Principles of Discrete Time Mechanics: II. Classical field Theory

We apply the principles discussed in an earlier paper to the construction of discrete time field theories. We derive the discrete time field equations of motion and Noether's theorem and apply them to the Schrodinger equation to illustrate the methodology. Stationary solutions to the discrete time Schrodinger wave equation are found to be identical to standard energy eigenvalue solutions except for a fundamental limit on the energy. Then we apply the formalism to the free neutral Klein Gordon system, deriving the equations of motion and conserved quantities such as the linear momentum and angular momentum. We show that there is an upper bound on the magnitude of linear momentum for physical particle-like solutions. We extend the formalism to the charged scalar field coupled to Maxwell's electrodynamics in a gauge invariant way. We apply the formalism to include the Maxwell and Dirac fields, setting the scene for second quantisation of discrete time mechanics and discrete time Quantum Electrodynamics.Comment: 23 pages, LateX, To be published in J.Phys.A: Math.Gen: contact email address: [email protected]

Extensions and degenerations of spectral triples

For a unital C*-algebra A, which is equipped with a spectral triple and an extension T of A by the compacts, we construct a family of spectral triples associated to T and depending on the two positive parameters (s,t). Using Rieffel's notation of quantum Gromov-Hausdorff distance between compact quantum metric spaces it is possible to define a metric on this family of spectral triples, and we show that the distance between a pair of spectral triples varies continuously with respect to the parameters. It turns out that a spectral triple associated to the unitarization of the algebra of compact operators is obtained under the limit - in this metric - for (s,1) -> (0, 1), while the basic spectral triple, associated to A, is obtained from this family under a sort of a dual limiting process for (1, t) -> (1, 0). We show that our constructions will provide families of spectral triples for the unitarized compacts and for the Podles sphere. In the case of the compacts we investigate to which extent our proposed spectral triple satisfies Connes' 7 axioms for noncommutative geometry.Comment: 40 pages. Addedd in ver. 2: Examples for the compacts and the Podle`s sphere plus comments on the relations to matricial quantum metrics. In ver.3 the word "deformations" in the original title has changed to "degenerations" and some illustrative remarks on this aspect are adde

Lack of consensus in social systems

We propose an exactly solvable model for the dynamics of voters in a two-party system. The opinion formation process is modeled on a random network of agents. The dynamical nature of interpersonal relations is also reflected in the model, as the connections in the network evolve with the dynamics of the voters. In the infinite time limit, an exact solution predicts the emergence of consensus, for arbitrary initial conditions. However, before consensus is reached, two different metastable states can persist for exponentially long times. One state reflects a perfect balancing of opinions, the other reflects a completely static situation. An estimate of the associated lifetimes suggests that lack of consensus is typical for large systems.Comment: 4 pages, 6 figures, submitted to Phys. Rev. Let

Reverberation Mapping Measurements of Black Hole Masses in Six Local Seyfert Galaxies

We present the final results from a high sampling rate, multi-month, spectrophotometric reverberation mapping campaign undertaken to obtain either new or improved Hbeta reverberation lag measurements for several relatively low-luminosity AGNs. We have reliably measured thetime delay between variations in the continuum and Hbeta emission line in six local Seyfert 1 galaxies. These measurements are used to calculate the mass of the supermassive black hole at the center of each of these AGNs. We place our results in context to the most current calibration of the broad-line region (BLR) R-L relationship, where our results remove outliers and reduce the scatter at the low-luminosity end of this relationship. We also present velocity-resolved Hbeta time delay measurements for our complete sample, though the clearest velocity-resolved kinematic signatures have already been published.Comment: 52 pages (AASTeX: 29 pages of text, 8 tables, 7 figures), accepted for publication in the Astrophysical Journa

A Revised Broad-Line Region Radius and Black Hole Mass for the Narrow-Line Seyfert 1 NGC 4051

We present the first results from a high sampling rate, multi-month reverberation mapping campaign undertaken primarily at MDM Observatory with supporting observations from telescopes around the world. The primary goal of this campaign was to obtain either new or improved Hbeta reverberation lag measurements for several relatively low luminosity AGNs. We feature results for NGC 4051 here because, until now, this object has been a significant outlier from AGN scaling relationships, e.g., it was previously a ~2-3sigma outlier on the relationship between the broad-line region (BLR) radius and the optical continuum luminosity - the R_BLR-L relationship. Our new measurements of the lag time between variations in the continuum and Hbeta emission line made from spectroscopic monitoring of NGC 4051 lead to a measured BLR radius of R_BLR = 1.87 (+0.54 -0.50) light days and black hole mass of M_BH = 1.73 (+0.55 -0.52) x 10^6 M_sun. This radius is consistent with that expected from the R_BLR-L relationship, based on the present luminosity of NGC 4051 and the most current calibration of the relation by Bentz et al. (2009a). We also present a preliminary look at velocity-resolved Hbeta light curves and time delay measurements, although we are unable to reconstruct an unambiguous velocity-resolved reverberation signal.Comment: 38 pages, 7 figures, accepted for publication in ApJ, changes from v1 reflect suggestions from anonymous refere

Investigating fusion plasma instabilities in the Mega Amp Spherical Tokamak using mega electron volt proton emissions (invited)a)

The proton detector (PD) measures 3 MeV proton yield distributions from deuterium-deuterium fusion reactions within the Mega Amp Spherical Tokamak (MAST). The PD's compact four-channel system of collimated and individually oriented silicon detectors probes different regions of the plasma, detecting protons (with gyro radii large enough to be unconfined) leaving the plasma on curved trajectories during neutral beam injection. From first PD data obtained during plasma operation in 2013, proton production rates (up to several hundred kHz and 1 ms time resolution) during sawtooth events were compared to the corresponding MAST neutron camera data. Fitted proton emission profiles in the poloidal plane demonstrate the capabilities of this new system

Disk-Jet Connection in the Radio Galaxy 3C 120

We present the results of extensive multi-frequency monitoring of the radio galaxy 3C 120 between 2002 and 2007 at X-ray, optical, and radio wave bands, as well as imaging with the Very Long Baseline Array (VLBA). Over the 5 yr of observation, significant dips in the X-ray light curve are followed by ejections of bright superluminal knots in the VLBA images. Consistent with this, the X-ray flux and 37 GHz flux are anti-correlated with X-ray leading the radio variations. This implies that, in this radio galaxy, the radiative state of accretion disk plus corona system, where the X-rays are produced, has a direct effect on the events in the jet, where the radio emission originates. The X-ray power spectral density of 3C 120 shows a break, with steeper slope at shorter timescale and the break timescale is commensurate with the mass of the central black hole based on observations of Seyfert galaxies and black hole X-ray binaries. These findings provide support for the paradigm that black hole X-ray binaries and active galactic nuclei are fundamentally similar systems, with characteristic time and size scales linearly proportional to the mass of the central black hole. The X-ray and optical variations are strongly correlated in 3C 120, which implies that the optical emission in this object arises from the same general region as the X-rays, i.e., in the accretion disk-corona system. We numerically model multi-wavelength light curves of 3C 120 from such a system with the optical-UV emission produced in the disk and the X-rays generated by scattering of thermal photons by hot electrons in the corona. From the comparison of the temporal properties of the model light curves to that of the observed variability, we constrain the physical size of the corona and the distances of the emitting regions from the central BH.Comment: Accepted for publication in the Astrophysical Journal. 28 pages, 21 figures, 2 table