arXiv subjects

R. F. Werner

Publications and source records attributed to R. F. Werner.

79 records · Page 5Linked to original sources

arXiv1995

The classical limit of quantum theory

For a quantum observable $A_\hbar$ depending on a parameter $\hbar$ we define the notion ``$A_\hbar$ converges in the classical limit''. The limit is a function on phase space. Convergence is in norm in the sense that $A_\hbar\to0$ is equivalent with $\Vert A_\hbar\Vert\to0$. The $\hbar$-wise product of convergent observables converges to the product of the limiting phase space functions. $\hbar^{-1}$ times the commutator of suitable observables converges to the Poisson bracket of the limits. For a large class of convergent Hamiltonians the $\hbar$-wise action of the corresponding dynamics converges to the classical Hamiltonian dynamics. The connections with earlier approaches, based on the WKB method, or on Wigner distribution functions, or on the limits of coherent states are reviewed.

quant-ph↗

arXiv1995

Quantum spin chains with quantum group symmetry

We consider actions of quantum groups on lattice spin systems. We show that if an action of a quantum group respects the local structure of a lattice system, it has to be an ordinary group. Even allowing weakly delocalized (quasi-local) tails of the action, we find that there are no actions of a properly quantum group commuting with lattice translations. The non-locality arises from the ordering of factors in the quantum group C*-algebra, and can be made one-sided, thus allowing semi-local actions on a half chain. Under such actions, localized quantum group invariant elements remain localized. Hence the notion of interactions invariant under the quantum group and also under translations, recently studied by many authors, makes sense even though there is no global action of the quantum group. We consider a class of such quantum group invariant interactions with the property that there is a unique translation invariant ground state. Under weak locality assumptions, its GNS representation carries no unitary representation of the quantum group.

cond-mat↗

arXiv1995

Ergodicity of quantum cellular automata

We define a class of dynamical maps on the quasi-local algebra of a quantum spin system, which are quantum analogues of probabilistic cellular automata. We develop criteria for such a system to be ergodic, i.e., to possess a unique invariant state. Intuitively, ergodicity obtains if the local transition operators exhibit sufficiently large disorder. The ergodicity criteria also imply bounds for the exponential decay of correlations in the unique invariant state. The main technical tool is a quantum version of oscillation norms, defined in the classical case as the sum over all sites of the variations of an observable with respect to local spin-flips.

cond-mat↗

arXiv1995

Ground states of the infinite q-deformed Heisenberg ferromagnet

We set up a general structure for the analysis of ``frustration-free ground states'', or ``zero-energy states'', i.e., states minimizing each term in a lattice interaction individually. The nesting of the finite volume ground state spaces is described by a generalized inductive limit of observable algebras. The limit space of this inductive system has a state space which is canonically isomorphic (as a compact convex set) to the set of zero-energy states. We show that for Heisenberg ferromagnets, and for generalized valence bond solid states, the limit space is an abelian C*-algebra, and all zero-energy states are translationally invariant or periodic. For the $q$-deformed spin-$1/2$ Heisenberg ferromagnet in one dimension (i.e., the XXZ-chain with S$_q$U(2)-invariant boundary conditions) the limit space is an extension of the non-commutative algebra of compact operators by two points, corresponding to the ``all spins up'' and the ``all spins down'' states, respectively. These are the only translationally invariant zero-energy states. The remaining ones are parametrized by the density matrices on a Hilbert space, and converge weakly to the ``all up'' (resp.\ ``all down'') state for shifts to $-\infty$ (resp.\ $+\infty$).

cond-mat↗

arXiv1993

Positive representations of general commutation relations allowing Wick ordering

We consider the problem of representing in Hilbert space commutation relations of the form $$ a_ia_j^*=δ_{ij}{\bold1} + \sum_{k\ell}T_{ij}^{k\ell} a_\ell^*a_k \quad,$$ where the $T_{ij}^{k\ell}$ are essentially arbitrary scalar coefficients. Examples comprise the $q$-canonical commutation relations introduced by Greenberg, Bozejko, and Speicher, and the twisted canonical (anti-)commutation relations studied by Pusz and Woronowicz, as well as the quantum group S$_ν$U$(2)$. Using these relations, any polynomial in the generators $a_i$ and their adjoints can uniquely be written in ``Wick ordered form'' in which all starred generators are to the left of all unstarred ones. In this general framework we define the Fock representation, as well as coherent representations. We develop criteria for the natural scalar product in the associated representation spaces to be positive definite, and for the relations to have representations by bounded operators in a Hilbert space. We characterize the relations between the generators $a_i$ (not involving $a_i^*$) which are compatible with the basic relations. The relations may also be interpreted as defining a non-commutative differential calculus. For generic coefficients $T_{ij}^{k\ell}$, however, all differential forms of degree 2 and higher vanish. We exhibit conditions for this not to be the case, and relate them to the ideal structure of the Wick algebra, and conditions of positivity. We show that the differential calculus is compatible with the involution iff the coefficients $T$ define a representation of the braid group. This condition is also shown to imply improved bounds for the positivity of the Fock representation. Finally, we study the KMS states of the group of gauge transformations defined by $a_j\mapsto\exp(it)a_j$.

funct-an↗

arXiv1993

The Free Quon Gas Suffers Gibbs' Paradox

We consider the Statistical Mechanics of systems of particles satisfying the $q$-commutation relations recently proposed by Greenberg and others. We show that although the commutation relations approach Bose (resp.\ Fermi) relations for $q\to1$ (resp.\ $q\to-1$), the partition functions of free gases are independent of $q$ in the range $-1<q<1$. The partition functions exhibit Gibbs' Paradox in the same way as a classical gas without a correction factor $1/N!$ for the statistical weight of the $N$-particle phase space, i.e.\ the Statistical Mechanics does not describe a material for which entropy, free energy, and particle number are extensive thermodynamical quantities.

hep-th↗

arXiv1993

Coherent States of the q--Canonical Commutation Relations

For the $q$-deformed canonical commutation relations $a(f)a^\dagger(g) = (1-q)\,\langle f,g\rangle{\bf1}+q\,a^\dagger(g)a(f)$ for $f,g$ in some Hilbert space ${\cal H}$ we consider representations generated from a vector $Ω$ satisfying $a(f)Ω=\langle f,ϕ\rangleΩ$, where $ϕ\in{\cal H}$. We show that such a representation exists if and only if $\Vertϕ\Vert\leq1$. Moreover, for $\Vertϕ\Vert<1$ these representations are unitarily equivalent to the Fock representation (obtained for $ϕ=0$). On the other hand representations obtained for different unit vectors $ϕ$ are disjoint. We show that the universal C*-algebra for the relations has a largest proper, closed, two-sided ideal. The quotient by this ideal is a natural $q$-analogue of the Cuntz algebra (obtained for $q=0$). We discuss the Conjecture that, for $d<\infty$, this analogue should, in fact, be equal to the Cuntz algebra itself. In the limiting cases $q=\pm1$ we determine all irreducible representations of the relations, and characterize those which can be obtained via coherent states.

funct-an↗