532 research outputs found
Solitons and nonsmooth diffeomorphisms in conformal nets
We show that any solitonic representation of a conformal (diffeomorphism
covariant) net on S^1 has positive energy and construct an uncountable family
of mutually inequivalent solitonic representations of any conformal net, using
nonsmooth diffeomorphisms. On the loop group nets, we show that these
representations induce representations of the subgroup of loops compactly
supported in S^1 \ {-1} which do not extend to the whole loop group.
In the case of the U(1)-current net, we extend the diffeomorphism covariance
to the Sobolev diffeomorphisms D^s(S^1), s > 2, and show that the
positive-energy vacuum representations of Diff_+(S^1) with integer central
charges extend to D^s(S^1). The solitonic representations constructed above for
the U(1)-current net and for Virasoro nets with integral central charge are
continuously covariant with respect to the stabilizer subgroup of Diff_+(S^1)
of -1 of the circle.Comment: 33 pages, 3 TikZ figure
Positive energy representations of Sobolev diffeomorphism groups of the circle
We show that any positive energy projective representation of Diff(S^1)
extends to a strongly continuous projective unitary representation of the
fractional Sobolev diffeomorphisms D^s(S^1) with s>3, and in particular to
C^k-diffeomorphisms Diff^k(S^1) with k >= 4. A similar result holds for the
universal covering groups provided that the representation is assumed to be a
direct sum of irreducibles.
As an application we show that a conformal net of von Neumann algebras on S^1
is covariant with respect to D^s(S^1), s > 3. Moreover every direct sum of
irreducible representations of a conformal net is also D^s(S^1)-covariant.Comment: 30 pages, 1 TikZ figur
On truncated -free Fock spaces: spectrum of position operators and shift-invariant states
The ergodic properties of the shift on both full and -truncated -free
-algebras are analyzed. In particular, the shift is shown to be uniquely
ergodic with respect to the fixed-point algebra. In addition, for every , the invariant states of the shift acting on the -truncated -free
-algebra are shown to yield a -dimensional Choquet simplex, which
collapses to a segment in the full case. Finally, the spectrum of the position
operators on the -truncated -free Fock space is also determined.Comment: 15 page
Low energy spectrum of the XXZ model coupled to a magnetic field
It is shown that, for a class of Hamiltonians of XXZ chains in an external
magnetic field that are small perturbations of an Ising Hamiltonian, the
spectral gap above the ground-state energy remains strictly positive when the
perturbation is turned on, uniformly in the length of the chain. The result is
proven for both the ferromagnetic and the antiferromagnetic Ising Hamiltonian;
in the latter case the external magnetic field is required to be small, and for
an even number of sites the two-fold degenerate ground-state energy of the
unperturbed Hamiltonian may split into two energy levels whose difference is
small. This result is proven by using a new, quite subtle refinement of a
method developed in earlier work and used to iteratively block-diagonalize
Hamiltonians of ever larger subsystems with the help of local unitary
conjugations. One novel ingredient of the method presented in this paper
consists of the use of Lieb-Robinson bounds.Comment: 6 figure
Weakly-monotone C*-algebras as Exel-Laca algebras
An abstract characterization of weakly monotone -algebras, namely the
concrete -algebras generated by creators and annihilators acting on the
so-called weakly monotone Fock spaces, is given in terms of (quotient of)
suitable Exel-Laca algebras. The weakly monotone -algebra indexed by
is shown to be a type-I -algebra and its representation
theory is entirely determined, whereas the weakly monotone -algebra
indexed by is shown not to be of type
Two Results in the Quantum Theory of Measurements
Two theorems with applications to the quantum theory of measurements are
stated and proven. The first one clarifies and amends von Neumann's Measurement
Postulate used in the Copenhagen interpretation of quantum mechanics. The
second one clarifies the relationship between ``events'' and ``measurements''
and the meaning of measurements in the -Approach to quantum mechanics.Comment: 14 pages, 0 figures, volume in memory of Giovanni Morchio, to be
published by Springer-Nature in 202
Freedman's theorem for unitarily invariant states on the CCR algebra
The set of states on , the CCR algebra of a separable Hilbert
space , is here looked at as a natural object to obtain a non-commutative
version of Freedman's theorem for unitarily invariant stochastic processes. In
this regard, we provide a complete description of the compact convex set of
states of that are invariant under the action of all
automorphisms induced in second quantization by unitaries of . We prove
that this set is a Bauer simplex, whose extreme states are either the canonical
trace of the CCR algebra or Gaussian states with variance at least .Comment: 22 page
Infinite index extensions of local nets and defects
Subfactor theory provides a tool to analyze and construct extensions of
Quantum Field Theories, once the latter are formulated as local nets of von
Neumann algebras. We generalize some of the results of [LR95] to the case of
extensions with infinite Jones index. This case naturally arises in physics,
the canonical examples are given by global gauge theories with respect to a
compact (non-finite) group of internal symmetries. Building on the works of
Izumi, Longo, Popa [ILP98] and Fidaleo, Isola [FI99], we consider generalized
Q-systems (of intertwiners) for a semidiscrete inclusion of properly infinite
von Neumann algebras, which generalize ordinary Q-systems introduced by Longo
[Lon94] to the infinite index case. We characterize inclusions which admit
generalized Q-systems of intertwiners and define a braided product among the
latter, hence we construct examples of QFTs with defects (phase boundaries) of
infinite index, extending the family of boundaries in the grasp of [BKLR16].Comment: 50 page
Comparison Theorems for Stochastic Chemical Reaction Networks
Continuous-time Markov chains are frequently used as stochastic models for
chemical reaction networks, especially in the growing field of systems biology.
A fundamental problem for these Stochastic Chemical Reaction Networks (SCRNs)
is to understand the dependence of the stochastic behavior of these systems on
the chemical reaction rate parameters. Towards solving this problem, in this
paper we develop theoretical tools called comparison theorems that provide
stochastic ordering results for SCRNs. These theorems give sufficient
conditions for monotonic dependence on parameters in these network models,
which allow us to obtain, under suitable conditions, information about
transient and steady state behavior. These theorems exploit structural
properties of SCRNs, beyond those of general continuous-time Markov chains.
Furthermore, we derive two theorems to compare stationary distributions and
mean first passage times for SCRNs with different parameter values, or with the
same parameters and different initial conditions. These tools are developed for
SCRNs taking values in a generic (finite or countably infinite) state space and
can also be applied for non-mass-action kinetics models. When propensity
functions are bounded, our method of proof gives an explicit method for
coupling two comparable SCRNs, which can be used to simultaneously simulate
their sample paths in a comparable manner. We illustrate our results with
applications to models of enzymatic kinetics and epigenetic regulation by
chromatin modifications.Comment: Compared to the first version, the Supplementary Information (SI)
file has been adde
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