Difference Between Theoretical and Mathematical Physics

[Neptulin]

Hey everyone

If you want the concise version, skip over the bit in italics.

First I'll quickly introduce myself. I am an undergraduate student, studying physics. I have been reading these forums for some time now (mostly just re-reading the stickies to be honest). In Australia, the first year the BSc is general study in science electives, you select your major in the second year, your specialisation (eg. astronomy and astrophysics, optics, nuclear physics ect.) in your third year and, if you want a pathway to graduate studies, your research topic in your fourth. I am only in the first year of this process, and have yet to have decided on a specialisation (I figure I'm going to wait until I have a far deeper understanding of what each area is about).

I have been looking at my university's website/prospectus almost every day for the last year looking at majors, minors, specialisations, courses, requisites, alternate programs etc., figuring out my options. Tonight, however, I came across something minor that confused me. I figure it still needs addressing, and at the same time I may as well introduce myself.

Now that I've made the most sidetracked OP ever (and FP, I believe), I think its due time to get on topic. On the website I noticed that theoretical physics and mathematical physics were implied to be separate. I never really thought of this, since one is a major and another a specialisation. I noticed this when a course page mentioned "theoretical physics or mathematical physics" and I wondered why a distinction was made between the two. I would think that, if it was truly theoretical, theoretical physics would be described by mathematics. A quick Google search just showed the two terms were commonly lumped together. Are they truly different, or is it just my uni separating them (eg. maybe they're different departments?).

In all likely-hood, I won't be doing either. I tend to enjoy making and designing experiments - I think I would prefer a specialisation that has theory and experiment. But who knows, maybe I will be more the wiser further into my degree.

So my question is this, what is the difference between theoretical physics and mathematical physics?

 

[Monocles]

These terms are not very well defined but I will try to give some feel as to what the difference is. Mathematical physicists are frequently (but not always) people who enjoy studying the mathematics that arises in physical scenarios without necessarily knowing or even caring about the physical implications. For example, in string theory, there are plenty of mathematicians who study geometrical invariants such as Donaldson-Thomas invariants and Gromov-Witten invariants, which have their roots in physics, but end up being of interest to mathematicians in their own right. Another example of mathematical physics that a theoretical physicist probably would not care to study is the free primon gas. Basically it is studying the statistical mechanics of an imaginary particle called a primon which have an energy spectrum equal to the natural log of the prime numbers. This creates some interesting connections between number theory and quantum field theory, though it is of no physical relevance.

But there are mathematical physicists that *do* have some training in physics and do care about actual physical results. It is a blurry line. Theoretical physicists tend to concentrate on predicting physical or explaining physical results, and tend to care a little less about mathematical rigor.

Finally, mathematical physicists tend to live in math departments, while theoretical physicists tend to live in physics departments. I also don't want to give the impression that only high energy stuff is done. There are plenty of people that attempt to apply rigorous methods to study e.g. condensed matter - such as the work of Bellissard on the use of non-commutative geometry to explain the integer quantum hall effect.

This is just the impression that I have gotten from working with mathematical physicists for the past couple of years - I am just a soon-to-be grad student. So I am sure that there are more qualified people to attempt to explain the subtle differences between the two (which are really overlapping labels).

 

[Dr Transport]

I think if you look at http://arxiv.org/ , you will be able to get a handle on Mathematical Physics and other areas of Theoretical Physics.

 

[deRham]

Mathematical physicists are frequently (but not always) people who enjoy studying the mathematics that arises in physical scenarios without necessarily knowing or even caring about the physical implications.

I liked this line. This has been my overall impression, too. Mathematical physicists seem to be interested in the mathematical constructions employed to produce significant advances in physics theory, and in their extensions, whether or not these are strictly motivated by experiment. In an era where there is a lot of physics that is hard to explain, increasingly the line can be blurred, as people even with great interest in physical applications seem to study the mathematics at least somewhat for its own sake, so as to get a clue as to where to go next.

I'd say the difference needn't be strict, but for the fact that mathematical physicists study the mathematical structures and constructions related to those appearing in physics, whereas theoretical physicists presumably are interested in physics, but acknowledge the need for sophisticated mathematics.

This is my attempt at a somewhat simplified version of what I think was a great explanation by Monocles, who gives some great examples too.

 

[Neptulin]

Thanks everyone for the detailed replies.If I'm understanding this correctly, then mathematical physicists find out the physical implications of a set of axioms and assumptions, without needing to worry about the validity of the axioms and assumptions. Meanwhile, theoretical physicists essentially do the same thing, but have to worry about whether their assumptions are actually correlated to reality. If a situation arises when the Mathematical Physicist's assumptions hold true, then their theory can be applied to that situation. Is this correct?

I'm also wondering if it's worth specialising in mathematical physics if your major is anything but theorietical physics or maths. For example, would it be worth specialising in mathematical physics if my major is, say, astrophysics (keeping in mind I can just do theoretical physics courses instead)?

 

[Nabeshin]

Thanks everyone for the detailed replies.


If I'm understanding this correctly, then mathematical physicists find out the physical implications of a set of axioms and assumptions, without needing to worry about the validity of the axioms and assumptions. Meanwhile, theoretical physicists essentially do the same thing, but have to worry about whether their assumptions are actually correlated to reality. If a situation arises when the Mathematical Physicist's assumptions hold true, then their theory can be applied to that situation. Is this correct?

Well, this is a little bit misguided. Quantum mechanics is derived assuming (among many other things) a non-commutation of variables like x,p, i.e. [x,p]=i. This is your axiom, but there's no way to verify (directly) that this is true. Rather, you use the theory based on this axiom to generate predictions, and if the predictions match reality, then the axiom obviously holds some weight. Such a prediction would be an uncertainty principle for x and p, which has great experimental verification. My point is that axioms are often cryptic and cannot be tested directly (an even better example might be simply positing some Lagrangian).

 

[Neptulin]

Well, this is a little bit misguided. Quantum mechanics is derived assuming (among many other things) a non-commutation of variables like x,p, i.e. [x,p]=i. This is your axiom, but there's no way to verify (directly) that this is true. Rather, you use the theory based on this axiom to generate predictions, and if the predictions match reality, then the axiom obviously holds some weight. Such a prediction would be an uncertainty principle for x and p, which has great experimental verification. My point is that axioms are often cryptic and cannot be tested directly (an even better example might be simply positing some Lagrangian).

I see. I understand what you mean (as Monocles said, its a blurry line), but I think I'm a few years off from it being relevant to my studies.

 

[SophusLies]

So my question is this, what is the difference between theoretical physics and mathematical physics?

Probably a couple orders of magnitude in career opportunities. Heck, theoretical physics don't have many job prospects let alone mathematical physics. I worked under a "mathematical physicist" for a couple of quarters and he had next to zero grant money coming for anything he was doing. Mathematical physicists seem to gain their ground on prizes, writing books, or going on TV shows and talking about crazy theories.

Good old theoretical physics, especially AMO, condensed matter and particle/accelerator, can pull in some serious grant money from NSF, military, and industry. I recently heard of a theoretical condensed matter grad student from last semester got a niiice job with IBM doing spintronics simulations. Meanwhile, the string, loop, mathematical physics graduates are off on their postdoc tours. I wish them luck but seeing that there aren't many jobs in those fields, luck won't help.