29 research outputs found
Nuclearity and Thermal States in Conformal Field Theory
We introduce a new type of spectral density condition, that we call
L^2-nuclearity. One formulation concerns lowest weight unitary representations
of SL(2,R) and turns out to be equivalent to the existence of characters. A
second formulation concerns inclusions of local observable von Neumann algebras
in Quantum Field Theory. We show the two formulations to agree in chiral
Conformal QFT and, starting from the trace class condition for the semigroup
generated by the conformal Hamiltonian L_0, we infer and naturally estimate the
Buchholz-Wichmann nuclearity condition and the (distal) split property. As a
corollary, if L_0 is log-elliptic, the Buchholz-Junglas set up is realized and
so there exists a beta-KMS state for the translation dynamics on the net of
C*-algebras for every inverse temperature beta>0. We include further
discussions on higher dimensional spacetimes. In particular, we verify that
L^2-nuclearity is satisfied for the scalar, massless Klein-Gordon field.Comment: 37 pages, minor correction
Bohrification of operator algebras and quantum logic
Following Birkhoff and von Neumann, quantum logic has traditionally been
based on the lattice of closed linear subspaces of some Hilbert space, or, more
generally, on the lattice of projections in a von Neumann algebra A.
Unfortunately, the logical interpretation of these lattices is impaired by
their nondistributivity and by various other problems. We show that a possible
resolution of these difficulties, suggested by the ideas of Bohr, emerges if
instead of single projections one considers elementary propositions to be
families of projections indexed by a partially ordered set C(A) of appropriate
commutative subalgebras of A. In fact, to achieve both maximal generality and
ease of use within topos theory, we assume that A is a so-called Rickart
C*-algebra and that C(A) consists of all unital commutative Rickart
C*-subalgebras of A. Such families of projections form a Heyting algebra in a
natural way, so that the associated propositional logic is intuitionistic:
distributivity is recovered at the expense of the law of the excluded middle.
Subsequently, generalizing an earlier computation for n-by-n matrices, we
prove that the Heyting algebra thus associated to A arises as a basis for the
internal Gelfand spectrum (in the sense of Banaschewski-Mulvey) of the
"Bohrification" of A, which is a commutative Rickart C*-algebra in the topos of
functors from C(A) to the category of sets. We explain the relationship of this
construction to partial Boolean algebras and Bruns-Lakser completions. Finally,
we establish a connection between probability measure on the lattice of
projections on a Hilbert space H and probability valuations on the internal
Gelfand spectrum of A for A = B(H).Comment: 31 page
New Concepts in Particle Physics from Solution of an Old Problem
Recent ideas on modular localization in local quantum physics are used to
clarify the relation between on- and off-shell quantities in particle physics;
in particular the relation between on-shell crossing symmetry and off-shell
Einstein causality. Among the collateral results of this new nonperturbative
approach are profound relations between crossing symmetry of particle physics
and Hawking-Unruh like thermal aspects (KMS property, entropy attached to
horizons) of quantum matter behind causal horizons, aspects which hitherto were
exclusively related with Killing horizons in curved spacetime rather than with
localization aspects in Minkowski space particle physics. The scope of this
modular framework is amazingly wide and ranges from providing a conceptual
basis for the d=1+1 bootstrap-formfactor program for factorizable d=1+1 models
to a decomposition theory of QFT's in terms of a finite collection of unitarily
equivalent chiral conformal theories placed a specified relative position
within a common Hilbert space (in d=1+1 a holographic relation and in higher
dimensions more like a scanning). The new framework gives a spacetime
interpretation to the Zamolodchikov-Faddeev algebra and explains its thermal
aspects.Comment: In this form it will appear in JPA Math Gen, 47 pages tcilate
Introduction to representations of the canonical commutation and anticommutation relations
Lecture notes of a minicourse given at the Summer School on Large Coulomb
Systems - QED in Nordfjordeid, 2003, devoted to representations of the CCR and
CAR. Quasifree states, the Araki-Woods and Araki-Wyss representations, and the
lattice of von Neumenn algebras in a bosonic/fermionic Fock space are discussed
in detail
CANONICAL EXTENSIONS OF SYMMETRIC LINEAR RELATIONS
The concept of canonical extension of Hermitian operators has been recently introduced by A. Kuzhel. This paper deals with a generalization of this notion to the case of symmetric linear relations. Namely, canonical regular extensions of symmetric linear relations in Hilbert spaces are studied. The main result is a characterization of canonical regular extensions in terms of a von-Neumann like formula