So, my background is that I have a bachelor's degree in electrical engineering, but I am doing my master's in engineering physics, theoretical track. (Both will be part of my "Civilingenjörsexamen", as it is called in Swedish, which I believe is usually translated as Master of Science in Engineering.)
When making my choice for master's track, the clear front-runners were applied mathematics and theoretical physics. I ended up choosing physics mostly because I was curious about relativity theory and quantum field theory, even though I am certainly stronger in mathematics than in physics. I also had some vague idea that I could transition into more general applied mathematics later on if I wanted to, while it might be more difficult to go the other way.
Now, a year and a half into my master's studies, I feel that, while there are many areas of physics that fascinate me, life as a physics researcher would likely be frustrating and unsatisfying for me. I am frequently annoyed with how loose and hand-wavy physical arguments can be, and I find some physical theories (say, the standard model) to be a bit too messy to be enjoyable. But more importantly I realize that, while I appreciate the applications, it is really the mathematics that I enjoy the most. I really empathized with Steven Strogatz in a recent interview when he said something to the effect that he found physics a wonderful source of problems, but that he could not care less about specific results.
So this leads me to the obvious question: is it feasible that I could get into pure or applied mathematics PhD programs with my background? I guess I could just apply and find out, but that takes significant work, so I would prefer to know if I am wasting my efforts. I am interested in both Swedish and international PhD programs.
I should probably mention that I have taken a few extra mathematics courses, specifically real analysis, abstract algebra (groups and rings), and discrete mathematics, in addition to the standard courses in calculus, linear algebra, differential equations, statistics, numerical methods etc. Also a couple of "mathematical methods for physics" courses, with a mix of topics like partial differential equations, tensor calculus, variational calculus, analytical mechanics, representation theory, differential geometry etc., though not to the depth of a dedicated mathematics course. Moreover, I have a little bit of research experience, with two summer research internships under my belt; though these were both related to my bachelor's subject of electrical engineering rather than math/physics. I am also under the impression that my institution, KTH, is pretty well regarded in Sweden. At least the physics program.