I often have the problem of what mathematical physics refers to, too, because it's like math is the framework of physics. If physics doesn't describe nature based on math, it seems not to be qualified to be called physics. When I was in high school, I found chemistry is too superficial because it often describes nature without giving the foundations underlying the description.
MarcusAgrippa said:
A mathematical physicist identifies, elucidates, develops and applies the mathematics to physical problems. His principal focus is on the mathematics.
MarcusAgrippa said:
A "theoretical physicist" for me is a physicist who constructs mathematical models and uses them to calculate what ought to be observed in a a laboratory or in an observatory. His principal focus is not the abstract mathematical structure of his models, but the practical application of them to make predictions and to provide explanations of physical phenomena.
According to your description, I seem to be characterized as a mathematical physicist to huge proportions, but in some sense I am also characterized as a theoretical physicist. My ex-boss, whose research field is the application aspect of quantum computation and quantum information, also said my interest is mathematical physics after I showed him what I am interested in. However, before he judged that my interest falls into the scope of mathematical physics, I kept considering my interest falls into the scope of theoretical physics, because I didn't know what exactly mathematical physics refers to but I deemed theoretical physics refers to the principles, theorems, and formulations used to explain nature. After he judged my interest belongs to mathematical physics, I often think over if that is really the case by considering what I am really interested in. I reckon I am indeed greatly interested in the math structure underlying physics, but I'm also interested in the prediction and explanation of physical phenomena by math models. I reckon though I am greatly interested in the math applied in physics, I usually don't have much interest in the math which has no application in physics though occasionally there is exception. On the other hand, though I'm interested in the application of math models to the prediction and explanation of physical phenomena, that is only to the level of the application to the theoretical physics, at most thought experiments: I am not so interested in the engineering required to put those physical principles into technology; I am also not so interested in analyzing experimental data.
I found in my undergraduate school time the subjects interesting me the most were theoretical mechanics and thermodynamics; I like those topics intensively containing analytical contents. Then in my graduate school I chose to take a research topic in General Relativity. Though I kept being greatly interested in all those material I need to learn during the process of the research, I finally felt it's like I was mainly engaged in math, not the physics I formerly considered to be, because General Relativity employs differential geometry, which is very math-oriented. But in retrospect, I found classical mechanics, quantum mechanics, statistical mechanics, classical electrodynamics I had studied, though didn't employ differential geometry, were also filled with math analyses though they were always sprinkled with a little bit of physical interpretations. The difference is that the physical interpretations in General Relativity are harder to be understood in an intuitive way. General Relativity discusses those geometrical quantities used to describe the curvedness of spacetime and formulates physical quantities in terms of those geometrical quantities, and thus gives me the feeling like it's all about math based on my conventional understanding of the discipline categorization. But afterwards I think in the past the physics I learned was the confined case of physics, the flat space setting in nonrelativistics; General Relativity just generalizes it to the relativistics in curved spacetime. And the spacetime is the venue where physical events take place, so it's justified to give a thorough investigation of it and see what influence the geometry of spacetime gives to the physical events taken place on it. Thus I think geometry should be a subfield of physics. A further rationalization for this is the gauge field theory, which describes an interaction field in terms of a field used to recover the symmetry under a certain group operation and gives Field Theory geometrical color.
