Uncover the Secrets of Studying Complex Subjects | Expert Tips

[ptabor]

Obtain the knowledge to study such subjects?

Heck, I'm having enough fun working through Jackson. What kind of class do you people take to learn all the math behind this stuff?

Boggled.

 

[marcus]

Obtain the knowledge to study such subjects?

Heck, I'm having enough fun working through Jackson. What kind of class do you people take to learn all the math behind this stuff?

Boggled.

It sounds like you are taking undergrad physics courses. If so it seems reasonable to just concentrate on that and ignore quantum gravity (QG). You didn't say where you are in the process. If you already have a solid physics major (or even an advanced degree) you could "take" Lee Smolin's video course INTRODUCTION TO QUANTUM GRAVITY. It is 25 lectures. Go to perimeterinstitute.com and select "streaming media" and find it on the menu. The video lectures are in split screen with blackboard stills and the lectures are searchable by the blackboard thumbnail menu, so you can jump into a lecture at any point.

QG is an unfinished field with several approaches being pursued with no guarantee as to which will make it.
I believe that Baez (one of the people working on QG) been brutally frank about this. If you study it, then do so because it is really interesting new mathematics and it MIGHT amazingly turn out to lead to the right physical answer, but don't get your hopes up too much about anyone particular approach.

Besides, or instead of, taking other people's advice you can also explore the current literature firsthand. For example, print the two most recent Baez papers of arxiv and SEE if there is anything in there you can understand and what more you would need to learn in order to understand it better.

then come back and tell me so I can learn it too

Probably some other people on this thread will have more specific and helpful suggestions.

If you want a BOOK that gives an overview, the best one is probably a collection by Daniele Oriti to be published this year by Cambridge Press, called "Toward Quantum Gravity". It will have essays by different people discussing their approaches.

Smolin's video course also covers several related but different approaches: Loop, Spinfoam, Group Field Theory, TQFTs (topological quantum field theories), form theories of gravity like BF theory. I am not getting these in order and may not have some terms right, but the 25 lectures do cover several related QGs.
It was nominally aimed at undergrads in a Canadian University (Uni Waterloo). But some grad students were taking the course.

 

[marcus]

the preceding post is the best I could do as direct answer to P Tabor's question. he wanted to know what courses to take. I answered relative to non-string QG. Other people may have other ideas for him.

this next is just some general observations about QG which might be helpful. I couldn't decide to leave them or to erase the post.

QG is getting more "topological".

the practical meaning for you, if you are an undergraduate physics major, is that you should probably make sure you take a course that introduces DIFFERENTIAL FORMS and things like exterior derivative and wedge product.

It might be NICE to also take an introductory topology course in the math department but that may not actually do much for you (depending on the course) and personally I don't think it is essential.

the main thing would be to get some familiarity with the manifolds equipment of differential geometry and in particular with diff. forms.

If you are just doing a physics major then you may not get around to QG for some years, but when you do knowing some of this differential geometry will be a good investment.

For starters, one concrete suggestion might be to take some kind of advanced calculus/diff geom class where you learn about differential forms.
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if you are a self-taught person there are online resources like one comes to mind is called
Preparation for Gauge Theory someone in Brazil wrote. It is 100 pages and covers a lot.
But you said what classes, not what online resources.

are you at a university? (you mentioned taking classes)
do you have the math department catalog handy, or the general course catalog?

what is the lowest number math class that says "differential geometry"?

also BTW what is the lowest number math class that says "topology", and the lowest that says "algebraic topology"?

tell me what the name and coursdescription says, maybe I or somebody else can help figure out

==============

the old basis of physics was foursquare euclidean geometry called Rⁿ. it was built on that, and elaborations, in the 19 century----a fixed flat space with a fairly rigid notion of the natural coordinates to use and a few set transformations between them.

and then there is the 1905 liberation with special relativity which means moving to "Minkowski" space----still fixed and flat and with a narrowly defined class of kosher coordinates.

then in 1915 the GRAVITISTS split off and built their gravity spacetime physics with DIFFERENTIAL GEOMETRY which means you can use curved coordinates and there is no idea of right ones, and you can do a lot of it in a "coordinate-free" fashion which means that your notation POSTPONES the choice of coordinates. the shape of space becomes a dynamic variable along with the other dynamically varying quantities.

but practical working physicists like Jackson did not immediately follow the Gravitists off into the fabled realms of diffy geom. They kept on working with primitive rigid space like euclidean or Minkowskispace of special relativity. After all effective macroscopic real space IS nearly flat and Minkowski is nearly right if gravity is negligible. Minkowski space is what EMERGES when you turn all the interesting stuff off.

What makes today's situation hard for us (who want to watch and learn) is that the (quantum) Gravitists NOW SEEM TO WANT TO GO SOMEWHERE BEYOND DIFFERENTIAL GEOMETRY and it is not obvious what that will be. maybe the fundamental picture of space they find at micro level and ultimately build the rest of physics on will NOT BE A MANIFOLD (not be a differentiable manifold, the standard idea of flexible continuum) maybe it will be a mess of purely topological information without any decent coordinates at all.

maybe you can abstract out all the topological relationships than can arise among things in a continuum and then throw away the continuum

and maybe then, contrary to all reasonable expectations, you can still do physics there

this would have conventional physicists tearing their hair out by handfuls and would be a big funny surprise to watch.

their familiar Minkowski space would be what emerges as the limit as you gradually turn the knobs and shut all the interesting stuff down.

and Minkowski space is what JACKSON ELECTRODYNAMICS IS BUILT ON so come to Papa baby.

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because we might NOT be in a gradual change situation but we could be in for a kind of rough JOLTING ride, maybe you should get a direct taste of what is going on. instead of my being able to tell you what courses to study, you maybe should look at some recent papers and try to guess for yourself. At least you get a taste of it now, which could gradually grow into insight later.

there is a baez paper with arxiv number "oh-fortyfortyforty" called "Quantum Quandaries"----F-H just mentioned it in that other thread----the Baez Perez thread. I would not have thought of it but F-H is probably right. he is doing his masters in this stuff right now and masters students usually know what to read.

Ah hah! that is what should happen. F-H should see this thread and suggest some things to you. If he wants, he could be much more helpful. My advice is keep on with jackson and do a standard Physics major and meanwhile just keep an eye on things.

sample arxiv links:
http://arxiv.org/find/gr-qc/1/au:+Baez_J/0/1/0/all/0/1

1. gr-qc/0605087 [abs, ps, pdf, other] :
Title: Quantization of strings and branes coupled to BF theory
Authors: John C. Baez, Alejandro Perez
Subj-class: General Relativity and Quantum Cosmology; Mathematical Physics

2. gr-qc/0603085 [abs, ps, pdf, other] :
Title: Exotic Statistics for Strings in 4d BF Theory
Authors: John C. Baez, Derek K. Wise, Alissa S. Crans
Comments: 41 pages, many figures. New version has minor corrections and clarifications, and some added references
Subj-class: General Relativity and Quantum Cosmology; Geometric Topology

3. quant-ph/0404040 [abs, ps, pdf, other] :
Title: Quantum Quandaries: a Category-Theoretic Perspective
Authors: John C. Baez
Comments: 21 pages, 2 encapsulated Postscript figures
Subj-class: Quantum Physics; Quantum Algebra

 

[George Jones]

What kind of class do you people take to learn all the math behind this stuff?

It is probably impossible for anyone person to learn all the mathematics useful for physics, so you have choose what mathematics you want to study and how much time you want to spend on it.

Typically, mathematical physics courses emphasize tecniques for solving differential equations, e.g., special functions, series solutions, Green's functions, etc. These techniques are still very important, but, over the last several decades, abstract mathematical structures have come to play an increasingly important role in fundamental theoretical physics. Consequentlly, useful courses include real/functional analysis, topology, differential geometry (from a modern perspective), abstract algebra, representation theory, etc., and, usually, should be taken from a math department, not a physics department.

These courses, supply vital background mathematics, and, just as importantly, facilitate a new way of thinking about mathematics that complements (but does not replace) the way one thinks about mathematics in traditional mathematical physics courses.

A number of good books on "modern" mathematics exist. Among these, my favourite is https://www.amazon.com/gp/sitbv3/reader/104-8106425-5831130?%5Fencoding=UTF8&asin=0226288625#reader-link"&tag=pfamazon01-20 by Robert Geroch. Geroch purposely and provocatively chose his title to indicate that, these days, mathematical physics includes topics other than those covered in more traditional mathematical physics courses. He starts with a few pages on category theory!

Geroch's book contains a broad survey of abstract algebra, topology, and functional analysis, and it does a wonderful job at motivating (mathematically) mathematical definitions and constructions. Surprisingly, since Geroch is an expert, it contains no differential geometry. Also, its layout is abominable.

At slightly lower levels are https://www.amazon.com/gp/sitbv3/reader/104-8106425-5831130?%5Fencoding=UTF8&asin=0521829607#reader-link"&tag=pfamazon01-20 by Chris Isham.

Geroch's book should be supplemented by more in-depth treatments of topics. For example, a good mathematical introduction to group theory is https://www.amazon.com/gp/sitbv3/reader/104-8106425-5831130?%5Fencoding=UTF8&asin=0521248701#reader-link"&tag=pfamazon01-20 by Shlomo Sternberg.

Also, none of the surveys that I listed treat fibre bundles, which are so important in modern gauge theories, and in other areas. Treatments include https://www.amazon.com/gp/product/9810220340/?tag=pfamazon01-20 by Chris Isham.

This is just the tip of the iceberg - there are many, many other good books including Nakahara, Choquet-Bruhat et al., Reed and Simon, Fulton and Harris, Naber, ...

Regards,
George