I am a Physics undergraduate, but over the past year, I have developed more attachment to Mathematics than to Physics. How common is it for a Physics major to go into research in Mathematics and what should be the steps that I should take. Would studying the specific area of maths I am interested in by myself suffice, or would I need a formal degree in Mathematics? Also, would I have to be well-versed and knowledgeable of all areas of Mathematics, or studying only the area that interest me suffice?
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The transition physics->mathematics is of course nothing unheard of (apart from the examples already given, see also Dan Shanks). I have met plenty of professional physicists with extremely strong background in mathematics, and I believe that they could change their field from physics to a suitable area of mathematics and become productive within a year or less if they so chose.
You certainly do not need a formal degree in Mathematics. Whether or not a degree is the most efficient way for you to acquire the necessary knowledge and skills, depends on how good you are at self-study versus guided study.
As for the last question, it all depends on what you mean by "well-versed", as well as on your own aspirations. It does not hurt to have a good overview of the different branches of mathematics, but that is something that comes over time. An undergraduate degree helps, but personally, I still don't feel that have a good overview of all of mathematics. Most mathematicians nowadays know two or three areas really well. Then they have a few other areas in which they would know how to phrase a question when approaching an expert, and might understand enough of the answer, when they need a specific result. Strictly speaking, if you want to work in combinatorics only, say, and do nothing else, then you would be able to find suitable problems. But firstly, it would limit your choice of problems, and secondly, you would find it unsatisfactory yourself. After all, I assume that the reason you want to go into mathematics is because you like mathematics, so a minimalist approach might not be the way to go.