A PDE: Between Physics and Mathematics

This is perhaps the single most important mathematical physics papers I have ever read; I think everyone - especially (theoretical) physicists - interested in theoretical physics should read it. In fact, read it now before reading the rest of the thread:
Klainerman 2010, PDE as a Unified Subject
Abstract said:
Given that one of the goals of the conference is to address the issue of the unity of Mathematics, I feel emboldened to talk about a question which has kept bothering me all through my scientific career: Is there really a unified subject of Mathematics which one can call PDE? At first glance this seems easy: we may define PDE as the subject which is concerned with all partial differential equations.

According to this view, the goal of the subject is to find a general theory of all, or very general classes of PDE’s. This “natural” definition comes dangerously close to what M. Gromov had in mind, I believe, when he warned us, during the conference, that objects, definitions or questions which look natural at first glance may in fact “be stupid”.

Indeed, it is now recognized by many practitioners of the subject that the general point of view, as a goal in itself, is seriously flawed. That it ever had any credibility is due to the fact that it works quite well for linear PDE’s with constant coefficients, in which case the Fourier transform is extremely effective.

It has also produced significant results for some general special classes of linear equations with variable coefficients. Its weakness is most evident in connection to nonlinear equations. The only useful general result we have is the Cauchy-Kowalevsky theorem, in the quite boring class of analytic solutions.

In the more restrictive frameworks of elliptic, hyperbolic, or parabolic equations, some important local aspects of nonlinear equations can be treated with a considerable degree of generality. It is the passage from local to global properties which forces us to abandon any generality and take full advantage of the special features of the important equations.
The author, Sergiu Klainerman, essentially argues that the premature christening of 'the theory of PDE' by mathematicians in the last century was completely unnatural, based on incorrect presuppositions and expectations which has consequently lead to an almost universally pervasive misunderstanding among all physicists and mathematicians about the very nature of both mathematics and physics and therefore also a misapprehension of the goal of mathematicians and physicists.

Klainerman's vision is a call for a basic reevaluation of how exactly mathematicians and physicists should naturally view their subject based upon what they do (and historically have always done) instead of what they think they (should) do; such a change of perspective has extremely nontrivial consequences for it directly quite literally shapes not only subjects (and so conjectures, research programmes, the boundaries of fields and careers) but literally also creates conventions of what is considered to be mathematics and what is considered to be physics.

Moreover, taking Klainerman's vision seriously also means that many popular criticisms and complaints by physicists and scientists leveled at string theory, multiverse theories and so on by Woit, Smolin, Hossenfelder et al. are indeed somewhat unfair premature mischaracterizations of the natural progression over time of theories which lay between physics and mathematics.

The artificial strict boundaries by which they judge theories, imposing upon the theories a completely artificial immediate necessity of either directly conforming to being 'physics' or 'mathematics', may be a wholly misguided endeavor, one which occurs due to modernly arisen incorrect idealized definitions of physics and mathematics, which reflect the modern conventional idea of what logically these subjects should be, not what they naturally are.

Indeed, if we look at history, in particular to the universalists who were both mathematicians and physicists starting in the 17th century with Newton & Leibniz, going on to Laplace, Lagrange, Euler, Fourier, Gauss et al. all the way up to and including Poincaré, we clearly see that their mathematics is indistinguishable from their physics. Today on the other hand we have so much specialization that not only does practically no one seem to be able to tell the forest from the trees, many often do not even realize that they may be studying the exact same mathematical structure as others in other branches are doing or have already done; worse still, often when this is pointed out to them many still do not recognize that this is actually so.

Prior to my formal introduction to physics, I was naturally a classical pure mathematician at heart; despite my abhorrence for engaging in modern formal mathematics, in many respects I still am. I think it is no accident that a very productive way of looking at almost any unsolved problem in pretty much any science is by comparing a potential solution, i.e. a mathematical model, to another already solved problem and its respective mathematical model; a return to our roots, like Klainerman is proposing here, therefore sounds like music to my ears.


Science Advisor
I do like the article, perhaps I should read it again to refresh my memory, but I think you are reading too much between the lines. I am not sure if Klainerman would agree with all of what you say that he says.
Perhaps I am, but I am merely carrying the presented view to its logical conclusion in order to compare it with other views; I believe such a framing is very useful because it is, as Feynman liked to say, a way of 'taking the world from another point of view'; such an alternative point of view might turn out to be very valuable for me or others for whatever reasons.

For example, some things I'm stating in the OP, e.g. that Woit, Smolin, Hossenfelder et al. may be judging the progress of some physics programmes unfairly based on non-existent ideals of what physics should be, most definitely is my own interpretation of Klainerman's view if one is to fully take Klainerman's message to heart; I am not personally acquainted with him so I can not speak about his actual views on such matters, but it doesn't seem to be inconsistent with them either.

To be clear about this, my above interpretation of Klainerman's message isn't my actual position, but - myself being another 'believer' in ideal distinctions - actually almost the exact opposite, with my views tending to be more or less aligned with Woit, Smolin, Hossenfelder et al.; I am actively trying to challenge my own point of view in case it might actually be an incorrect view and at the same time exposing it and opening it up to criticism by posting it in this forum.
Mathematicians are a strange bunch is all I got after reading this. Math does not predict anything, all it does is state some truths based off the logical conclusions we can make from some axioms. That's all, it just states truths. No where in math does "throwing a stick" come about. It's only when someone asks a questions, "I wonder how far this rock will go if I throw it" and actually want to predict how far it will go BEFORE I measure it, will I need math. Otherwise, I just throw it and go "Wow, that's pretty far!" Do mathematicians actually think their abstractions are any different than me describing a unicorn in English? Does anyone think Hilbert Spaces exist in nature? Mathematics is useful in physics because it is the language system needed to paint the picture.


The author then states on page 23 of the PDF (301 in the book?) " THE SITUATION OF SINGULARITIES IN GENERAL RELATIVITY IS RADICALLY DIFFERENT TO THE TYPE OF SINGULARITIES EXPECTED HERE IS SUCH THAT NO CONTINUATION OF THE SOLUTIONS IS POSSIBLE WITHOUT ALTERING THE PHYSICAL THEORY ITSELF THE PREVALING OPINION IN THIS RESPECT IS THAT ONLY A QUANTUM FIELD THEORY OF GRAVITY COULD ACHIEVE THIS". Right, something that isn't stopped by mathematics. Only when we consider the physical aspect of the theory do we stop... but naturally, a mathematician has no reason to stop.

To restate my first paragraph: there is a fundamental difference between the two subjects. Physics is a predictive science. Math is a logical system built up on axioms. No different than grammar structures set forth by languages. Do we also need to set the standard between linguists and physicists as well?!

Now my view is expressed by a part of the preface of "Supermanifolds" by Bryce DeWitt- "A previous generation of theoretical physicists could function adequately with a knowledge of the theory of ordinary manifolds and ordinary Lie groups. With the discovery of Bose-Fermi supersymmetry all this changed. Nowadays the theorist must know about supermanifolds and super Lie groups. The purpose of the present volume is to provide him with an easily accessible account of these mathematical structures. Mathematicians will find much of this book incomplete and expressed in a language that they have nowadays passed beyond, but it is probably pitched about right for the average physicist."

The quote shows my feelings towards the issue because it shows what physicists do: we learn new math if it helps describe reality. The way we learn the math is molded to how it helps us describe reality. Mathematicians won't like how we present the math. (To keep up the comparison with people use "text-speak" as English to communicate with each other, a notion I'm sure a lot of linguists wouldn't like.)
It's only when someone asks a questions, "I wonder how far this rock will go if I throw it" and actually want to predict how far it will go BEFORE I measure it, will I need math.
Actually this simply isn't true; the utility of a mathematical model is often present regardless of the intent of why the model was constructed in the first place: the direct utility is that it offers a coherent description of some known concept, therefore potentially increasing our understanding of said concept.

The practice of classical pure mathematics (which confusingly is practiced today by applied mathematicians) busies itself with building models for their own sake or for the sake of exploring what kinds of models can be demonstrated to exist mathematically; very often these 'abstract' models turn out to have a reflection in the real world, whether that be in physics as a natural description of some physical phenomenon or as a model of some phenomenon in any other human endeavor.
Do mathematicians actually think their abstractions are any different than me describing a unicorn in English?
As a physicist, I will fully agree with the sentiment - especially w.r.t. physics - that engaging in the construction not of mathematical models of phenomena or of concepts but instead axiomatic or logical models of the theory of mathematics itself is far less useful, and that such abstractions should possibly be seen - just like works of art and literature - as merely abstract works of fiction.

It is easy to spot such mathematicians (or physicists engaged in the same activity of fruitless fantasizing): they more often than not cannot easily reduce their ramblings to any (semi)concrete examples which might pass for a description of something in the real world, but more importantly they do not seem to be interested in learning more from the real world in order to more accurately construct their models.

Regardless of the fact that mathematical constructs and linguistic constructs are in many cases only that - constructs - it is patently ridiculous to state or to think that all mathematical abstractions are equally useful or that they are all only as useful or as real as e.g. literature or music.
Does anyone think Hilbert Spaces exist in nature?
I think there would be many who would picture it the other way around, i.e. that it is nature which exists in or even be an abstract mathematical realm, e.g. Tegmark's mathematical universe. It seems that most people, including professional mathematicians, simply have never actually taken the time to think about such questions in depth.
Mathematics is useful in physics because it is the language system needed to paint the picture.
Most practitioners of mathematics at all levels would probably agree or have agreed at least once during their lifetimes that many things in reality are reflections of mathematical things, i.e. quantities are reflections of numbers and shapes are reflections of geometry; this point of view is quite typical and is called Platonism.
Don't laugh too hard, this statement is obviously true; we physicists often do not look beyond very surface level descriptions, sometimes even fully eschewing sophisticated mathematics in favor of solvability (e.g. small angle approximation, perturbative methods, etc).

The fact of the matter is that some mathematicians are able to take a deeper look at nature than physicists, because of their knowledge - not only of the models but also of the deeper mathematical properties underlying the models - e.g. smoothness, analyticity, holomorphicity, etc.

Studying a natural phenomenon at such a deeper mathematical level than is required for the physicists' experimental comparison often has lead to the discovery of novel mathematical objects i.e. lead to new unexpected results or fields in mathematics; many models of phenomenon once considered to be uniquely part of physics were shown to be much more generally applicable to describe many things in the world and so became instead to be regarded as applied mathematics instead.

Moreover, any newly discovered mathematical objects gained by mathematically studying physical models may or may not be reflected in nature as well, whether that be directly in the modelled phenomenon or indirectly in any other phenomenon which has some type of similarity with the modelled phenomenon, e.g. similar descriptive equations.
we learn new math if it helps describe reality. The way we learn the math is molded to how it helps us describe reality.
This is just a red herring. It is obvious that new math should be learned if it is already known and/or has already been demonstrated that some particular form of mathematics is useful for describing some (aspect of) physical phenomena, and that this new math need not be presented or explicated in the highly abstract formalist manner that mathematicians tend to use.

However, what is not obvious - and what the author is alluding to - is that older but unconventional math for physics (which might even already be known to some smaller subset of physicists, but typically remains unknown among physicists generally) may already be capable of describing some poorly understood aspects of physics in a manner superior to what the conventional math already adopted by physicists is capable of doing w.r.t. the specific subject in question; the fact that the elucidation hasn't occurred yet is then a mere accident of convention.
Mathematicians won't like how we present the math.
I could care less about what they like or do not like about our presentation of mathematics depending on what they are aiming to achieve:
- if the mathematician-in-question's goal is aligned with that of the theoretician i.e. to directly create more mathematically sophisticated models of some phenomena or concepts, based on a less conventional and/or more sophisticated analysis of said phenomena, then we should listen
- if the mathematician's goal is simply to create abstruse propositions, definitions and axioms in order to be able to generalize and/or delineate what can be considered to be the 'true' or necessary formal properties of such models generally, then we should ignore them
I want to share a passage from a pretty recent monograph by a mathematical physicist; it seems to me that the confusion of the goal or intentions of the practitioners of mathematical physics seems to be rife with widespread misunderstandings.

Both the phraseology and the intent of the author here seems to be typical, so I hope this passage is able to illustrate some of the differences between the person who identifies as a physicist first versus one who identifies as a mathematician first, independent of what his degree says:
R.W. Carroll said:
We want to make a few remarks about writing style. With a mathematical background my writing style has acquired a certain flavor based on equations and almost devoid of physical motivation or insights. I have tried to temper this when writing about physics but it is impossible to achieve the degree of physical insight characteristic of natural born physicists. I am comforted by the words of Dirac who seemed to be guided and motivated by equations (although he of course possessed more physical insight than I could possibly claim). In any case the “meaning” of physics for me lies largely in beautiful equations (Einstein, Maxwell, Schrodinger, Dirac, Hamilton-Jacobi, etc.) and the revelations about the universe presumably therein contained.

I hope to convey this spirit in writing and perhaps validate somewhat this approach. Physics is a vast garden of delights and we can only gape at some of the wonders (as expressed here via equations in directions of personal esthetic appeal). We refrain from rashly suggesting that nature is simply a manifestation of underlying mathematical structure (i.e. symmetry, combinatorics, topology, geometry etc.) played out on a global stage with energy and matter as actors. On the other hand we are aware that much of physics, both experimental and theoretical, apparently has little to do with equations.

Many phenomena are now recognized as emergent (cf. [594, 860]) and one has to deal with phase transitions, self organized criticality, chaos, etc. I know little about such matters and hope that the mathematical approach is not mistaken for arrogance; it is only a dominant reverence for that beauty which I am able to perceive. The book of course is not finished; it probably cannot ever be finished since there is new material appearing every weekday on the electronic bulletin boards. Hence we have had to declare it finished as of April 10, 2005, it has reached the goal of roughly 450 pages and includes whatever I conceived of as most important about the quantum potential and Bohmian mechanics.

I have learned a lot in writing this and there is some original material (along with over 1000 references). Philosophy has not been treated with much respect since I can only sense meaning in physics through the equations. The fact that one can envision and manipulate “composite” and abstract concepts or entities such as EM fields, energy, entropy, force, gravity, mass, pressure, spacetime, spin, temperature, wave functions, etc. and that there should turn out to be equations and relations among these “creatures” has always staggered my imagination. So does the fact that various combinations of large and small numbers (such as ##c##, ##e##, ##G##, ##h##, ##H## etc.) can be combined into dimensionless form. Abstract mathematics has its own garden of concepts and relations also of course but physics seems so much more real (even though I have resisted any desire to experiment with these beautiful concepts mainly because I seem to have difficulty even in setting an alarm clock for example).

We will present a number of ideas which lack establishment deification, either because they are too new or, in heuristic approaches of considerable import, because of obvious conceptual flaws requiring further consideration. There is also some speculative material, clearly labeled as such, which we hope will be productive. The need for crazy ideas has been immortalized in a famous comment of N. Bohr so there is no shame attached to exploratory ventures, however unconventional they may seem at first appearance. The idea of a quantum potential as a link between classical and quantum phenomena, and the related attachment to Bohmian type mechanics, seems compelling and is pursued throughout the book; that it should also have links to quantum fluctuations (and information) and Weyl curvature makes it irresistible.

There is no attempt to present a final version of anything. For a time in the 1990's the connection of string theory to soliton mathematics (e.g. in Seiberg-Witten theory) seemed destined to solve everything but it was only preliminary. Some of this is summarized nicely in a lovely paper of A. Morozov [662] where connections to matrix models and special functions (and much more) are indicated and the need for mathematical development in various directions is emphasized (cf. [191, 662] and references there along with a few comments in Chapter 8). The arena of noncommutative geometry and quantum groups is personally very appealing (see e.g. [192, 258, 589, 615, 619, 620]) and we refer to [157, 158, 159, 160, 161, 162, 260, 580, 581, 582, 722] for some fascinating work on Feynman diagrams, quantum groups, and quantum field theory (some of this is sketched in Chapter 8). We have not gone into here (and know little about) several worlds of phenomenological physics regarding e.g. QCD, quarks, gluons, etc. (whose story is partly described in [997] by F. Wilczek in his Nobel lecture).

We have also been fascinated by the mathematics of superfluids à la G. Volovik ([968, 969]) and the mathematics is often related to other topics in this book; however our understanding of the physics is very limited. There are also a few remarks about cosmology and here again we know little and have only indicated a few places where Dirac-Weyl geometry, and hence perhaps the quantum potential, play a role.

As to the book itself we mainly develop the theme of the quantum potential and with it the Bohmian trajectory representation of QM. The quantum potential arises most innocently in the Bohmian theory and the Schrödinger equation (SE) as an expression ##Q = -(\hbar^2 /2m)(\Delta|\psi| / |\psi|)## where ##\psi## is the wave function and ##Q## appears then in the corresponding Hamilton-Jacobi (HJ) equation as a potential term. It is possible to relate this term to Fisher information, entropy, and quantum fluctuations in a natural manner and further to hydrodynamics, stress tensors, diffusion, Weyl-Ricci scalar curvatures, fractal velocities, osmotic pressure, etc.

It arises in relativistic form via ##Q = (h^2/m^2c^2)(\square|\psi| / |\psi|)## and in field theoretic models as e.g. ##Q = -(h^2/2|\Psi|)(\delta^2|\Psi|/\delta|\Psi|2)##. In terms of lapse and shift functions one has a WDW version in the form ##Q = h^2NqG_{ijkl}(\delta^2|\Psi|/(\delta q_{ij}\delta q_{kl})## (##q## is a surface metric). There is also a way in which the quantum potential can be considered as a mass generation term (cf. Remark 2.2.1 and [110]) and this is surely related to its role in Weyl-Dirac geometry in determining the matter field.

In purely Weyl geometry one can use the quantum matter field (determined by ##Q##) as a metric multiplier to create the conformal geometry. In Chapter 7 we give a resume of various aspects of the quantum potential contained in Chapters 1-7, followed by further information on the quantum potential. We had hoped to include material on the mathematics of acoustics and superfluids a la Volovik and thermo-field-theory à la Blassone, Celeghini, Das, Iorio, Jizba, Rasetti, Vitiello, Umezawa, et al. (see e.g. [122, 231, 493, 519, 912, 913, 963, 964, 965]). This however is too much (and no new insights into the quantum potential were visible); in any case there are already books [286, 950, 968] available.

Aside from a few short sketches we have also felt that there is not enough room here to properly cover the nonlinear Schrödinger equation (NLSE). (cf [223, 280, 292, 311, 312, 313, 413, 691. 692, 693, 1028, 1029, 1030, 1031, 1032, 1033, 1034]).We have had recourse to quantum field theory (QFT) at several places in the book, referring to [120, 457, 935, 1015] for example, and we provide in Chapter 8 a survey article written in 2003-2004 (updated a bit) on quantum field theory (QFT), tau functions, Hopf algebras, and vertex operators.

Although this does not touch on the quantum potential (and delves mainly into taming the combinatorics of QFT via quantum groups for example), some of this material seems to be relevant to the program suggested in [662] and should be of current interest (cf. [191, 192] for more background). We recently became aware of important work in various directions by Brown, Hiley, deGosson, Padmanabhan, and Smolin (see [741, 1018 1019, 1022, 1023, 1024] and cf. also [1020, 1021]).
I'll respond to your post directly when I have more time, but I think this interview with Michael Atiyah has a few bits (there are more) that are related to this thread:

The last one (one on magnetic monopoles) is an interesting one.

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