# Fibre bundle formulation of nonrelativistic quantum mechanics. IV. Mixed states and evolution transport's curvature

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Bozhidar Zakhariev Iliev, Jul 30, 2015 Available from:-
- "The present investigation can be regarded as a continuation of the fibre bundle formulation of quantum physics begun in [1] [2] [3] [4] [5]. Here we have applied a slightly different approach to the (canonical) quantum field theory in Heisenberg picture. "

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**ABSTRACT:**The paper contains a differential-geometric foundations for an attempt to formulate Lagrangian (canonical) quantum field theory on fibre bundles. In it the standard Hilbert space of quantum field theory is replace with a Hilbert bundle; the former playing a role of a (typical) fibre of the letter one. Suitable sections of that bundle replace the ordinary state vectors and the operators on the system's Hilbert space are transformed into morphisms of the same bundle. In particular, the field operators are mapped into corresponding field morphisms.10/2010; DOI:10.1063/1.3582756 -
- "In the series of papers [2] [3] [4] [5] [6] [7] we have reformulated nonrelativistic quantum mechanics in terms of fibre bundles. The mathematical base for this was the Schrödinger equation i dψ(t) dt = H(t)ψ(t), (2.1) where i ∈ C is the imaginary unit, (= h/2π) is the Plank constant divided by 2π, ψ is the system's state vector belonging to an appropriate Hilbert space F, and H is the system's Hamiltonian. "

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**ABSTRACT:**We propose a new fibre bundle formulation of the mathematical base of relativistic quantum mechanics. At the present stage the bundle form of the theory is equivalent to its conventional one, but it admits new types of generalizations in different directions. In the present first part of our investigation we consider the time-dependent or Hamiltonian approach to bundle description of relativistic quantum mechanics. In it the wavefunctions are replaced by (state) liftings of paths or sections along paths of a suitably chosen vector bundle over space-time whose (standard) fibre is the space of the wavefunctions. Now the quantum evolution is described as a linear transportation (by means of the evolution transport along paths in the space-time) of the state liftings/sections in the (total) bundle space. The equations of these transportations turn to be the bundle versions of the corresponding relativistic wave equations. Comment: 16 standard LaTeX pages. The packages AMS-LaTeX and amsfonts are required. The paper continuous the application of fibre bundle formalism to quantum physics began in the series of works quant-ph/9803083, quant-ph/9803084, quant-ph/9804062, quant-ph/9806046, quant-ph/9901039, quant-ph/9902068, and quant-ph/0004041. For related papers, view http://theo.inrne.bas.bg/~bozho/Physica Scripta 05/2001; 68(1). DOI:10.1238/Physica.Regular.068a00022 · 1.30 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**The theory of linear transports along paths in vector bundles, generalizing the parallel transports generated by linear connections, is developed. The normal frames for them are defined as ones in which their matrices are the identity matrix or their coefficients vanish. A number of results, including theorems of existence and uniqueness, concerning normal frames are derived. Special attention is paid to the important case when the bundle's base is a manifold. The normal frames are defined and investigated also for derivations along paths and along tangent vector fields in the last case. It is proved that normal frames always exist at a single point or along a given (smooth) path. On other subsets normal frames exist only as an exception if (and only if) certain additional conditions, derived here, are satisfied. Gravity physics and gauge theories are pointed out as possible fields for application of the results obtained.10/1998;