Untill now most of the posts inthese blog have been expository. It is time for an expeculative one. I´ll try to give a few musings about the role of math in modern physic.
Before anything else I must say that it is totally ridiculous that while theoretical physics goes into more abstract mathemathics graduate programs of the universities goes towards less level in maths in favour of informatic skills. I don´t meanthat the abbility to program would be not important, but I guess a physics student can learn to program anywhere else and t it totally unnecsary to cope an asignature for that purpose. If somene thinks that the informatic in a physics or mathemathic faculty is anything special all I can say is that I learened to program out of any faculty and when I needed tto pass an exam (in the math faculty) I got a 10 note althought I didn´t go to the classes of the asignature, didn´t study a single minut of it, althoguth when I did the exam I had not programmed a single code line in C (the actual language asked in the asignature) for more than a year. So I am certain that not studing a math asignature to study informatic is a totall mess.
Well, afther these break about informatic let´s go with the actual topic. When I was studiying physic at the faculty I had the good sense to look what people was doing inn research. And the most obvious thing that i noticed was that the level in math was very far beyond that was beeing teached in the faculty. That, together with the fact that I hate the lack of rigour of physics in some aspects lend me to ampliate my math studies. I, firstly, studied some of it by myself (begining by set topology), later I simultaneated studies in the physics and math faculties. In particular I got all the geometry and topology related assignatures. In fact I later beguined to study both geometry and topology to an upper level that the graduate asignatures. Specially I loved diferential and algebraic topology. I also studied somewhat about funtional analisis an related topics (that is, measure theory) but with not too much enthsuaism and only to be able to understand a book, the Galdindo and Pascual two volume exposure of quantum mechanics which uses the full aparatous of hilbert spaces as studied by mathematicians. However I didn´t studied another asignatures which I didn´t find necesary for physics so I didn´t end maths at that time. I prefered, instead, to study things such like nonlinear dynais, which had not place in the curriculum of my faculty. A separate chapter deserves group theory. It was inteh physics faculty curriculum and it was teached along the lines of books such as the hammermesch and similar ones. I just can say that fore someone used to the manifold theory I found so terribly poor the exposition that I had to studi by myself ll the manifold part of the theory, it was a pitty that as cause of it I didn´t get such as good familiarity with things like representations, young tableaux and similars and I needed to relearn it later.
The price to study all these math was that I didn´t get preciselly a brilliant expediente in physicst. I never have got disapointed with that fact because I have found a lot more usefull to learn modern math that having a good and deepd knowledge of such things as electronic, optics, nuclear physics and such that.
But said all these in favour of maths I must point also some negative. The spirit with math and physics at the end of the seventies and the beguining of the eigthies seemed to be that it was neccesary to recover the lost time and that physic could win a lot using the modern math. I totally agree that formulation of general gravity in the old fashioned way, i.e. the tensor calculus of Levy-Civita is a total mess. The language of difernetial manifolds makes a crucial diference at the level of understanding the physical ideas behind general relativity.
Somewhat diferent for me is the formulation of gauge theories as conections in principal fibre bundles. Ok, it is a good tool for actual calculatons of monopole or instanton solutions, but personally I don´t see that it gives too much physical insight, if any, which you couldn´t get in the traditional formulations common to phyisicans. I have not gone too much into the moduli space stuff, beyond that teached in the string theory books, so I can´t say anything about it, butwith that exception I personally don´t find find as fastanstic these formulations as many people seems to think.
Well, in the previous two parrafes I have treated mainly the reformulation of stablished theories in a new language. But that is not all what physics could expect to gain from maths. Afther all the very born of physic such as we know it is intimately related to the born of infinitesimal calculus. The Newton laws simply couldn´t have existed if calculus wouldn´t have beeen created first. A few centuries later there was another physical theory whcih required of a, by the epoch, new area of maths. I speak, of course, of general relativity. Albert Einstein didn´t like when Minkowsky reformulated his special theory of relativity in geometrical terms. But without that reformulation it would have been probably imposible to have reached the formulation of general relativity in terms of tensor calculus, i.e. diferential geometry. That was the second case in history where a radically new branch of math played an important role in physics. But for around tw centuries physic could work with the stairght develoments of the math which had raised it to the existence, calculus. The third case where a new branch of math played an important role in physics was the matrice formulation of quantum mechanics.But said these the truth is that the schröedinger formulations in terms of wavefuntions and diferential equations was mcuh more important. It could be said that hilbert spaces, which for many old fashioned physics is not much more that a combination of linear álgebra and sturn liouville theory, play a crucial role in quantum physics. Well, sure, but still most introductory books in quantum mechanics don´t actually explain what a hilbert space is (for a mathematician taste, I mean).
The next big even where math was crucial for physical developments came from the hand of Murray Gellman and his "eight fold way". The aplication of group theory to make sense of the hadronic zoo had a crucial practical impact. I am no sure of how important group theory was before that. Now most people like to relate the Lorentz group to quantum field theory in an absolutely crucial way. But I guess that in fact It didn´t play a very important role and that the diferential equationsprocedure was mos relevant in the development. I.e. People had the klein gordon equation, and later the Dirac equation, and separatelly the Maxwell equations for electromagnetsim. The fact that they were related to spin 0, 1/2 and 1 representations of the Lorentz group was probably something very secondary. It was not until the introduction of grout theory in the flavour stuff of haronic physicis, and later in yang mills theories that group theory was realised as something important and usefull, but my particular viewpoint is that is importance is somehwat exagerated. In particular I think that grout theory allows some quick calculations that let people to play not as much atention to some aspects of the theories and probably something is lost in the process.
With the rise of string theory modern maths became crucial. At the begining math made a diference. Most physics didn´t know modern math and simply couldn´t follow the results, less to say to participate in the developent. But that times went and now in greater or less extend everybody is familaar with modern maths. These means that wht decides if some can make important contributions to string theory (or other aproachs to quantum gravity) depend more in the usual physical intuition and less in familiarity with abstract math. Althougth I must say that I am very skeptic about how appropiately some of the string theorist have learned modern math. Undoubtly Witten did it (his field medaill proves it) but not everybody is Witten.
I have invoked the name of Witten and I must say some more things about him. Altought some of his works are of aundoubtly physical utility many of them are mostly mathematical. For example topoligcal field theories (which I learened beofre string theory, remember, my faouvourite branch of maths was topology ;-) ) are mainly an aplication of the path integral to a topological problem. Althought TFT are a beatifull theory it´s physical utility somewhat dissapointed to me. But TFT´s are a somewhat special topci, what about the rest of physics? I am begining to belive that people still are in the initial beief that by simply appliying new branchs of math they would automathically get new physics. And it is not working, as somewhat would expect. In these point I clearly disagree with Lubos Motl. Le´ts go with an example. algebraic gemoetry. People didn´t learn algebraic gemetry and sudenlly decided to searchwhere to use it. It worked somewhat in the reveres direction. There was a proble, to make proper sense of compactifications in orbifold gometries and reomve some kind of singularities. And them algebraic geometry, and blowing up of singularities came to the rescue. B.T.W. if am skeptic of how properly physicans have learend some branchs of math, topolgy, difeential geometry, my doubts increase when we go to algebraic geometry (by algebraic geometry I understand it inhis full formalism, varieties in arbitrary fields, scheme theory, and not only the special case of complex diferential geometry, or it´s still most reduced subjecto of Rienman surfaces) . By now my understanding of it comes from what is teached in string theory books and some clarifications that a friend of me, who is doing a thesis in number theory (which requires hughs amounts of algebraic geometry) did to me of some aspects.
Anyway, the thing is that physics have almost exausted all the branchs of mathematics (they are using even some absturse areas such as p-adic numbers in things such as p-adic-or adelic-strings or topological geometrodynamics). I don´t think that looking towards the few remote areas which, maybe, still have not been exploited will make any diference (particularly I seriously doubt that category theory would be something which will give anything relevant to physics). Perhaps the only exception would be some areas of math which are somehwat beyond the usual scope of theoretical physicians, nonlinear systems and complexity (in a broad sense which covers things such as markov/stocasthic process, grahp theory, etc) could become relevant. These rise agian the role of group theory. Group theory is importan when symmetry is the key ingrediente. But in nonlinear sciences symmetry is not such important. Maybe playing more atention to nonlinear sciences could force to an small change of paradigm to theorethical phyisics or maybe not.
A place where certianly complexity should play a role (I am aware that some papers ahve already gone trought these line) is in the landscape problem. I totally agree with the skeptics about the utilitie of the anthropic principle. If there is no way to remove the landscape the apropiate tools to investigate it whould be complexity theories and not any kind of anthropic principle. I have had the luck to teach math to people working in biology and for sure all their lines of reasoning are much more addequat to trate all the landscape questions that that stupid anthropic principle. But I hope that someone would find, and if possilble soon, a diferent solution to the cosmological constant problem that the landscape idea.
But with that possible exception I gues that it is time for physicist triying to actually do physics and not relay on looking into mathemathcians to search for their new "revolution". Afther all there was a short epoch where the game worked in the opposite direction. I am talking about Dirac and it´s extensevely used "Dirac´s funtion" which when formalised by the mathematicians became the distirbution theory and about the Feynman path integrals whose proper formalization suposed a lot of hard work in measure theory. Physicians could use these math in the non formal treatement which mathemathicians developed later with a lot of success. In fact many still do it and don´t care at all about the more sophisticated versions. B.T.W. I mentined before that the fibre theory formulation of gauge theories wasn´t, in my opinion, too relevant for physics. But it has been very usefull for mathemathicians (there are a lot of docotrants doing his thesis about those topics). Also, seemengly, applies with some aspects of stiring theory. But I am not sure that these cases are the same that dirac delta funtion or path integrals. Another important aspect is whether most mathemathicians could understand relatively well classical physics and evenquantum mechanics (and certainly general relativty) but I seriously doubt they understand properly gauge theories, the QFT aspects of it, or string theory so the relation, or relevance of the interplay betwen these theories and maths, from the mathemathicals viewpoint is more obscure.
I have not been as organized in the exposition of the ideas that I wanted to express as I would have liked, but hope it sitll there is some coherence in the post.