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Becoming a mathematically mature physicist.

  1. Aug 14, 2011 #1
    I've just acquired a copy of Principles of Mathematical Analysis by Ruden (3e) and, I must say, it is unlike any text I've ever read. Indeed, the level of abstraction and the amount of 'blanks' in the proofs have caused me to invest more time than usual into a book.

    Reading this has caused to me re-evaluate my command of mathematics. By this, I don't mean to convey the usual "I thought I was good, but realized I am not." Rather, I've found a whole new field for exploration, where I can become mathematically mature!

    Now, I have no delusions - I understand that this maturity only follows from dedicated study of the material. However, as I am self-taught, I don't really have any guidance in this matter, and do not wish to waste my time by practicing 'incorrectly,' or acting redundantly. It is my hope that the members of this forum can help me to develop an action plan, by answering the following questions:

    1. What texts, at the same level of Ruden's book, could help cultivate mathematical maturity?
    2. How exactly should I go about studying these books - in essence, how can I milk them for all they're worth?
    3. Do you have any tips on writing proofs?

    Naturally, you will need to know my aspirations, and my current level of understanding. My main field of interest lies at the intersection of Differential Geometry and Differential Topology, though I also enjoy Projective Geometry and Algebraic Topology. My main aim is to have a profound, intuitive, and mathematically rigorous understanding of these topics (and, eventually, all of mathematics).

    Regarding my present ability, I understand all the mathematics necessary for graduate level physics, specifically in QFT and General Relativity. My strongest field is Riemannian geometry and Geometric Algebra, my weakest Real Analysis (for which I have only a basic exposure).

    Thank you in advance!
     
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  3. Aug 14, 2011 #2

    atyy

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    Step 1: Spell the author's name correctly :tongue2:
     
  4. Aug 14, 2011 #3
    Huh, if you've done a lot of differential geometry, I would imagine real analysis wouldn't be very unfamiliar. How did you go about learning it without doing proofs? Is it the same Riemann geometry as this: http://www.maths.lth.se/matematiklu/personal/sigma/Riemann.pdf" [Broken] ? I don't see how you could really understand Riemann geometry without picking up a little bit of what Rudin is talking about (really, it seems like Rudin or some other introduction to real analysis book would be a prerequisite).

    ETA: That said, maybe you could look at Royden's Real Analysis along with http://people.whitman.edu/~gordon/higher_math.pdf" [Broken] on proofs. As an autodidact, google is your friend. I'm pretty good at finding information online about all of the stuff that interests me, and it's important to develop that skill so you can find multiple explanations of some subject matter from different perspectives.

    Also, having a "profound, intuitive understanding" of any area in mathematics means, in my book, being able to do original work in it. So unless you're Riemann reincarnated, I wouldn't worry about having even the knowledge of a standard mathematician in more than one field**; if you had a deep, intuitive understanding of everything closely related to, say, Algebraic geometry alone, you would be very unusual.

    **[meaning, essentially, the combined knowledge of two or more average mathematicians specializing in two distinct areas; as a caveat I should mention that it is perfectly feasible to have superficial knowledge across a broad variety of domains, even while having truly deep knowledge in only one]
     
    Last edited by a moderator: May 5, 2017
  5. Aug 14, 2011 #4
    @atyy Hahaha. Thank you, I meant Rudin.

    @Bourbaki Yes, as I mentioned, I have been exposed to Real Analysis (by basic, I meant the equivalent of a third year course, although I paid far too little attention to anything which was not essential for my understanding of physics). Regarding Differential Geometry, I am familiar with the abstract notion of Manifolds (having read Wald's text), but, as I see it, the main part of it is the geometric aspects of curvature, metrics, tangent spaces, Grassman algebra, etc.

    Moving on, as I see it (and Wikipedia vouches for me), the basis of mathematics is set theory, mathematical logic, and category theory. Given the amount of work that has been put into these fields, I understand that all of mathematics may be beyond my grasp. I suppose, then, what I seek is more akin to your caveat, though I do intend to grasp a bit more than 'superficial' knowledge.

    And yes, I do hope to do original work in mathematics, which is one of the reasons I want to develop my mathematical maturity! As an additional reason, being mathematically mature allows me to be more rigorous in my physics, whereby I can 'easily' perform a detailed analysis of the material involved. As an example, I am at present investigating Hawking radiation, where, from my current command of mathematics, I have encountered some inconsistencies. Becoming more mathematically mature would certainly aid my deeper investigations.

    Accordingly, I respectfully direct you to my three original questions - and, once again, thank you!
     
  6. Aug 14, 2011 #5
    TomCurious,

    I've spent about 20 years working on that problem: how to become more mathematically mature from math foundations to QFT, econophysics, etc. I recently put together a Word document outlining what worked for me: texts, why, what sequence, and so on. It's in three parts. There are sections below listing mathematics texts and physics texts. The audience ranges from advanced undergraduates to post doctoral research fellows. See attachment Word document.

    Excerpt of group theory and physics.

    Abstract (or Modern) Algebra (or simply Algebra): First the books:

    1. Sophomore level mathematics (recommended for the physicist who may never take such a course). “Modern Algebra, An Introduction,” by John R. Durbin, 2nd ed. It is very readable and easy to do the homework problems. The whole concept of elaborating on the subgroups of a group is very important to the physicist who uses group theory. This takes up the first four chapters of Durbin, and the physicist will get some ideas of the pure mathematics approach.
    2. “Groups, Representations and Physics,” by H. F. Jones, 2nd ed. This should be read by the physicists concurrently, or shortly after the one year series in graduate quantum mechanics. For reinforcing Jones, I strongly recommend “Modern Quantum Mechanics” by Sakurai.
    3. For those physicist looking for deeper applications in physics, I then recommend “Lie Algebras in Particle Physics, From Isospin to Unified Theories,” by Georgi, 2nd ed. I found this text difficult to read, but it can be done once you have mastered Jones. In fact, reconciling Georgi to Jones is a great exercise.
    4. For the mathematician looking to see what Sophus Lie was up to regarding applications of group theory and topology to partial differential equations, I recommend “Lie Groups, Lie Algebras, and Some of their Applications,” by Robert Gilmore. A MUST READ for the physicist. This is fundamental stuff for the exploration of physics from a very general language: A Lagrangian for a universe, the particle spectra, the local and global topological properties,…
    5. Along the lines of exploring physics using very general language, I strongly recommend “Geometry, Topology, and Physics” by Nakahara. It’s very readable, and shows how gauge field theories can be expressed in terms of connections and fiber bundles. This motivates the use of differential forms, a far more general theory than vector and tensor calculus. (For a quick, but to-the-point introduction on differential forms see, “Introduction to Differential Forms,” by Donu Arapura; I honestly can’t recommend any good physics or math book for a good introduction into forms.)
    6. To the physicist, “Quantum Field Theory,” by Lewis Ryder would make the efforts in 1 through 5 above worth the pain, especially the material that relates differential geometry (General Relativity) with Lie Groups/Algebras—taking an infinitesimal loop in the underlying space (connection in GR, commutator in Lie Groups/Algebras. I also recommend “A First Course in String Theory,” by Barton Zweibach, 1st or 2nd eds. A great tease full of history and ideas for further study is “Knots, Mathematics With a Twist,” by Alexei Sossinsky—you’ll see that the knot theory built up by Vortex atom physicists in the 19th century resembles today’s string theory work.

    I strongly recommend algebra to the physicist, not so much the engineer. The physicist will appreciate aspects of quantum mechanics far better with a solid foundation in Abstract Algebra. Since this is not recommended for all majors, I’ll elaborate on its importance here in Part 1. I took a junior and senior level course in algebra, and years later a full year of graduate classes in a mathematics department. The incredible dryness and ivory ‘towers’ of detachment modern mathematicians have brought to this powerful field is what made me quit pursuing a doctoral program in mathematics and switch to physics. One of the founding fathers of this field, Evariste Galois, died at twenty after a duel. He spent the night before his duel penning his mathematical thoughts. Galois theory solved an old problem, but this first requires some history. Babylonians and Egyptians, and probably earlier peoples, knew how to solve quadratic equations. The derivation of a formula to solve cubic equations had to wait until the early 16th century. Working at this, it may have been the Italian Girolamo Cardano who first ran into what today we call the complex numbers. Cardano’s assistant, or student, Ferrari found a formula for the general quartic, which was published in Cardano’s Ars Magna in 1545. Naturally, mathematicians then went on to search for a formula for the general quintic and higher order polynomials. No dice. Galois put an end to this pursuit in 1832 before a bullet put an end to him. I should say that by the time Galois penned his notes, another ill-fated young man, Abel of Norway had already proved the impossibility of solving the quintic. Abel died at twenty-seven. Is pursuing modern algebra ill-fated?

    Methods of modern algebra were used to classify all 230 possible three-dimensional types of crystals (symmetry groups). In the plane, there are 17 symmetry groups. You have seen them all on stained glass windows in ancient cathedrals, in Roman mosaics, and beneath your feet on bathroom floor tessellations (tiling patterns). Today, circa 2011, the first hard x-ray free electron laser has started making images of the crystallographic structures of whole viruses unperturbed by any preparation techniques required by earlier methods. The crystal structure of table salt was the first resolved by x-ray crystallography in 1914. The first organic crystal structure was solved in 1923, cholesterol in 1937, and by the end of 2010, just over 70,000 protein structures have been worked out by this method. We’re about to cross 75,000 in August of 2011.

    The modest Jewish physicist Eugene Wigner (who’s brief biography is a delight to read) was one of the earlier promoters of group theory to physics early in the 20th century. Many physicists reviled him for bringing this incomprehensible “gruppen-pest” to quantum physics, a mathematical tool which now underlies one of the most basic paradigms through which we describe existence. Group theory underpins our most advanced description of all that we see in the universe, the so-called Standard Model, which we know is likely not a complete theory, as it is too rife with parameters we must put in by hand from experimental results, and it does not include gravity. Algebra also underpins our work to move beyond the Standard Model in the exploration of Grand Unified Theories (GUTs) and Theories of Everything (TOEs) which do include gravity. An honest-to-goodness surfer dude, Garrett Lisi, stirred up the world of physics in 2007 with his TOE based on the group E8. This work is now defunct, but it may serve as a model for further investigations.

    My breakout from mathematics to physics in part lays with the textbook on Abstract Algebra by Lang. The moon has more atmosphere. Thankfully, I eventually ran into “Groups, Representations and Physics,” 2nd ed. by H. F. Jones, Institute of Physics Publishing, Bristol and Philadelphia, 1998. There is an old, high school book which can serve as a great introduction: “Groups and Their Graphs” by Israel Grossman and Wilhelm Magnus, The Mathematical Association of America, 9th printing, 1964. A sophomore level book that can also serve as a great introduction is “Modern Algebra, An Introduction,” 2nd ed. by John R. Durbin, John Wiley & Sons, 1985. A much harder read, to follow the Jones text is by the physicist Howard Georgi: “Lie Algebras in Particle Physics, From Isospin to Unified Theories,” Frontiers in Physics, 1999; it is worth the time if you are willing to fill in the steps. The Georgi text is one of perhaps thirty books in mathematics and physics from which I’ve extracted cleaned up notes which many a fellow graduate student has used to make copies of for their own studies. Lastly, there is “Lie Groups, Lie Algebras, and some of their applications” by Robert Gilmore, Dover Publications, Inc., 1974, 2002. Many of the results of mathematical physics (this subject discussed below) are tied together by Lie groups and Lie algebra. Check out “Symmetry Methods for Differential Equations, A Beginner’s Guide” by Peter E. Hydon, Cambridge Texts in Applied Mathematics, 2000.

    It's definitely worth it to improve mathematical maturity!

    A. Alaniz
     

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  7. Aug 14, 2011 #6
    As I mentioned before, Royden is a nice book. Specifically Royden's fourth edition, which is a significant revision. Also, Rudin's Real and Complex Analysis is a good sequel to Principles of Mathematical Analysis (often referred to as "baby Rudin", Papa Rudin being the former).

    I used Churchill and Brown for my undergraduate complex analysis course, you might consider tackling that and then moving on to Lars Alfhors' text.

    If you have an interest in mathematically rigorous algebra, you could check out Michael Artin's Algebra as well as Sternbergs' delightful "Group Theory and Physics" (which ought to fit your interests quite well).
     
  8. Aug 27, 2011 #7
    Thank you everyone! This was enormously helpful. Sorry about the late reply - I've been on vacation!
     
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