Quantization: History and problems

Is time a cyclic dimension? Canonical quantization implicit in classical cyclic dynamics

“…This article has a mathematical character but we will use a formalism familiar to physicists. Our strategy will be the following: i) we introduce a Dirac constraint of intrinsic periodicity, first in Lagrangian form and then in Hamiltonian form [18,19], which projects ordinary Hamiltonian dynamics (non compact manifold) into related intrinsically cyclic dynamics (compact manifold); ii) we prove, by using theorems of Geometric Quantization (GQ) [1, [20][21][22][23][24], that the resulting intrinsically cyclic dynamics naturally satisfies Dirac's rules of canonical quantization -without postulating them. In short, the canonical quantization is equivalent to a local transformation from ordinary non-compact manifolds into corresponding intrinsically compact manifolds, see also [25,26].…”

“Internal times” and how to second-quantize fields by means of periodic boundary conditions

“…The answer has been given in the form of theorem in recent [3] by using results of Geometric Quantization [14][15][16][17] and without interpretational ambiguities. Under the very general hypothesis of Hamiltonian systems, i.e., generic symplectic manifolds of even dimensions equipped with a non-degenerate, closed 2-form, and U(1) symmetry, the result of such PBCs imposed as constraint is a quantization formally equivalent to the canonical quantization of the system.…”

Mathematics Behind the Heisenberg Uncertainty Principle

“…(2) Recalling the Broglie relation, 𝜆 = ℎ/𝑝 then gives 𝐿 = 𝑝𝑟 = 𝑛ℏ which is just Bohr's quantization condition (Carosso, 2022;Dolce, 2023).…”