8,015 research outputs found
Weak Hopf Algebras II: Representation theory, dimensions and the Markov trace
If A is a weak C^*-Hopf algebra then the category of finite dimensional
unitary representations of A is a monoidal C^*-category with monoidal unit
being the GNS representation D_eps associated to the counit \eps. This category
has isomorphic left dual and right dual objects which leads, as usual, to the
notion of dimension function. However, if \eps is not pure the dimension
function is matrix valued with rows and columns labelled by the irreducibles
contained in D_eps. This happens precisely when the inclusions A^L < A and A^R
< A are not connected. Still there exists a trace on A which is the Markov
trace for both inclusions. We derive two numerical invariants for each C^*-WHA
of trivial hypercenter. These are the common indices I and \delta, of the Haar,
respectively Markov conditional expectations of either one of the inclusions
A^{L/R} \delta. In the
special case of weak Kac algebras we show that I=\delta is an integer.Comment: 45 pages, LaTeX, submitted to J. Algebr
Weak Hopf Algebras I: Integral Theory and C^*-structure
We give an introduction to the theory of weak Hopf algebras proposed recently
as a coassociative alternative of weak quasi-Hopf algebras. We follow an
axiomatic approach keeping as close as possible to the "classical" theory of
Hopf algebras. The emphasis is put on the new structure related to the presence
of canonical subalgebras A^L and A^R in any weak Hopf algebra A that play the
role of non-commutative numbers in many respects. A theory of integrals is
developed in which we show how the algebraic properties of A, such as the
Frobenius property, or semisimplicity, or innerness of the square of the
antipode, are related to the existence of non-degenerate, normalized, or Haar
integrals. In case of C^*-weak Hopf algebras we prove the existence of a unique
Haar measure h in A and of a canonical grouplike element g in A implementing
the square of the antipode and factorizing into left and right algebra
elements. Further discussion of the C^*-case will be presented in Part II.Comment: 40 pages, LaTeX, to appear in J. Algebr
Symmetry Representations in the Rigged Hilbert Space Formulation of Quantum Mechanics
We discuss some basic properties of Lie group representations in rigged
Hilbert spaces. In particular, we show that a differentiable representation in
a rigged Hilbert space may be obtained as the projective limit of a family of
continuous representations in a nested scale of Hilbert spaces. We also
construct a couple of examples illustrative of the key features of group
representations in rigged Hilbert spaces. Finally, we establish a simple
criterion for the integrability of an operator Lie algebra in a rigged Hilbert
space
Irreversible Quantum Mechanics in the Neutral K-System
The neutral Kaon system is used to test the quantum theory of resonance
scattering and decay phenomena. The two dimensional Lee-Oehme-Yang theory with
complex Hamiltonian is obtained by truncating the complex basis vector
expansion of the exact theory in Rigged Hilbert space. This can be done for K_1
and K_2 as well as for K_S and K_L, depending upon whether one chooses the
(self-adjoint, semi-bounded) Hamiltonian as commuting or non-commuting with CP.
As an unexpected curiosity one can show that the exact theory (without
truncation) predicts long-time 2 pion decays of the neutral Kaon system even if
the Hamiltonian conserves CP.Comment: 36 pages, 1 PostScript figure include
Misleading signposts along the de Broglie-Bohm road to quantum mechanics
Eighty years after de Broglie's, and a little more than half a century after
Bohm's seminal papers, the de Broglie--Bohm theory (a.k.a. Bohmian mechanics),
which is presumably the simplest theory which explains the orthodox quantum
mechanics formalism, has reached an exemplary state of conceptual clarity and
mathematical integrity. No other theory of quantum mechanics comes even close.
Yet anyone curious enough to walk this road to quantum mechanics is soon being
confused by many misleading signposts that have been put up, and not just by
its detractors, but unfortunately enough also by some of its proponents.
This paper outlines a road map to help navigate ones way.Comment: Dedicated to Jeffrey Bub on occasion of his 65th birthday. Accepted
for publication in Foundations of Physics. A "slip of pen" in the
bibliography has been corrected -- thanks go to Oliver Passon for catching
it
The density matrix in the de Broglie-Bohm approach
If the density matrix is treated as an objective description of individual
systems, it may become possible to attribute the same objective significance to
statistical mechanical properties, such as entropy or temperature, as to
properties such as mass or energy. It is shown that the de Broglie-Bohm
interpretation of quantum theory can be consistently applied to density
matrices as a description of individual systems. The resultant trajectories are
examined for the case of the delayed choice interferometer, for which Bell
appears to suggest that such an interpretation is not possible. Bell's argument
is shown to be based upon a different understanding of the density matrix to
that proposed here.Comment: 15 pages, 4 figure
Hypersurface Bohm-Dirac models
We define a class of Lorentz invariant Bohmian quantum models for N entangled
but noninteracting Dirac particles. Lorentz invariance is achieved for these
models through the incorporation of an additional dynamical space-time
structure provided by a foliation of space-time. These models can be regarded
as the extension of Bohm's model for N Dirac particles, corresponding to the
foliation into the equal-time hyperplanes for a distinguished Lorentz frame, to
more general foliations. As with Bohm's model, there exists for these models an
equivariant measure on the leaves of the foliation. This makes possible a
simple statistical analysis of position correlations analogous to the
equilibrium analysis for (the nonrelativistic) Bohmian mechanics.Comment: 17 pages, 3 figures, RevTex. Completely revised versio
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