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High-Order Adiabatic Approximation for Non-Hermitian Quantum System and Complexization of Berry's Phase
In this paper the evolution of a quantum system drived by a non-Hermitian
Hamiltonian depending on slowly-changing parameters is studied by building an
universal high-order adiabatic approximation(HOAA) method with Berry's phase
,which is valid for either the Hermitian or the non-Hermitian cases. This
method can be regarded as a non-trivial generalization of the HOAA method for
closed quantum system presented by this author before. In a general situation,
the probabilities of adiabatic decay and non-adiabatic transitions are
explicitly obtained for the evolution of the non-Hermitian quantum system. It
is also shown that the non-Hermitian analog of the Berry's phase factor for the
non-Hermitian case just enjoys the holonomy structure of the dual linear bundle
over the parameter manifold. The non-Hermitian evolution of the generalized
forced harmonic oscillator is discussed as an illustrative examples.Comment: ITP.SB-93-22,17 page
Consequences of 't Hooft's Equivalence Class Theory and Symmetry by Large Coarse Graining
According to 't Hooft (Class.Quantum.Grav. 16 (1999), 3263), quantum gravity
can be postulated as a dissipative deterministic system, where quantum states
at the ``atomic scale''can be understood as equivalence classes of primordial
states governed by a dissipative deterministic dynamics law at the ``Planck
scale''. In this paper, it is shown that for a quantum system to have an
underlying deterministic dissipative dynamics, the time variable should be
discrete if the continuity of its temporal evolution is required. Besides, the
underlying deterministic theory also imposes restrictions on the energy
spectrum of the quantum system. It is also found that quantum symmetry at the
``atomic scale'' can be induced from 't Hooft's Coarse Graining classification
of primordial states at the "Planck scale".Comment: 12 papge, Late
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