Does the concept of quantum correspond to a mathematical description? For example, could a combination of the concepts of symplectic geometry (which is the fundamental component in a mathematical formulation in mechanics) and algebraic geometry give an idea of the quantum concept?
This is an important topic.
A good introduction to the relationship between geometry and quanta is given in
Geometry and the Quantum: Basics
https://arxiv.org/pdf/1411.0977.pdf
Starting on page 25, see the presentation of a particle picture in quantum gravity.
More to the point, see
Quantum Behaviors on an Excreting Black Hole
https://arxiv.org/abs/0810.4515
In this paper, J. Lindesay observes that geometries with horizons offer insights into the intricate relationships between general relativity and quantum physics.
This is an important topic.
A good introduction to the relationship between geometry and quanta is given in
Geometry and the Quantum: Basics
https://arxiv.org/pdf/1411.0977.pdf
Starting on page 25, see the presentation of a particle picture in quantum gravity.
More to the point, see
Quantum Behaviors on an Excreting Black Hole
https://arxiv.org/abs/0810.4515
In this paper, J. Lindesay observes that geometries with horizons offer insights into the intricate relationships between general relativity and quantum physics.
The historical book which presents the fundamental mathematics of Quantum Mechanics was written by von Neumann many years ago and this year 2018 re-edited as one of the jewels of the scientific literature
J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 2018.
Dear Daniel,
Many thanks for your excellent observation and the reference to J. von Neumann's book.
Dear James,
Your references were also very important and in fact are within the question where is suggested the symplectic formalism. Elie Cartan introduced it and it was employed more for mathematicians generalizing it to Kähler manifolds, but unfortunatally this geometry was not too successful in Physics. What was very successful was the geometry associted to the functional spaces and in this area von Neumann was a real master (at least for me and for generations of physicists).
Dear Professor Peters,
Thank you for sharing the documents and experiences as a multi-published author in Engineering and for your generous contributions.
Dear all,
As a supplement to the reference mentioned by Daniel Baldomir, I would like to add the article “Von Neumann’s Contributions to Quantum Theory”, by Leon Van Hove, which is available at https://pdfs.semanticscholar.org/74fd/cf3b1b128bc6db3eda9183c8a0c5a647000c.pdf.
Dear Spiros,
The another mathematical formalism which has produced a good progress in Quantum Mechanics is the classical Dirac's book
http://digbib.ubka.uni-karlsruhe.de/volltexte/wasbleibt/57355817/57355817.pdf
where the wave function can be represented as a vector.
Dear Daniel,
Thank you. I think that "The Principles of Quantum Mechanics" is maybe the most classical of all classical books in quantum mechanics.
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