Recently, Li Qin, associate researcher of the Institute of Quantum Science and Engineering of Southern University of Science and Technology, and professors Liang Yicong and Chen Guowei of Hong Kong Chinese University have made important progress in the field of geometric quantization and morphological variable subtification in mathematical physics. They proposed the concept of non-formal quantization and constructed concrete examples of such quantization on Cayley manifolds, laying the foundation for further research on geometric quantization and problems related to high-dimensional quantum field theory. The results were published in Advances in Mathematics.

**Research diagram Courtesy of SUSTech**

In mathematical physics , the mathematical foundations of quantum mechanics are based on Hilbert spaces satisfying the Dirac–von Neumann axiom system and the corresponding operator algebras. Since phase space in classical mechanics corresponds to symplectic manifolds in geometry, quantization has become one of the central problems in the study of symplectic geometry.

Guided by the Dirac-von Neumann axiom , the study of quantization of symplectic manifolds mainly includes two branches: geometric quantization and morphological quantization , focusing on the algebra A composed of Hilbert spaces and operators in axiom systems, respectively. Since the two theories of geometric quantization and morphological variable suburbanization are relatively independent in the development process, how algebra A acts on Hilbert space H is an unknown problem. One of the technical difficulties is that the shape-variable sub-algebra A includes in its definition the formal variable h corresponding to Planck’s constant in physics.

In this regard, the key to the research team’s previous technical problems and breakthroughs was to find a dense subalgebra D in algebra A so that the formal variable h could be assigned in it. Further, the team placed the breakthrough point in a class of symplectic manifolds with better geometric properties, namely Cayley manifolds, and proposed the concept of non-formal quantization.

In the course of the research, the research team used a series of new techniques and methods, first, there is a special L infinite algebraic structure on the Cayley manifold, and by deforming and expanding this algebraic structure, a flat connection on such geometric objects is constructed. This special flat connection can satisfy the excellent symmetry property, so that it can not only be used to construct a deformation algebra A with a formal variable h, but also assign the formal variable to the value h=1/k in the subalgebra D, and obtain the specific implementation of the non-formal quantized algebra D proposed by the team.

Subsequently, the team generalized the past Fedosov quantization method to construct the Hilbert space H. Building on the previous two steps, the research team found that this algebra D has a Bargmann-Fock effect on the Hilbert space H when k is taken as a positive integer, and proved that this effect gives an isomorphism of the differential operator on the Hilbert space for non-formal algebras. This series of achievements completely solves and answers the key problems in the quantization of symplectic manifolds, and brings enlightenment to related problems in high-dimensional quantum field theory. (Source: China Science News, Diao Wenhui)

Related paper information:https://doi.org/10.1016/j.aim.2023.109293