270 research outputs found
Entropy on effect algebras with the Riesz decomposition property I: Basic properties
summary:We define the entropy, lower and upper entropy, and the conditional entropy of a dynamical system consisting of an effect algebra with the Riesz decomposition property, a state, and a transformation. Such effect algebras allow many refinements of two partitions. We present the basic properties of these entropies and these notions are illustrated by many examples. Entropy on MV-algebras is postponed to Part II
Metric operators, generalized hermiticity and lattices of Hilbert lpaces
A quasi-Hermitian operator is an operator that is similar to its adjoint in
some sense, via a metric operator, i.e., a strictly positive self-adjoint
operator. Whereas those metric operators are in general assumed to be bounded,
we analyze the structure generated by unbounded metric operators in a Hilbert
space. It turns out that such operators generate a canonical lattice of Hilbert
spaces, that is, the simplest case of a partial inner product space
(PIP-space). We introduce several generalizations of the notion of similarity
between operators, in particular, the notion of quasi-similarity, and we
explore to what extend they preserve spectral properties. Then we apply some of
the previous results to operators on a particular PIP-space, namely, a scale of
Hilbert spaces generated by a metric operator. Finally, motivated by the recent
developments of pseudo-Hermitian quantum mechanics, we reformulate the notion
of pseudo-Hermitian operators in the preceding formalism.Comment: 51pages; will appear as a chapter in \textit{Non-Selfadjoint
Operators in Quantum Physics: Mathematical Aspects}; F. Bagarello, J-P.
Gazeau, F. H. Szafraniec and M. Znojil, eds., J. Wiley, 201
Clifford geometric parameterization of inequivalent vacua
We propose a geometric method to parameterize inequivalent vacua by dynamical
data. Introducing quantum Clifford algebras with arbitrary bilinear forms we
distinguish isomorphic algebras --as Clifford algebras-- by different
filtrations resp. induced gradings. The idea of a vacuum is introduced as the
unique algebraic projection on the base field embedded in the Clifford algebra,
which is however equivalent to the term vacuum in axiomatic quantum field
theory and the GNS construction in C^*-algebras. This approach is shown to be
equivalent to the usual picture which fixes one product but employs a variety
of GNS states. The most striking novelty of the geometric approach is the fact
that dynamical data fix uniquely the vacuum and that positivity is not
required. The usual concept of a statistical quantum state can be generalized
to geometric meaningful but non-statistical, non-definite, situations.
Furthermore, an algebraization of states takes place. An application to physics
is provided by an U(2)-symmetry producing a gap-equation which governs a phase
transition. The parameterization of all vacua is explicitly calculated from
propagator matrix elements. A discussion of the relation to BCS theory and
Bogoliubov-Valatin transformations is given.Comment: Major update, new chapters, 30 pages one Fig. (prev. 15p, no Fig.
Gravity as a four dimensional algebraic quantum field theory
Based on a family of indefinite unitary representations of the diffeomorphism
group of an oriented smooth -manifold, a manifestly covariant
dimensional and non-perturbative algebraic quantum field theory formulation of
gravity is exhibited. More precisely among the bounded linear operators acting
on these representation spaces we identify algebraic curvature tensors hence a
net of local quantum observables can be constructed from -algebras
generated by local curvature tensors and vector fields. This algebraic quantum
field theory is extracted from structures provided by an oriented smooth
-manifold only hence possesses a diffeomorphism symmetry. In this way
classical general relativity exactly in dimensions naturally embeds into a
quantum framework.
Several Hilbert space representations of the theory are found. First a
"tautological representation" of the limiting global -algebra is
constructed allowing to associate to any oriented smooth -manifold a von
Neumann algebra in a canonical fashion. Secondly, influenced by the
Dougan--Mason approach to gravitational quasilocal energy-momentum, we
construct certain representations what we call "positive mass representations"
with unbroken diffeomorphism symmetry. Thirdly, we also obtain "classical
representaions" with spontaneously broken diffeomorphism symmetry corresponding
to the classical limit of the theory which turns out to be general relativity.
Finally we observe that the whole family of "positive mass representations"
comprise a dimensional conformal field theory in the sense of G. Segal.Comment: LaTeX, 22 pp, no figures. The final, revised and published versio
Scattering theory for Klein-Gordon equations with non-positive energy
We study the scattering theory for charged Klein-Gordon equations:
\{{array}{l} (\p_{t}- \i v(x))^{2}\phi(t,x) \epsilon^{2}(x,
D_{x})\phi(t,x)=0,[2mm] \phi(0, x)= f_{0}, [2mm] \i^{-1} \p_{t}\phi(0, x)=
f_{1}, {array}. where: \epsilon^{2}(x, D_{x})= \sum_{1\leq j, k\leq
n}(\p_{x_{j}} \i b_{j}(x))A^{jk}(x)(\p_{x_{k}} \i b_{k}(x))+ m^{2}(x),
describing a Klein-Gordon field minimally coupled to an external
electromagnetic field described by the electric potential and magnetic
potential . The flow of the Klein-Gordon equation preserves the
energy: h[f, f]:= \int_{\rr^{n}}\bar{f}_{1}(x) f_{1}(x)+
\bar{f}_{0}(x)\epsilon^{2}(x, D_{x})f_{0}(x) - \bar{f}_{0}(x) v^{2}(x) f_{0}(x)
\d x. We consider the situation when the energy is not positive. In this
case the flow cannot be written as a unitary group on a Hilbert space, and the
Klein-Gordon equation may have complex eigenfrequencies. Using the theory of
definitizable operators on Krein spaces and time-dependent methods, we prove
the existence and completeness of wave operators, both in the short- and
long-range cases. The range of the wave operators are characterized in terms of
the spectral theory of the generator, as in the usual Hilbert space case
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