Differential Geometry for General Relativity and Yang-Mills Theories

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Discussion Overview

The discussion centers around recommendations for texts on differential geometry, particularly in relation to quantum field theory (QFT) and general relativity (GR). Participants share their experiences and preferences for various books that could aid in self-learning the mathematical foundations necessary for these advanced topics.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant seeks recommendations for accessible texts on differential geometry applicable to QFT and GR.
  • Another participant suggests "Elementary Differential Geometry" by Pressley as a suitable primer.
  • A different participant recommends Mikio Nakahara's book, providing a link for reference.
  • One participant prefers O'Neill's book for its focus on the basics of differential geometry of surfaces.
  • Another participant advocates for John M. Lee's books, noting their quality and relevance, while also mentioning Isham's book for gauge theories, despite its lack of rigor and depth.
  • Further recommendations include Fecko and Frankel for deeper exploration, although one participant admits to not having read them thoroughly.
  • A participant shares their impression of Frankel's text, noting its utility as a reference but expressing a preference for more exposition in learning materials.

Areas of Agreement / Disagreement

Participants present a variety of recommendations without a clear consensus on a single best text. Multiple competing views on preferred resources remain evident.

Contextual Notes

Some recommendations may depend on the reader's background and learning style, with varying levels of rigor and accessibility noted among the suggested texts.

Who May Find This Useful

Individuals interested in self-learning differential geometry with applications to quantum field theory and general relativity may find these discussions and recommendations beneficial.

boltzman1969
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I have been teaching myself QFT and General Relativity. The mathematics of those fields is daunting, and I find that what I have come across is very difficult to master. Of course it will take work, but can someone recommend a good text for self-leaning differential geometry with application to QFTs (particularly non-abelian gauge theories) and GR? One that is clearly written and accessible to an amateur like myself. Thank you in advance.
 
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I would recommend Elementary Differential Geometry by Pressley as a primer to differential geometry. And maybe supplement it with Differential Geometry by Kreyszig.
 
I like O'Neill for the basics of differential geometry of surfaces, which is the easiest place to start.
 
I think the best place to learn the basics of differential geometry has to be the books by Lee. They are close to perfect. Isham's book is a good place to get an introduction to the stuff that you need for gauge theories (but not the stuff you need for GR). If I remember correctly, it's not entirely rigorous, and it doesn't go deep enough, but it's a good place to start.

Those are the books I have actually studied. If you want to go deeper, I think Fecko looks very interesting, and so does Frankel. But I haven't actually read them.

John M. Lee: Introduction to smooth manifolds
John M. Lee: Riemannian manifolds: an introduction to curvature
Isham: Modern differential geometry for physicists
Fecko: Differential geometry and Lie groups for physicists
Frankel: The geometry of physics: an introduction
 
I have Frankel's text. I flipped through it to skim certain material; my general impression: if you can gain insight from just mathematical expressions , then it is great. I like a little more exposition myself. It is a better reference text once one has learned from a more accessible text.
 

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