Path to quantum geometry for mathematicians?

Question

I am looking for an introduction of Quantum Geometry (math-subject) for mathematicians. This paper presents a survey however I am looking for something with more mathematical depth and motivation (for a no-physicist). Something like the book Quantum Geometry: A Framework for Quantum General Relativity but for mathematicians.

Just to be clear: I am aware of the these questions on quantum physics for mathematicians and this question on quantum mechanics. I am asking for a specific kind of geometry (quantum geometry) from the perspective of geometry (mathematics). This is not a general question asking about the geometry used in quantum physics.

Answer 1

Here are some references from the physicists' perspective: https://www.quora.com/What-is-the-most-standard-prescribed-book-for-understanding-quantum-physics-in-universities

However, as a mathematician trying to dig into quantum theory as presented in phycisits' literature, you will encounter conjectures/ad hoc arguments that may not convince you (and probably should not). For a discussion of this issue from a perspective of mathematical physics, consider this recent paper by Maik Reddiger. It also comes with a readable introduction on the history of attempts to formalize "quantization" beginning with Dirac.

The main mathematical tools come from differential geometry (and, of course, as always in Physics, ODE and PDE theory). For some references to differential geometry, see here: https://mathoverflow.net/questions/395/reading-list-for-basic-differential-geometry

Answer 2

It's not 100% clear to me that "quantum geometry" is actually a well-defined term; based on the paper linked in the OP, it appears to be something of an ad hoc collection of related techniques. But at the core of those techniques seems to be the theory of Noncommutative Geometry, so one should certainly begin with Alain Connes's book of that title. (See https://books.google.com/books?id=QofCkAJjreYC&lpg=PP1&pg=PP1#v=onepage&q&f=false; apparently also available as a PDF from the author's website, at www.alainconnes.org/docs/book94bigpdf.pdf). This text (now almost 25 years old -- boy do I feel old!) is a thoroughly mathematical development of the theory of noncommutative spaces, which seems to be the foundation on which "quantum geometry" is based.