180,750 research outputs found
Bound state energies and phase shifts of a non-commutative well
Non-commutative quantum mechanics can be viewed as a quantum system
represented in the space of Hilbert-Schmidt operators acting on non-commutative
configuration space. Within this framework an unambiguous definition can be
given for the non-commutative well. Using this approach we compute the bound
state energies, phase shifts and scattering cross sections of the non-
commutative well. As expected the results are very close to the commutative
results when the well is large or the non-commutative parameter is small.
However, the convergence is not uniform and phase shifts at certain energies
exhibit a much stronger then expected dependence on the non-commutative
parameter even at small values.Comment: 12 pages, 8 figure
Non-commutative Hilbert modular symbols
The main goal of this paper is to construct non-commutative Hilbert modular
symbols. However, we also construct commutative Hilbert modular symbols. Both
the commutative and the non-commutative Hilbert modular symbols are
generalizations of Manin's classical and non-commutative modular symbols. We
prove that many cases of (non-)commutative Hilbert modular symbols are periods
in the sense on Kontsevich-Zagier. Hecke operators act naturally on them.
Manin defines the non-commutative modilar symbol in terms of iterated path
integrals. In order to define non-commutative Hilbert modular symbols, we use a
generalization of iterated path integrals to higher dimensions, which we call
iterated integrals on membranes. Manin examines similarities between
non-commutative modular symbol and multiple zeta values both in terms of
infinite series and in terms of iterated path integrals. Here we examine
similarities in the formulas for non-commutative Hilbert modular symbol and
multiple Dedekind zeta values, recently defined by the author, both in terms of
infinite series and in terms of iterated integrals on membranes.Comment: 50 pages, 5 figures, substantial improvement of the article
arXiv:math/0611955 [math.NT], the portions compared to the previous version
are: Hecke operators, periods and some categorical construction
Commutative Quaternion Matrices
In this study, we introduce the concept of commutative quaternions and
commutative quaternion matrices. Firstly, we give some properties of
commutative quaternions and their Hamilton matrices. After that we investigate
commutative quaternion matrices using properties of complex matrices. Then we
define the complex adjoint matrix of commutative quaternion matrices and give
some of their properties
Duality and Non-Commutative Gauge Theory
We study the generalization of S-duality to non-commutative gauge theories.
For rank one theories, we obtain the leading terms of the dual theory by
Legendre transforming the Lagrangian of the non-commutative theory expressed in
terms of a commutative gauge field. The dual description is weakly coupled when
the original theory is strongly coupled if we appropriately scale the
non-commutativity parameter. However, the dual theory appears to be
non-commutative in space-time when the original theory is non-commutative in
space. This suggests that locality in time for non-commutative theories is an
artifact of perturbation theory.Comment: 7 pages, harvmac; a typo fixe
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