7,798 research outputs found
On the Particle Definition in the presence of Black Holes
A canonical particle definition via the diagonalisation of the Hamiltonian
for a quantum field theory in specific curved space-times is presented. Within
the provided approach radial ingoing or outgoing Minkowski particles do not
exist. An application of this formalism to the Rindler metric recovers the
well-known Unruh effect. For the situation of a black hole the Hamiltonian
splits up into two independent parts accounting for the interior and the
exterior domain, respectively. It turns out that a reasonable particle
definition may be accomplished for the outside region only. The Hamiltonian of
the field inside the black hole is unbounded from above and below and hence
possesses no ground state. The corresponding equation of motion displays a
linear global instability. Possible consequences of this instability are
discussed and its relations to the sonic analogues of black holes are
addressed. PACS-numbers: 04.70.Dy, 04.62.+v, 10.10.Ef, 03.65.Db.Comment: 44 pages, LaTeX, no figures, accepted for publication in Phys. Rev.
The Sampling Theorem and Coherent State Systems in Quantum Mechanics
The well known Poisson Summation Formula is analysed from the perspective of
the coherent state systems associated with the Heisenberg--Weyl group. In
particular, it is shown that the Poisson summation formula may be viewed
abstractly as a relation between two sets of bases (Zak bases) arising as
simultaneous eigenvectors of two commuting unitary operators in which geometric
phase plays a key role. The Zak bases are shown to be interpretable as
generalised coherent state systems of the Heisenberg--Weyl group and this, in
turn, prompts analysis of the sampling theorem (an important and useful
consequence of the Poisson Summation Formula) and its extension from a coherent
state point of view leading to interesting results on properties of von Neumann
and finer lattices based on standard and generalised coherent state systems.Comment: 20 pages, Late
Generalised Cesaro Convergence, Root Identities and the Riemann Hypothesis
We extend the notion of generalised Cesaro summation/convergence developed
previously to the more natural setting of what we call "remainder" Cesaro
summation/convergence and, after illustrating the utility of this approach in
deriving certain classical results, use it to develop a notion of generalised
root identities. These extend elementary root identities for polynomials both
to more general functions and to a family of identities parametrised by a
complex parameter \mu. In so doing they equate one expression (the derivative
side) which is defined via Fourier theory, with another (the root side) which
is defined via remainder Cesaro summation. For \mu a non-positive integer these
identities are naturally adapted to investigating the asymptotic behaviour of
the given function and the geometric distribution of its roots. For the Gamma
function we show that it satisfies the generalised root identities and use them
to constructively deduce Stirling's theorem. For the Riemann zeta function the
implications of the generalised root identities for \mu=0,-1 and -2 are
explored in detail; in the case of \mu=-2 a symmetry of the non-trivial roots
is broken and allows us to conclude, after detailed computation, that the
Riemann hypothesis must be false. In light of this, some final direct
discussion is given of areas where the arguments used throughout the paper are
deficient in rigour and require more detailed justification. The conclusion of
section 1 gives guidance on the most direct route through the paper to the
claim regarding the Riemann hypothesis
Stability of Correction Procedure via Reconstruction With Summation-by-Parts Operators for Burgers' Equation Using a Polynomial Chaos Approach
In this paper, we consider Burgers' equation with uncertain boundary and
initial conditions. The polynomial chaos (PC) approach yields a hyperbolic
system of deterministic equations, which can be solved by several numerical
methods. Here, we apply the correction procedure via reconstruction (CPR) using
summation-by-parts operators. We focus especially on stability, which is proven
for CPR methods and the systems arising from the PC approach. Due to the usage
of split-forms, the major challenge is to construct entropy stable numerical
fluxes. For the first time, such numerical fluxes are constructed for all
systems resulting from the PC approach for Burgers' equation. In numerical
tests, we verify our results and show also the advantage of the given ansatz
using CPR methods. Moreover, one of the simulations, i.e. Burgers' equation
equipped with an initial shock, demonstrates quite fascinating observations.
The behaviour of the numerical solutions from several methods (finite volume,
finite difference, CPR) differ significantly from each other. Through careful
investigations, we conclude that the reason for this is the high sensitivity of
the system to varying dissipation. Furthermore, it should be stressed that the
system is not strictly hyperbolic with genuinely nonlinear or linearly
degenerate fields
Analysis as a source of geometry: a non-geometric representation of the Dirac equation
Consider a formally self-adjoint first order linear differential operator
acting on pairs (2-columns) of complex-valued scalar fields over a 4-manifold
without boundary. We examine the geometric content of such an operator and show
that it implicitly contains a Lorentzian metric, Pauli matrices, connection
coefficients for spinor fields and an electromagnetic covector potential. This
observation allows us to give a simple representation of the massive Dirac
equation as a system of four scalar equations involving an arbitrary two-by-two
matrix operator as above and its adjugate. The point of the paper is that in
order to write down the Dirac equation in the physically meaningful
4-dimensional hyperbolic setting one does not need any geometric constructs.
All the geometry required is contained in a single analytic object - an
abstract formally self-adjoint first order linear differential operator acting
on pairs of complex-valued scalar fields.Comment: Edited in accordance with referees' recommendation
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