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Background

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.

Attempted Solution

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.

Actual Question

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?

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    Just in case someone is thinking about suggesting migrating this to Physics: it's not on topic there. – David Z Nov 13 '16 at 4:48
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    The famous representation theorist Harish Chandra did a thesis in physics at Cambridge, under the direction of Paul Dirac. – Dan Fox Nov 13 '16 at 10:10
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    The statement that mathematics is "probably useless to create new knowledge" is unbecoming of a physicist. – Dan Fox Nov 13 '16 at 10:10
  • @DanFox I also found this statement absurd when I began studying physics... Somehow they made me believe it. – QuantumBrick Nov 13 '16 at 15:09
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I have a somewhat similar background: I primarily studied physics but also did roughly a bachelor in mathematics (a “Vordiplom” to be precise). I never liked regular theoretical physics much because the mathematical background was never properly explained, if it was mentioned at all. I did my master’s-thesis equivalent and PhD in complex systems / chaos theory.

So far, my mathematical background was an enormous benefit to me. I published three papers which clearly wouldn’t have been possible without it, and in the rest of my work, it often helped me to understand things much quicker. The main impact of mathematics on my work is that it allows me to identify and formulate problems, draw connections, know the keywords to find the relevant literature and understand it. While I did prove things at times, this was only the icing on the cake – and I would consider most of my proofs not very challenging. The main exception was the proof of a number-theoretical statement that I had already confirmed very well empirically. If I had not eventually found the proof, I would probably have consulted the next-best expert on number theory at my university. Still, I am more proud of finding the right statement to prove than of proving it. Almost everything that I proved is published.

So, I really do not see a problem with your interests. There is almost certainly a workgroup that is interested in the same questions, though it does not seem to be the one you are in right now. However, if your supervisor accepts your mathematical inclination, they may also be able to connect you to the people who care about these issues to give you feedback.

Bringing a new background to your workgroup is an asset that is very likely to make you see new connections or identify and solve new problems, which can eventually lead to new physics. You may need a proof on the way, but I would not worry that this will consume most of your time. At the end of the day, even most mathematicians do not spend all day at their desks trying to prove something, but also have to understand existing concepts, find connections, and so on. And should you stumble upon something that you cannot prove, you can still collaborate with mathematicians or leave it as an open problem/conjecture.

Another perspective could be to simply establish a mathematical framework for what your workgroup does (which then may contain open questions for you or others to fill). Depending on the relevance and extent of what we are talking about, this may already be worthy of a PhD or even enough work for an entire research career.

Finally, some random asides:

  • You probably are at a stage of your career and a field, where few people care about whether you are actually a mathematician or a physicist.

  • Beware of the impostor syndrome. If your master’s thesis could be turned into a paper, that’s already a good start. That somebody else managed to publish it earlier, is bad luck for you, but still shows that you produced publishable work.

  • If you are doing a lot of programming work, it can be very helpful to understand more about the methods you are using. Moreover, you may find yourself developing new numerical methods – where a mathematical background is extremely helpful.

  • a physicist should know enough math to be able to do his research, and no more

    While there is some truth to this for some areas of physics, most physicists do not actually know enough math to do their research, or they are limited in what they can research by the math they know. Most importantly, in the more mathematically inclined fields of physics, I disagree with this due to the reasons stated above.

  • I couldn't be more grateful for your answer. It really helped me think I'm not doing anything wrong. Comparing the mathematical physics done by Brazil and by, say, Poland, I can see that we are very far from doing real mathematical physics here... it just looks like the theoretical one that, as you said, many times makes little sense. Maybe this contributed to making me believe that to do physics knowing mathematics was not a good idea. I hope I can get rid of this thought and what you said was very motivational. Than you again. – QuantumBrick Nov 13 '16 at 15:18
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So far, you've been reacting. You sign up for a course, maybe it's required, maybe it's not. You're given a homework assignment. The hand waving approach in the lectures bothers you and you spend a lot of time building up the mathematical background, and you fill in the blanks. If you successfully managed to keep your instructors happy enough to give you good grades while you were engaged in this intensive self-study, good for you!

At some point in the not too distant future, you are going to want to start taking the initiative to give shape to your studies. Here are some ways to do that:

  1. Search for papers that really, really intrigue you. One of these might give you an idea for a direction you'd like to go; at the very least, such a paper can give you a mental image of what a math-friendly physics project can look like.

  2. Find out how you feel about other aspects of physics. You mentioned that you enjoy working out the mathematical underpinnings of the assignments you're given, and you enjoy programming. But I wonder if you have figured out how you feel about working in a lab, designing experiments, carrying them out, analyzing experimental results, writing programs to analyze experimental results, choosing and installing new equipment, designing a new or modified experimental apparatus, communicating with the engineer and the machine shop technician, testing a new experimental set-up ("commissioning").... This is good information to have about yourself.

Please don't forget that you are studying in the Third World, not that one should look down on science in the Third World, but just to reassure you that it is forgivable that your instructors might do more hand waving and less rigor than is at least sometimes found more commonly in the developed world.

My spouse is an experimental physicist, who uses math very, very frequently, and sometimes writes a theoretical paper, using mathematics to explain, as you have been doing so far, but also to create, as you are dreaming of doing.

Note 1: At some point you may want to correspond with an author of a paper you've identified in your search. Science as become incredibly international.

Note 2: I wasn't sure from your post, whether you might sometimes have a bit of trouble keeping your eye on the big picture (getting the assignments done, without getting so sucked into the mathematical underpinnings that you might have trouble making your deadlines). If so, take a look at Is there a place in academia for someone who compulsively solves every problem on their own?, which is a rather extreme example of what I was describing. If not -- have fun in your explorations!

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    In my undergrad I had 4 advanced physics labs and worked with experimental physics in my first project. I liked doing experiments, but I hate analysing data. I really tried working a bit with experiments, but I think I'm more comfortable with the abstract aspect of physics. Right now I'm trying to get a collaboration with an experimental group going. They look interested... let's see if it gets somewhere. – QuantumBrick Nov 13 '16 at 18:02
  • Also, I don't compulsively solve problems. I just solve what I need such that my vision is clear. My grades are pretty high, but I sense that if I used some conclusions without proving them I would get the same grades with much less work... the thing is that I'm already convinced that maybe this extra work might lead somewhere, thanks to the answers received. I can only thank you for your suggestions and answer :) – QuantumBrick Nov 13 '16 at 18:05
  • @QuantumBrick - Good! I did have the impression you were probably holding the reins pretty comfortably. Forget I mentioned it. – aparente001 Nov 13 '16 at 19:39
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Do not worry about leaving the field, there is plenty of work to be done in physics using a more mathematical approach. From my opinion, finding and applying new mathematical techniques in physics help us advance - great advances in physics in the past have come by using new ideas and concepts from mathematics. Take the famous example of general relativity, or new modern theories of gravitation. Without mathematical techniques it would be impossible to find new ways to describe the universe. Developing and introducing new mathematical concepts in physics could potentially redefine modern physics. Hence, pass the 8a/8a+ grade and go beyond!

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