What is the point of geometric quantization?

[andresB]

I studied the basics of geometric quantization for a recent work in quantum-classical hybrid systems¹. It was an easy application of the method of gometric quantization (prequantization + polarization in $\mathbb{R}^{3}$).
The whole topic seems interesting since I want to learn more of symplectic geometry, but (outside the aforementioned half-quantization to get a quantum-classical theory) I fail to see the point of the whole endeavor. For example, particles constrained to move on a surface are treated with the formalism of the geometric potential of Jensen, Koppe, and Da Costa² , and not from a quantization of the related classical situation.

What actual result from geometric quantization is important and useful outside the mathematical dicipline of geometric quantization itself?[1]https://arxiv.org/abs/2107.03623
[2]https://arxiv.org/abs/1602.00528

 

[A. Neumaier]

What actual result from geometric quantization is important and useful outside the mathematical dicipline of geometric quantization itself?

It is important for the understanding of quantization in general, for integrable quantum systems and for the construction of irreducible unitary representations of groups.

 

[andresB]

It is important for the understanding of quantization in general, for integrable quantum systems and for the construction of irreducible unitary representations of groups.

Well, the first part does not sound convincing. Does geometric quantization reproduces the quantum mechanic of a partice constrained to move on a surface embbeded in $\mathbb{R}^{3}$?
Does it produce testable results that can't be obtained just by using the usual tools of QM?The unitary representation of groups does sound interesting. My work was about the Galilei Group in classical and "half"-quantized systems. Any good reference on group representation and geometric quantization?

 

[A. Neumaier]

Does geometric quantization reproduces the quantum mechanic of a partice constrained to move on a surface

Understanding and reproducing details are very different issues.

Any good reference on group representation and geometric quantization?

https://link.springer.com/chapter/10.1007/978-3-662-06791-8_2
https://arxiv.org/pdf/math-ph/0208008.pdf
https://www.math.columbia.edu/~woit/QM/qmbook.pdf
https://arxiv.org/pdf/1801.02307.pdf
https://link-springer-com.uaccess.univie.ac.at/content/pdf/10.1007/BF02097053.pdf
https://s3.cern.ch/inspire-prod-files-6/6869199e89f197300faf2f93e55dc112

 

[CelHolo]

Woodhouse is the standard reference on Geometric Quantization and quite well written.