air drag is different from using a quadratic model of air drag, and so on. The underlying theory of physics is the same (classical mechanics), but different choices are made about the model.

So while I might agree that the Newtonian, Lagrangian, and Hamiltonian formulations of classical mechanics are simply different formulations of the same theory (since they are equivalent and this designation is consistent with the general use of the word theory in all levels of science), I don't think I would say they are different formulations of the same model. The word model has too many uses in science to be dogmatic that it always have exactly the same meaning as theory. The Venn diagrams of usage do not completely overlap.

 

[Auto-Didact]

Your Feynman quote touches on something different, namely if there is a point to having several ways of thinking about the same model - there is, because there may be a deeper model that makes different predictions that more closely conforms to one or the other.

That is exactly what I am alluding to as well.

No, you are simply wrong here. I never said there is no point in thinking differently about a theory, but as a theory, two theories that make exactly the same predictions are equivalent and should be considered the same theory. This has absolutely nothing to do with uncertainties in measurements. It is just about whether two theories are, even in principle, distinguishable.

My usage of the term certainty does not refer to measurement uncertainties, I am instead talking about the possibility that one of the mathematical models, let's say A, hasn't been pushed to its natural limits yet and the assignment of an equivalence with another model, B, will then constitute a premature closure, which enables that it can later turn out that B is actually merely a numerical approximation of the completed form of A with actual physical qualitative differences w.r.t. the prediction of objects and their properties in nature. Feynman addresses this exact point in the same text, immediately following the earlier quote:

That reminds me of another point, that the philosophy or ideas around a theory may change enormously when there are very tiny changes in the theory. For instance, Newton’s ideas about space and time agreed with experiment very well, but in order to get the correct motion of the orbit of Mercury, which was a tiny, tiny difference, the difference in the character of the theory needed was enormous. The reason is that Newton’s laws were so simple and so perfect, and they produced definite results. In order to get something that would produce a slightly different result it had to be completely different. In stating a new law you cannot make imperfections on a perfect thing; you have to have another perfect thing. So the differences in philosophical ideas between Newton’s and Einstein’s theories of gravitation are enormous.

What are these philosophies? They are really tricky ways to compute consequences quickly. A philosophy, which is sometimes called an understanding of the law, is simply a way that a person holds the laws in his mind in order to guess quickly at consequences. Some people have said, and it is true in cases like Maxwell’s equations, ‘Never mind the philosophy, never mind anything of this kind, just guess the equations. The problem is only to compute the answers so that they agree with experiment, and it is not necessary to have a philosophy, or argument, or words, about the equation’. That is good in the sense that if you only guess the equation you are not prejudicing yourself, and you will guess better. On the other hand, maybe the philosophy helps you to guess. It is very hard to say.

The point I am trying to make to you, is the following: To assign an equivalence in practice before all experiments are done and/or before the models have been theoretically pushed to their natural completion is to put the matter at rest prematurely. In such cases - and we are almost always working in such a case, or practically unable to tell that we are not in such a case - this assignment of equivalence gives psychologically speaking, a false sense of absolute certainty about what must be, i.e. it treats the physics as if it was actually mathematics. Doing this, i.e. letting yourself become absolutely sure about something in science by treating it as mathematics, is an unwarranted philosophical step and it is a very fallable subjective human psychological error that you and many other physical scientists, all of course being human, seem to be willing to make by talking about the matter in principle. These unwarranted philosophical steps that scientists, especially physicists, tend to make can and literally do hold back the progression of science because their logically convincing nature confims to our human bias i.e. our psychological need for certainty, which incidentally Feynman also adressed:

What is necessary ‘for the very existence of science’, and what the characteristics of nature are, are not to be determined by pompous preconditions, they are determined always by the material with which we work, by nature herself. We look, and we see what we find, and we cannot say ahead of time successfully what it is going to look like. The most reasonable possibilities often turn out not to be the situation. If science is to progress, what we need is the ability to experiment, honesty in reporting results, the results must be reported without somebody saying what they would like the results to have been, and finally an important thing the intelligence to interpret the results. An important point about this intelligence is that it should not be sure ahead of time what must be. It can be prejudiced, and say ‘That is very unlikely; I don’t like that’. Prejudice is different from absolute certainty. I do not mean absolute prejudice just bias. As long as you are only biased it does not make any difference, because if your bias is wrong a perpetual accumulation of experiments will perpetually annoy you until they cannot be disregarded any longer. They can only be disregarded if you are absolutely sure ahead of time of some precondition that science has to have. In fact it is necessary for the very existence of science that minds exist which do not allow that nature must satisfy some preconceived conditions, like those of our philosopher.

 

[Orodruin]

That is exactly what I am alluding to as well.

I am not merely talking about measurement uncertainties, I am talking about the possibility that one of the mathematical models, let's say A, hasn't been pushed to its natural limits yet and the assignment of an equivalence with another model, B, will then constitute a premature closure, which enables that it can later turn out that B is actually merely a numerical approximation of the completed form of A with actual physical qualitative differences w.r.t. the prediction of objects and their properties in nature.

I am sorry, but then you are breaking the first and only requirement I stated for two models being the same, namely that they make exactly the same predictions. If you could, in principle, tell A from B using experiments, then they are different models. I never placed any requirement that this should be done already. A ”completed form of A” generally makes different predictions from A and is therefore a different model than A.

 

[Auto-Didact]

I am sorry, but then you are breaking the first and only requirement I stated for two models being the same, namely that they make exactly the same predictions. If you could, in principle, tell A from B using experiments, then they are different models. I never placed any requirement that this should be done already. A ”completed form of A” generally makes different predictions from A and is therefore a different model than A.

The problem is not whether I agree with that in principle statement or not, the problem is whether taking the conclusion of such an in principle statement and reasoning further upon the basis thereof is scientifically justified; for physics the answer is no, while for mathematics and philosophy the answer is yes.

 

[Auto-Didact]

No, this is simply not the case. Distinguishing between LET and standard SR is physically irrelevant. It is a purely philosophical question that has nothing to do with physics, which is an empirical science. In an empirical science, a model is nothing more than the sum of its predictions.

Actually it is not necessarily physically irrelevant, but experimentally or scientifically irrelevant. The fact is that physics is an amalgamation of mathematics and philosophy and is moreover an empirical science as well. More carefully stated, physics is the practice of natural philosophy through the usage of mathematical language wherein the theories can produce numerical predictions which can be scientifically checked in an empirical observation or experiment. This is reflected in the two main branches of physics, namely theoretical physics and experimental physics, which have different criteria of scientific standards which do not fully overlap. This guy named Newton explained this pretty well in a book which even has it in the title when he practically invented, you know, the science which we call physics.

Incidentally, Feynman as always already warned us against the danger of completely disregarding the philosophical aspect of physical theories and only focusing on mathematical formalisms, i.e. the contemporarily very popular 'shut up and calculate' attitude among physicists. I quote:

For those people who insist that the only thing that is important is that the theory agrees with experiment, I would like to imagine a discussion between a Mayan astronomer and his student. The Mayans were able to calculate with great precision predictions, for example, for eclipses and for the position of the moon in the sky, the position of Venus, etc. It was all done by arithmetic. They counted a certain number and subtracted some numbers, and so on. There was no discussion of what the moon was. There was no discussion even of the idea that it went around. They just calculated the time when there would be an eclipse, or when the moon would rise at the full, and so on. Suppose that a young man went to the astronomer and said, ‘I have an idea. Maybe those things are going around, and there are balls of something like rocks out there, and we could calculate how they move in a completely different way from just calculating what time they appear in the sky’. ‘Yes’, says the astronomer, ‘and how accurately can you predict eclipses ?’ He says, ‘I haven’t developed the thing very far yet’. Then says the astronomer, ‘Well, we can calculate eclipses more accurately than you can with your model, so you must not pay any attention to your idea because obviously the mathematical scheme is better’. There is a very strong tendency, when someone comes up with an idea and says, ‘Let’s suppose that the world is this way’, for people to say to him, ‘What would you get for the answer to such and such a problem ?’ And he says, ‘I haven’t developed it far enough’. And they say, ‘Well, we have already developed it much further, and we can get the answers very accurately’. So it is a problem whether or not to worry about philosophies behind ideas.

This text is also the immediate followup from the previous text I quoted; maybe this should be compulsory literature for all physicists?

 

[Orodruin]

The problem is not whether I agree with that in principle statement or not, the problem is whether taking the conclusion of such an in principle statement and reasoning further upon the basis thereof is scientifically justified; for physics the answer is no, while for mathematics and philosophy the answer is yes.

Then you have not really read my posts, sorry. I never claimed that there was no merit to further reasoning and speculation based on different descriptions of the same model. There is even a merit in doing so for furthering physics in order to find different possible completions of the model.

 

[Auto-Didact]

Then you have not really read my posts, sorry. I never claimed that there was no merit to further reasoning and speculation based on different descriptions of the same model. There is even a merit in doing so for furthering physics in order to find different possible completions of the model.

I am not saying that you did, but I am saying that the mere act of acceptance of such a purely in principle logical or mathematical equivalences in practice in the context of physics often is a very real impediment to scientific thinking and therefore to science. This is because scientists like all humans have a psychological preference for certainty and tend to spontaneously form groups wherein if such unjustifiable arguments becomes generally accepted i.e. conventional wisdom, it is almost impossible for some other scientist whether inside or outside that group to question it, regardless of whether or not the statement was actually scientifically justified in the first place.

Acting as if it is completely scientifically legitimate to argue based upon such purely logical/mathematical in principle arguments in the context of physics is to contribute directly to the above stated problem, whether or not you intended to do so, because others who are also in the discussion or merely reading the discussion and who are also thinking about it, also have this psychological need for certainty and seeing someone else repeat their subjective stance on the matter and passing it off as some necessary objective 'truth', directly feeds into their confirmation bias on top of that need; this problem of course is way worse depending on if such an argument comes from an authority figure within the community.

 

[Orodruin]

I am not saying you did, but I am saying that the mere acceptance of such a purely in principle logical or mathematical equivalences in practice often is a very real impediment to science.

I am going to disagree with that what I said is an impediment to science. In no way did I make such a statement that two models being equivalent should prevent people from thinking about completions using both of them or to stop thinking completely. If this is not your point, then I do not see the need for you to loudly exclaim that what I said is incorrect.

 

[Auto-Didact]

I am going to disagree with that what I said is an impediment to science. In no way did I make such a statement that two models being equivalent should prevent people from thinking about completions using both of them or to stop thinking completely. If this is not your point, then I do not see the need for you to loudly exclaim that what I said is incorrect.

I get the feeling you are taking my admonition of your original argument personally, that isn't my intention. Make no mistake however: if you are a fellow practicing scientist then you should take it very seriously for what you proclaim has actual consequences, which is why you have a responsibility to both the public and your colleagues to not lightly make such wide sweeping statements and pass them off as scientifically legitimate; if you aren't a scientist then be aware that my intention isn't to criticize you personally, but to criticize your argument:

If two mathematical frameworks lead to the same testable predictions, then they are for all purposes (apart from computational) to be regarded as the same model. You cannot tell them apart by empirical observation

I stand by my point that this very sweeping statement, regardless of its logical validity or whether scientists in practice act as if it were legitimate practice, is neither a scientifically legitimate, academically justifiable nor physically valid argument to make for all of the reasons I stated in the above posts. In any case, I hope you and/or other participants or readers of this thread have learned something from this exchange.

I will try to end on a more positive note, by referring once again to the words of Feynman:

The scientist has a lot of experience with ignorance and doubt and uncertainty, and this experience is of very great importance, I think. When a scientist doesn’t know the answer to a problem, he is ignorant. When he has a hunch as to what the result is, he is uncertain. And when he is pretty darn sure of what the result is going to be, he is still in some doubt. We have found it of paramount importance that in order to progress we must recognize our ignorance and leave room for doubt. Scientific knowledge is a body of statements of varying degrees of certainty — some most unsure, some nearly sure, but none absolutely certain.
Now, we scientists are used to this, and we take it for granted that it is perfectly consistent to be unsure, that it is possible to live and not know. But I don’t know whether everyone realizes this is true. Our freedom to doubt was born out of a struggle against authority in the early days of science. It was a very deep and strong struggle: permit us to question — to doubt — to not be sure. I think that it is important that we do not forget this struggle and thus perhaps lose what we have gained.

 

[Dr. Courtney]

Incidentally, Feynman as always already warned us against the danger of completely disregarding the philosophical aspect of physical theories and only focusing on mathematical formalisms, i.e. the contemporarily very popular 'shut up and calculate' attitude among physicists. I quote:

This text is also the immediate followup from the previous text I quoted; maybe this should be compulsory literature for all physicists?

Love the Feynman quotes, keep them coming. Incidentally, I elevated my son's high school physics course (home schooled him) from "regular" to honors, by having him watch Feyman videos once a week. He started with "The Character of Physical Law" and then went on to other material that struck his fancy. See: http://www.richard-feynman.net/videos.htm

In any case, over time (without me prodding, but obviously hoping) his interest and intended college major shifted from Mechanical Engineering to Physics, and I think the Feynman videos played an essential role in that.

Feynman had a great grasp of the essential epistemology of physics (and all science), and he often expressed it in ways that make the forest very clear for those who have spent too much time looking at their own few trees and wanting to generalize their own small view and apply it to physics or science more broadly.

I am not saying that you did, but I am saying that the mere act of acceptance of such a purely in principle logical or mathematical equivalences in practice in the context of physics often is a very real impediment to scientific thinking and therefore to science. This is because scientists like all humans have a psychological preference for certainty and tend to spontaneously form groups wherein if such unjustifiable arguments becomes generally accepted i.e. conventional wisdom, it is almost impossible for some other scientist whether inside or outside that group to question it, regardless of whether or not the statement was actually scientifically justified in the first place.

Acting as if it is completely scientifically legitimate to argue based upon such purely logical/mathematical in principle arguments in the context of physics is to contribute directly to the above stated problem, whether or not you intended to do so, because others who are also in the discussion or merely reading the discussion and who are also thinking about it, also have this psychological need for certainty and seeing someone else repeat their subjective stance on the matter and passing it off as some necessary objective 'truth', directly feeds into their confirmation bias on top of that need; this problem of course is way worse depending on if such an argument comes from an authority figure within the community.

Well said. Since Feynman, there seems to be a trend in pure theorists losing touch with how theories are actually connected to experiments. Lots of choices get made in the process of making actual quantitative predictions regarding the outcome of real experiments. Some of these choices have physical meanings (such as coordinate systems and basis sets) and others may only be computational (such as choices on integration and matrix diagonalization teechniques.) But insofar as they can lead to differences in the actual predictions, they are differences in the model. And as Feynman pointed out, there may be cases where the theory itself is more correct at some conceptual level (though that may not be known yet), but the present challenges in yielding complete model predictions may not yet allow that to be known.

Not only are we not usually dealing with mature theories (like classical or quantum mechanics), often we are only guessing at the level of maturity of the framework with which we are dealing. In the history of science, it is a frequent error to assume that a given framework is more complete/general/mature than it really is.

 

[bobob]

I'm not sure I agree with this. A model is a story with its associated mathematical description. I might model relativity using the usual Einstein method or use the LET (Lorentz Ether Theory) model. The story and the mathematics may be different different but the predictions of each model are the same.

That is not true. If you were a physicist who finds ether theory appealing, you would be doing research on the ether to determine what properties ether particles must have (mass, size, etc.) to reproduce results predicted by lorentz transforms. If you were doing research in relativity, your research would be entirely different. You would be studying geometry (which is what einstein did for general theory). In addition, there are transformations which are just fine in special relativity, since it's geometry, but I can't imagine how you would interpret in an ether theory. For example, t+ = t+z, t- = t-z (light front coordinates). Different models predict different things and ultimately have different mathematics if you take the model seriously enough to pursue it beyond some superficial similarity.

 

[Orodruin]

I stand by my point that this very sweeping statement, regardless of its logical validity or whether scientists in practice act as if it were legitimate practice, is neither a scientifically legitimate, academically justifiable nor physically valid argument to make for all of the reasons I stated in the above posts. In any case, I hope you and/or other participants or readers of this thread have learned something from this exchange.

I think you are reading way too much meaning into the ”for all purposes” part. By ”for all purposes” I think it should be pretty clear that what is intended is related to the scientific process of empirically telling two models apart. As I have (tried to) made clear, there is clearly still merit in developing different ways of thinking about the same model, but for all purposes in terms of making testable predictions the models are still the same. Whether or not one or the other lies closer at heart for a completing model is irrelevant for this.

The fact remains that unless there are different predictions from the models, there is no way of empirically telling them apart. Or are you opposing this statement?

I get the feeling you are taking my admonition of your original argument personally, that isn't my intention.

So I am not to take it personally unless I should take it personally? (Yes, I am a scientist. Yes, I would consider myself successful in my field.) You may need to work on how you communicate such things online. That is fine, I have realized that need for myself too.

 

[Auto-Didact]

I think you are reading way too much meaning into the ”for all purposes” part. By ”for all purposes” I think it should be pretty clear that what is intended is related to the scientific process of empirically telling two models apart. As I have (tried to) made clear, there is clearly still merit in developing different ways of thinking about the same model, but for all purposes in terms of making testable predictions the models are still the same. Whether or not one or the other lies closer at heart for a completing model is irrelevant for this.

It isn't pretty clear, definitely not to those who aren't deeply familiar with doing particular kinds of physics experiments, such as mathematicians and computer scientists, inexperienced physics students, high school students who are aspiring physicists, let alone other kinds of scientists.

As it stands you are explicitly qualifying 'for all purposes (apart from computational)', which can and does come off as if the matter is already settled once and for all. This is pretty careless, regardless if other practicing scientists do the same among each other, which needless to say they do. The difference is this is a public forum, i.e. your statements are open to public scrutiny, meaning you need to be able to justify your argument, be able to defer the issue to some widely accepted physics protocol or resource, or admit defeat which isn't a personal failure but a laudable public display of character; this is the very purpose of an online discussion.

The fact remains that unless there are different predictions from the models, there is no way of empirically telling them apart. Or are you opposing this statement?

Of course I'm familiar with what you are trying to say and I believe it is a good rule of thumb, let's call it the theory equivalence rule (N.B. Feynman dedicated almost an entire lecture to this topic arguing your point far more carefully). If we were in the ideal situation to have completed models, such as in the exemplary case of Newtonian, Hamiltonian and Lagrangian mechanics, upon which entire edifices of mathematics have been constructed over the course of centuries allowing us easily through posterity to establish equivalence, then nothing is wrong.

Having said that, I starkly recognize that when comparing other theories we are almost never in such an ideal situation, where it is actually valid to apply the rule. Of course, by thinking about the matter purely formally we can easily convince ourselves into thinking that we are in such a case, by being careless and so consciously or subconsciously fudging it just because we want to make some headway. This means we need to proceed with caution and reserve our right to have a necessary amount of doubt, especially w.r.t. comparing two models of which their completeness is often unknown, as Dr. Courtney wonderfully illustrates here:

Not only are we not usually dealing with mature theories (like classical or quantum mechanics), often we are only guessing at the level of maturity of the framework with which we are dealing. In the history of science, it is a frequent error to assume that a given framework is more complete/general/mature than it really is.

i.e. it is almost never really clear whether or not some model is actually carried to completion or if it is even possible to do so given the contemporary state of mathematics, because obviously not every theorist is a Newton. To give an example, if differential geometry/tensor calculus was not yet invented by the time Einstein was formulating general relativity, he more likely than not would simply not have been able to naturally complete the theory; a preliminary version of his theory however may still have come out as equivalent to some other theory under the rule.

I suspect many if not most new theories may suffer from being naturally completable by similarly lacking the discovery of their natural mathematical settings (whether that be due to the mathematics not yet being discovered or that branch of mathematics simply being unfamiliar to the physicist), yet in their preliminary form still be capable of being judged using the rule inappropriately to be equivalent to perhaps some already falsified theory and then discarded prematurely. There simply doesn't seem to be any criteria for telling when a theory is naturally completed, meaning strict adherence to the rule in practice may not merely give misleading results due to careless misuse but even actively be harmful to science.

So I am not to take it personally unless I should take it personally? (Yes, I am a scientist. Yes, I would consider myself successful in my field.) You may need to work on how you communicate such things online. That is fine, I have realized that need for myself too.

I said don't take it personally, but do take it seriously; I'm not saying any of this to spite you, but - being the hopeless optimist that I am - in the hope that it will actually help you (and others reading who might also share your cavalier attitude w.r.t. in principle statements) to try to better your character. In other words, I'm doing this for the same fraternal/tribal reason some tribesman would reprimand his fellow tribesmen if he sees they are not conforming to some mutually agreed upon code of conduct which they both swore an oath to uphold.

It doesn't hurt to be more careful and reflective when thinking, arguing and talking about ideas in the context of physics, especially when discussing controversial and unsettled subject matters; on the contrary, I think it only makes us think more clearly and communicate more honestly, both with ourselves and with others, making us better scientists than what we already are.

I recognize your complaints about my disregard of etiquette/political correctness, but I think the topic is too important to let self-censorship in the form of political correctness get in our way of proceeding further. Also, I'm a scientist, not a politician nor someone looking to score publicity points; I care about the science and I'm very happy you and others also take the time to contribute to this important discussion.

What I've learned in my time in academia is that unsettled matters and open questions probably aren't as simple they might appear at first glance if I approach them uncritically as I would approach any regular already settled scientific matter; to act as if these matters are in any way simple or obvious is to pompously disregard all the hard and serious work of the legions of brilliant thinkers who worked on the problem before you without solving it, which of course implies that you place your own intellect (way) above theirs. During my more arrogant days as an undergraduate when I did think that way, luckily I myself was reprimanded and I learned to stop doing this; now I tend to act and approach such difficult questions as if everyone in the room either knows more than me or knows something valuable and essential that I have yet to learn, even when this isn't likely.

This has led me to take quite serious the stance that 'all that it takes for evil to prevail is for good men to do nothing', evil being ignorance in this context. In other words, I think any practicing scientist can and should call out other scientists if he suspects they are making some correctable mistake in the context of science and so doing a disservice to not only themselves but to their colleagues, their students, the public at large and to the science itself. What he definitely shouldn't do though is attack the other personally through the usage ad hominem e.g. see Lubos Motl's attitude w.r.t. to colleagues which is nothing short of disgraceful.

 

[atyy]

I disagree. If two models make the same predictions, then they are by definition the same model.

But don't you think a model (or theory, ignoring Dr. Courtney's distinctions for the moment) is not simply its predictions, but also the internal relations among predictions? If we take all the predictions of classical physics and remove the internal logic, then we lose the simplicity that a good theory gives. This is a little bit like the difference in how renormalization was understood before and after Wilson - with respect to say, QED, he changed nothing about the predictions, but improved our understanding.

 

[crastinus]

So, here is another argument from the math side that the PLA lacks physical meaning. It is given in reference to classical systems.

This one is worth discussing because the argument, unlike most, actually defines both "physical meaning" and "physically possible"; it does so in terms of dynamical systems theory.

1. For a variational principle like PLA to have physical meaning, it would have to be the case that we find the system minimizing its action with respect to trajectories that are physically possible for the system.
2. What is physically possible for a system are the ways in which that system can evolve in a given time (its set of true trajectories).
3. The phase space S represents all the possible states of the system, but it is the phase portrait P that represents the possible states of the system, the set of times, and the ways in which the systems can evolve from one state to another in a given time.
4. What a variational principle like PLA does is vary the trajectories through the phase space S in order to discover the phase portrait P of the system.
5. But the system only ever evolves according to the phase portrait P (its set of true trajectories); so, the trial trajectories through phase space S are not physically possible for the system; the only trajectories in the phase space S that are possible are those that are mapped in the phase portrait P.
6. And thus we do not find the system minimizing its action with respect to what is physically possible for it; we only find minimization with respect to our mathematical construction, i.e. the phase space S.
7. And so, since we do not find minimization with respect to the phase portrait P, PLA does not have physical meaning; it is, rather, a powerful mathematical tool that we use to discover the ways in which physical systems evolve over time (their phase portraits).

I broke it up into numbered premises to make it easier to object to, argue about, etc.

One merit of the argument is that it's clear! --even if unsound somehow.

 

[crastinus]

To add one perhaps final point of literature (recently pointed out to me):

Mario Bunge argued in American Journal of Physics that we should not interpret Lagrangian formulations in a physical way. There is a response to his piece in the letters to the editor. Below are the references if you're interested.

That's about all I can find on this question!

Thanks to everyone for your input.

References

Bunge, Mario. “Lagrangian Formulation and Mechanical Interpretation.” American Journal of Physics 25, no. 4 (April 1, 1957): 211–18. https://doi.org/10.1119/1.1934403.

Hart, John B. “Dr. Bunge’s Lagrangian Formulation.” American Journal of Physics 25, no. 7 (October 1, 1957): 495–495. https://doi.org/10.1119/1.1934528.

 

[Herman Trivilino]

Someone recently argued to me that PLA is not fundamental.

Being fundamental and being physically meaningful are two different things. Do you have a reference for a claim that the PLA is not physically meaningful?

 

[sophiecentaur]

We could talk forever about what Maths is. But, if you regard Maths as just the 'best' description of many Physical processes, then it will be the 'first' thing in many instances. When a mathematical process produces a prediction about some physical situation and that is confirmed by some subtle and non-intuitive physical measurement, which came first then?
I think our brains often prefer to interpret the mathematical with a more simple and 'intuitive' physical model. That model is what we hold in our heads because it will often involve far fewer steps and can be based on some really (over-)simple rules. That's the way our Consciousness seems to work about most things. It's like 'the boss' who takes very simple inputs from the rest of the company and makes decisions (or thinks he/she does) on that basis.
The one thing that the Consciousness really has to do is to present itself with a 'reasonable' picture of things. That, for many or even most people will often not actually be a mathematical picture. But there are examples when we actually do involve The Maths as a shorthand explanation for some phenomena.
I have a personal 'paradox in understanding' and that is how Statistics (mathematical) seems to be what drives Thermodynamics (physical). Also, when a beginner tries to discuss Electricity in physical terms, PF quickly slaps them down and tells them to use V=IR.

 

[Stephen Tashi]

Being fundamental and being physically meaningful are two different things. Do you have a reference for a claim that the PLA is not physically meaningful?

Back in post #21 there is a link to a paper that points out that "the principle of least action" is an ambiguous phrase. The paper distinguishes two versions of the principle of least action. The two versions have different physical meanings. So, yes, there is a source that says the PLA "is not phyically meaningful" in the sense that it does not have a unique physical interpretation.