Since quite a while ago, all physics has been mathematical physics. Galileo famously proclaimed in his book The Assayer (1623) that the book of the universe is written in the language of mathematics, and that those who do not know it are doomed to wander in a dark labyrinth. Still, someone who calls themselves a mathematical physicist has concerns that are somewhat different from most physicists.
Firstly, they will usually not have direct contact with experiments. So to the extent that they are physicists, they will be theoretical physicists. Secondly, they are typically much more concerned than even theoretical physicists with rigorous argumentation. What does this mean? A simplified version of the work of physicists might go like this. Some natural phenomena require explanation. The physicist proposes that there is a theory that explains it. This means that if you assume some simple rules about nature--the formulation of such rules is what require's genuine insight--a sequence of logical deductions from them will lead to the observed phenomena, that is, predict them. I should say that the term 'predict' is being used a bit casually here. This is because many such predictions are observed before the theory. Thus, they become consistency checks, obviously important. But it's most impressive when the theory predicts unknown phenomena that are subsequently observed. In any case, it's convenient to call both kinds of deductions 'predictions from theory.' For example, the prediction of black holes from the general theory of relativity is a genuine prediction of the latter kind. On the other hand, a classic example of prediction of known phenomena comes out of Newton's law of motion and his theory of gravity. Starting from them, one can use calculus to deduce that the orbits of the planets about the sun must be ellipses as had been observed by Kepler. Because such predictive power is so astounding, when this happens in a fairly reliable way, people say the theory is correct. It is at this stage that mathematical physics comes in. You see, unfortunately, in most of the cutting-edge theories of fundamental physics today, the logical steps from the theory to the predictions are carried out quite loosely. So some people feel nervous about concluding at this point that the theory is correct. To illustrate the distinction, I remark that the predictions of general relativity can, by and large, be deduced by way of rigorous argumentation.
On the other hand, the predictions of the main theory of particle physics, the so-called standard model, are based on fairly shaky, even if intuitively plausible, logic. So many mathematicians work very hard on the still unsolved problem of making this process of deduction rigorous. These are examples of what I would call 'hard-core' mathematical physicists. I find this activity admirable, but do not have the precise thinking power and stamina necessary to engage in it myself. Instead, I'm a breed of mathematical physicist that emerged en masse only in the late 1980s. We try to use ideas of physics to make interesting discoveries in mathematics. In other words, instead of the well-known process of using mathematics for physics, we use physics for mathematics. So sometimes, a person like me is called a 'physical mathematician.' There have been truly spectacular advances in the areas of algebraic geometry and topology that came out of physical mathematics. For the past 5 years or so, I have been advocating the view that physics is also useful for arithmetic geometry. However, arithmetic geometers are a tough crowd. It's not so easy to convince them this idea has substance.
When I was studying for my Ph.D., in fact, I worked on mathematical physics for the first three-and-half years or so, advised mostly by Igor Frenkel, a great expert on the algebraic structures used in string theory. Then, in the last one-and-half years, I switched to arithmetic geometry, supervised by Serge Lang and Barry Mazur. This didn't stop me from continuing to learn physics, mostly through conversations with my old friend John Baez, who is the most patient explainer of anything under the sun. (I really don't know why he puts up with me.) Then, as the years went by, I got help from people like Dan Freed, Jeff Harvey, Sergei Gukov, Philip Candelas, Xenia de la Ossa, Chris Beem, Tudor Dimofte, Theo Johnson-Freyd, Philsang Yoo, Kevin Costello, Brian Williams, Ingmar Saberi, Alex Schekochihin, Yang-Hui He, Bernd Schroers, Neil Turok,... even some experimental physicists like Alan Barr, Sangwook Cheong, and Philip Kim. Anything non-trivial I know about the subject is entirely thanks to them. Quite recently, I started publishing in actual physics journals. However, my goal is not to write papers. Truly, I would just like to understand physics.
