19 research outputs found
Modular Structures on Trace Class Operators and Applications to Landau Levels
The energy levels, generally known as the Landau levels, which characterize
the motion of an electron in a constant magnetic field, are those of the
one-dimensional harmonic oscillator, with each level being infinitely
degenerate. We show in this paper how the associated von Neumann algebra of
observables display a modular structure in the sense of the Tomita-Takesaki
theory, with the algebra and its commutant referring to the two orientations of
the magnetic field. A KMS state can be built which in fact is the Gibbs state
for an ensemble of harmonic oscillators. Mathematically, the modular structure
is shown to arise as the natural modular structure associated to the Hilbert
space of all Hilbert-Schmidt operators
Multi Matrix Vector Coherent States
A class of vector coherent states is derived with multiple of matrices as
vectors in a Hilbert space, where the Hilbert space is taken to be the tensor
product of several other Hilbert spaces. As examples vector coherent states
with multiple of quaternions and octonions are given. The resulting generalized
oscillator algebra is briefly discussed. Further, vector coherent states for a
tensored Hamiltonian system are obtained by the same method. As particular
cases, coherent states are obtained for tensored Jaynes-Cummings type
Hamiltonians and for a two-level two-mode generalization of the Jaynes-Cummings
model.Comment: 24 page