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The long and winding road that leads to your TOE

  1. Apr 27, 2013 #1
    I am hoping to get some guidance in this thread from those who have traveled the path before me. For the past couple of years I have been on a tear to educate myself in mathematical physics for a variety of reasons I won't bore you with here (some personal, some professional). In any case, to make a long story short I want to get to a point where I can speak and write intelligently on popular cosmology (TOE) models, L-CDM, string, LQG/LQC, Quantum field theory, etc. I don't know if I'll ever get there or how close I'll get, but shoot for a star and hit a bird, shoot for a bird and hit a rock, right?

    To put it simply, I'm kind of looking to do what Lenny Susskind is trying to put together with his "theoretical minimum" campaign. In fact, at one point I got real excited about this thinking, hey, I can get where I want to go with some one-stop shopping at Lenny's. Just watch all his classes on you tube and there you go. Unfortunetely, it wasn't so easy. In fact, I personally got very little out of his lectures other than a great deal of entertainment. He is a master showman and a master at his craft. However, for me watching him lecture is like watching performance art. It's like watching an episode of Iron Chef America. I can see how the iron chef is preparing his meal and get caught up in the excitement, but at the end of the day I have no idea why he picked the ingredients he did or used the proportions he did, so I learned almost nothing about how to prepare those dishes--evidenced by the fact that I get nowhere when I attempt to tackle the problem myself.

    Other than the Susskind lectures, I have been hard at work rekindling my math skills which have laid dormant for two decades and this is where I could use some advise. I started at the beginning with general math, algebra, geomerty, trig, matrix algebra, calculus 1,2, and 3, and differential equations, and now I'm kind of stuck. I'm stuck because this is the traditional basic progression of maths classes that I am aware of, and I don't know where to go from here to get to my quest for theoretical minimum, or TOE, street-cred. There seems to be a hundred different areas of study, and I don't know 1) which ones are necessary to get me TOE street-cred, and 2) what order I need to learn these mathematical disciplines.

    To put it simply, what in your opinion is the essential maths that you feel a student needs to learn and in what order in their attempt to begin to understand contemporary TOE models. I know its best to learn and know all of them, but the idea is to get an idea of a sane and doable curriculum. RIght now, I'm standing at a crossroads with 100 different roads to take and I have no idea how to proceed. Here's a short list of the roadsigns--unitary groups, differential geometry, Lie algebra, Spinors, Legendre polynomials, Fourier analysis, Laplace transforms, tensor calculus, Special unitary groups, Cauchy integrals, Christoffel symbols, The residue theorem, gauge theory, Phasors, yada yada yada, I could go on for a while here.

    So right now I'm stuck, I don't know which of these I need to study, or in what order. I'm looking for a workable path here, a way to organize my approach here. It would be great if they had a TOE service pack instruction kit. Service pack one is to study this, this and that in this order. Then move on to service pack 2, etc.

    I know this is as much of an art as a science, as far as what to study and how to go about it, but that's kind of what I'm looking for, what your personal path or approach was and what you would recommend to those following behind.

    BTW, here's some background music while you're thinking it over:

    Last edited: Apr 27, 2013
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  3. Apr 27, 2013 #2


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    We have an Academic Guidance section of PF which includes advice for Self-study
    There is a specific thread on Advice for self-study of physics:
    This thread already has a lot of suggestions and shared experience posted. You might find it useful. You could even ask these questions there.
  4. Apr 27, 2013 #3


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    Thread moved.

  5. Apr 27, 2013 #4
    Thanks for the link, Marcus. Great information there. However, that is basically a thread for self-study of physics in general, which is definitely what I'm doing, but I'm looking more for a curriculum path to the study/understanding of TOE models specifically. Since most of that seems to be related to quantum gravity, I'm thinking that intense study in classical physics might not be the best usage of time and mental resources. In any case, maybe there is no answer to my question. Maybe what I'm really looking for is suggestions on books or textbooks that may serve as the "service packs" I mentioned in the OP.

    I guess what I'm looking for is a "theoretical minimum" program or book(s) like Susskinds, but one that spends alot more time to actually teaching the student/reader the concepts and the maths in detail, starting from the beginning and working methodically upwards. Susskind has a book out recently entitled, "The theoretical minimum." However, it only covers classical mechanics and hasn't done much more for me than his you-tube lectures.
  6. Apr 27, 2013 #5


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    Umm, it's not obvious (to me anyway) what stage you're at right now (which is obviously a prerequisite for suggesting texts). Can you solve the Einstein Field Equations to get (eg) the Schwarzschild solution? Can you work with the Friedman-type metrics in cosmology? Can you do concrete 1-loop QFT calculations to derive scattering cross-sections in QED and/or electroweak scenarios? Do you understand the Bergman (geometric) approach to quantizing a classical system? Can you derive the ordinary QM angular momentum spectrum from first principles? Do you understand Wignerian group-theoretic representations as a foundation for constructing quantum fields? Do you understand the "forms of dynamics" introduced by Dirac, and used in (eg) Weinberg's text to construct particular interaction terms in field theories? Do you understand the extra challenges involved when trying to construct ordinary quantum fields on curved spacetime? Can you derive the Hawking/Unruh effects?

    I could go on, but... you probably get the idea... :smile:
  7. Apr 27, 2013 #6
    You have to understand classical physics before you can understand quantum physics. Get a copy of Goldstein or something and work through it.

    As for mathematics, two natural next steps would be linear algebra (abstract vector spaces). Then functional analysis, differential topology (you might need general topology for this), differential geometry and analysis on manifolds (learn some de Rham cohomology while you're at it).

    If you dget the above under your belt, then you are very well prepared to tackle advanced topics in quantum mechanics and general relativty. From here you can study quantum field theory, and for the math you will need here, just google "mathematics of quantum field theory". Normally this kind of advanced math is not neccesary for physics students, but if you want to understand how the theory is buildt up and how to generalize it and merge it with GR, you have to know the mathematics that it is based on.
  8. Apr 27, 2013 #7


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    Well, after you've taken the basic ODE, and linear algebra and calculus it really depends on you which route to go by.

    But if you want to take a sane approach then now you should learn:

    1. Functional analysis + group theory and representation of groups+ Lie groups and algebra (it goes together).

    2. PDE, measure theory, point set topology, differential geometry more analysis...

    Now the first part is good to learn alongside QM and the geometry part is good for GR; After that you're on your own.

    As for the sane approach, maths and physics is not for the sane people, though I am not aware who is really sane.
  9. Apr 27, 2013 #8
    Sorry Strangerep, but I don't get the idea. I think I was pretty clear in my original post what my background was and where I was wanting to go. I'm not sure where you were going with that. Maybe I could have been clearer but espen180 and MathematicalPhysicist seem to have understood it pretty well.


    Thanks for the Goldstein reference, I keep hearing about that one, I'll check it out


    Thank you too, that's exactly what I was looking for. Looks like you and espen180 are on pretty much on the same page there, and that's the kind of consensus that helps me.
  10. Apr 27, 2013 #9


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    You've picked a good bunch of roadsigns. I think those are all useful in either relativity or quantum mechanics, or both. Stuff like Spinors I did not even learn in undergraduate... I don't know if this means you are maybe skipping over some of the more basic physics, or if you have already gone over it. For example, have you learned about electrodynamics, and the Heaviside form of Maxwell's equations? And about continuum mechanics, like inertia tensor of rigid objects and the continuity equation? Also, the basic parts of quantum mechanics, like simple wavefunctions, and hydrogenic atoms. I feel like all this stuff is pretty necessary before going on to more advanced physics.

    A good way to learn more physics is probably just to look through textbooks which are aimed at undergraduates. For example, the book "mathematical methods for physics and engineering" Is a great book for the kind of maths required for undergraduate physics. Although it is maybe not so advanced. For example, if I remember correctly, I don't think it includes any stuff on contour integration. It might be useful if you want to fill in some of the gaps, so to speak...

    Learning maths beyond this level (for application to physics), really I would say you should look into the particular physics subject a bit, then you will see something about the kind of maths it uses. And then you can go off and learn a bit more about that maths subject. For example, you might want to learn a bit more about relativistic quantum mechanics, and then you see spinors being talked about a lot, so then you go off and learn a bit about spinors, so that you can understand the physics a bit better.

    So what I'm trying to say is that the mathematics which is taught in an undergraduate physics course, you should pretty much learn most of this. But any maths further than that, I would say you can just learn it when you find that it is useful for a particular branch of physics that you are interested in. I am not a professor or anything, so maybe I am not 100% familiar with how the academic world works. But this is how I think it works.
  11. Apr 27, 2013 #10


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    BTW, this road never really ends, and sometimes even repeat itself...
  12. Apr 27, 2013 #11


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    One would need all that and more. One could add Green's functions, calculus of variations, . . . .

    A good place to start is browsing the Mathematics and Physics sections the Science Textbook Discussion forum

    For example, the following texts are provide a reasonable structure

    Mathematical Methods in the Physical Sciences by Mary L. Boas

    Mathematical Methods for Physicists by Arfken and Weber
  13. Apr 27, 2013 #12


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    I see a lot of math topics. What physics have you studied, and at what level?
  14. Apr 27, 2013 #13
  15. Apr 27, 2013 #14


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  16. Apr 27, 2013 #15


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    yeah, that's a good point too. I guess it is a good idea to study most physics topics up to about the undergraduate level, as well as most of the maths topics that have science applications.

    edit: I'm saying it is a good idea to study most undergraduate physics because there is often recurring themes, and it is just good practice. For example, if someone is talking about angular momentum operator in quantum mechanics, and you haven't learned about angular momentum in classical physics, then it could be difficult to grasp what the person is saying.
    Last edited: Apr 27, 2013
  17. Apr 27, 2013 #16


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    Try http://www.amazon.co.uk/Road-Reality-Complete-Guide-Universe/dp/0099440687

    Whether or not you agree with Penrose is beside the point. Of course one book won't teach you "all" the details of "everything" you need to follow his arguments, but at least it's a (very thick) one-volume road-map.
  18. Apr 27, 2013 #17


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    Let's slow down here. If I recall from an age old thread, you had a hard time distinguishing between the geodesic equation and the equation of geodesic deviation. I'm just saying that you are throwing out a flurry of advanced topics when you really should be focusing on the basics before even looking at these things. You have written down a massive expanse of mathematics that will need a firm foundation in the basics or else everything will crumble. For example before learning about lie algebras and lie groups you need to know smooth manifold theory but before that you need to know topology and before that you need to know, at the very least, how to write proofs. Tensor calculus is a whole nother (not to be confused with noether :biggrin:) subject area on its own that usually tends to overlap with Riemannian geometry thus leading down another road to the needed basics (well in my case I learned tensor calculus through general relativity texts like Wald but the foundations are always needed!). spinor theory is something else that can be presented using a good amount of differential geometry and algebraic topology (e.g. homotopy theory) and again you are led down the same road map to the fundamentals.

    You said you got up to differential equations and then stopped. Unfortunately the standard calc 1-3 curriculum + standard linear algebra + DEs are still a long way from where you want to get to. If you really want to have a thorough understanding of the math needed for things like mathematical physics then the most natural next step would be rigorous linear algebra (e.g. Axler) and then real analysis or, at the least, rigorous calculus e.g. Spivak.

    I would have to say that popular books similar to the one by Susskind might leave you very unsatisfied. I know I would be if I didn't know the proper mathematics behind the physics. What's the fun in learning physics if you can't appreciate the beautiful connections (no pun intended :tongue:) with mathematics?
  19. Apr 27, 2013 #18
    I agree here. You need to study a theoretical Linear Algebra text now. It will be very useful in physics and mathematics.

    Introductory real analysis is also something you need to cover once. The problem with this is that it is practically useless in physics. I don't know any physics where a knowledge of introductory real analysis is very helpful (or at least: significantly more helpful than just knowing the relevant calculus).
    However, the thing is that introductory real analysis will be used very extensively in later math such as functional analysis, topology, differential geometry, etc. So while the topic in itself is not very useful, it is something that you absolutely need to know.

    Both topics require you to understand and write proofs. So that's also something you should look at.
  20. Apr 27, 2013 #19

    Ben Niehoff

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    The standard graduate physics texts are Goldstein, Jackson, Sakurai, Arfken. Start there. Walk before you run.
  21. Apr 27, 2013 #20


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    Aren't we already running if we're talking about Jackson and Goldstein :tongue2:
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