70 research outputs found
Local Operations and Completely Positive Maps in Algebraic Quantum Field Theory
Einstein introduced the locality principle which states that all physical
effect in some finite space-time region does not influence its space-like
separated finite region. Recently, in algebraic quantum field theory, R\'{e}dei
captured the idea of the locality principle by the notion of operational
separability. The operation in operational separability is performed in some
finite space-time region, and leaves unchanged the state in its space-like
separated finite space-time region. This operation is defined with a completely
positive map. In the present paper, we justify using a completely positive map
as a local operation in algebraic quantum field theory, and show that this
local operation can be approximately written with Kraus operators under the
funnel property
Quantized reduction as a tensor product
Symplectic reduction is reinterpreted as the composition of arrows in the
category of integrable Poisson manifolds, whose arrows are isomorphism classes
of dual pairs, with symplectic groupoids as units. Morita equivalence of
Poisson manifolds amounts to isomorphism of objects in this category.
This description paves the way for the quantization of the classical
reduction procedure, which is based on the formal analogy between dual pairs of
Poisson manifolds and Hilbert bimodules over C*-algebras, as well as with
correspondences between von Neumann algebras. Further analogies are drawn with
categories of groupoids (of algebraic, measured, Lie, and symplectic type). In
all cases, the arrows are isomorphism classes of appropriate bimodules, and
their composition may be seen as a tensor product. Hence in suitable categories
reduction is simply composition of arrows, and Morita equivalence is
isomorphism of objects.Comment: 44 pages, categorical interpretation adde
Topos quantum theory with short posets
Topos quantum mechanics, developed by Isham et. al., creates a topos of
presheaves over the poset V(N) of abelian von Neumann subalgebras of the von
Neumann algebra N of bounded operators associated to a physical system, and
established several results, including: (a) a connection between the
Kochen-Specker theorem and the non-existence of a global section of the
spectral presheaf; (b) a version of the spectral theorem for self-adjoint
operators; (c) a connection between states of N and measures on the spectral
presheaf; and (d) a model of dynamics in terms of V(N). We consider a
modification to this approach using not the whole of the poset V(N), but only
its elements of height at most two. This produces a different topos with
different internal logic. However, the core results (a)--(d) established using
the full poset V(N) are also established for the topos over the smaller poset,
and some aspects simplify considerably. Additionally, this smaller poset has
appealing aspects reminiscent of projective geometry.Comment: 14 page
Perturbations of nuclear C*-algebras
Kadison and Kastler introduced a natural metric on the collection of all
C*-subalgebras of the bounded operators on a separable Hilbert space. They
conjectured that sufficiently close algebras are unitarily conjugate. We
establish this conjecture when one algebra is separable and nuclear. We also
consider one-sided versions of these notions, and we obtain embeddings from
certain near inclusions involving separable nuclear C*-algebras. At the end of
the paper we demonstrate how our methods lead to improved characterisations of
some of the types of algebras that are of current interest in the
classification programme.Comment: 45 page
Epistemic and Ontic Quantum Realities
Quantum theory has provoked intense discussions about its interpretation since its pioneer days. One of the few scientists who have been continuously engaged in this development from both physical and philosophical perspectives is Carl Friedrich von Weizsaecker. The questions he posed were and are inspiring for many, including the authors of this contribution. Weizsaecker developed Bohr's view of quantum theory as a theory of knowledge. We show that such an epistemic perspective can be consistently complemented by Einstein's ontically oriented position
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