[Physics] Prerequisites to start the study of non-commutative geometry in physics


What are prerequisites (in mathematics and physics), that one should know about for getting into use of ideas from non-commutative geometry in physics?

Best Answer

I am not a fan of Alain Connes's ideas of non-commutative geometry, I prefer Michael Artin. Nor do I think he was Dixmier's best student, although Dixmier thought so. Nevertheless, to be fair, you should understand the physics of General Relativity and Quantum Field Theory first. I recommend Einstein's original papers still, and Yvonne Choquet-Bruhat's two books, Géométrie Différentielle et Systèmes Extérieurs, short and to the point and mathematically clear...obviously it is still commutative geometry, and her much longer, too long, Analysis, Manifolds, and Physics with two excellent co-authors. I also recommend Dirac's own short physics book on the subject. Then for QFT, Pierre Ramond is accessible and standard but very far from the point of view of Connes, which is why it would be good for you to read so you do not stay way too isolated. Also for QFT read the work of Irving Segal (sometimes with Roe Goodman) on relativistic quantum fields from the operator algebra point of view, but he is not a physicist so you should also look at Streater and Wightman PCT, Spin and Statistics, And All That... to see how physicists do operator algebra approaches to QFT.

If you do not yet even know Quantum Mechanics, I can recommend Sudbery's Quantum Mechanics and the Particles of Nature.

I assume from the way you phrase your question that you already are familiar with Connes's work itself. If not, you would need to prepare by studying Dixmier's book on von Neumann algebras or Guichardet or Dixmier, Les $C^*$-algebres et leurs représentations. Hope this helps.