Continuous Symmetries of Difference Equations

Lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations. In this article we review the results of a much more recent program: the use of Lie groups to study difference equations. We show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners. One is to use generalized symmetries acting on solutions of difference equations, but leaving the lattice invariant. The other is to restrict to point symmetries, but to allow them to also transform the lattice.Comment: Review articl

A New Perspective On Time And Physical Laws

Callender claims that `time is the great informer' (Callender 2017, chapter 7), meaning that the direction(s) in which our `best' physical theories inform are temporal. This is intended to be a metaphysical claim, and as such expresses a relationship between the physical world and information-gathering systems such as ourselves. This paper gives two counterexamples to this claim, illustrating the fact that time and informative strength doubly dissociate, so the claim cannot be about physical theories in general. The first is a case where physical theories inform in directions that we have no reason to regard as temporal. The second is a case where our best physical theories fail to inform in directions that we have independent (pre-theoretic) reasons to regard as temporal. Taking these two cases into account suggests that the connection Callender makes between time and informativeness is perspectival. The second case demonstrates that, although scientists often seek information in temporal directions, the behaviour of the physical world can present serious difficulties for finding it. In response, this paper proposes a perspectival reading of Callender's claim, according to which the connection between time and informative strength has more to do with the aims and objectives of science than the workings of the physical world

Spectral Invariants of Operators of Dirac Type on Partitioned Manifolds

We review the concepts of the index of a Fredholm operator, the spectral flow of a curve of self-adjoint Fredholm operators, the Maslov index of a curve of Lagrangian subspaces in symplectic Hilbert space, and the eta invariant of operators of Dirac type on closed manifolds and manifolds with boundary. We emphasize various (occasionally overlooked) aspects of rigorous definitions and explain the quite different stability properties. Moreover, we utilize the heat equation approach in various settings and show how these topological and spectral invariants are mutually related in the study of additivity and nonadditivity properties on partitioned manifolds.Comment: 131 pages, 9 figure

Wavelets in Banach Spaces

We describe a construction of wavelets (coherent states) in Banach spaces generated by ``admissible'' group representations. Our main targets are applications in pure mathematics while connections with quantum mechanics are mentioned. As an example we consider operator valued Segal-Bargmann type spaces and the Weyl functional calculus. Keywords: Wavelets, coherent states, Banach spaces, group representations, covariant, contravariant (Wick) symbols, Heisenberg group, Segal-Bargmann spaces, Weyl functional calculus (quantization), second quantization, bosonic field.Comment: 37 pages; LaTeX2e; no pictures; 27/07/99: many small correction

Spin geometry and conservation laws in the Kerr spacetime

In this paper we will review some facts, both classical and recent, concerning the geometry and analysis of the Kerr and related black hole spacetimes. This includes the analysis of test fields on these spacetimes. Central to our analysis is the existence of a valence $(2,0)$ Killing spinor, which we use to construct symmetry operators and conserved currents as well as a new energy momentum tensor for the Maxwell test fields on a class of spacetimes containing the Kerr spacetime. We then outline how this new energy momentum tensor can be used to obtain decay estimated for Maxwell test fields. An important motivation for this work is the black hole stability problem, where fields with non-zero spin present interesting new challenges. The main tool in the analysis is the 2-spinor calculus, and for completeness we introduce its main features.Comment: 30 pages. To appear in the volume "The Centenary of General Relativity" in "Surveys in Differential Geometry", edited by Lydia Bieri and Shing-Tung Yau, in the series "Surveys in Differential Geometry

Open Problems for Painlevé Equations

In this paper some open problems for Painlevé equations are discussed. In particular the following open problems are described: (i) the Painlevé equivalence problem; (ii) notation for solutions of the Painlevé equations; (iii) numerical solution of Painlevé equations; and (iv) the classification of properties of Painlevé equations

Causal perturbation theory in terms of retarded products, and a proof of the Action Ward Identity

In the framework of perturbative algebraic quantum field theory a local construction of interacting fields in terms of retarded products is performed, based on earlier work of Steinmann. In our formalism the entries of the retarded products are local functionals of the off shell classical fields, and we prove that the interacting fields depend only on the action and not on terms in the Lagrangian which are total derivatives, thus providing a proof of Stora's 'Action Ward Identity'. The theory depends on free parameters which flow under the renormalization group. This flow can be derived in our local framework independently of the infrared behavior, as was first established by Hollands and Wald. We explicitly compute non-trivial examples for the renormalization of the interaction and the field.Comment: 76 pages, to appear in Rev. Math. Phy