Mathematicians' contributions to physics

[fresh_42]

I'm just curious: how do you make physics without infinitesimal quantities? do you have another way?

No. I just do not behave as if they were tiny little creatures. In most cases, they are real numbers $\left. \dfrac{dy}{dx}\right|_{x=x_0}$ and in the rest of the cases they are differential forms or basis vectors.

I love Weierstraß's formula for derivatives: $f(x+v)=f(x)+D_x(f)\cdot v + o(v).$ No tiny little creastures necessary except for the remainder function.

 

[coquelicot]

No. I just do not behave as if they were tiny little creatures. In most cases, they are real numbers $\left. \dfrac{dy}{dx}\right|_{x=x_0}$ and in the rest of the cases they are differential forms or basis vectors.

I love Weierstraß's formula for derivatives: $f(x+v)=f(x)+D_x(f)\cdot v + o(v).$ No tiny little creastures necessary except for the remainder function.

So, how should the derivation of the propagation of heat be written according to you?

 

[fresh_42]

So, how should the derivation of the propagation of heat be written according to you?

I do not suggest another concept, just another wording. Do not use the word "infinitesimal". It creates more confusion than it solves. There is no need. The only case I can think of that has a bit of a justification is the chain rule, but even this is cheating. There is no problem in using $dx$. There is a problem with treating it as a small quantity.

 

[coquelicot]

I do not suggest another concept, just another wording. Do not use the word "infinitesimal". It creates more confusion than it solves. There is no need. The only case I can think of that has a bit of a justification is the chain rule, but even this is cheating. There is no problem in using $dx$. There is a problem with treating it as a small quantity.

I don't see what you mean. $dx$ is actually supposed to represent the limit of $\Delta x$, so what is the problem to name it "infinitesimal quantity"?

 

[fresh_42]

I don't see what you mean. $dx$ is actually supposed to represent the limit of $\Delta x$, so what is the problem to name it "infinitesimal quantity"?

$dx$ is a differential form. $dy/dx$ is a limit, and it is not "something small", especially not for undergraduates. How would you rigorously define the word infinitesimal without falling to the seventeenth century? And, of course, without using hyperreals or surreals. Wikipedia writes

In mathematics, a positive infinitesimal is an object which is greater than zero in the order of the real numbers but less than any positive real number, however small.

I consider this pure nonsense as long as we teach calculus traditionally (meanwhile) with epsilontic. This definition is a crime.

 

[coquelicot]

$dx$ is a differential form. $dy/dx$ is a limit, and it is not "something small", especially not for undergraduates. How would you rigorously define the word infinitesimal without falling to the seventeenth century? And, of course, without using hyperreals or surreals. Wikipedia writes

I consider this pure nonsense as long as we teach calculus traditionally (meanwhile) with epsilontic. This definition is a crime.

yes I agree with you that such a definition in Wikipedia is rather ridiculous, except in the context of non standard analysis. But the use of the wording "infinitesimal" is valid in my opinion, as an intuitive and very useful abbreviation for limit processes. Once you have specified what is boiling down mathematically, it's perfectly OK to use such a jargon, which has a natural and intuitive meaning. Everyone with sufficient maturity in calculus is then able to translate, if he wishes, a demonstration exposed this way into a demonstration with epsilons (even thought he would never do that as this complicates the matter uselessly).

 

[malawi_glenn]

Speaking of wikipedia, you are always welcome to edit it...

 

[fresh_42]

Speaking of wikipedia, you are always welcome to edit it...

I would delete it.

 

[malawi_glenn]

I would delete it.

Well that is a subset of editing :D

 

[romsofia]

I think this is a very interesting question and depends on which field of physics you're in.

I like how fresh_42 put it where math is more akin to philosophy, and physics is more akin to nature. In the 60s-80s, some fields saw those who could philosophize nature succeed the most. Wheeler, DeWitt, Penrose, Choquet-Bruhat, Unruh, etc

In the fields where people like this thrive, I would think a mathematician (although, they better be a good one!) could find the supposed holes of the field. But the closer we get to actual experimental data, say designing the simulation of a binary black hole merger, or calculating the velocity of a black hole kick, it's less likely that a mathematician (even a great one) could be of that much help, even less that they'd be able to find the supposed holes in it.

You need actual physics for that, you can't just "abstractize" the experimental set up. Nor should we put experimental design in a "definition: corollary: proposition: remark: " fashion. Now, it was talked about back in the 80s (one of my favorite essays of all time "What have we learned from Quantum field Theory in curved space-time?" by Fulling talks a little about this at the start), and should be emphasized: the foundational community *does* need to transition to that point. There is way too much overlap and usage of common words, and each subfield of this overall community has their own definitions, and it gets confusing fast. Heck, I'd wager that at least 40% of the posts on the interpretation forum is about someone's usage of a word in a way they don't use it. Imagine how fast a Bourbakian style would clean up that field and the potential progress we would see if those great minds talked to each other properly! But does that mean a mathematician in the current state could find the problems the physicists aren't seeing? Maybe.

To put it more bluntly: If we're talking about topological censorship, then yes, I could see how a mathematician could see some conceptual junk the physicist is unaware of, and be able to break the problem down to a more abstract view which could lead to some connections, and that will patch some holes in the framework. If we're talking about experimental design, simulations of events, and analyzing real world data, I don't see why a mathematician would be any better than the common human in moving the field forward. You need actual scientific protocol, and training.

 

[fresh_42]

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If we're talking about experimental design, simulations of events, and analyzing real world data, I don't see why a mathematician would be any better than the common human in moving the field forward. You need actual scientific protocol, and training.

I think this is a bit one-dimensional. As long as SUSYs and Branes are on the table, as long there are serious mathematical aspects to be considered. E.g., mathematicians could find theorems that would allow actual experiments to test those theories. I guess that would help a lot! And I can't get these slime molds out of my mind. If they really represent a valid model for the distribution of DM, then mathematics could well be a first step to connect the two, possibly easier than the very different physical entities can.

I like the, admittedly nostalgic view of scientists who do not distinguish so sharply as we do these days and consider physics and mathematics as two aspects of the same thing. I mean, a circle is a circle:

• It does not exist. (Nature)
• Except as an idea of Plato. (Philosophy)
• $re^{it}$ however, is good enough to be useful. (Physics)
• $re^{i\varphi }$ however, is good enough to be useful. (Mathematics)

The only difference is time (t) versus angle ($\varphi$). And everyone far enough from QM does not really care about this different perspective.

 

[apostolosdt]

We, who studied physics, are not in a war with mathematicians—-for goodness sake! But please, PLEASE, let us remember that physics is an experimental science, and yes it’s full of errors, unanswered questions… perhaps the worst is yet to come!

Speaking of experiments, an “experiment” is going on for more than forty years now, superstring theory. Mathematicians’ contribution is unquestionable; let’s await to witness progress.

 

[Astronuc]

What are the "real" problems in physics? Can these problems we solved solely by mathematicians? What are your thoughts on that?

Knowing that one does not know what one does not know. It applies to a current situation of some physics (a natural phenomenon) that has been overlooked, probably because it was/is unexpected.

The tendency to present a jungle of equations as a well established theory. The problem is real: you have to spend two years to understand the equation jungle, just to realize that there is no clear directing idea behind them. And the worse is that you can even not criticize the theory, because you will be considered as an idiot for not understanding the (alleged) idea behind the equations. The book of de Groot I am trying to study may be an example of that (not sure, I'm just beginning). That's bad for the physics, because some problems may be considered to be solved while they are actually not

Not so much 'well-establish', but perhaps established enough to solve a problem (behavior of a physical system in which a multitude of phenomena are occurring, and one would like to describe different aspects in an economical way). Economical can mean solving a problem on a workstation (multicore), or a small cluster or work stations, and not a supercomputer with 100,000 cores, and I want an answer within 24 hour (or less ideally) and not a week, or weeks, or longer. And the solution shouldn't take MWh (MJ) of energy, but maybe many kWh (KJ).

Is the argument about how physicists use mathematics or describe the mathematics (or mathematical tools) employed in solving a problem, or describing a problem.

When I began studying physics in earnest, in high school then university, I quickly realized that I needed a firm background in mathematics, at least calculus, and then advanced calculus, but I didn't seen the need for subjects like group theory.

Consider some 'small' problems, or pieces of physics, involving photon interactions in matter, and simultaneously, interactions of electrons (and positrons) in the same matter, and the response of that matter to the population of photons, electrons and positrons moving through it (I haven't introduced neutrons yet). I have dozens of elements (atoms) jiggling about. Ideally they stay near there initial location before the photons, electrons, positrons (and neutrons) starting flying about. And there's more physics to add - transmutation of atoms (one element to another) and cations wanting to meet up with anions.

Consider the mathematics in this text describing the theory applied in the PENELOPE code.
https://www.oecd-nea.org/upload/doc...mulation_of_electron_and_photon_transport.pdf

It only covers part of the physics of photons, electrons, positrons (and doesn't involve neutrons, atomic displacements/diffusion, nuclear transmutations, nuclear fission, chemical reactions). The output is generally the energy deposition (dose) in some structure of matter, which could be simply a volume of water, air, human tissue, medical plastics, a detector material (NaI, GeLi, . . . ), semiconductors, ceramics (e.g., SiC) stainless steels, Zr-alloys, nuclear fuel (UO₂). When high Z materials are involved, the problem takes hours to run on a small PC (Intel I7-6600U CPU @ 2.60 GHz).

PENELOPE has been incorporated into GEANT4.
https://twiki.cern.ch/twiki/bin/genpdf/Geant4/LowePenelope
https://twiki.cern.ch/twiki/bin/view/Geant4/LowePenelope

Another code similar to PENELOPE is EGS5 (5th version of EGS, Electron-Gamma Shower).
https://rcwww.kek.jp/research/egs/egs5_manual/slac730-160113.pdf

Just one example of application - https://www.sciencedirect.com/science/article/pii/S027288422030883X

I'm interested in the effects of radiation on the evolution of structural materials and their performance in rather aggressive environments, and in particularly, how high energy gamma rays (and coincident/consequential electrons/positrons) change the atomic microstructure (crystal lattices and grain structure) of structural materials. The above codes PENELOPE does not yet consider photoneutron or photofission reactions, which only become significant in the presence of very light atoms (isotopes of Z=1 through Z=8) and heavy atoms (Z≥92), nor have the codes. EGS5 does appear to do photoneutron reactions.

Nuclear reactor core simulation codes do not consider photonuclear reactions necessarily, and in some cases, the cross section data libraries are incomplete.

Now if we turn to simulating the evolution of all the stars (do we consider the influence/perturbations of planets?) in all the galaxies in the universe over say a billion years - that's a much bigger problem. Or we could pick a small problem and just do all the stars in the galaxy in which we find ourselves. Or we pick a piece of a large problem and see if our mathematical models are sufficiently rigorous and robust to predict the behavior and outcome (describe the physics) we observe now and in the future.

 

[Astronuc]

So, how should the derivation of the propagation of heat be written according to you?

As simply as possible without sacrificing too much accuracy. Note that there are competing effects: conduction, convection and radiation.

 

[Astronuc]

How about a simpler problem looking to be solved.

EXISTENCE AND SMOOTHNESS OF THE NAVIER–STOKES EQUATION
https://www.claymath.org/sites/default/files/navierstokes.pdf

from
https://www.claymath.org/millennium-problems/navier–stokes-equation

It's not so simply give that turbulence is a bit chaotic.

A lot has been written about the challenge problem. Yet - not much progress in a solution.

https://www.quantamagazine.org/mathematicians-find-wrinkle-in-famed-fluid-equations-20171221/

 

[malawi_glenn]

Yes, once we physicsists know for sure that the Navier-Stokes equation has solutions, we can finally start using it to calculate stuff like wind and water flows :) Oh wait...

 

[apostolosdt]

I checked the book "Neoclassical Theory of Electromagnetic Interactions" by Babin & Figotin. I'm not opening a discussion of the book itself, but of two mathematicians' viewpoints on a well-established physical theory. After all, that's how both threads started.

The authors claim that it is feasible to use the classical interpretation of the EM phenomena down to atomic scales. No problem so far if we, open-minded, are willing to read about a new approach to EM. But it looks as if some "Kuhnian losses" are necessary in order to found the new theory; from the book's preface:

[..] Our theory, though similar to QM in some respects, is markedly different from it. In particular, (i) there is no need, in our theory, for the correspondence principle and consequent quantization procedure to obtain the wave equation; (ii) the Heisenberg uncertainty principle, though quite often applicable, is not a universal principle; (iii) there is no configuration space; (iv) there is no probabilistic interpretation of the wave function. [..]

I thought that adds to the present discussion constructively.

 

[coquelicot]

I checked the book "Neoclassical Theory of Electromagnetic Interactions" by Babin & Figotin. I'm not opening a discussion of the book itself, but of two mathematicians' viewpoints on a well-established physical theory. After all, that's how both threads started.

The authors claim that it is feasible to use the classical interpretation of the EM phenomena down to atomic scales. No problem so far if we, open-minded, are willing to read about a new approach to EM. But it looks as if some "Kuhnian losses" are necessary in order to found the new theory; from the book's preface:

[..] Our theory, though similar to QM in some respects, is markedly different from it. In particular, (i) there is no need, in our theory, for the correspondence principle and consequent quantization procedure to obtain the wave equation; (ii) the Heisenberg uncertainty principle, though quite often applicable, is not a universal principle; (iii) there is no configuration space; (iv) there is no probabilistic interpretation of the wave function. [..]

I thought that adds to the present discussion constructively.

Thank you for addressing the thread I opened previously.

Regarding your comment on the book, I think their authors does not claim to offer an alternative theory to quantum mechanics (e.g. if my understanding is OK, they do not pretend to be able to explain the double slits experiments), but their aim is to offer a theory which bridges the gap between microscopic and macroscopic phenomenons, at a point where it is even able to explain some microscopic phenomenons usually dealt with quantum mechanics. I may have not sufficiently understood though.

Could you explain in what sense you think the fragment you quoted adds to the present discussion constructively.

 

[coquelicot]

How about a simpler problem looking to be solved.

EXISTENCE AND SMOOTHNESS OF THE NAVIER–STOKES EQUATION
https://www.claymath.org/sites/default/files/navierstokes.pdf

from
https://www.claymath.org/millennium-problems/navier–stokes-equation

It's not so simply give that turbulence is a bit chaotic.

A lot has been written about the challenge problem. Yet - not much progress in a solution.

https://www.quantamagazine.org/mathematicians-find-wrinkle-in-famed-fluid-equations-20171221/

I think the original question of Malawi-Glenn was not a listing of the problems in physics that everybody knows about them, but rather what I meant when I wrote in another forum that "physicists are often unaware of the real problems in physics", in parentheses, as an answer to someone that presented physics books by mathematicians as (probably) bad.

 

[Astronuc]

I think the original question of Malawi-Glenn was not a listing of the problems in physics that everybody know about them, but rather what I meant when I wrote in another forum that "physicists are often unaware of the real problems in physics",

The title of the thread is "Mathematicians' contributions to physics." Malawi-Glenn also asked "What are the real problems?" to which one alluded in the statement that most many "physicists are often unaware of the real problems in physics". That seems a rather gross generalization, and as of yet, unsubstantiated. I'm curious as to "what real problems in physics that some, many or most physicists are unaware."

I provided the example of the PENELOPE and EGS5 codes, since they, or at least the PENELOPE author discusses some of the theory and mathematics involved, and how those equations are transformed into numerical methods.

At university, I took courses in mathematics, physics, mathematical physics, numerical analysis/methods, as well as a variety of engineering courses. We were tackling complex physics problems, and for some, looking for the best mathematical models to solve efficiently. I don't recall anyone being 'unaware', but rather many times, we knew there were approximations and simplifications, because some problems are highly non-linear, or highly coupled, for which there are no simple, nice solutions.

One could find plenty of examples on particle transport theory, whether it be neutrons (and photons, electrons) in a nuclear reactor, or charged particles (protons, deuterons, tritons, alphas, nuclei) in fusion plasmas, stars or large clouds.

So

What are the REAL problems in physics?

 

[apostolosdt]

[...] a theory which bridges the gap between microscopic and macroscopic phenomenons, at a point where it is even able to explain some microscopic phenomenons usually dealt with quantum mechanics. I may have not sufficiently understood though.

Could you explain in what sense you think the fragment you quoted adds to the present discussion constructively.

I'm afraid, in the modern physical approach, there is no gap between microscopic and macroscopic phenomena, for quantum mechanics along with relativistic invariance principles are equipped well enough to describe both phenomena. The so-called classical approach is a bunch of approximations which, one, saves us a tremendous amount of mathematical analysis and numerical work, and, two, eventually offers similar results from the point of view of experimental verification. That's a point again raised and, once again, clearly answered recently in another thread.

My own belief is this. The poorly mathematically supported, far from complete, Standard Model must be correct because no other concept in human history enjoys that verification of one part in a hundred million. So, why don't we leave physics to do its thing and mathematics to do its thing?

Regarding your last question, I meant my post, not the quoted passage---sorry for not having been clearer.

 

[coquelicot]

I'm afraid, in the modern physical approach, there is no gap between microscopic and macroscopic phenomena, for quantum mechanics along with relativistic invariance principles are equipped well enough to describe both phenomena. The so-called classical approach is a bunch of approximations which, one, saves us a tremendous amount of mathematical analysis and numerical work, and, two, eventually offers similar results from the point of view of experimental verification. That's a point again raised and, once again, clearly answered recently in another thread.

My own belief is this. The poorly mathematically supported, far from complete, Standard Model must be correct because no other concept in human history enjoys that verification of one part in a hundred million. So, why don't we leave physics to do its thing and mathematics to do its thing?

Regarding your last question, I meant my post, not the quoted passage---sorry for not having been clearer.

Thank you.
I would like to ask you a practical question, as you seem to know the matter very well. I already know EM theory at the level of most graduate students (I believe). Nevertheless, my mathematician intuition tells me that the way the properties in dielectrics are justified in most classical books (e.g. Jackson, Griffith, Landau and Lifshitz) is hardly a diagrammatic persuasion, or a superposition of equations with some context but at the least insufficiently convincing. So, I am currently trying to understand the macroscopic Maxwell equations deduced from the so called microscopic one, in books by authors that dedicated a part of their life to this task, namely the books of De Groot and that of Robinson. To put flesh on bones, I would like, at the very least, a convincing explanation of the following facts: when a dielectric is inserted inside a uniform field (say), surface charges appear. If on the other hand there is a div field, then volume charges appear inside the dielectric (in addition to the charges that caused the field). This may seem elementary to most students, but it is not. The book of De Groot is somewhat hard to follow, and it will take me a while to read it. The book of Robinson is easier, but I dislike some aspects of his theory (I also like some other aspects like his idea that the macroscopic smoothing comes from the high pass filtering of the high frequencies in space). Actually, Robinson criticizes the book of De Groot, after having noted that he is more or less the only person that has dealt extensively with this topic since Lorenz. I also had a look at more recent articles etc.

Now I come to my point: what is boiling down is that there is no clear "main stream" in this domain, a theory about which most physicists would agree (I may be wrong and hope you will correct me).
Of course, I expect some persons will post unintelligent and anoying answers like "QM is the theory", without addressing the real and practical question I've just raised: The problem does not consist in invoking a theory that is supposed to solve every microscopic and macroscopic problem, but to actually provide the solution of the questions with the theory.
This is why the idea of a "Bourbaki" like group has popped up in this thread.
And again, I see a lot of theories, but I don't see a main stream theory for the question I raised above.

You seem to say that the Standard Model provides the deal. So, are you aware of a widely accepted work, which provides the solution to the questions above?

 

[dextercioby]

@coquelicot Which chapters in de Groot's book do you refer to? Which de Groot book, actually? What about Robinson?

 

[coquelicot]

@coquelicot Which chapters in de Groot's book do you refer to? Which de Groot book, actually? What about Robinson?

De Groot S.R. Suttorp - Foundations of Electrodynamics (all the chapters are relevant).
Robinson F.N.H. - Macroscopic Electromagnetism

 

[dextercioby]

How can Robinson comment on de Groot, since Robinson's book appeared in 1972, while de Groot's in 1973? Unless it was a 2nd edition published more recently.

 

[coquelicot]

How can Robinson comment on de Groot, since Robinson's book appeared in 1972, while de Groot's in 1973? Unless it was a 2nd edition published more recently.

Sorry, you are probably right. If my memory is exact, Robinson criticizes the whole work of de Groot from 1950 (you can imagine that the book of de Groot has not pop up from nowhere).

 

[Astronuc]

How can Robinson comment on de Groot, since Robinson's book appeared in 1972, while de Groot's in 1973? Unless it was a 2nd edition published more recently.

Sorry, you are probably right. If my memory is exact, Robinson criticizes the whole work of de Groot from 1950 (you can imagine that the book of de Groot has not pop up from nowhere).

There is a chapter entitled "Statistical Foundations of Electrodynamic Theory in the book
Physics in the Making by AA. Sarlemijn and M.J. Sparnaay, 1989

The author of the chapter is L.G. Suttorp, who cites two books by S.R. de Groot, one coauthored with Suttorp
Groot, S.R. de, 1969, The Maxwell Equations (North-Holland, Amsterdam)
Groot, S.R. de and L.G. Suttorp, 1972, Foundations of Electrodynamics (North-Holland, Amsterdam)

Perhaps Robinson reviewed the draft/proof, or he received a pre-print.

Suttorp also cites three book by F.N.H. Robinson,
Robinson, F.N.H, 1971, Physics
Robinson, F.N.H, 1973, Macroscopic Electromagnetism (Pergamon Press, Oxford)
Robinson, F.N.H, 1975, Phys. Rep.

Ref: https://staff.science.uva.nl/l.g.suttorp/articles/making89.pdf

There is this article - "V Foundations of the Macroscopic Electromagnetic Theory of Dielectric Media,"
J Van Kranendonk, JE Sipe, Progress in Optics, Volume 15, 1977, Pages 245-350
"This chapter is concerned with the derivation of the macroscopic Maxwell equations and associated constitutive relations from the underlying microscopic equations describing the dynamics of the constituent particles and the electromagnetic fields created by these particles."
Ref: https://www.sciencedirect.com/science/article/abs/pii/S0079663808704803

 

[coquelicot]

There is this article - "V Foundations of the Macroscopic Electromagnetic Theory of Dielectric Media,"
J Van Kranendonk, JE Sipe, Progress in Optics, Volume 15, 1977, Pages 245-350
"This chapter is concerned with the derivation of the macroscopic Maxwell equations and associated constitutive relations from the underlying microscopic equations describing the dynamics of the constituent particles and the electromagnetic fields created by these particles."

Actually, there are many other articles, and the book of Babin and Figotin cited in this thread and elsewhere is another view. But what is the main stream?

And you have another book that processes the subject in a very particular and (probably) interesting manner:
F.W Hehl, Y.L. Obukhov, fundations of classical electrodynamics.

But what is the main stream ?

 

[apostolosdt]

De Groot S.R. Suttorp - Foundations of Electrodynamics (all the chapters are relevant).
Robinson F.N.H. - Macroscopic Electromagnetism

OK, coquelicot, I've got access to both books, but I can't promise I will read them front-to-end. To be honest, I had never heard of Robinson or de Groot. Is de Groot the senior author of the "Non-Equilibrium Thermodynamics" book as well? He is referred to as a theoretical physicist; I take it to be him, although I came across a number of mathematicians with that name.

[...] as you seem to know the matter very well. I already know EM theory at the level of most graduate students [...] You seem to say that the Standard Model provides the deal. So, are you aware of a widely accepted work, which provides the solution to the questions above?

Now, thanks for your kind words, but that's hardly the case; there are more proficient members in these forums than me. But, I'll bite and see what I can do with your question, although, having read about EM thru the standard texts years ago, your question sounds more like a puzzle than a physical one.

 

[malawi_glenn]

I am very glad that the discussion is still going on! However, the book thread is re-opened https://www.physicsforums.com/threa...ic-interactions-by-babin-and-figotin.1050807/ ;)

 

[coquelicot]

I am very glad that the discussion is still going on! However, the book thread is re-opened https://www.physicsforums.com/threa...ic-interactions-by-babin-and-figotin.1050807/ ;)

Actually, that's not the same book(s), and my question is much wider. You can see it as an example of what real problems a real mathematician can somewhat worry about, if you wish.

 

[malawi_glenn]

Actually, that's not the same book(s), and my question is much wider. You can see it as an example of what real problems a real mathematician can somewhat worry about, if you wish.

I know, it was just a heads-up ;)

 

[Frabjous]

To put flesh on bones, I would like, at the very least, a convincing explanation of the following facts: when a dielectric is inserted inside a uniform field (say), surface charges appear. If on the other hand there is a div field, then volume charges appear inside the dielectric (in addition to the charges that caused the field). This may seem elementary to most students, but it is not.

This discusses it a little.



Smith-White (1949). I found the reference in the Herring Archive which is an interesting collection of old references. http://large.stanford.edu/herring/a/a1/a12/

You might also like this arXiv article by Griffiths.
https://arxiv.org/abs/1506.02590

 

[coquelicot]

This discusses it a little.

View attachment 323877View attachment 323878View attachment 323879

Smith-White (1949). I found the reference in the Herring Archive which is an interesting collection of old references. http://large.stanford.edu/herring/a/a1/a12/

You might also like this arXiv article by Griffiths.
https://arxiv.org/abs/1506.02590

I've read the article of Griffiths. Thank you so many, very interesting (it also provides all the relevant derivation of the formulae in one place.

 

[Vanadium 50]

Of course, I expect some persons will post unintelligent and anoying answer

I would not go there.

I have had and continue to have a successful career in physics. Almost 1700 papers, almost a quarter-million cites, and an h-index over 200.

Do some of the theories I use have inconsistencies. Yup. Even QED has a Landau pole. Do I care? Nope.
Do some of the empirical laws I use have unphysical regions? Yup. Do I care? Nope. Don't use them there.
Are some of the calculation tools I use less than rigorous? Yup. Do I care? Nope.

I realize that this gets some mathematicians' goat, Not my problem. If I can measure x and compare it to theory, I'm good.

Could I stop measuring things and wait for the mathematical rigor to catch up? I could - but that would stop progress. So I don't.

Physicists use mathematics. Doesn't make them mathematicians.