Quantum mechanics with a time-dependent random unitary Hamiltonian: A perturbative study of the nonlinear Keldysh sigma-model

We analyze the perturbative series of the Keldysh-type sigma-model proposed recently for describing the quantum mechanics with time-dependent Hamiltonians from the unitary Wigner-Dyson random-matrix ensemble. We observe that vertices of orders higher than four cancel, which allows us to reduce the calculation of the energy-diffusion constant to that in a special kind of the matrix \phi^4 model. We further verify that the perturbative four-loop correction to the energy-diffusion constant in the high-velocity limit cancels, in agreement with the conjecture of one of the authors.Comment: 27 pages, 15 figures; typos corrected, one reference adde

Dyson Pairs and Zero-Mass Black Holes

It has been argued by Dyson in the context of QED in flat spacetime that perturbative expansions in powers of the electric charge e cannot be convergent because if e is purely imaginary then the vacuum should be unstable to the production of charged pairs. We investigate the spontaneous production of such Dyson pairs in electrodynamics coupled to gravity. They are found to consist of pairs of zero-rest mass black holes with regular horizons. The properties of these zero rest mass black holes are discussed. We also consider ways in which a dilaton may be included and the relevance of this to recent ideas in string theory. We discuss accelerating solutions and find that, in certain circumstances, the `no strut' condition may be satisfied giving a regular solution describing a pair of zero rest mass black holes accelerating away from one another. We also study wormhole and tachyonic solutions and how they affect the stability of the vacuum.Comment: 41 pages LaTex, 5 figure

The Future Evolution of White Dwarf Stars Through Baryon Decay and Time Varying Gravitational Constant

Motivated by the possibility that the fundamental ``constants'' of nature could vary with time, this paper considers the long term evolution of white dwarf stars under the combined action of proton decay and variations in the gravitational constant. White dwarfs are thus used as a theoretical laboratory to study the effects of possible time variations, especially their implications for the future history of the universe. More specifically, we consider the gravitational constant $G$ to vary according to the parametric relation $G = G_0 (1 + t/t_\ast)^{-p}$, where the time scale $t_\ast$ is the same order as the proton lifetime. We then study the long term fate and evolution of white dwarf stars. This treatment begins when proton decay dominates the stellar luminosity, and ends when the star becomes optically thin to its internal radiation.Comment: 12 pages, 10 figures, accepted to Astrophysics and Space Scienc

Survival of contact processes on the hierarchical group

We consider contact processes on the hierarchical group, where sites infect other sites at a rate depending on their hierarchical distance, and sites become healthy with a constant recovery rate. If the infection rates decay too fast as a function of the hierarchical distance, then we show that the critical recovery rate is zero. On the other hand, we derive sufficient conditions on the speed of decay of the infection rates for the process to exhibit a nontrivial phase transition between extinction and survival. For our sufficient conditions, we use a coupling argument that compares contact processes on the hierarchical group with freedom two with contact processes on a renormalized lattice. An interesting novelty in this renormalization argument is the use of a result due to Rogers and Pitman on Markov functionals.Comment: Minor changes compared to previous version. Final version. 30 pages. 1 figur

Bosonic Excitations in Random Media

We consider classical normal modes and non-interacting bosonic excitations in disordered systems. We emphasise generic aspects of such problems and parallels with disordered, non-interacting systems of fermions, and discuss in particular the relevance for bosonic excitations of symmetry classes known in the fermionic context. We also stress important differences between bosonic and fermionic problems. One of these follows from the fact that ground state stability of a system requires all bosonic excitation energy levels to be positive, while stability in systems of non-interacting fermions is ensured by the exclusion principle, whatever the single-particle energies. As a consequence, simple models of uncorrelated disorder are less useful for bosonic systems than for fermionic ones, and it is generally important to study the excitation spectrum in conjunction with the problem of constructing a disorder-dependent ground state: we show how a mapping to an operator with chiral symmetry provides a useful tool for doing this. A second difference involves the distinction for bosonic systems between excitations which are Goldstone modes and those which are not. In the case of Goldstone modes we review established results illustrating the fact that disorder decouples from excitations in the low frequency limit, above a critical dimension $d_c$, which in different circumstances takes the values $d_c=2$ and $d_c=0$. For bosonic excitations which are not Goldstone modes, we argue that an excitation density varying with frequency as $\rho(\omega) \propto \omega^4$ is a universal feature in systems with ground states that depend on the disorder realisation. We illustrate our conclusions with extensive analytical and some numerical calculations for a variety of models in one dimension

Banking union in historical perspective: the initiative of the European Commission in the 1960s-1970s

This article shows that planning for the organization of EU banking regulation and supervision did not just appear on the agenda in recent years with discussions over the creation of the eurozone banking union. It unveils a hitherto neglected initiative of the European Commission in the 1960s and early 1970s. Drawing on extensive archival work, this article explains that this initiative, however, rested on a number of different assumptions, and emerged in a much different context. It first explains that the Commission's initial project was not crisis-driven; that it articulated the link between monetary integration and banking regulation; and finally that it did not set out to move the supervisory framework to the supranational level, unlike present-day developments

The Free Energy of the Quantum Heisenberg Ferromagnet at Large Spin

We consider the spin-S ferromagnetic Heisenberg model in three dimensions, in the absence of an external field. Spin wave theory suggests that in a suitable temperature regime the system behaves effectively as a system of non-interacting bosons (magnons). We prove this fact at the level of the specific free energy: if $S \to \infty$ and the inverse temperature $\beta \to 0$ in such a way that $\beta S$ stays constant, we rigorously show that the free energy per unit volume converges to the one suggested by spin wave theory. The proof is based on the localization of the system in small boxes and on upper and lower bounds on the local free energy, and it also provides explicit error bounds on the remainder.Comment: 11 pages, pdfLate

Moments of vicious walkers and M\"obius graph expansions

A system of Brownian motions in one-dimension all started from the origin and conditioned never to collide with each other in a given finite time-interval $(0, T]$ is studied. The spatial distribution of such vicious walkers can be described by using the repulsive eigenvalue-statistics of random Hermitian matrices and it was shown that the present vicious walker model exhibits a transition from the Gaussian unitary ensemble (GUE) statistics to the Gaussian orthogonal ensemble (GOE) statistics as the time $t$ is going on from 0 to $T$. In the present paper, we characterize this GUE-to-GOE transition by presenting the graphical expansion formula for the moments of positions of vicious walkers. In the GUE limit $t \to 0$, only the ribbon graphs contribute and the problem is reduced to the classification of orientable surfaces by genus. Following the time evolution of the vicious walkers, however, the graphs with twisted ribbons, called M\"obius graphs, increase their contribution to our expansion formula, and we have to deal with the topology of non-orientable surfaces. Application of the recent exact result of dynamical correlation functions yields closed expressions for the coefficients in the M\"obius expansion using the Stirling numbers of the first kind.Comment: REVTeX4, 11 pages, 1 figure. v.2: calculations of the Green function and references added. v.3: minor additions and corrections made for publication in Phys.Rev.

Dynamic Evolution Model of Isothermal Voids and Shocks

We explore self-similar hydrodynamic evolution of central voids embedded in an isothermal gas of spherical symmetry under the self-gravity. More specifically, we study voids expanding at constant radial speeds in an isothermal gas and construct all types of possible void solutions without or with shocks in surrounding envelopes. We examine properties of void boundaries and outer envelopes. Voids without shocks are all bounded by overdense shells and either inflows or outflows in the outer envelope may occur. These solutions, referred to as type $\mathcal{X}$ void solutions, are further divided into subtypes $\mathcal{X}_{\rm I}$ and $\mathcal{X}_{\rm II}$ according to their characteristic behaviours across the sonic critical line (SCL). Void solutions with shocks in envelopes are referred to as type $\mathcal{Z}$ voids and can have both dense and quasi-smooth edges. Asymptotically, outflows, breezes, inflows, accretions and static outer envelopes may all surround such type $\mathcal{Z}$ voids. Both cases of constant and varying temperatures across isothermal shock fronts are analyzed; they are referred to as types $\mathcal{Z}_{\rm I}$ and $\mathcal{Z}_{\rm II}$ void shock solutions. We apply the `phase net matching procedure' to construct various self-similar void solutions. We also present analysis on void generation mechanisms and describe several astrophysical applications. By including self-gravity, gas pressure and shocks, our isothermal self-similar void (ISSV) model is adaptable to various astrophysical systems such as planetary nebulae, hot bubbles and superbubbles in the interstellar medium as well as supernova remnants.Comment: 24 pages, 13 figuers, accepted by ApS

Vicious walk with a wall, noncolliding meanders, and chiral and Bogoliubov-deGennes random matrices

Spatially and temporally inhomogeneous evolution of one-dimensional vicious walkers with wall restriction is studied. We show that its continuum version is equivalent with a noncolliding system of stochastic processes called Brownian meanders. Here the Brownian meander is a temporally inhomogeneous process introduced by Yor as a transform of the Bessel process that is a motion of radial coordinate of the three-dimensional Brownian motion represented in the spherical coordinates. It is proved that the spatial distribution of vicious walkers with a wall at the origin can be described by the eigenvalue-statistics of Gaussian ensembles of Bogoliubov-deGennes Hamiltonians of the mean-field theory of superconductivity, which have the particle-hole symmetry. We report that the time evolution of the present stochastic process is fully characterized by the change of symmetry classes from the type $C$ to the type $C$I in the nonstandard classes of random matrix theory of Altland and Zirnbauer. The relation between the non-colliding systems of the generalized meanders of Yor, which are associated with the even-dimensional Bessel processes, and the chiral random matrix theory is also clarified.Comment: REVTeX4, 16 pages, 4 figures. v2: some additions and correction