arXiv subjects
Publications and source records attributed to R. F. Streater.
So far only quasifree fields have been shown to satisfy the Haag-Araki axioms for local algebras of observables; we show from a model in 1 + 1 dimensions that there can be representations in which two ingoing free particles produce a pair of out-going solitons with positive probability, which can be computed. This happens when the experiment is designed to observe this outcome. It is proposed that the same idea will work in four dimensions.
A start is made to redefining the topology of the spaces of normal states (density operators) by a new norm which is finite only for states of finite entropy. It is shown that a symmetrized version of the free energy difference between states can be used as a quantum version of the cosh Young function used in the theory of Orlicz space. The results form a quantum version of the classical treatment of nonparametric estimation by information geometry, in the work of Pistone and Sempi. We succeed in constructing the Luxemburg norm, the tangent space carrying the (+1)-affine structure, and the cotangent space carrying the (-1) affine structure, and we demonstrate the Holder-Orlicz inequality.
We derive the Euler equations as the hydrodynamic limit of a stochastic model of a hard-sphere gas on a lattice. We show that the system does not produce entropy.
We consider the quantum information manifold whose underlying set M consists of density operators rho with the extra property that some fractional power of rho is of trace class. The topology is defined by defining a neighbourhood of a point rho to be all density operators that dominate rho and are dominated by rho. We show that this is the same set as that of all states whose relative Hamiltonian X in the sense of Araki is bounded, and such that X(t) is holomorphic in the circle |t| less than 1/2. Here, X(t) is the time evolution of X determined by the modular automorphism defined by rho. We show that M is a Banach manifold in Araki's norm, and that both the canonical and the mixture affine connections can be defined. These are dual relative to the Kubo-Mori metric, and so generalise Amari's dual theory to quantum theory in infinite dimensions.
We show that the general Lévy process can be embedded in a suitable Fock space, classified by cocycles of the real line regarded as a group, ${\bf R}$. The formula of de Finetti corresponds to coboundaries. Kolmogorov's processes correspond to cocycles of which the derivatives are cocycles of the Lie algebra of ${\bf R}$. Lévy's formula gives the most general cocycle possible.
We show that a Galilean invariant version of fluid dynamics can be derived by the methods of statistical dynamics using Maxwell's balance equations. The basic equation is non-local, and might replace Boltzmann's equation if the latter turns out not to have global smooth solutions in general. As an approximation, a local form of the equation of motion is derived. It turns out to be a version of the compressible Navier-Stokes system with temperature, obeying Stokes's relation, and with viscosity rising as the square-root of the temperature. The new feature is the presence of a Dufour effect for a gas of a single component.
We show that, in finite dimensions, the only monotone metrics for which the (+1) and (-1) affine connections are mutually dual are constant multiples of Bogoliubov-Kubo-Mori metric
The methods of statistical dynamics are applied to a fluid with 5 conserved fields (the mass, the energy, and the three components of momentum) moving in a given external potential. When the potential is zero, we recover a previously derived system of parabolic differential equations, called "corrections to fluid dynamics".
The act of measuring a physical signal or field suggests a generalization of the wavelet transform that turns out to be a windowed version of the Radon transform. A reconstruction formula is derived which inverts this transform. A special choice of window yields the "Analytic--Signal transform" (AST), which gives a partially analytic extension of functions from R^n to C^n. For n =1, this reduces to Gabor's classical definition of "analytic signals." The AST is applied to the wave equation, giving an expansion of solutions in terms of wavelets specifically adapted to that equation and parametrized by real space and imaginary time coordinates (the "Euclidean region").
We study coupled non-linear parabolic equations for a fluid described by a material density and a temperature, both functions of space and time. In one dimension, we find some stationary solutions corresponding to fixing the temperature on the boundary, with no-escape boundary conditions for the material. For the special case, where the temperature on the boundary is the same at both ends, the linearised equations for small perturbations about a stationary solution are derived; they are subject to the boundary conditions, Dirichlet for the temperature and no-flow conditions for the material. The spectrum of the generator L of time-evolution, regarded as an operator on the Hilbert space of square-integrable functions on [0,1], is shown to be real, discrete and non-positive, even though L is not self-adjoint. This result is necessary for the stability of the stationary state, but might not be sufficient. The problem lies in the fact that L is not a sectorial operator; its numerical range is the whole of the complex plane.
We survey recent work on the Cramer-Rao inequality by Hasagawa and Petz; the notion of information manifold in infinite-dimensional Hilbert spaces is introduced, and the extension by Grasselli and the author to quadratic form perturbations is summarised.
We survey the development of probability from 1900, starting with Bachelier's theory of speculation. Fisher information appears in the theory of estimation. We touch on Brownian motion, and the Wiener integral. The Ito calculus, and its relation to to the heat equation, is mentioned. Quantum theory is introduced as a generalisation of probability, rather than of mechanics. The weakness of attempts to describe quantum theory in terms of hidden variables is explained, by a simple proof of Bell's inequality. Quantum versions of the Langevin equation are discussed, and the theory of continuous tensor products is used to give a possible quantum version. The quantum stochastic calculus of Barnett, Wilde and the author, as well as that of Parthasarathy and Hudson, is introduced.
We construct a Banach manifold of states, which are Gibbs states for potentials that are form-bounded in the sense of Kato relative to the free Hamiltonian. We construct the (+1)-affine structure and the (+1)-affine connection in the sense of Amari.
Let H be a self-adjoint operator such that exp(-aH) is of trace class for some a<1. Let V be a symmetric operator, Kato bounded relative to H. We show that log Tr[exp(-H+xV)] is a real analytic function of x in a hood of x=0. We show that the Gibbs states of H+xV form a real analytic Banach manifold. This work has been extended in math-ph/9910031.
We set up a discrete space-time dynamical model of molecules with thermalised kinetic energy and repulsive cores, in an external potential. The state is specified by a probability on the sample space. One time-step is given by a bistochastic map, followed by a local thermalising map. The model obeys the first and second laws of thermodynamics. The continuum limit, obtained using a MAPLE program, gives rise to coupled nonlinear reaction-diffusion equations for the density and temperature fields. The system obeys Onsager symmetry and exhibits the Soret and Dufour effects.
Let H be a self-adjoint operator bounded below by 1, and let V be a small form perturbation such that RVS has finite norm, where R is the resolvent at zero to the power 1/2 +epsilon, and S is the resolvent to the power 1/2-epsilon. Here, epsilon lies between 0 and 1/2. If the Gibbs state defined by H is sufficiently regular, we show that the free energy is an analytic function of V in the sense of Frechet, and that the family of density operators defined in this way is an analytic manifold modelled on a Banach space.