Prerequisites to understand Theoretical Physics

[Coelum]

Dear all,
I want to be able to read and understand the basic concepts of Theoretical Physics (String Theory, LQG, Supersymmetry, AdS/CFT) at the level of:
1- A First Course in String Theory by Barton Zwiebach
2- Quantum Gravity by Carlo Rovelli.
I plan to study the following textbooks (in fact, I already began...).
Mathematics:
1- Mathematics for Physics and Physicists by Walter Appel
2- Nonlinear Partial Differential Equations for Scientists and Engineers by Lokenath Debnath
3- Geometrical Methods of Mathematical Physics by B. Schutz.
Physics:
1- A First Course in General Relativity by Bernard F. Schutz
2- An Introduction to Quantum Field Theory by M Dasgupta
3- Introduction to Elementary Particles by D. Griffiths.
You may safely assume that I have the background to study all of those texts - both in physics and mathematics.
My main question is: are there unfilled gaps in my planned preparation?
A subordinate question is: are there better books for introductory self-study (rather than those by Zwiebach and Rovelli)?.
Thanks a lot in advance,

Francesco

P.S. In case you need to check the book's contents: all the cited books are available from Amazon except for the textbook by M. Dasgupta, which can be downloaded at http://hepwww.rl.ac.uk/hepsummerschool/Dasgupta%2008%20Intro%20to%20QFT.pdf.

 

[Kevin_Axion]

In terms of Mathematics for understanding Superstring Theory you'll need a basis of the following subjects: http://superstringtheory.com/math/index.html.

Here are topics in Superstring Theory that are also of great interest: http://en.wikipedia.org/wiki/List_of_string_theory_topics.

 

[Coelum]

Hi Kevin_Axion,
you got one of the main points behind my question.

If you look at Gerard't Hooft's page on HOW to BECOME a GOOD THEORETICAL PHYSICIST (http://www.phys.uu.nl/~thooft/theorist.html), some of the topics cited in the Superstring Theory page are not listed, namely:
1- Homology
2- Cohomology
3- Homotopy
4- Fiber Bundles
5- Characteristics Classes
6- Index Theorems
7- Graded Lie Algebras, Spinors, Grassman Numbers, etc.
It look like most of the topology is not needed - at least as a first approximation. Of course, those topics may be implicit in the physics, but I need an explicit point of view .

At this point, my (new) questions are:
1- do you think one needs to study those topics before attacking String Theory and LQG?
2- adding to my list "Geometry, Topology and Physics" by Mikio Nakahara does fill the gap?
Thanks,

Francesco

 

[tom.stoer]

You definately do not need all this stuff to study LQG which is a rather pedestrian approach to quantize gravity compared with string theory.

 

[sheaf]

I agree, I've been reading about LQG for the first time recently, and haven't done any serious mathematical physics for about 25 years. You need surprisingly little fancy mathematics for LQG - just a reasonable amount of Lie group theory. If you want to follow the story to see where LQG is motivated from, then a reasonable amount of differential geometry and GR is also necessary.

I can't comment on string theory because I haven't had time to study it yet - I also have Zwiebach, which I want to start with.

 

[marcus]

Dear all,
I want to be able to read and understand the basic concepts..
...
2- Quantum Gravity by Carlo Rovelli.
...

You definately do not need all this stuff to study LQG which is a rather pedestrian approach to quantize gravity compared with string theory.

I agree, I've been reading about LQG for the first time recently, and haven't done any serious mathematical physics for about 25 years. You need surprisingly little fancy mathematics for LQG - ...

Dear Coelum, I think Tom and Sheaf are right. It's good you have started your self-study program with that list of books on QM and GR basics. But they are still right in saying that there is a big difference in the amount and kind of prerequisites.

There is a free online version of Rovelli's book. And Chapter 1 is not mathematical. It gives some philosophical and historical perspective. How Einstein found his way into GR and how the vision of the world in GR was different in essential ways (from, say, in special relativity, or in conventional quantum field theory).
You could start now by reading the non-math Chapter 1 of Rovelli's book. Begin a process of getting used to a different perspective.

 

[Kevin_Axion]

Homology, and Cohmology should be learned in Group Theory, and Fiber Bundles should be learned in Toplogy, the entire list of Mathematical ideas you stated are sub-topics of the list I provided. Good luck in your studies and don't be discouraged if you don't understand a certain idea people spend almost a decade of education to learn the basics and even after fifty years of research in Superstring Theory and slightly less in LQG we still don't understand the theories fully.

 

[unusualname]

I recommend getting hold of a few of the most important papers in the subjects, and just reading through them to get some familiarity with the math/physics being used even if you hardly understand a word or an equation at first.

If your math/physics grounding is good you can probably get up to speed in one or two areas by directly skipping to the relevant sections in any textbooks you have.Top Cited Articles of All Time (2009 edition) in gr-qc

Top Cited Articles of All Time (2009 edition) in hep-th

 

[bcrowell]

I agree, I've been reading about LQG for the first time recently, and haven't done any serious mathematical physics for about 25 years. You need surprisingly little fancy mathematics for LQG - just a reasonable amount of Lie group theory. If you want to follow the story to see where LQG is motivated from, then a reasonable amount of differential geometry and GR is also necessary.

Really? I have the strong impression that you can't do string theory without spending about 5 years studying stringy mathematics. Had no idea that the situation was so different in LQG.

Suppose I want to learn some LQG -- just enough LQG to think I understand something when I actually understand nothing :-) I have a PhD in physics (as an experimentalist in low-energy nuclear physics), and I understand GR reasonably well, but my understanding of field theory is at the level of "I did the calculations required on the homework and exams in my one-semester graduate course in field theory," i.e., I never really learned it. Could I actually learn some LQG without multi-year full-time remediation? If so, where would I start? Please insult my intelligence by suggesting a ridiculously easy place to start.

 

[atyy]

Really? I have the strong impression that you can't do string theory without spending about 5 years studying stringy mathematics. Had no idea that the situation was so different in LQG.

Suppose I want to learn some LQG -- just enough LQG to think I understand something when I actually understand nothing :-) I have a PhD in physics (as an experimentalist in low-energy nuclear physics), and I understand GR reasonably well, but my understanding of field theory is at the level of "I did the calculations required on the homework and exams in my one-semester graduate course in field theory," i.e., I never really learned it. Could I actually learn some LQG without multi-year full-time remediation? If so, where would I start? Please insult my intelligence by suggesting a ridiculously easy place to start.

This seems friendly http://arxiv.org/abs/1007.0402 .

I also like http://arxiv.org/abs/0909.4221 .

 

[bcrowell]

There is a free online version of Rovelli's book.

Is that this? http://www.livingreviews.org/lrr-2008-5

 

[bcrowell]

This seems friendly http://arxiv.org/abs/1007.0402 .

Hmm...I think it wants to be your friend more than it want to be mine :-) The first equation involves notation I don't know (the script D in equation 1.1) and the first page has terms I've never heard of (DeDonder gauge) or have heard of but couldn't define without looking them up (Wick rotation).

 

[marcus]

Suppose I want to learn some LQG -- just enough LQG to think I understand something when I actually understand nothing :-)

Is that this? http://www.livingreviews.org/lrr-2008-5

That LivingReview 2008 source is good! It is not the same as Rovelli's book though.

A 2003 draft of Rovelli's book is available at his website free (by arrangement with Cambridge Press)
www.cpt.univ-mrs.fr/~rovelli/book.pdf
I am not suggesting the book for you, Crowell, because there is a more condensed intro. But it does have some interesting philosophical/historical perspective e.g. chapter 1 and it is a useful reference work.
============================================

The goal "to think you understand" would be to see why the constructions in these papers are simple and natural:
1. general review of current formulation (April) 1004.1780
2. first cosmological spinfoam calculation of transition amplitude (March) 1003.3483

These two are probably not the place to start but more the concise criteria for "think you understand". They crystallize the elegant manifoldless development and show where the field is at present. In a certain sense these two explain how the theory is constructed and so are self-contained, but not introductory.

To achieve that goal (of reading the March and April papers with appreciation and satisfaction) one should not begin with those papers. One should begin with rudimentary tutorials like these:

Rovelli Upadhya 1998 (short paper devoted to LQG intro)
Rovelli Gaul 1999 (only the parts developing LQG at intro level)

I would say to read R-U first---it is short. Then find the parts of R-G which go over the same material. R-G also has a short section (pp 43-46) on spinfoam. That links to some 1997 writing by John Baez on spinfoams. Out-of-date but still helpful. You might also sample Lewandowski et al 0909.9939 on spinfoam, to know what up-to-date treatment is there if you need it.

Then see if you can make the jump to the April paper. There are parts of the March paper which cover the same material in a practical straightforward fashion without commentary.
So it may help to read alternately in both the March and April papers.

The March and April papers give an elegant economical way to do what the other three papers do in a more intuitive commonsense brick and mortar way.

Suggested Prep:
Rovelli Upadhya http://arxiv.org/abs/gr-qc/9806079 (easy)
Rovelli Gaul http://arxiv.org/abs//gr-qc/9910079 (just relevant parts, including section on spinfoam pp 43-46)
Lewandowski et al http://arxiv.org/abs/0909.0939 (hard, but reliable up-to-date treatment of spinfoam, at least skim to know it's there)

Suggested Target:
Bianchi Rovelli Vidotto http://arxiv.org/abs/1003.3483
Rovelli http://arxiv.org/abs/1004.1780

References (there is no up-to-date reference but these are useful):
http://www.livingreviews.org/lrr-2008-5
www.cpt.univ-mrs.fr/~rovelli/book.pdf

The March and April papers use the spinfoam vertex recipe given by Lewandowski. But Lewandowski is harder to read because he uses a spacetime manifold and embeds the spin-networks (graphs) and spinfoams (2-complex "histories" of graphs) in the manifold. March and April papers are cleaner and actually easier. But Lewandowski's treatment is thorough and detailed and in some sense more traditional. How to get an introduction to spinfoams? I don't know. Maybe someone will have a suggestion.

Afterthought: John Baez has a 1997 spinfoam paper, written when it was early days. It might serve as a tutorial. He writes clearly.
Maybe one could find a few pages of that 1997 and add them to the "prep" list. Or the spinfoam section out of Rovelli's book might serve. It is always the Lewandowski 2009 paper that brings you up to date, but there should be some "light reading" to give background and motivation before you look at Lewandowski et al, and right now I can't think what would be best.

Afterthought: Baez paper is http://arXiv.org/abs/gr-qc/9709052 and you could try pages 1-4 and the short paragraph at the bottom of page 33 and top of page 34. That paragraph points in the direction which the field has taken by moving to abstract spinfoams instead of ones embedded in a manifold.

Also I have a gathering of occasionally useful LQG source links in this other thread:
https://www.physicsforums.com/showthread.php?p=2868468#post2868468

 

[Coelum]

Dear all,
I am struck by the number and breadth of your answers. Now I have a much clearer picture of the path I need to follow. The good news is that I can attack LQG as soon as I feel confident with QFT and GR. The bad news is that I need to study a lot of geometry and topology before String Theory. But geometry was my first love, so that is not bad news either... :!)
Thanks again to all those who offered their point of view on my question (and many interesting links).

Francesco

 

[bcrowell]

Thanks, Marcus, for being so generous with your time in preparing such a helpful guide to the introuctory literature!

I took a stab at the Rovelli-Upadhya paper. What was really unclear to me was both the mathematical meaning and the physical significance of the SU(2) connection.

Mathematically, I understand the connection to be a rule for parallel transport, and I've seen it specified using either a metric or Christoffel symbols. The paper seems to be defining it using a smooth vector field, which is unfamiliar to me. Is this covered somewhere in texts like Wald and MTW? The reason for giving it values in su(2) is also unclear to me, and I guess that's a separate issue. Since they're talking about a three-dimensional space, I would think that the tangent space would be R³, not su(2)...??

Physically, I don't understand the motivation for introducing all the spin algebra. Is it basically because they want to be able to describe fundamental particles as existing at different places on this graph? Would there be spin-2 gravitons?

I also didn't understand the physical motivation for introducing the graphs. Is the manifold primary and the graphs secondary? Or vice-versa? What do the graphs represent physically?

Why is this all done in a three-dimensional space? Does this three-dimensional space relate to the quantum state of spacetime on some surface of simultaneity? Or does it not even relate to three dimensions of actual spacetime? I'd thought that the dimensionality of spacetime was an emergent property in LQG, and there was some difficulty in even showing that something like Minkowski space was a solution.

 

[marcus]

Crowell, these strike me as material for a discussion of LQG appropriate to have in Beyond forum. It has less to do with academic guidance and more to do with questions about the content of LQG that might be of general interest there. So unless someone objects I will copy your questions and start a thread in BTSM.

...
I took a stab at the Rovelli-Upadhya paper. What was really unclear to me was both the mathematical meaning and the physical significance of the SU(2) connection.

Mathematically, I understand the connection to be a rule for parallel transport, and I've seen it specified using either a metric or Christoffel symbols. The paper seems to be defining it using a smooth vector field, which is unfamiliar to me. Is this covered somewhere in texts like Wald and MTW? The reason for giving it values in su(2) is also unclear to me, and I guess that's a separate issue. Since they're talking about a three-dimensional space, I would think that the tangent space would be R³, not su(2)...??

Physically, I don't understand the motivation for introducing all the spin algebra. Is it basically because they want to be able to describe fundamental particles as existing at different places on this graph? Would there be spin-2 gravitons?

I also didn't understand the physical motivation for introducing the graphs. Is the manifold primary and the graphs secondary? Or vice-versa? What do the graphs represent physically?

Why is this all done in a three-dimensional space? Does this three-dimensional space relate to the quantum state of spacetime on some surface of simultaneity? Or does it not even relate to three dimensions of actual spacetime? I'd thought that the dimensionality of spacetime was an emergent property in LQG, and there was some difficulty in even showing that something like Minkowski space was a solution.

Here is the new thread in Beyond forum, https://www.physicsforums.com/showthread.php?t=426900 , to consider just these basic questions you raised from the Rovelli-Upadhya "LQG Primer" article.