Contextual unification of classical and quantum physics

Following an article by John von Neumann on infinite tensor products, we develop the idea that the usual formalism of quantum mechanics, associated with unitary equivalence of representations, stops working when countable infinities of particles (or degrees of freedom) are encountered. This is because the dimension of the corresponding Hilbert space becomes uncountably infinite, leading to the loss of unitary equivalence, and to sectorization. By interpreting physically this mathematical fact, we show that it provides a natural way to describe the "Heisenberg cut", as well as a unified mathematical model including both quantum and classical physics, appearing as required incommensurable facets in the description of nature.Comment: 13 pages, no figure. In v2 some clarifications adde

Random Network Models and Quantum Phase Transitions in Two Dimensions

An overview of the random network model invented by Chalker and Coddington, and its generalizations, is provided. After a short introduction into the physics of the Integer Quantum Hall Effect, which historically has been the motivation for introducing the network model, the percolation model for electrons in spatial dimension 2 in a strong perpendicular magnetic field and a spatially correlated random potential is described. Based on this, the network model is established, using the concepts of percolating probability amplitude and tunneling. Its localization properties and its behavior at the critical point are discussed including a short survey on the statistics of energy levels and wave function amplitudes. Magneto-transport is reviewed with emphasis on some new results on conductance distributions. Generalizations are performed by establishing equivalent Hamiltonians. In particular, the significance of mappings to the Dirac model and the two dimensional Ising model are discussed. A description of renormalization group treatments is given. The classification of two dimensional random systems according to their symmetries is outlined. This provides access to the complete set of quantum phase transitions like the thermal Hall transition and the spin quantum Hall transition in two dimension. The supersymmetric effective field theory for the critical properties of network models is formulated. The network model is extended to higher dimensions including remarks on the chiral metal phase at the surface of a multi-layer quantum Hall system.Comment: 176 pages, final version, references correcte

Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries

The article reviews the current status of a theoretical approach to the problem of the emission of gravitational waves by isolated systems in the context of general relativity. Part A of the article deals with general post-Newtonian sources. The exterior field of the source is investigated by means of a combination of analytic post-Minkowskian and multipolar approximations. The physical observables in the far-zone of the source are described by a specific set of radiative multipole moments. By matching the exterior solution to the metric of the post-Newtonian source in the near-zone we obtain the explicit expressions of the source multipole moments. The relationships between the radiative and source moments involve many non-linear multipole interactions, among them those associated with the tails (and tails-of-tails) of gravitational waves. Part B of the article is devoted to the application to compact binary systems. We present the equations of binary motion, and the associated Lagrangian and Hamiltonian, at the third post-Newtonian (3PN) order beyond the Newtonian acceleration. The gravitational-wave energy flux, taking consistently into account the relativistic corrections in the binary moments as well as the various tail effects, is derived through 3.5PN order with respect to the quadrupole formalism. The binary's orbital phase, whose prior knowledge is crucial for searching and analyzing the signals from inspiralling compact binaries, is deduced from an energy balance argument.Comment: 109 pages, 1 figure; this version is an update of the Living Review article originally published in 2002; available on-line at http://www.livingreviews.org

On a Spectral Method for Calculating the Electrical Resistivity of a Low Temperature Metal from the Linearized Boltzmann Equation

While it is well known that transport equations may be derived diagrammatically, both this approach and that of Boltzmann inevitably encounter an integral equation that both is difficult to solve and, for the most part, has yielded only to uncontrolled approximations. Even though the popular approximations, which are typically either variational in nature or involve dropping memory effects, can be expected to capture the temperature scaling of the kinetic coefficients, it is desirable to exactly obtain the prefactor by way of a mathematically justifiable approximation. For the purpose of so precisely resolving the distribution function that governs the elementary excitations of a metal perturbed by an externally applied static electric field, a spectral method was developed that makes use of the temperature as a control parameter to facilitate an asymptotic inversion of the collision operator; the technique leverages a singularity that is inherent to the Boltzmann equation in the low temperature limit, i.e. when the dissipating Boson bath is all but frozen out. This present dissertation is mainly concerned with the anomalous transport behavior that is commonly observed in quantum magnets; throughout a wide range of their phase diagram, materials such as the metallic ferromagnet ZrZn$_2$ display a power law behavior of the electrical resistivity $\rho \propto T^s$ with $s 2$ at temperatures $T \approx 10K$ (where ZrZn$_2$ exhibits $1.5 < s < 1.7$). After preliminarily investigating the electron-phonon system by way of rigorous reasoning, I will argue that the observed scaling of the residual resistivity $\rho \propto T^{3/2}$ in metallic ferromagnets can be attributed to interference between two scattering mechanisms: ferromagnons and static impurities. This dissertation includes previously published co-authoredmaterial

Quantum algorithms

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Physics, 1999.Includes bibliographical references (leaves 89-94).by Daniel S. Abrams.Ph.D

Dimensions of Timescales in Neuromorphic Computing Systems

This article is a public deliverable of the EU project "Memory technologies with multi-scale time constants for neuromorphic architectures" (MeMScales, https://memscales.eu, Call ICT-06-2019 Unconventional Nanoelectronics, project number 871371). This arXiv version is a verbatim copy of the deliverable report, with administrative information stripped. It collects a wide and varied assortment of phenomena, models, research themes and algorithmic techniques that are connected with timescale phenomena in the fields of computational neuroscience, mathematics, machine learning and computer science, with a bias toward aspects that are relevant for neuromorphic engineering. It turns out that this theme is very rich indeed and spreads out in many directions which defy a unified treatment. We collected several dozens of sub-themes, each of which has been investigated in specialized settings (in the neurosciences, mathematics, computer science and machine learning) and has been documented in its own body of literature. The more we dived into this diversity, the more it became clear that our first effort to compose a survey must remain sketchy and partial. We conclude with a list of insights distilled from this survey which give general guidelines for the design of future neuromorphic systems