1,194 research outputs found
Noncommmutative theorems: Gelfand Duality, Spectral, Invariant Subspace, and Pontryagin Duality
We extend the Gelfand-Naimark duality of commutative C*-algebras, "A
COMMUTATIVE C*-ALGEBRA -- A LOCALLY COMPACT HAUSDORFF SPACE" to "A
C*-ALGEBRA--A QUOTIENT OF A LOCALLY COMPACT HAUSDORFF SPACE". Thus, a
C*-algebra is isomorphic to the convolution algebra of continuous regular Borel
measures on the topological equivalence relation given by the above mentioned
quotient. In commutative case this reduces to Gelfand-Naimark theorem.
Applications: 1) A simultaneous extension, to arbitrary Hilbert space
operators, of Jordan Canonical Form and Spectral Theorem of normal operators 2)
A functional calculus for arbitrary operators. 3) Affirmative solution of
Invariant Subspace Problem. 4) Extension of Pontryagin duality to nonabelian
groups, and inevitably to groups whose underlying topological space is
noncommutative.Comment: 10 page
Hyperoctahedral Chen calculus for effective Hamiltonians
The algebraic structure of iterated integrals has been encoded by Chen.
Formally, it identifies with the shuffle and Lie calculus of Lyndon, Ree and
Sch\"utzenberger. It is mostly incorporated in the modern theory of free Lie
algebras. Here, we tackle the problem of unraveling the algebraic structure of
computations of effective Hamiltonians. This is an important subject in view of
applications to chemistry, solid state physics, quantum field theory or
engineering. We show, among others, that the correct framework for these
computations is provided by the hyperoctahedral group algebras. We define
several structures on these algebras and give various applications. For
example, we show that the adiabatic evolution operator (in the time-dependent
interaction representation of an effective Hamiltonian) can be written
naturally as a Picard-type series and has a natural exponential expansion.Comment: Minor corrections. Some misleading notations and typos in the first
version have been fixe
Monopoles, noncommutative gauge theories in the BPS limit and some simple gauge groups
For three conspicuous gauge groups, namely, SU(2), SU(3) and SO(5), and at
first order in the noncommutative parameter matrix h\theta^{\mu\nu}, we
construct smooth monopole --and, some two-monopole-- fields that solve the
noncommutative Yang-Mills-Higgs equations in the BPS limit and that are formal
power series in h\theta^{\mu\nu}. We show that there exist noncommutative BPS
(multi-)monopole field configurations that are formal power series in
h\theta^{\mu\nu} if, and only if, two a priori free parameters of the
Seiberg-Witten map take very specific values. These parameters, that are not
associated to field redefinitions nor to gauge transformations, have thus
values that give rise to sharp physical effects.Comment: 30 pages, no figure
State Vector Reduction as a Shadow of a Noncommutative Dynamics
A model, based on a noncommutative geometry, unifying general relativity with
quantum mechanics, is further develped. It is shown that the dynamics in this
model can be described in terms of one-parameter groups of random operators. It
is striking that the noncommutative counterparts of the concept of state and
that of probability measure coincide. We also demonstrate that the equation
describing noncommutative dynamics in the quantum gravitational approximation
gives the standard unitary evolution of observables, and in the "space-time
limit" it leads to the state vector reduction. The cases of the spin and
position operators are discussed in details.Comment: 20 pages, LaTex, no figure
Dequantization via quantum channels
For a unital completely positive map ("quantum channel") governing the
time propagation of a quantum system, the Stinespring representation gives an
enlarged system evolving unitarily. We argue that the Stinespring
representations of each power of the single map together encode the
structure of the original quantum channel and provides an interaction-dependent
model for the bath. The same bath model gives a "classical limit" at infinite
time in the form of a noncommutative "manifold" determined by the
channel. In this way a simplified analysis of the system can be performed by
making the large- approximation. These constructions are based on a
noncommutative generalization of Berezin quantization. The latter is shown to
involve very fundamental aspects of quantum-information theory, which are
thereby put in a completely new light
Unification of SU(2)xU(1) Using a Generalized Covariant Derivative and U(3)
A generalization of the Yang-Mills covariant derivative, that uses both
vector and scalar fields and transforms as a 4-vector contracted with Dirac
matrices, is used to simplify and unify the Glashow-Weinberg-Salam model. Since
SU(3) assigns the wrong hypercharge to the Higgs boson, it is necessary to use
a special representation of U(3) to obtain all the correct quantum numbers. A
surplus gauge scalar boson emerges in the process, but it uncouples from all
other particles.Comment: 12 pages, no figures. To be published in Int. J. Mod. Phys.
- …