27,682 research outputs found
Poisson vertex algebras in the theory of Hamiltonian equations
We lay down the foundations of the theory of Poisson vertex algebras aimed at
its applications to integrability of Hamiltonian partial differential
equations. Such an equation is called integrable if it can be included in an
infinite hierarchy of compatible Hamiltonian equations, which admit an infinite
sequence of linearly independent integrals of motion in involution. The
construction of a hierarchy and its integrals of motion is achieved by making
use of the so called Lenard scheme. We find simple conditions which guarantee
that the scheme produces an infinite sequence of closed 1-forms \omega_j, j in
Z_+, of the variational complex \Omega. If these forms are exact, i.e. \omega_j
are variational derivatives of some local functionals \int h_j, then the latter
are integrals of motion in involution of the hierarchy formed by the
corresponding Hamiltonian vector fields. We show that the complex \Omega is
exact, provided that the algebra of functions V is "normal"; in particular, for
arbitrary V, any closed form in \Omega becomes exact if we add to V a finite
number of antiderivatives. We demonstrate on the examples of KdV, HD and CNW
hierarchies how the Lenard scheme works. We also discover a new integrable
hierarchy, which we call the CNW hierarchy of HD type. Developing the ideas of
Dorfman, we extend the Lenard scheme to arbitrary Dirac structures, and
demonstrate its applicability on the examples of the NLS, pKdV and KN
hierarchies.Comment: 95 page
Completeness and Incompleteness of Synchronous Kleene Algebra
Synchronous Kleene algebra (SKA), an extension of Kleene algebra (KA), was
proposed by Prisacariu as a tool for reasoning about programs that may execute
synchronously, i.e., in lock-step. We provide a countermodel witnessing that
the axioms of SKA are incomplete w.r.t. its language semantics, by exploiting a
lack of interaction between the synchronous product operator and the Kleene
star. We then propose an alternative set of axioms for SKA, based on Salomaa's
axiomatisation of regular languages, and show that these provide a sound and
complete characterisation w.r.t. the original language semantics.Comment: Accepted at MPC 201
The Proca Field in Curved Spacetimes and its Zero Mass Limit
We investigate the classical and quantum Proca field (a massive vector
potential) of mass in arbitrary globally hyperbolic spacetimes and in the
presence of external sources. We motivate a notion of continuity in the mass
for families of observables and we investigate the
massless limit . Our limiting procedure is local and covariant and it
does not require a choice of reference state. We find that the limit exists
only on a subset of observables, which automatically implements a gauge
equivalence on the massless vector potential. For topologically non-trivial
spacetimes, one may consider several inequivalent choices of gauge equivalence
and our procedure selects the one which is expected from considerations
involving the Aharonov-Bohm effect and Gauss' law. We note that the limiting
theory does not automatically reproduce Maxwell's equation, but it can be
imposed consistently when the external current is conserved. To recover the
correct Maxwell dynamics from the limiting procedure would require an
additional control on limits of states. We illustrate this only in the
classical case, where the dynamics is recovered when the Lorenz constraint
remains well behaved in the limit.Comment: 35 page
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