501 research outputs found

### UHF flows and the flip automorphism

A UHF flow is an infinite tensor product type action of the reals on a UHF
algebra $A$ and the flip automorphism is an automorphism of $A\otimes A$
sending $x\otimes y$ into $y\otimes x$. If $\alpha$ is an inner perturbation of
a UHF flow on $A$, there is a sequence $(u_n)$ of unitaries in $A\otimes A$
such that $\alpha_t\otimes \alpha_t(u_n)-u_n$ converges to zero and the flip is
the limit of \Ad u_n. We consider here whether the converse holds or not and
solve it with an additional assumption: If $A\otimes A\cong A$ and $\alpha$
absorbs any UHF flow $\beta$ (i.e., $\alpha\otimes\beta$ is cocycle conjugate
to $\alpha$), then the converse holds; in this case $\alpha$ is what we call a
universal UHF flow.Comment: 18 page

### Cauchy Problem and Green's Functions for First Order Differential Operators and Algebraic Quantization

Existence and uniqueness of advanced and retarded fundamental solutions
(Green's functions) and of global solutions to the Cauchy problem is proved for
a general class of first order linear differential operators on vector bundles
over globally hyperbolic Lorentzian manifolds. This is a core ingredient to
CAR-/CCR-algebraic constructions of quantum field theories on curved
spacetimes, particularly for higher spin field equations.Comment: revised version: typos; reordering of sec 2; results unchange

### Validity and failure of some entropy inequalities for CAR systems

Basic properties of von Neumann entropy such as the triangle inequality and
what we call MONO-SSA are studied for CAR systems.
We show that both inequalities hold for any even state. We construct a
certain class of noneven states giving counter examples of those inequalities.
It is not always possible to extend a set of prepared states on disjoint
regions to some joint state on the whole region for CAR systems.
However, for every even state, we have its `symmetric purification' by which
the validity of those inequalities is shown.
Some (realized) noneven states have peculiar state correlations among
subsystems and induce the failure of those inequalities.Comment: 14 pages, latex, to appear in JMP. Some typos are correcte

### Endomorphism Semigroups and Lightlike Translations

Certain criteria are demonstrated for a spatial derivation of a von Neumann
algebra to generate a one-parameter semigroup of endomorphisms of that algebra.
These are then used to establish a converse to recent results of Borchers and
of Wiesbrock on certain one-parameter semigroups of endomorphisms of von
Neumann algebras (specifically, Type III_1 factors) that appear as lightlike
translations in the theory of algebras of local observables.Comment: 9 pages, Late

### The Measure of a Measurement

While finite non-commutative operator systems lie at the foundation of
quantum measurement, they are also tools for understanding geometric iterations
as used in the theory of iterated function systems (IFSs) and in wavelet
analysis. Key is a certain splitting of the total Hilbert space and its
recursive iterations to further iterated subdivisions. This paper explores some
implications for associated probability measures (in the classical sense of
measure theory), specifically their fractal components.
We identify a fractal scale $s$ in a family of Borel probability measures
$\mu$ on the unit interval which arises independently in quantum information
theory and in wavelet analysis. The scales $s$ we find satisfy $s\in
\mathbb{R}_{+}$ and $s\not =1$, some $s 1$. We identify these
scales $s$ by considering the asymptotic properties of $\mu(J) /| J| ^{s}$
where $J$ are dyadic subintervals, and $| J| \to0$.Comment: 18 pages, 3 figures, and reference

### Separability for lattice systems at high temperature

Equilibrium states of infinite extended lattice systems at high temperature
are studied with respect to their entanglement. Two notions of separability are
offered. They coincide for finite systems but differ for infinitely extended
ones. It is shown that for lattice systems with localized interaction for high
enough temperature there exists no local entanglement. Even more quasifree
states at high temperature are also not distillably entangled for all local
regions of arbitrary size. For continuous systems entanglement survives for all
temperatures. In mean field theories it is possible, that local regions are not
entangled but the entanglement is hidden in the fluctuation algebra

### Endomorphisms of B(H). II. Finitely Correlated States on On

AbstractWe identify sets of conjugacy classes of ergodic endomorphisms of B(H) where H is a fixed separable Hilbert space. They correspond to certain equivalence classes of pure states on the Cuntz algebras Onwherenis the Powers index. These states, called finitely correlated states, and strongly asymptotically shift invariant states, are defined and characterized. The subsets of these states defining shifts will in general be identified in a later work, but here an interesting cross section for the conjugacy classes of shifts called diagonalizable shifts is introduced and studied

### Wavelets in mathematical physics: q-oscillators

We construct representations of a q-oscillator algebra by operators on Fock
space on positive matrices. They emerge from a multiresolution scaling
construction used in wavelet analysis. The representations of the Cuntz Algebra
arising from this multiresolution analysis are contained as a special case in
the Fock Space construction.Comment: (03/11/03):18 pages; LaTeX2e, "article" document class with
"letterpaper" option An outline was added under the abstract (p.1),
paragraphs added to Introduction (p.2), mat'l added to Proofs in Theorems 1
and 6 (pgs.5&17), material added to text for the conclusion (p.17), one add'l
reference added [12]. (04/22/03):"number 1" replace with "term C" (p.9),
single sentences reformed into a one paragraph (p.13), QED symbol moved up
one paragraph and last paragraph labeled as "Concluding Remarks.

### Microscopic Conductivity of Lattice Fermions at Equilibrium - Part I: Non-Interacting Particles

We consider free lattice fermions subjected to a static bounded potential and
a time- and space-dependent electric field. For any bounded convex region
$\mathcal{R}\subset \mathbb{R}^{d}$ ($d\geq 1$) of space, electric fields
$\mathcal{E}$ within $\mathcal{R}$ drive currents. At leading order, uniformly
with respect to the volume $\left| \mathcal{R}\right|$ of $\mathcal{R}$ and
the particular choice of the static potential, the dependency on $\mathcal{E}$
of the current is linear and described by a conductivity distribution. Because
of the positivity of the heat production, the real part of its Fourier
transform is a positive measure, named here (microscopic) conductivity measure
of $\mathcal{R}$, in accordance with Ohm's law in Fourier space. This finite
measure is the Fourier transform of a time-correlation function of current
fluctuations, i.e., the conductivity distribution satisfies Green-Kubo
relations. We additionally show that this measure can also be seen as the
boundary value of the Laplace-Fourier transform of a so-called quantum current
viscosity. The real and imaginary parts of conductivity distributions satisfy
Kramers-Kronig relations. At leading order, uniformly with respect to
parameters, the heat production is the classical work performed by electric
fields on the system in presence of currents. The conductivity measure is
uniformly bounded with respect to parameters of the system and it is never the
trivial measure $0\,\mathrm{d}\nu$. Therefore, electric fields generally
produce heat in such systems. In fact, the conductivity measure defines a
quadratic form in the space of Schwartz functions, the Legendre-Fenchel
transform of which describes the resistivity of the system. This leads to
Joule's law, i.e., the heat produced by currents is proportional to the
resistivity and the square of currents

### Correlations in Free Fermionic States

We study correlations in a bipartite, Fermionic, free state in terms of
perturbations induced by one party on the other. In particular, we show that
all so conditioned free states can be modelled by an auxiliary Fermionic system
and a suitable completely positive map.Comment: 17 pages, no figure

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