Hamiltonian aspects of quantum theory

Abstract

Listen

Translate article

This paper considers Hamiltonian structures related to quantum mechanics. We do not discuss structures used in the quantization of classical Hamil� tonian or Lagrangian systems (which was first done by Heisenberg and Feynman); on the contrary, we are mainly interested in structures which make it possible to consider quantum systems as classical (although infinitedimensional in the most natural cases) Hamiltonian (or Lagrangian) systems. The introduc� tion of such structures can be called the dequantiza� tion of a quantum system. We emphasize that, whereas the quantization of a classical (Hamiltonian or Lagrangian) system sup� poses the introduction of fundamentally new mathe� matical structures (usually related to the passage from ordinary differential equations to partial differential equations), dequantization means only the passage to an equivalent description of the same quantum sys� tem. To obtain the result of two successive operations (usual quantization and subsequent dequantization), it suffices to know only properties of the initial classi� cal system; therefore, it can be said that the passage from the initial classical Hamiltonian (or Lagrangian) system to the Hamiltonian system being the result of the corresponding dequantization is yet another quan� tization method of the initial system, different from quantizations in the sense of Schrodinger, Heisenberg, and Feynman. We refer to this quantization as Hamil� tonian quantization (precise definitions are given below).