I decided to start living dangerously right away and write my opening post on a “landmine field” topic, namely a semantic discussion with little personal acquaintance on its history. I won’t apologise, though – may the flames come to me! ๐

(Of course, I *do* apologise in advance for historical imprecisions in what follows, and welcome any corrections…)

The meaning of the neologistic title above is that I’ll endeavor into giving an answer of my own to the question:

*“What is Mathematical Physics?”*

Since this is an area I claim to be part of as a researcher, I’m supposed to know what I’m doing for a living, right? Ahn… wrong, it seems. For I don’t (nor should) propose this post to be the final nail into the coffin.

Anyone who has lived for a while inside the theoretical physics community (and, for those who haven’t, Wikipedia will do, though with a bias of its own) is lileky to have met several definitions of the term above. I’ll list the ones I’ll address here below:

- Mathematical Physics = study of
*mathematical methods* employed in the analysis of *physical phenomena*;
- Mathematical Physics = study of
*problems in theoretical physics* emphasising *mathematical rigour*;
- Mathematical Physics = pursuit of
*new mathematical results and concepts* related to problems in theoretical physics *by means of physical intuition*.

Definition 1 is probably the oldest, and could be applied as such to practically all theoretical physicists and mathematical analysts between Newton and Einstein, going through Leibniz, Bernoulli, Euler, Lagrange, Laplace, Legendre, Fourier, Dirichlet, D’Alembert, Poisson, Jacobi, Green, Gauss, Stokes, Neumann, Kelvin, Maxwell, Riemann, Poincarรฉ, Lorentz, Minkowski, Hilbert, Weyl, Wigner, Whittaker, Lyapunov (non exaustive list!)… Point to be made: both disciplines were *practically indistinguishable* and evolved *practically together* during this long period, as they were largely *born together* around Newton’s times (*Principia*). Although mathematical analysis started as an independent discipline really in the 19th century, mainly with the work of Cauchy and Weierstrass, the true “divorce” only came in the 20th century, after some of the creators of modern functional analysis (Hilbert, von Neumann) having laid the mathematical foundations of quantum mechanics, when many mathematicians, under the influence of the Bourbaki group, started systematising mathematics and, as a result, seeing it *as an end in itself. *Hence, it perhaps can be said that this definition, whether whenever valid or not, is *no more*.

Definition 2 was born with this name a bit after this divorce, when some physicists and mathematicians tried to apply the new mathematical tools and rigorous grounds developed until around WWII to tackle some hard fundamental questions that could not be addressed either by direct calculation, either by the approximation methods developed so far (perturbation theory, etc.). Among these, we may cite the problems of *stability of the solar system*, *stability of atoms and molecules*, the *mathematical meaning of renormalization in perturbative quantum electrodynamics* and the *nature* *of gravitational collapse in general relativity*. The line of attack which led to a total or partial success in answering these questions was to make physically reasonable but sufficiently robust hypotheses about the principles underlying these phenomena (or simplified versions of it) and then apply only *mathematically rigorous* methods. It often came as a surprise that, for such an approach to succeed, not only the hypotheses but also the methods employed had to comply with physical intuition. This is starkly true in the case of the proof of stability of matter given by Lieb and Thirring, where Heisenberg’s uncertainty principle and Pauli’s exclusion principle play a pivotal role. Thus, Definition 2 may be summarised in the following aphorism:

*“Whereas a theoretical physicist makes calculations, a mathematical physicist proves theorems.”*

As an apology to such a definition, I recommend Joรฃo C. A. Barata’s writing (in Portuguese) on the very theme of this post.

Definition 3 is the most recent, being advocated specially by Ludwig D. Faddeev. It characterises, for instance, the physical feedback to mathematical developments such as, for instance, the theory of bundles and connections forged by Cartan, Ehresmann, Chern, Weil and others, which have shown to form the basis of (classical) gauge theories which were developed independently in physics by Weyl, Yang and Mills, and also the theory of moduli of varieties in modern algebraic geometry, specially in the context of quantum field theory and, more recently, string theory. To be more precise, the extrapolation of these mathematical concepts to areas well beyond their original scope, guided by physical ideas, albeit also invoking other, mathematically ill-founded objects such as (certain) functional integrals, ended up unraveling *both* new physics and new mathematics. This powerful and highly seductive means of discovery have speeded up the flow of new results in both areas to an unprecedented rate nowadays. To the point that some scholars, such as Eric Zaslow, also proposed the neologism *“Physmatics”* I used (i.e. borrowed from him!) in the title of this post as a more appropriate name for this interdisciplinary trend.

Before saying more about the third definition, I want to recall as well the closely related term *“Physical Mathematics”*, which is found to be defined as:

- Physical Mathematics =
*realistic* mathematical modelling of scientific experiments ๐ ;
- Physical Mathematics = Mathematical Physics (which one from the above three? See below! ๐ );
- Physical Mathematics Mathematical Physics (so…? ๐ ).

I have no idea of how representative is the percentage of the community of physicists which employs Definition 1 – it *is* a set of measure greater than zero among mathematicians, though, as it can be seen here. I’ll make no further comments on this definition. ๐ฟ

Now I come with another teaser: unlike “Mathematical Physics”, the term “Physical Mathematics” *does* seem to have a more agreed upon definition *among physicists* (such as, for instance, Albert Schwarz), which is *nothing more than Definition 3 of “Mathematical Physics”* when restricted to new *mathematical* results and concepts. What the heck is *this*? One may ask…

In this form, one can trace the origins of “Physical Mathematics” also way back to Newton, but, as far as *strictly* mathematical results are concerned, this path of discovery was first stepped by Riemann and F. Klein in the context of Riemann surfaces. Riemann’s original “proof” of his celebrated mapping theorem was based on reasoning around electrostatic lines of force, which is nothing more than a *physical* way of building conformal mappings (this is related to the fact that the electostatic potential is an *harmonic* function). This line of reasoning was further pursued by F. Klein. A modern exponent of “Physical Mathematics” in this sense is Edward Witten (as acknowledged by the maths community, which gave him the Fields medal), whose most important results include an approach to Morse theory and the positive mass conjecture in general relativity by supersymmetry, the finding of new topological (Floer, Seiberg-Witten) invariants of 3- and 4-manifolds by functional integral methods and, very recently, the use of electromagnetic duality in the understanding of the Geometric Langlands Program. This brings us to another important question, which serves to pinpoint what, to me, seems to be the answer to the first one:

*“Which is the ***aim** of ‘Mathematical Physics'”?

It’s here, though, that our waters become really muddy, for one may broaden or collimate the answer to the second question in the absence of an *a priori* answer to the first one. My view here is that mathematical physics as a science (or at least as a way of doing theoretical physics) should ultimately test *physical principles*, or, in maths jargon, *hypotheses* – to test models is really an *intermediate* step, for any model is an *approximate* description of nature, no matter how complex, and, at the same time, a *particular instance* of certain principles. Both are bounded by experimental knowledge, but principles can be more easily extrapolated to possibly unrealistic models which are nevertheless more amenable to precise mathematical scrutiny. So, which is the boundary line between Definitions 2 and 3 for *both* “Mathematical Physics” and “Physical Mathematics”? We have boiled it down to *two* matters:

- The demand of mathematical precision on the methods used;
- The demand on scientific aims pursued.

Since we’ve seen that success for the approach advocated by Definition 2 of “Mathematical Physics” is dictated by an underlying physical intuition, this shows a substantial overlap with Definition 3 – the true difference lies precisely on the first matter: which comes first? Rigour or Intuition? Does the former imply (partial) sacrifice of the latter? At this point, we’re faced with an Apollonian/Dionysian debate with no definitive answer. And one is faced with a choice – to end this perilous journey with a fitting and perhaps disappointing (haha!) hiatus, I’ll just say that I stick to Apollo. Or, perhaps, we should just drop the discussion altogether and use a melting pot – “Mathephysmatics”??? Perhaps this would be at least a nice future name for my blog…