Mathematicians' contributions to physics

[Demystifier]

But their theory seem interesting, and being myself a mathematician, I think this is only good for physics (most physicists seem to be unaware of the real problems in physics. They just choose to ignore them and consider that everything is "clear" for them).

Are you suggesting that mathematicians are sometimes better than physicists in spotting the real problems in physics? (Just asking.)

 

[malawi_glenn]

A spin-off discussion in this thread https://www.physicsforums.com/threa...ic-interactions-by-babin-and-figotin.1050807/
A quote from member @coquelicot (now deleted) was suggesting that most physicsts are unaware of the "real" problems with physics and that they [the physicsts] choose to ignore them.

Apart from the logical problem that you can not choose to ignore something that you are unaware of, I thought that here we could discuss if/how mathematicians are needed in physics. For intance list examples of where mathematicians have entered the field of physics and made significant contributions. Also examples where such endevaours have not been fruitful. Also examples in which the original idea was nonsense physically, but was still fruitful - like Weyl which was mentioned in a post by vanhees (see below).
Also we could discuss areas in physics which (still) lacks mathematical rigour.

This is, of course, nonsense. There are brillant books on theoretical physics written by mathematicians. Historical examples are Weyl's, Raum, Zeit, Materie and von Neumann's book on the mathematical foundations of quantum mechanics.

What these example also demonstrate is that you must be careful when it comes to the physics part. Weyl had the superficially brillant idea to gauge the scale invariance of the free gravitational field in GR to describe the electromagnetic field as the corresponding gauge field. The only disadvantage is that it's physically impossible, because it contradicts the simple fact that the spatial and temporal scales of charged matter doesn't depend on its electromagnetic history, as both Einstein and Pauli immediately pointed out to Weyl. Nevertheless the idea is indeed brillant, because the principle of making global symmetries local lead to a tremendous success in model building in connection with relativistic quantum field theory and the understanding of the fundamental interactions in terms of the Standard Model of elementary particle physics, which heavily builds on this idea of "gauge invariance".

As the thread starter, one should always contribute something yourself. I will do so with this quote by Hilbert
Physics is becoming too difficult for physicists.

What are your thoughts on this?

 

[coquelicot]

A spin-off discussion in this thread https://www.physicsforums.com/threa...ic-interactions-by-babin-and-figotin.1050807/
A quote from member @coquelicot (now deleted) was suggesting that most physicsts are unaware of the "real" problems with physics and that they [the physicsts] choose to ignore them.

Actually, I've edited the question and replaced "most physicists" by "many physicists", which is closer to my own thoughts.

 

[malawi_glenn]

Actually, I've edited the question and replaced "most physicists" by "many physicists", which is closer to my own thoughts.

What are the REAL problems in physics?

 

[Swamp Thing]

Emmy Noether ?

 

[Haborix]

Aside from the usual problem with discussions like these, i.e., that the physics being discussed is really a very narrow subset of physics, it seems the problem is more a lack of new physical insight combined with a lag in experimental data. Mathematicians have often come in later and made the physics richer and more coherent. That's a long-winded way of saying I'm skeptical that mathematics is the primary problem for progress in fundamental physics.

 

[vanhees71]

I think, it's just a matter of specialization. A mathematician has different interests and thus aims at answering different questions than physicists.

An example is relativistic quantum field theory. The physicists have a framework, which they find pretty "successful", because it describes all (known) elementary particles and their interactions in terms of the Standard Model with the exception of the gravitational interaction. The latter is not (yet?) implemented in this "physicists' QFT framework" in a satisfactory way.

Now nobody can deny that there are serious mathematical problems with how physicists treat relativistic quantum field theories by just introducing operator-valued distributions and mutliply them without asking, which sense this makes, because it's an ill-defined operation. The result is that almost from the beginning one gets divergent quantities. E.g., already the total energy of non-interacting fields in the vacuum state is divergent, but this is obviously due to the indefiniteness of the ordering of the field operators when writing down the formal expression for the energy density, and this divergence can be subtracted and one gets a well-defined finite energy for all physical states of the field, and usually one makes the energy eigenvalue of the vacuum state 0.

Now the physicists go on and look at interacting field theories. Not being able to solve the corresponding non-linear field-operator equations exactly they do "perturbation theory", introducing the "interaction picture". This ends in a desaster, because going beyond tree level in the corresponding Feynman diagrams, i.e., to higher-order loop corrections, which are the most interesting calculations needed to get the high-precision results in, e.g., QED (anomalous magnetic moment of electrons or muons, the lamb shift of hydrogen spectral lines,...). Also there it turned out that with the right prescription of how to get rid of the divergences, lumping them into unobservable "bare field-normalization factors, masses, and coupling constants) and working with the corresponding finite physical (renormalized) quantities, leads to a consistent scheme, which is empirically highly successful.

On this "perturbative level" of definition of relativistic QFTs one can make things quite rigorous by being more careful when mutliplying field operators by using "smeared field operators" that can be multiplied in a meaningful way. That's the Epstein-Glaser approach, and well explained in the book by Scharf, Finite Quantum Electrodynamics.

On the other hand for mathematicians all this is, of course, very unsatisfactory, because it's not clear, whether quantum field theories can be formulated in a proper clean way without all the tricks and beyond perturbation theory. This lead to the development of what's called "axiomatic quantum field theory" with various branches. The problem with this is that it seems that not much has been achieved, which goes beyond the free fields and toy models in lower space-time dimensions. There are, however, interesting results like the fact that the physicists's beloved interaction picture does not exist from this rigorous mathematical point of view, and indeed the physicists intuitively know that and first define things in a finite-volume box and at the very end take this volume to infinity in a specific way for the quantities (like S-matrix elements), where this procedure leads to plausible results.

In some sense there exist now two camps in the QFT community: the pragmatic physicists, being satisfied (maybe with a bad conciousness by some) with the "robust mathematics" leading to empirically successful descriptions of observable quantities like scattering cross sections, particle lifetimes, etc. and the "axiomatic-QFT mathematicians", which are not at all satisfied with this.

In any case, in my opinion, it's worth while to have in mind both points of view, because on the one hand, one wants to understand the phenomenology from a theoretical-physicists'-point-of view using mathematically somewhat "mediocre" techniques of calculation, but on the other hand, thinking deeper about the mathematical problems with this naive approach and maybe finding some remedy of all the problems, may lead to better theories, which one day may be also applicable to "realistic" interacting-field theories in 4 spacetime dimensions and maybe finally also lead to a satisfactory description also of gravitation (general relativity) within a unified quantum (field?) theoretical framework.

 

[coquelicot]

To begin with, I think that:
Many physicists are also excellent mathematicians, and many mathematicians are quite good in physics.

As a mathematician, I could witness that when I was learning at the university, there were many physicists in the class learning with us topology, group theory etc.

These two disciplines always gone together. At the beginning, there were no mathematicians and no physicists, they were most often both of them. Archimedes, Fermat, Descartes, Pascal, Robertval, Newton, Lagrange, Legendre, Gauss, and many, many other. In a letter of Abel, I could read once: "I'm the only one in Europe that deals only with mathematics".

Admittedly, by our days, you have to specialize in a sub-sub-sub discipline of these sciences.
Nevertheless, the links are sufficiently strong to allow mathematicians to contribute to physics, especially fundamental and original ideas (that's the domain were we are really good after all).

This being said, in addition to the reply of Vanhees71, I would like to give an example at a very, very elementary level, where most physicists are unaware of a hole (here I say most and not many). I mean the notion of capacitance matrix. When I read once elementary books in EM, I immediately felt the hole (I call that my mathematician intuition). Then I asked several physicists (even in this forum if my memory is good), and they reply to me that for them, it is evident that the charges of the conductors depend linearly on the potentials, invoking the "linearity of the equations of EM". After all, that's what is written in Jackson, Landau and Lipschitz etc. etc.
Then I tried and succeeded to answer by myself to the question for conductors in free space, which involves a uniqueness theorem by no way trivial, that I found in some lectures by Dr Konoplisky. I couldn't believe that this is not dealt somewhere, and I continued to seek in books. I finally found that Griffith was aware of the problem and of its solution (but he writes nowhere the solution), and that Jefimenko book is perhaps the only book that contains an (almost) valid proof by induction. Regarding Maxwell, I think his proof is incomplete (he proves only the multiplicative linearity, which is trivial, but not the additive linearity).
And all of that concerns only conductors in free space, but what about conductors in a mixture of dielectrics whose properties are described a tensor?
Only recently were two articles published on the question. Unfortunately, in some aspect, they are misleading (I am currently writing an article where I develop the theory in a general framework, even if it is an extremely modest contribution to physics.)
The point I wanted to illustrate is that holes exist even at such an elementary level. How is it possible that elementary physics has not been duly formalized, akin to what the Bourbaki school did in mathematics ?
I would like to cite the problem of microscopic to macroscopic EM field, where I feel that something big has been missed as well; but I am currently trying to study the books of Robinson and De Groot who have dealt extensively with the subject. So, I will abstain.

Edit: actually, Jefimenko does provide a valid proof for the elastance matrix, but assert without proof that it is invertible (its inverse is the capacitance matrix). Nevertheless, the tools he provides are sufficient to establish the capacitance matrix as well.

 

[malawi_glenn]

Thank your for this contribution @coquelicot
Though the notion of a capacitance matrix is not that important for physics as a whole.
What are the "real" problems in physics? Can these problems we solved solely by mathematicians? What are your thoughts on that?

 

[coquelicot]

Thank your for this contribution @coquelicot
Though the notion of a capacitance matrix is not that important for physics as a whole.
What are the "real" problems in physics? Can these problems we solved solely by mathematicians? What are your thoughts on that?

Well, many high level problems are well known (see Vanhee71, and all the insight articles of this forum). I don't think these problems can solely be solved by mathematicians, because, as I said above, physicists are actually mathematicians as well, sometimes at a very high level. I say that it is possible that one day, a pure mathematician propose an original idea that proves to be excellent and fruitful for the development of physics.
Obviously, the unification of GR with QFT is one of the most fundamental problem in physics. But I disagree with you about the fact that "smaller" problems (and even very small ones like the capacitance matrix) should be neglected.

 

[malawi_glenn]

Still this "sloppy" treatmeant of operator-valued distrubtions have given us some of the most accurately known physical models which are tested to extreme high precision. Is it then a "problem" that such mathematical operation is not rigoursly defined? Or perhaps physicsts were just lucky... ?

the unification of GR with QFT is one of the most fundamental problem in physics.

We might be working with the wrong premisse: that it IS possible to unify these two frameworks at all. This might be wrong.

 

[coquelicot]

In addition to what I said above, I would like to point two possible problems of the physics today. I'm not sure I am right, and I am not sufficiently good to defend these ideas. So, please, just consider them as "feelings" from a non professional physicist.

• The use of Lagrangian and the action principle as a "magical tool" to produce theories. As a person who try to read a lot of articles, I feel that there are as many physical theories as there are Lagrangians. Everyone can "tinker" a Lagrangian to produce (almost :-) ) all what he wants, and propose a new theory. Again, I may well be wrong, but from the exterior side, this looks like a jungle of Lagrangian derived theories, almost magical, hiding the real physical concepts.
• 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

 

[coquelicot]

Is it then a "problem" that such mathematical operation is not rigoursly defined? Or perhaps physicsts were just lucky... ?

We all know that they are not "just lucky". I think that yes, there is a problem because mathematical operations must be rigorously defined.
Fortunately, historically, mathematicians have always succeeded in establishing on rigorous bases the ideas of their physicist colleagues.

Examples are:

• distribution theory, that was used by Heaviside and Dirac and established rigorously by Schartz.
• path integral of Feynman that was established by specialists of integration theory.
• General relativity where Einstein was helped by Levy-civita, Grossman and in some sense by Minkowski, Hilbert and Noether.

 

[malawi_glenn]

, there is a problem because mathematical operations must be rigorously defined.

Can you give an example what consequences of this problem give, other than an itch in mathematicians brains? E.g. practical results

path integral of Feynman that was established by specialists of integration theory.

This is off topic, but do know of any good path integrals for mathematician style textbook?

 

[BillTre]

To me, not a physicist, physics seems to be doing OK.

A quote from member @coquelicot (now deleted) was suggesting that most physicsts are unaware of the "real" problems with physics and that they [the physicsts] choose to ignore them.

This is why I am entering this tread.

This seems to be a common complaint in more fields than just physics.

As a biologist, I could say the same thing about origin of life issues. While basic to biology, many ignore these unresolved issues "at the base of biology" and carry on with what ever they are studying.
It is a common sense approach to dealing with issues in a complex field.

If this weren't done, generations of biologists would have wasted their careers on trying to figure this kind of thing out before the field was in a state where the subject could be realistically approached.
These are individual decisions each person makes for themselves.

 

[coquelicot]

Can you give an example what consequences of this problem give, other than an itch in mathematicians brains? E.g. practical results
This is off topic, but do know of any good path integrals for mathematician style textbook?

q1: Obviously, distribution theory allows rigorous thinking in physics. I am not sufficiently good to give you examples, but I am pretty sure that it is possible to find articles with minimal effort.

q2: Yes, for example Henstock book "A general theory of integration" (not sure this is the exact name but this is close to).

 

[coquelicot]

To me, not a physicist, physics seems to be doing OK.This is why I am entering this tread.

This seems to be a common complaint in more fields than just physics.

As a biologist, I could say the same thing about origin of life issues. While basic to biology, many ignore these unresolved issues "at the base of biology" and carry on with what ever they are studying.
It is a common sense approach to dealing with issues in a complex field.

If this weren't done, generations of biologists would have wasted their careers on trying to figure this kind of thing out before the field was in a state where the subject could be realistically approached.
These are individual decisions each person makes for themselves.

A good point.
I'm not sure that my sentence quoted by Malawi-Glenn reflects exactly what I think (it was after all a short sentence inside parentheses, written quickly and carelessly to reply to the underlying claim that physics books by mathematician are bad). My opinion is that mathematicians may be very qualified to discover flaws and holes inside physics, and even to propose new and original theories.

 

[malawi_glenn]

q1: Obviously, distribution theory allows rigorous thinking in physics. I am not sufficiently good to give you examples, but I am pretty sure that it is possible to find articles with minimal effort.

That is ok, but don't you think that physics has "enough" rigorous thinking considering all the success physics has made so far? Not sure why there is an urgent need for making sense of operator-valued distributions will lead to any practical result. Let's see if someone can chime in.

I'm not sure that my sentence quoted by Malawi-Glenn reflects exactly what I think (it was after all a short sentence inside parentheses, written quickly and carelessly to reply to the underlying claim that physics books by mathematician are bad)

Which is why I created this thread, to have a better place to discuss these matters.

 

[fresh_42]

Physics is about nature, mathematics is about philosophy. Nevertheless, they developed pretty much in parallel, especially after the physicist Newton (Or was he a mathematician?) and the mathematician Leibniz (Or was he a philosopher?) discovered differential calculus. When I read about the history of these sciences between the seventeenth century and the great eruptions in both fields at the beginning of the twentieth century, then nobody ever really made a difference: Gauß, the Bernoullis, the French mathematicians, they never made a distinction. Calculus and linear algebra seemed to be just the language to describe and solve physical problems.

Things changed a bit in the early decades of the twentieth century. Physics had its determinism crisis, mathematics its logical crisis. Whereas the issue in physics isn't really settled until today, as you can see in our QM interpretation forum, mathematicians shrugged over Russell and Gödel and proceeded as usual. The real switch came with Bourbaki when mathematics was all of a sudden written in patterns [: definition - example - lemma - proposition - theorem - corollary - example :|]) instead of prose. This makes a real difference today. I call it the Bourbakian transition. It also led to the fact that mathematics nowadays follows the route of inventing some crude environment by making up some technical definitions, often looking as if they were artificial and at random, and then playing around with them like a boy does with his toy trains.

This decoupling of physics and mathematics didn't serve either in my opinion. Maybe Noether was the last big result of both fields together.

 

[coquelicot]

The real switch came with Bourbaki when mathematics was all of a sudden written in patterns [: definition - example - lemma - proposition - theorem - corollary - example :|]) instead of prose. This makes a real difference today. I call it the Bourbakian transition. It also led to the fact that mathematics nowadays follows the route of inventing some crude environment by making up some technical definitions, often looking as if they were artificial and at random, and then playing around with them like a boy does with his toy trains.

I agree about everything you said but this claim. Of course, the pattern "definition, theorem , problem" was not invented by Bourbaki. This is the merit of the ancient Greeks to invent it, and they come to it by the way of philosophy: the idea of the philosopher Thales was that in order to reach the knowledge, one has to start with the clearest principles, and to deduce from them less clear, or totally unclear, principles, in a deductive way. This method set by the ancient Greeks, and marvelously illustrated in Euclid's elements, is one of the greatest progress, if not the greatest, for the mankind.
On the other hand, the merit of the Bourbaki school is that they have thought and established mathematics on deep and solid bases.
I would like to see such a revolution in physics.

 

[Haborix]

I would like to see such a revolution in physics.

Let's not. There are a few mathematical physicists and applied mathematicians running around. That's enough. Do you want to Bourbaki-ize chemistry? How about biology? It seems like an incredible waste of time in an experimental science.

 

[coquelicot]

That is ok, but don't you think that physics has "enough" rigorous thinking considering all the success physics has made so far? Not sure why there is an urgent need for making sense of operator-valued distributions will lead to any practical result. Let's see if someone can chime in.

The situation can be compared to the situation of infinitesimal calculus in mathematics before the development of "limits" by Cauchy. Mathematicians had reached a high level in infinitesimal calculus, but everthing was rather intuitive and not rigorous. Near this time, Abel complained that most reckonings in this domain are not valid, and that Cauchy is the only man that knows how to write mathematics (he himself invented several series convergence methods to improve the situation). Because of that, many results were erroneous. Thanks to the revolution of Cauchy, everything became clear and rigorous, and mistakes were eliminated from the domain. The same can be said about everything in physics that is based on intuitive principles whose limits have still not be determined. Eventually, this will necessarily produce erroneous statements and theoretical mistakes, if this is not already the case. Not everything can be easily checked by experiments.

 

[coquelicot]

Let's not. There are a few mathematical physicists and applied mathematicians running around. That's enough. Do you want to Bourbaki-ize chemistry? How about biology? It seems like an incredible waste of time in an experimental science.

Bourbaki a waste of time? are you serious?

 

[Haborix]

Bourbaki a waste of time? are you serious?

For physics, yes.

 

[coquelicot]

For physics, yes.

This precisely shows that this revolution has to come, in order for people
like you realize how necessary it was.

 

[coquelicot]

For physics, yes.

Let try a test: please, explain to me what is the macroscopic E.M field, and why a surface charge appears when a dielectric is exposed to an exterior field, and why a volume charge appears inside the dielectric when there is a div field. Elementary question isn't it? you have probably learnt that at your beginning. But think well, think very well.

 

[fresh_42]

I agree about everything you said but this claim. Of course, the pattern "definition, theorem , problem" was not invented by Bourbaki.

That does not change the fact that mathematics was written in prose before Bourbaki, cp. Kurosh, van der Waerden, Courant, et al., and written in the pattern above after Bourbaki, cp. Lang and basically every modern textbook. Their explicit goal was to deduce and present mathematics on an axiomatical base. That wasn't the case before, possibly except for the ancient Greeks. Well, you could note that the idea was based on Hilbert's program, but Bourbaki turned the idea into books.

 

[coquelicot]

That does not change the fact that mathematics was written in prose before Bourbaki, cp. Kurosh, van der Waerden, Courant, et al., and written in the pattern above after Bourbaki, cp. Lang and basically every modern textbook. Their explicit goal was to deduce and present mathematics on an axiomatical base. That wasn't the case before, possibly except for the ancient Greeks. Well, you could note that the idea was based on Hilbert's program, but Bourbaki turned the idea into books.

But the fact that it was sometimes written in prose did not reduce the rigor of the exposition, nor its beauty. Have you read the "Disquisitiones" of Gauss?

 

[coquelicot]

That does not change the fact that mathematics was written in prose before Bourbaki, cp. Kurosh, van der Waerden, Courant, et al., and written in the pattern above after Bourbaki, cp. Lang and basically every modern textbook. Their explicit goal was to deduce and present mathematics on an axiomatical base. That wasn't the case before, possibly except for the ancient Greeks. Well, you could note that the idea was based on Hilbert's program, but Bourbaki turned the idea into books.

And here is a page of Van der Waerden book you cite as an example, demonstrating that you are wrong in this case.

 

[fresh_42]

Have you read the "Disquisitiones" of Gauss?

Only took a look (after download).