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

Department Mathematical Modeling, Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Boul. Tzarigradsko chaussée 72, 1784 Sofia, Bulgaria
International Journal of Modern Physics A (Impact Factor: 1.7). 01/2012; 17(02). DOI: 10.1142/S0217751X02005669
Source: arXiv


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 section along paths in this bundle. The evolution of a pure state is determined through the bundle (analog 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 work. The present fourth part of this series is devoted mainly to the fibre bundle description of mixed quantum states. We show that to the conventional density operator there corresponds a unique density lifting of paths for which the corresponding equations of motion are derived. It is also investigated the bundle description of mixed quantum states in the different pictures of motion. We calculate the curvature of the evolution transport and prove that it is curvature free iff the values of the Hamiltonian operator at different moments commute.

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Available from: Bozhidar Zakhariev Iliev, Sep 29, 2015
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    • "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; 1340(1). DOI:10.1063/1.3582756
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    • "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
    Physica Scripta 05/2001; 68(1). DOI:10.1238/Physica.Regular.068a00022 · 1.13 Impact Factor
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    • "In both works the quantum evolution is described via appropriate transport along paths: In [1, see, e.g., equations (3.54) and (4.53)] this is an 'Abelian' parallel transport along curves, whose holonomy group is U (1) [1, equation (4.38)], while in the investigation [2] [3] [4] [5] [6] is employed a transport along paths uniquely determined by the Hamiltonian (see Sect. I.5) which, generally, need not to be a parallel translation. "
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