Mathematical Prerequisites For Understanding String Theory

[CMoore]

Please forgive me if this question has been posted before, but I was wondering if anyone could provide a semi-detailed and sequential mathematical syllabus that, once mastered, would allow one to follow development of string theory.

So, assuming basic undergraduate mathematics such as real analysis, general topology and abstract/linear algebra, what would come next? I have several mathematical physics books including Nakahara, Frankel, Lam, etc. Do texts such as these prepare one for understanding string theory? Should the bulk of material in these texts be considered as prerequiste for ST?

Thank You.

 

[Kalimaa23]

I would say Nakahara covers it quite nicely.

 

[selfAdjoint]

I would say Nakahara covers it quite nicely.

Oh, I don't think so. You need a serious prep in complex variables as well, including holomorphic and meromorphic functions, Laurent series, etc. plus an intro to Riemann surfaces. I don't think this could be described as a prerequisite for Nakahara. He assumes you've taken all the undergraduate calculus courses that are offered, plus linear algebra, and teaches you differential geometry, fiber bundles, and such, all good stuff for string math.

For more advanced string/brane theory you will need K-theory, which pretty much means a grad course in algebraic geometry. And that in turn will entail some kind of undergraduate "higher algebra", groups rings, ideals, etc.

Not a math course, but a year of quantum field theory would really help; you don't want to hit operator product expansions for the first time in your string studies! Plus it will give you the context in which string theory was erected in the first place.

 

[Kalimaa23]

Oh, I don't think so. You need a serious prep in complex variables as well, including holomorphic and meromorphic functions, Laurent series, etc. plus an intro to Riemann surfaces. ... And that in turn will entail some kind of undergraduate "higher algebra", groups rings, ideals, etc.

True. I didn't think about mentioning it because it's standard undergraduate stuff for physicists over here.

 

[Cexy]

See http://www.superstringtheory.com/math/index.html and just keep clicking next!

Edit: I'm reading Nakahara at the moment as a complement to my courses in QFT, differential geometry and string theory and I have to say that it's pretty damn good. Are there other books in a similar vein that people can recommend - ones that take a more mathematical line, but make frequent references to their applications to physics?

 

[CMoore]

The reference at the superstringtheory website is excellent; thank you. I did, however, receive a disouraging email. The author claimed that string theory was a sinking ship and by the time someone learned enough math to understand it that it would be dead. I have read threads on here that discussed some problems that ST was having, but I did not get the impression that the theory was in that bad of shape. Are things really so dire?

 

[FSC729]

Be Careful

Here is a thread that might help answer your question:

https://www.physicsforums.com/showthread.php?t=105871

John G.

 

[CMoore]

John,

Thanks for your reference. Unfortunately, however, that thread did little to clear things up. I don't know enough about string theory to know how valid it may or may not be. I do know that any reasonable scientific theory must have some means of experimental verification; however, I also have a hard time believing that some of the most brilliant minds in physics are devoted to psuedoscience. ?!?

 

[FSC729]

Its not that the most brilliant minds in physics are ignorant of the philosophy of "real science", rather they are following the well tested methods of theoretical physics.

In theoretical physics you construct the model/theory first and then develop it sufficiently so that your theory can make testable predictions. One can always argue that such activity is "pseudoscience" or "metaphysics" but that does not imply that it will always be so, or that it is unnecessary. In fact without such speculation advancement in physics is impossible. The difference between "pseudoscience" and "science" is a fine line called testability.

Just because they are brilliant minds doesn't mean they won't be misled. Imagine how many brilliant physicsts worked on the ether theory of space, or on Rutherford's model of the atom, or on Neil's Bohr model of the atom, all of those theories appeared valid and potentially fruitful, only to be disproven later. Even though they may have been wrong, each provided the foundation for more "accurate" theories to come about. We learn as much from our mistakes as we do from our successes.

Anyway here is another thread that you might find useful:

https://www.physicsforums.com/showthread.php?t=105978

John G.

 

[ZapperZ]

Its not that the most brilliant minds in physics are ignorant of the philosophy of "real science", rather they are following the well tested methods of theoretical physics.

In theoretical physics you construct the model/theory first and then develop it sufficiently so that your theory can make testable predictions.

This might be close to being off-topic, but I'll interject here to correct one thing. This statement isn't correct. If you look carefully, a lot of "theories" started as an attempt to describe or explain an already existing phenomenon. This is true with Quantum Mechanics, Special Relativity, Superconductivity, Quantum Hall effect, etc...etc. The Standard Model came into existence as a purely phenomenological model as some sort of a "periodic table" for the elementary particles that have already been discovered.

Now in all cases, the theory makes OTHER predictions as well that are later verified (or not verified) by experiments. But in the majority of cases (Dirac's anti-electron is an exception), the impetus has always been the experimental observation first, theoretical description later.

This is what makes the development of String Theory different, and why theorists like Phil Anderson, David Pines, and Bob Laughlin are seriously criticizing its "popularity".

Zz.

 

[FSC729]

This statement isn't correct. If you look carefully, a lot of "theories" started as an attempt to describe or explain an already existing phenomenon.

This is true, but my statement does not exclude this possibility:

In theoretical physics you construct the model/theory first and then develop it sufficiently so that your theory can make testable predictions.

Nowhere did I say that the theories "do not" attempt to explain already existing phenomenon, or that the theories are not based on experimentation. I guess I should have made it more clear. You are reading things into my statement that just aren't there.

In my first drafts of the previous reply, I was more clear as to the origin of such theories. All I was saying is that in order to explain experimental data and previous theories you need to develop a theory first and then test the theory.

John G.

 

[selfAdjoint]

To get back to the state of string-brane-M-theory, when it was first developed in the 1980's it made some seemingly important breakthroughs. It LOOKED like the graviton it predicted would provide the long desired quantization of Einstein's gravity, and it LOOKED like it could almost-but-not-quite do the Standard Model. These results attracted a lot of brilliant workers to the field and they are now the old guard of string physics, ensconced as senior faculty in all the leading institutions.

String theory then went into one of its periodic doldrums, waiting for the next big insight. This came with superstring theory, which at first LOOKED like it provided a single theory-of-everything (Feynmann danced!), and new graduate students appeared. But then turned out to provide several theories with no apparent connection to each other. More doldrums, but now with still more brilliant people busy deriving details. At this point lots of conections to current mathematical research became apparent and Witten was awarded the Fields medal, the math community's equivalent of the Nobel prize.

More doldrums until the dualities were discovered in the 1990's and M-theory gave a notional unification of the several superstring theories. Onward to the T-O-E! A new generation of very smart people comes on board. But M-theory is inherently non-perturbative, i.e. maximum tough to compute, and no clear overall picture of what it contains has ever emerged. And in searching for that they discovered the "landscape" which contains inconceivably many, maybe infinite supersting solutions that describe different worlds, and no valid way to distinguish which solution describes our world. So Susskind and others have turned to the Anthropic Principle to try to bring order out of this chaos, and in doing so they have been roundly criticised by many physicists, including prominent string physicsts, for doing philosophy, if not religion, in the name of physics.

This is the position today and you yourself must decide whether the cup is half full or half empty. On the one hand SST is in a pretty confused state. On the other hand it's had down periods before, always ended by a striking new insight.

 

[CMoore]

Thank you SA, that was a very coherent and informative explanation.