58 research outputs found
Quasistationary quaternionic Hamiltonians and complex stochastic maps
We show that the complex projections of time-dependent -quasianti-Hermitian quaternionic Hamiltonian dynamics are complex stochastic
dynamics in the space of complex quasi-Hermitian density matrices if and only
if a quasistationarity condition is fulfilled, i. e., if and only if is
an Hermitian positive time-independent complex operator. An example is also
discussed.Comment: Submitted to J. Phys. A on October 25 200
Quantum Bi-Hamiltonian systems, alternative Hermitian structures and Bi-Unitary transformations
We discuss the dynamical quantum systems which turn out to be bi-unitary with
respect to the same alternative Hermitian structures in a infinite-dimensional
complex Hilbert space. We give a necessary and sufficient condition so that the
Hermitian structures are in generic position. Finally the transformations of
the bi-unitary group are explicitly obtained.Comment: Note di Matematica vol 23, 173 (2004
Alternative Algebraic Structures from Bi-Hamiltonian Quantum Systems
We discuss the alternative algebraic structures on the manifold of quantum
states arising from alternative Hermitian structures associated with quantum
bi-Hamiltonian systems. We also consider the consequences at the level of the
Heisenberg picture in terms of deformations of the associative product on the
space of observables.Comment: Accepted for publication in Int. J. Geom. Meth. Mod. Phy
The Quantum-Classical Transition: The Fate of the Complex Structure
According to Dirac, fundamental laws of Classical Mechanics should be
recovered by means of an "appropriate limit" of Quantum Mechanics. In the same
spirit it is reasonable to enquire about the fundamental geometric structures
of Classical Mechanics which will survive the appropriate limit of Quantum
Mechanics. This is the case for the symplectic structure. On the contrary, such
geometric structures as the metric tensor and the complex structure, which are
necessary for the formulation of the Quantum theory, may not survive the
Classical limit, being not relevant in the Classical theory. Here we discuss
the Classical limit of those geometric structures mainly in the Ehrenfest and
Heisenberg pictures, i.e. at the level of observables rather than at the level
of states. A brief discussion of the fate of the complex structure in the
Quantum-Classical transition in the Schroedinger picture is also mentioned.Comment: 19 page
Exact PT-Symmetry Is Equivalent to Hermiticity
We show that a quantum system possessing an exact antilinear symmetry, in
particular PT-symmetry, is equivalent to a quantum system having a Hermitian
Hamiltonian. We construct the unitary operator relating an arbitrary
non-Hermitian Hamiltonian with exact PT-symmetry to a Hermitian Hamiltonian. We
apply our general results to PT-symmetry in finite-dimensions and give the
explicit form of the above-mentioned unitary operator and Hermitian Hamiltonian
in two dimensions. Our findings lead to the conjecture that non-Hermitian
CPT-symmetric field theories are equivalent to certain nonlocal Hermitian field
theories.Comment: Few typos have been corrected and a reference update
On the pseudo-Hermitian nondiagonalizable Hamiltonians
We consider a class of (possibly nondiagonalizable) pseudo-Hermitian
operators with discrete spectrum, showing that in no case (unless they are
diagonalizable and have a real spectrum) they are Hermitian with respect to a
semidefinite inner product, and that the pseudo-Hermiticity property is
equivalent to the existence of an antilinear involutory symmetry. Moreover, we
show that a typical degeneracy of the real eigenvalues (which reduces to the
well known Kramers degeneracy in the Hermitian case) occurs whenever a
fermionic (possibly nondiagonalizable) pseudo-Hermitian Hamiltonian admits an
antilinear symmetry like the time-reversal operator . Some consequences and
applications are briefly discussed.Comment: 22 page
Alternative Descriptions in Quaternionic Quantum Mechanics
We characterize the quasianti-Hermitian quaternionic operators in QQM by
means of their spectra; moreover, we state a necessary and sufficient condition
for a set of quasianti-Hermitian quaternionic operators to be anti-Hermitian
with respect to a uniquely defined positive scalar product in a infinite
dimensional (right) quaternionic Hilbert space. According to such results we
obtain two alternative descriptions of a quantum optical physical system, in
the realm of quaternionic quantum mechanics, while no alternative can exist in
complex quantum mechanics, and we discuss some differences between them.Comment: 16 page
Complex Projection of Quasianti-Hermitian Quaternionic Hamiltonian Dynamics
We characterize the subclass of quasianti-Hermitian quaternionic Hamiltonian dynamics such that their complex projections are one-parameter semigroup dynamics in the space of complex quasi-Hermitian density matrices. As an example, the complex projection of a spin-½ system in a constant quasianti-Hermitian quaternionic potential is considered
Pseudohermitian Hamiltonians, time-reversal invariance and Kramers degeneracy
A necessary and sufficient condition in order that a (diagonalizable)
pseudohermitian operator admits an antilinear symmetry T such that T^{2}=-1 is
proven. This result can be used as a quick test on the T-invariance properties
of pseudohermitian Hamiltonians, and such test is indeed applied, as an
example, to the Mashhoon-Papini Hamiltonian.Comment: 6 page
Pseudo-Hermitian Hamiltonians, indefinite inner product spaces and their symmetries
We extend the definition of generalized parity , charge-conjugation
and time-reversal operators to nondiagonalizable pseudo-Hermitian
Hamiltonians, and we use these generalized operators to describe the full set
of symmetries of a pseudo-Hermitian Hamiltonian according to a fourfold
classification. In particular we show that and are the generators of
the antiunitary symmetries; moreover, a necessary and sufficient condition is
provided for a pseudo-Hermitian Hamiltonian to admit a -reflecting
symmetry which generates the -pseudounitary and the -pseudoantiunitary
symmetries. Finally, a physical example is considered and some hints on the
-unitary evolution of a physical system are also given.Comment: 20 page
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