1998/03/31 by Bozhidar Z. Iliev, Bozhidar Z Iliev
Mathematics · Physics and Astronomy · #Quantum Mechanics and Non-Hermitian Physics #Quantum, superfluid, helium dynamics #Topological Materials and Phenomena #gr-qc #hep-th #math.DG #math.GT #quant-ph
paper · pdf · doi:10.1088/0305-4470/34/23/308
published as J.Phys.A34:4887-4918,2001 · 26 standard (11pt, A4) LaTeX 2e pages. The packages AMS-LaTeX and amsfonts are required. Revised: new material, references, and comments are added. Minor style chages. Continuation of quan-ph/9803083. For continuation of the this series see http://www.inrne.bas.bg/mathmod/bozhome/
arxiv created 1998/10/16 · openalex publication_date 2001/05/31 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/31
We propose a new systematic fibre bundle formulation of nonrelativistic quantum mechanics. The new form of the theory is equivalent to the usual one and is in harmony with the modern trends in theoretical physics and potentially admits new generalizations in different directions. In it the Hilbert space of a quantum system (from conventional quantum mechanics) is replaced with an appropriate Hilbert bundle of states and a pure state of the system is described by a lifting of paths or sections along paths in this bundle. The evolution of a pure state is determined through the bundle (analogue of the) Schrödinger equation. Now the dynamical variables and density operators are described via liftings of paths or morphisms along paths in suitable bundles. The mentioned quantities are connected by a number of relations derived in this paper. The present, first, part of this investigation is devoted to the introduction of basic concepts on which the fibre bundle approach to quantum mechanics rests. We show that the evolution of pure quantum mechanical states can be described as a suitable linear transport along paths, called evolution transport, of the state liftings in the Hilbert bundle of states of a considered quantum system.