13 research outputs found
From quantum cellular automata to quantum lattice gases
A natural architecture for nanoscale quantum computation is that of a quantum
cellular automaton. Motivated by this observation, in this paper we begin an
investigation of exactly unitary cellular automata. After proving that there
can be no nontrivial, homogeneous, local, unitary, scalar cellular automaton in
one dimension, we weaken the homogeneity condition and show that there are
nontrivial, exactly unitary, partitioning cellular automata. We find a one
parameter family of evolution rules which are best interpreted as those for a
one particle quantum automaton. This model is naturally reformulated as a two
component cellular automaton which we demonstrate to limit to the Dirac
equation. We describe two generalizations of this automaton, the second of
which, to multiple interacting particles, is the correct definition of a
quantum lattice gas.Comment: 22 pages, plain TeX, 9 PostScript figures included with epsf.tex
(ignore the under/overfull \vbox error messages); minor typographical
corrections and journal reference adde
The Real Symplectic Groups in Quantum Mechanics and Optics
text of abstract (We present a utilitarian review of the family of matrix
groups , in a form suited to various applications both in optics
and quantum mechanics. We contrast these groups and their geometry with the
much more familiar Euclidean and unitary geometries. Both the properties of
finite group elements and of the Lie algebra are studied, and special attention
is paid to the so-called unitary metaplectic representation of .
Global decomposition theorems, interesting subgroups and their generators are
described. Turning to -mode quantum systems, we define and study their
variance matrices in general states, the implications of the Heisenberg
uncertainty principles, and develop a U(n)-invariant squeezing criterion. The
particular properties of Wigner distributions and Gaussian pure state
wavefunctions under action are delineated.)Comment: Review article 43 pages, revtex, no figures, replaced because
somefonts were giving problem in autometic ps generatio
Product rule for gauge invariant Weyl symbols and its application to the semiclassical description of guiding center motion
We derive a product rule for gauge invariant Weyl symbols which provides a
generalization of the well-known Moyal formula to the case of non-vanishing
electromagnetic fields. Applying our result to the guiding center problem we
expand the guiding center Hamiltonian into an asymptotic power series with
respect to both Planck's constant and an adiabaticity parameter already
present in the classical theory. This expansion is used to determine the
influence of quantum mechanical effects on guiding center motion.Comment: 24 pages, RevTeX, no figures; shortened version will be published in
J.Phys.
Uncertainty Relations in Deformation Quantization
Robertson and Hadamard-Robertson theorems on non-negative definite hermitian
forms are generalized to an arbitrary ordered field. These results are then
applied to the case of formal power series fields, and the
Heisenberg-Robertson, Robertson-Schr\"odinger and trace uncertainty relations
in deformation quantization are found. Some conditions under which the
uncertainty relations are minimized are also given.Comment: 28+1 pages, harvmac file, no figures, typos correcte
Cotangent bundle quantization: Entangling of metric and magnetic field
For manifolds of noncompact type endowed with an affine connection
(for example, the Levi-Civita connection) and a closed 2-form (magnetic field)
we define a Hilbert algebra structure in the space and
construct an irreducible representation of this algebra in . This
algebra is automatically extended to polynomial in momenta functions and
distributions. Under some natural conditions this algebra is unique. The
non-commutative product over is given by an explicit integral
formula. This product is exact (not formal) and is expressed in invariant
geometrical terms. Our analysis reveals this product has a front, which is
described in terms of geodesic triangles in . The quantization of
-functions induces a family of symplectic reflections in
and generates a magneto-geodesic connection on . This
symplectic connection entangles, on the phase space level, the original affine
structure on and the magnetic field. In the classical approximation,
the -part of the quantum product contains the Ricci curvature of
and a magneto-geodesic coupling tensor.Comment: Latex, 38 pages, 5 figures, minor correction
Dual symmetric Lagrangians and conservation laws
By using a complex field with a symmetric combination of electric and magnetic fields, a first-order covariant Lagrangian for Maxwell's equations is obtained. This leads to a dual-symmetric quantum-field theory with an infinite set of local conservation laws. The dual symmetry is shown to correspond to a helical phase, conjugate to a conserved helicity number. [S1050-2947(99)50809-3]