14 research outputs found
Brownian motion on a smash line
Brownian motion on a smash line algebra (a smash or braided version of the
algebra resulting by tensoring the real line and the generalized paragrassmann
line algebras), is constructed by means of its Hopf algebraic structure.
Further, statistical moments, non stationary generalizations and its diffusion
limit are also studied. The ensuing diffusion equation posseses triangular
matrix realizations.Comment: Latex, 6 pages no figures. Submitted to Journal of Nonlinear
Mathematical Physics. Special Issue of Proccedings of NEEDS'9
Pseudo Memory Effects, Majorization and Entropy in Quantum Random Walks
A quantum random walk on the integers exhibits pseudo memory effects, in that
its probability distribution after N steps is determined by reshuffling the
first N distributions that arise in a classical random walk with the same
initial distribution. In a classical walk, entropy increase can be regarded as
a consequence of the majorization ordering of successive distributions. The
Lorenz curves of successive distributions for a symmetric quantum walk reveal
no majorization ordering in general. Nevertheless, entropy can increase, and
computer experiments show that it does so on average. Varying the stages at
which the quantum coin system is traced out leads to new quantum walks,
including a symmetric walk for which majorization ordering is valid but the
spreading rate exceeds that of the usual symmetric quantum walk.Comment: 3 figures include
Chord diagrams and BPHZ subtractions
The combinatorics of the BPHZ subtraction scheme for a class of ladder graphs
for the three point vertex in theory is transcribed into certain
connectivity relations for marked chord diagrams (knots with transversal
intersections). The resolution of the singular crossings using the equivalence
relations in these examples provides confirmation of a proposed fundamental
relationship between knot theory and renormalization in perturbative quantum
field theory.Comment: 12 pages, 5 Postscript figures, LaTex 2
Twisted Quantum Affine Superalgebra , Invariant R-matrices and a New Integrable Electronic Model
We describe the twisted affine superalgebra and its quantized
version . We investigate the tensor product representation
of the 4-dimensional grade star representation for the fixed point
subsuperalgebra . We work out the tensor product decomposition
explicitly and find the decomposition is not completely reducible. Associated
with this 4-dimensional grade star representation we derive two
invariant R-matrices: one of them corresponds to and the
other to . Using the R-matrix for , we
construct a new invariant strongly correlated electronic model,
which is integrable in one dimension. Interestingly, this model reduces, in the
limit, to the one proposed by Essler et al which has a larger, ,
symmetry.Comment: 17 pages, LaTex fil
Region Operators of Wigner Function: Transformations, Realizations and Bounds
An integral of the Wigner function of a wavefunction |psi >, over some region
S in classical phase space is identified as a (quasi) probability measure (QPM)
of S, and it can be expressed by the |psi > average of an operator referred to
as the region operator (RO). Transformation theory is developed which provides
the RO for various phase space regions such as point, line, segment, disk and
rectangle, and where all those ROs are shown to be interconnected by completely
positive trace increasing maps. The latter are realized by means of unitary
operators in Fock space extended by 2D vector spaces, physically identified
with finite dimensional systems. Bounds on QPMs for regions obtained by tiling
with discs and rectangles are obtained by means of majorization theory.Comment: 16 pages, 4 figures. Hurst Bracken Festschrift, Reports of
Mathematical Physics, Feb 2006, to appea
Remarks on Q-oscillators representation of Hopf-type boson algebras
We present a method of constructing known deformed or undeformed oscillators as quotients of certain models of Hopf-type oscillator algebras, using similar techniques to those of determining fix point sets of the adjoint action of a Hopf algebra. Moreover we give a characterization of these models in terms of these quotients coupled to Euclidean Clifford algebra. A theorem is proved which provides representations of the models, induced from those of a certain type of quotient algebra
Anyonic Quantum Walks
The one dimensional quantum walk of anyonic systems is presented. The anyonic
walker performs braiding operations with stationary anyons of the same type
ordered canonically on the line of the walk. Abelian as well as non-Abelian
anyons are studied and it is shown that they have very different properties.
Abelian anyonic walks demonstrate the expected quadratic quantum speedup.
Non-Abelian anyonic walks are much more subtle. The exponential increase of the
system's Hilbert space and the particular statistical evolution of non-Abelian
anyons give a variety of new behaviors. The position distribution of the walker
is related to Jones polynomials, topological invariants of the links created by
the anyonic world-lines during the walk. Several examples such as the SU(2)
level k and the quantum double models are considered that provide insight to
the rich diffusion properties of anyons.Comment: 17 pages, 10 figure