3,185 research outputs found
An Analogue of the Kac-Wakimoto Formula and Black Hole Conditional Entropy
A local formula for the dimension of a superselection sector in Quantum Field
Theory is obtained as vacuum expectation value of the exponential of the proper
Hamiltonian. In the particular case of a chiral conformal theory, this provides
a local analogue of a global formula obtained by Kac and Wakimoto within the
context of representations of certain affine Lie algebras. Our formula is model
independent and its version in general Quantum Field Theory applies to black
hole thermodynamics. The relative free energy between two thermal equilibrium
states associated with a black hole turns out to be proportional to the
variation of the conditional entropy in different superselection sectors, where
the conditional entropy is defined as the Connes-Stoermer entropy associated
with the DHR localized endomorphism representing the sector. The constant of
proportionality is half of the Hawking temperature. As a consequence the
relative free energy is quantized proportionally to the logarithm of a rational
number, in particular it is equal to a linear function the logarithm of an
integer once the initial state or the final state is taken fixed.Comment: AMS-TeX v1.1c, minor grammatical correction
The Bisognano-Wichmann Theorem for Charged States and the Conformal Boundary of a Black Hole
Recent applications of Operator Algebras to Quantum Field Theory on a Curved
Spacetime show that the incremental entropy associated with a quantum black
hole, due the addition of a short range charge, is quantized proportionally to
the logarithm of an integer. This talk first reviews the case of a Rindler
black hole and then sketches the case of a spacetime with bifurcate Killing
horizon and charges localizable on the horizon. An important tool is the
construction of the conformal symmetries on the horizon.Comment: AMS-LaTeX, 7 pages. talk delivered at the conference on
``Mathematical Physics and Quantum Field Theory'', Berkeley, 11-13 June 1999.
Electronic Journal of Differential Equations (to appear
Conformal Subnets and Intermediate Subfactors
Given an irreducible local conformal net A of von Neumann algebras on the
circle and a finite-index conformal subnet B of A, we show that A is completely
rational iff B is completely rational. In particular this extends a result of
F. Xu for the orbifold construction. By applying previous results of Xu, many
coset models turn out to be completely rational and the structure results in
[KLM] hold. Our proofs are based on an analysis of the net inclusion B in A;
among other things we show that, for a fixed interval I, every von Neumann
algebra R intermediate between B(I) and A(I) comes from an intermediate
conformal net L between B and A with L(I)=R. We make use of a theorem of
Watatani (type II case) and Teruya and Watatani (type III case) on the
finiteness of the set I(N,M) of intermediate subfactors in an irreducible
inclusion of factors N in M with finite Jones index [M:N]. We provide a unified
proof of this result that gives in particular an explicit bound for the
cardinality of I(N,M) which depends only on [M:N].Comment: 29 pages; AMS-Latex2
Covariant Sectors with Infinite Dimension and Positivity of the Energy
We consider a Moebius covariant sector, possibly with infinite dimension, of
a local conformal net of von Neumann algebras on the circle. If the sector has
finite index, it has automatically positive energy. In the infinite index case,
we show the spectrum of the energy always to contain the positive real line,
but, as seen by an example, it may contain negative values. We then consider
nets with Haag duality on the real line, or equivalently sectors with
non-solitonic extension to the dual net; we give a criterion for irreducible
sectors to have positive energy, namely this is the case iff there exists an
unbounded Moebius covariant left inverse. As a consequence the class of sectors
with positive energy is stable under composition, conjugation and direct
integral decomposition.Comment: 25 pages, Latex2
A Converse Hawking-Unruh Effect and dS^2/CFT Correspondance
Given a local quantum field theory net A on the de Sitter spacetime dS^d,
where geodesic observers are thermalized at Gibbons-Hawking temperature, we
look for observers that feel to be in a ground state, i.e. particle evolutions
with positive generator, providing a sort of converse to the Hawking-Unruh
effect. Such positive energy evolutions always exist as noncommutative flows,
but have only a partial geometric meaning, yet they map localized observables
into localized observables.
We characterize the local conformal nets on dS^d. Only in this case our
positive energy evolutions have a complete geometrical meaning. We show that
each net has a unique maximal expected conformal subnet, where our evolutions
are thus geometrical.
In the two-dimensional case, we construct a holographic one-to-one
correspondence between local nets A on dS^2 and local conformal non-isotonic
families (pseudonets) B on S^1. The pseudonet B gives rise to two local
conformal nets B(+/-) on S^1, that correspond to the H(+/-)-horizon components
of A, and to the chiral components of the maximal conformal subnet of A. In
particular, A is holographically reconstructed by a single horizon component,
namely the pseudonet is a net, iff the translations on H(+/-) have positive
energy and the translations on H(-/+) are trivial. This is the case iff the
one-parameter unitary group implementing rotations on dS^2 has
positive/negative generator.Comment: The title has changed. 38 pages, figures. To appear on Annales H.
Poincare
Classification of Local Conformal Nets. Case c < 1
We completely classify diffeomorphism covariant local nets of von Neumann
algebras on the circle with central charge c less than 1. The irreducible ones
are in bijective correspondence with the pairs of A-D_{2n}-E_{6,8} Dynkin
diagrams such that the difference of their Coxeter numbers is equal to 1. We
first identify the nets generated by irreducible representations of the
Virasoro algebra for c<1 with certain coset nets. Then, by using the
classification of modular invariants for the minimal models by
Cappelli-Itzykson-Zuber and the method of alpha-induction in subfactor theory,
we classify all local irreducible extensions of the Virasoro nets for c<1 and
infer our main classification result. As an application, we identify in our
classification list certain concrete coset nets studied in the literature.Comment: 30 pages, LaTeX2
Localization in Nets of Standard Spaces
Starting from a real standard subspace of a Hilbert space and a
representation of the translation group with natural properties, we construct
and analyze for each endomorphism of this pair a local, translationally
covariant net of standard subspaces, on the lightray and on two-dimensional
Minkowski space. These nets share many features with low-dimensional quantum
field theory, described by corresponding nets of von Neumann algebras.
Generalizing a result of Longo and Witten to two dimensions and massive
multiplicity free representations, we characterize these endomorphisms in terms
of specific analytic functions. Such a characterization then allows us to
analyze the corresponding nets of standard spaces, and in particular compute
their minimal localization length. The analogies and differences to the von
Neumann algebraic situation are discussed.Comment: 34 pages, 1 figur
Natural Energy Bounds in Quantum Thermodynamics
Given a stationary state for a noncommutative flow, we study a boundedness
condition, depending on a positive parameter beta, which is weaker than the KMS
equilibrium condition at inverse temperature beta. This condition is equivalent
to a holomorphic property closely related to the one recently considered by
Ruelle and D'Antoni-Zsido and shared by a natural class of non-equilibrium
steady states. Our holomorphic property is stronger than the Ruelle's one and
thus selects a restricted class of non-equilibrium steady states. We also
introduce the complete boundedness condition and show this notion to be
equivalent to the Pusz-Woronowicz complete passivity property, hence to the KMS
condition.
In Quantum Field Theory, the beta-boundedness condition can be interpreted as
the property that localized state vectors have energy density levels increasing
beta-subexponentially, a property which is similar in the form and weaker in
the spirit than the modular compactness-nuclearity condition. In particular,
for a Poincare' covariant net of C*-algebras on the Minkowski spacetime, the
beta-boundedness property, for beta greater equal than 2 pi, for the boosts is
shown to be equivalent to the Bisognano-Wichmann property. The Hawking
temperature is thus minimal for a thermodynamical system in the background of a
Rindler black hole within the class of beta-holomorphic states. More generally,
concerning the Killing evolution associated with a class of stationary quantum
black holes, we characterize KMS thermal equilibrium states at Hawking
temperature in terms of the boundedness property and the existence of a
translation symmetry on the horizon.Comment: 28 pages, LaTeX. Minor modifications in the abstract, to appear in
Commun. Math. Phy
Graded KMS Functionals and the Breakdown of Supersymmetry
It is shown that the modulus of any graded or, more generally, twisted KMS
functional of a C*-dynamical system is proportional to an ordinary KMS state
and the twist is weakly inner in the corresponding GNS-representation. If the
functional is invariant under the adjoint action of some asymptotically abelian
family of automorphisms, then the twist is trivial. As a consequence, such
functionals do not exist for supersymmetric C*-dynamical systems. This is in
contrast with the situation in compact spaces where super KMS functionals occur
as super-Gibbs functionals.Comment: 9 pages, Latex, note added about a generalization of Corollary
- …