I'm a non-mathematician who is self-studying mathematics. Although I'm very interested in mathematics, my main purpose is to apply math in theoretical physics. The problem is that when I read a mathematics books, I can't see a clear way to apply these math in a concrete setting. I want to apply higher math in my study of theoretical physics (not mathematical physics). I'm not looking to put physics on a rigourous basis (e.g axiomatic field theory). I want to use math (e.g. category theory and algebraic geometry) in order to discover new ways of thinking about physics, generalizing concepts and to calculate stuff. I'm completely self taught in math. Should I read pure mathematics textbooks aimed at mathematicians? What's your advice on this?
[Math] Using mathematics in theoretical physics
physicssoft-question
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As a graduate student of pure mathematics, I question how valid the statement that a proper appreciation of pure math requires some knowledge of applied mathematics and theoretical physics. Perhaps part of this comes from: where is the line that separates applied mathematics and pure mathematics? If we include the whole of calculus within applied mathematics, than this is without a doubt true. But I do a lot of work in analytic number theory, and I have a hard time coming up with theoretical physics that I employ. I also have a hard time separating theoretical physics from pure mathematics - both are largely mystical, loosely formatted, and open. One might argue that theoretical physicists concern themselves more with descriptions of the natural world while mathematicians only concern themselves with what is consistent rather than what is possible... but I don't know how I feel about that either.
I will say, however, that I think both applied math and theoretical physics rely heavily on 'pure math.' I think of fields such as Lie Theory, which I consider a pure math. Lie Groups are one of the fundamental tools used in theoretical quantum physics these days - the existence of many particles and symmetries is often suspected because of mathematics and thoughts birthed within Lie Algebras.
But perhaps it is nice to know the awesome power of pure mathematics sometimes. When learning about Lie Algebra, or any sort of Abstract Algebra, I think it might be very enriching to learn about the solutions to the quantum harmonic oscillator and/or particle in a box situations. While these can be done analytically (or through divine inspiration, as seems to be what my old professor expected of us), Dirac employed a very cool use of algebra to solve these things. This included group-like behavior and the creation of 'ladder operators,' and more can be found at wikipedia. In this sense, I do appreciate pure mathematics more because of this knowledge of applied mathematics.
A large amount of group theory can also be applied to quantum mechanics. The astounding properties of the Pauli spin matrices might seem flukelike, but they can be predicted and analyzed from a group-theoretic context as well. I suspect many find is vastly satisfying to be able to predict experimental results from entirely theoretical pursuits.
As a very pure mathematician, one of the questions I am often asked is 'Why do people care about what you do?' While I might come up with something good to say, I think it is reasonable to say that the majority of the work that pure mathematicians do will never find an application that people would declare 'useful.' Thus whenever a bit of pure mathematics is 'applied,' this re-convinces people that funding the study of pure mathematics is a worthwhile endeavor. I rely on that belief for my own funding, so in that I appreciate applied mathematics a whole lot :P.
I will end with one more example. I again refer to Dirac, because I happen to know a lot about his life and how he went about his research. Dirac championed the use of projective geometry (and what I think we would now classify as differential geometry) to discover physics. Here is a transcript of a talk he gave about this subject at one time. Geometry is an interesting thing, because it's often visually based or well-grounded on intuition. Although I am not a geometer, I nonetheless am very pleased whenever I can use a physical situation (even quantum physical, slightly less intuitive) to better interpret some sort of geometric situation. Similarly, it is nice to be able to apply a geometric intuition to an apparently non-geometric problem from the applied sciences. I really encourage a quick glance through the transcript.
This is what I have for now.
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I happened across a few links that I think talk about this subject nicely. One is a site that addresses how physics does not come from math alone - I take this as an example of how math is too general (in general) for physics. The second is a paper that talks about three very high-level math things that arose out of physical interpretations. A nice addendum, I hope.
Since most of book covers on research mathematics is intentionally kept sober, to make a poster you may want to keep it simple yet not too boring. Also, it depends on the subject, audience, medium and budget.
Some tips:
keep it simple or to borrow the term: minimalistic
Visual wordplay or pun (as long as it is not overdone) OR, allusion to artwork such as Magritte or Escher if you are dealing with a topic that is self-referential, but avoid being cliche. Since mathematics is hotbed for symbols and logos, you can alter the font to mold into an object. The zero-sharp, zero-dagger, club suit, diamond suit, etc. Example, a bird morphing from "w" (the smallercase Greek omega letter)
You can also use an equation in a Rebus style such as the much abused meme of $i$, complex number and irrationality. Although memes can be rather cheap humor, you can browse to keep ideas flowing. This cartoon was quite interesting. (It can also be a simple Euler's identity with the tagline: Thus God exists.) Also, try to make a campaign - to borrow advertising term- so that you keep one constraint, eg: Thus God exist and you show mathematical equations such as the the one already mentioned, Kurt Goedel's proof, etcetera. Only caveat: keep it simple and connect in a non-sequiter manner. If you show the image of two balls from one of Banach-Tarski paradox, your tagline could be something of the nature such as: Never a boring day at the classroom.
Probably a historical image in a monotone shade either a sketch of the mathematician if he is obscure or of the university or locale
Keep in mind of that if it is posted on a bullet inboard, passerby will have very short span to notice it, so it cannot be too deep to "get it"
Another idea could be "ambient"-a term in advertising, where you use the surrounding to prove a point, such as a life-size ballerina image around revolving door. Stickers on calendars, numpad on phone, mirror, trees can be ways to spread the message. Taglines like: Average person sees a tree, a number theorist sees the sequence: 1 1 2 3 5... or, Average person sees an coffee, but a topologist sees a a donut. (The last one is cliche, but for illustration purpose).
Some examples:
The famous book cover wittily shows and tells the theme
A Cantor one I found online that shows but does not tell the theme
Best Answer
While studying physics as a graduate student, I took a course at the University of Waterloo by Achim Kempf titled something like Advanced Mathematics for Quantum Physics. It was an extraordinary introduction to pure mathematics for physicists. For example, in that course we showed that by taking the Poisson bracket (used in Hamiltonian mechanics) and enforcing a specific type of non-commutativity on the elements, one will get Quantum Mechanics. This was Paul Dirac's discovery. After taking his course I left physics and went into graduate school in pure mathematics.
(I don't believe he published a book or lecture notes, unfortunately, though I just emailed him.)
In transitioning from physics to mathematics, I learned that the approach to mathematics is different in a pure setting than in a physics setting. Mathematicians define and prove everything. Nothing is left unsaid or stated. There is an incredible amount of clarity. Even in theoretical physics, I found there to be a lot of hand-waving and ill-defined statements and lack of rigor (which hilariously caused me a lot of anxiety). Overall, though, Mathematicians are focused on understanding and proving relationships between abstractions, whereas physicists are more interested in using these abstractions as tools. Therefore, the approach is very different: mathematicians don't care what the application is, they only want to understand the object under consideration.
Nevertheless, for a theoretical physicist looking to get a firm background in mathematics, you want to have the following core mathematical concepts, which will provide a foundation to explore any avenue:
But the real list is something like:
Set, Group, and Ring theory are used extensively in physics, especially in Hamiltonian mechanics (see Poisson Bracket). Real Analysis and Linear Algebra are needed as a foundation for Functional Analysis. Functional Analysis could be described as an extension or marriage of Ring Theory, Group Theory, Linear Algebra, and Real Analysis. Therefore, many concepts in functional analysis are extended or used directly from Real Analysis and Linear Algebra. Measure Theory is important for the theory of integration, which is used extensively in applied physics and mathematics, probability theory (used in quantum mechanics), condensed matter physics, statistical physics, etc.
Topology and Operator Algebras are used extensively in advanced quantum mechanics and Relativity. Specifically, Algebraic Geometry is studied extensively in String Theory, whereas Topology is used extensively in General Relativity. Operator Algebras are an important area for understanding advanced Quantum Mechanics (ever heard someone talk about a Lie Group before?)
Some canonical text-books I would recommend:
Those are some decent text-books. I would say: give yourself two years to digest that material. Don't be hasty. Remember: mathematics is about definitions and proofs. Do not expect to see "applications" in any of those books. Just understand that the concepts are needed in advanced physics.
Unfortunately, though, I don't know of any text-book that forms a direct bridge between the two. If Achim Kempf had published his lecture notes, those may have worked, as essentially, he was doing just that.
Good luck!