382 research outputs found
Passive States for Essential Observers
The aim of this note is to present a unified approach to the results given in
\cite{bb99} and \cite{bs04} which also covers examples of models not presented
in these two papers (e.g. -dimensional Minkowski space-time for ).
Assuming that a state is passive for an observer travelling along certain
(essential) worldlines, we show that this state is invariant under the isometry
group, is a KMS-state for the observer at a temperature uniquely determined by
the structure constants of the Lie algebra involved and fulfills (a variant of)
the Reeh-Schlieder property. Also the modular objects associated to such a
state and the observable algebra of an observer are computed and a version of
weak locality is examined.Comment: 27 page
A separability criterion for density operators
We give a necessary and sufficient condition for a mixed quantum mechanical
state to be separable. The criterion is formulated as a boundedness condition
in terms of the greatest cross norm on the tensor product of trace class
operators.Comment: REVTeX, 5 page
A Topos Foundation for Theories of Physics: II. Daseinisation and the Liberation of Quantum Theory
This paper is the second in a series whose goal is to develop a fundamentally
new way of constructing theories of physics. The motivation comes from a desire
to address certain deep issues that arise when contemplating quantum theories
of space and time. Our basic contention is that constructing a theory of
physics is equivalent to finding a representation in a topos of a certain
formal language that is attached to the system. Classical physics arises when
the topos is the category of sets. Other types of theory employ a different
topos. In this paper, we study in depth the topos representation of the
propositional language, PL(S), for the case of quantum theory. In doing so, we
make a direct link with, and clarify, the earlier work on applying topos theory
to quantum physics. The key step is a process we term `daseinisation' by which
a projection operator is mapped to a sub-object of the spectral presheaf--the
topos quantum analogue of a classical state space. In the second part of the
paper we change gear with the introduction of the more sophisticated local
language L(S). From this point forward, throughout the rest of the series of
papers, our attention will be devoted almost entirely to this language. In the
present paper, we use L(S) to study `truth objects' in the topos. These are
objects in the topos that play the role of states: a necessary development as
the spectral presheaf has no global elements, and hence there are no
microstates in the sense of classical physics. Truth objects therefore play a
crucial role in our formalism.Comment: 34 pages, no figure
String-- and Brane--Localized Causal Fields in a Strongly Nonlocal Model
We study a weakly local, but nonlocal model in spacetime dimension
and prove that it is maximally nonlocal in a certain specific quantitative
sense. Nevertheless, depending on the number of dimensions , it has
string--localized or brane--localized operators which commute at spatial
distances. In two spacetime dimensions, the model even comprises a covariant
and local subnet of operators localized in bounded subsets of Minkowski space
which has a nontrivial scattering matrix. The model thus exemplifies the
algebraic construction of local observables from algebras associated with
nonlocal fields.Comment: paper re-written with a change of emphasis and new result
Groupoid normalizers of tensor products
We consider an inclusion B [subset of or equal to] M of finite von Neumann algebras satisfying B′∩M [subset of or equal to] B. A partial isometry vset membership, variantM is called a groupoid normalizer if vBv*,v*Bv[subset of or equal to] B. Given two such inclusions B<sub>i</sub> [subset of or equal to] M<sub>i</sub>, i=1,2, we find approximations to the groupoid normalizers of [formula] in [formula], from which we deduce that the von Neumann algebra generated by the groupoid normalizers of the tensor product is equal to the tensor product of the von Neumann algebras generated by the groupoid normalizers. Examples are given to show that this can fail without the hypothesis [formula], i=1,2. We also prove a parallel result where the groupoid normalizers are replaced by the intertwiners, those partial isometries vset membership, variantM satisfying vBv*[subset of or equal to] B and v*v,vv*[set membership, variant] B
Leibniz Seminorms and Best Approximation from C*-subalgebras
We show that if B is a C*-subalgebra of a C*-algebra A such that B contains a
bounded approximate identity for A, and if L is the pull-back to A of the
quotient norm on A/B, then L is strongly Leibniz. In connection with this
situation we study certain aspects of best approximation of elements of a
unital C*-algebra by elements of a unital C*-subalgebra.Comment: 24 pages. Intended for the proceedings of the conference "Operator
Algebras and Related Topics". v2: added a corollary to the main theorem, plus
several minor improvements v3: much simplified proof of a key lemma,
corollary to main theorem added v4: Many minor improvements. Section numbers
increased by
Transition amplitudes and sewing properties for bosons on the Riemann sphere
We consider scalar quantum fields on the sphere, both massive and massless.
In the massive case we show that the correlation functions define amplitudes
which are trace class operators between tensor products of a fixed Hilbert
space. We also establish certain sewing properties between these operators. In
the massless case we consider exponential fields and have a conformal field
theory. In this case the amplitudes are only bilinear forms but still we
establish sewing properties. Our results are obtained in a functional integral
framework.Comment: 33 page
Localization via Automorphisms of the CARs. Local gauge invariance
The classical matter fields are sections of a vector bundle E with base
manifold M. The space L^2(E) of square integrable matter fields w.r.t. a
locally Lebesgue measure on M, has an important module action of C_b^\infty(M)
on it. This module action defines restriction maps and encodes the local
structure of the classical fields. For the quantum context, we show that this
module action defines an automorphism group on the algebra A, of the canonical
anticommutation relations on L^2(E), with which we can perform the analogous
localization. That is, the net structure of the CAR, A, w.r.t. appropriate
subsets of M can be obtained simply from the invariance algebras of appropriate
subgroups. We also identify the quantum analogues of restriction maps. As a
corollary, we prove a well-known "folk theorem," that the algebra A contains
only trivial gauge invariant observables w.r.t. a local gauge group acting on
E.Comment: 15 page
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