I’m 25 years old and have recently started the second year in my PhD. I’m from Brazil, where most people start their PhDs at 24 and finish them at 28, so I’d say I’m following the classical schedule quite closely. I’ve graduated in physics and did my masters in a field of mathematical physics called quantum chaos. I’m currently working in the same field. I had my first article almost ready from my masters, but then found out another guy had already published something very similar.
When I identified the Problem
During my undergrad years I took courses on measure and integration and differential equations in the math department, having studied real analysis and linear algebra on my own. On several occasions I almost left physics to go to math.
During my master’s I had to take physics courses again, and they made me feel exactly the same way I felt when I was in undergrad: I didn’t understand what those people were talking about, basically because most of them were not interested in defining well enough what they were doing. People were talking about S matrices and Feynman diagrams, while I was extremely uncomfortable with the fact that no one had even defined what an inner product was; people were taking limits of series they didn’t prove were Cauchy in spaces they didn’t prove were closed and applying opperators they didn’t prove were continous to get results they didn’t prove were unique. I couldn’t figure out what the hell they were doing.
I turned myself to functional-analysis and spectral-theory books, studied a lot, and was able to make sense of some things they did, correct some logical mistakes, understand that some of those problems they were ignoring are actually very hard and at least grasp what they were doing.
I then resumed my usual undergrad behavior: I quit going to classes, studied the subject on my own and only met the class to deliver exercise lists. While most student’s lists had x pages, mine had 3 x: I needed to prove everything I was doing made sense, and this consumed a lot of time and effort, and was – probably – ignored by the teacher and useless to everyone but me. I then talked to some teachers and they said that studying the maths behind physics was very useful, but that I shouldn’t spend too much time on it, otherwise I wouldn’t be able to do research. In their words:
a physicist should know enough math to be able to do his research, and no more
I had a very hard time processing this, since 90 % of my time was devoted to study mathematics. I thought about leaving academia.
Leave physics and go to math. Do what I was reluctant to do in my undergrad. If I like math so much, why am I still working with physics? I spent almost six months thinking about this, unable to do my work, after all I was apparently a useless piece of gear in the non-Cauchy, non-continuous and non-unique mechanism of physics research.
But then I noticed even though I liked studying math from math books – that is, working on exercises and proving theorems –, I would be very unhappy if I had to prove theorems for a living. I only used mathematics to be able to clear my view regarding physics, and although it did allow my to see deep into many physics problems (“rigour clears the window through which intuition shines”), I think this clear, rigorous view might lead me to a bitter, unfruitful place inside academia: a place where I understand very well what is done, but cannot create new physics by myself.
I might just not have been born for research, only for studying. My advisor is not at all interested in math, although he does respect and see some advantages in having a student that is. My work with him is mathematically ill-defined and has a lot of programming (which I learned to like), but when I attempted to try to make it rigorous he only cared about the end conclusion and didn’t pay attention to the process. In a word, he doesn’t care about rigour, but allows it.
I would like to work exactly in the same field when I finish my PhD, but touching more profound problems which could only be accessed with the use of heavy maths – which I still cannot actually fathom.
Sometimes I feel terrible about my interest in mathematics, since it is not very well accepted between people in my field, besides being probably useless to create new knowledge. I would like to apply mathematics to physics from inside physics, that is, I like rigour to understand, but I don’t like rigour to explain (I don’t want to prove complex theorems, I want to see new problems in mathematical physics and expose – perhaps study – them). Is this possible? Is there a place for someone like me inside academia?