Journal of Mathematical Physics, since 1960, publishes some of the best papers from outstanding mathematicians and physicists. It was the first journal in the field of mathematical physics and publishes research that connects the application of mathematics to problems in physics, as well as illustrates the development of mathematical methods suitable for such applications and for the formulation of physical theories.
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In this paper, we study the quantum generalisation of the skew divergence, which is a dissimilarity measure between distributions introduced by Lee in the context of natural language processing. We provide an indepth study of the quantum skew divergence, including its relation to other state distinguishability measures. Finally, we present a number of important applications: new continuity inequalities for the quantum JensenShannon divergence and the Holevo information, and a new and short proof of Bravyi's Small Incremental Mixing conjecture.

We consider the scattering theory for discrete Schrödinger operators on with longrange potentials. We prove the existence of modified wave operators constructed in terms of solutions of a HamiltonJacobi equation on the torus .

Several quantum gravity approaches and field theory on an evolving lattice involve a discretization changing dynamics generated by evolution moves. Local evolution moves in variational discrete systems (1) are a generalization of the Pachner evolution moves of simplicial gravity models, (2) update only a small subset of the dynamical data, (3) change the number of kinematical and physical degrees of freedom, and (4) generate a dynamical (or canonical) coarse graining or refining of the underlying discretization. To systematically explore such local moves and their implications in the quantum theory, this article suitably expands the quantum formalism for global evolution moves, constructed in Paper I [P. A. Höhn, “Quantization of systems with temporally varying discretization. I. Evolving Hilbert spaces,” J. Math. Phys.55, 083508 (2014); eprint arXiv:1401.6062 [grqc]], by employing that global moves can be decomposed into sequences of local moves. This formalism is spelled out for systems with Euclidean configuration spaces. Various types of local moves, the different kinds of constraints generated by them, the constraint preservation, and possible divergences in resulting state sums are discussed. It is shown that nontrivial local coarse graining moves entail a nonunitary projection of (physical) Hilbert spaces and “fine grained” Dirac observables defined on them. Identities for undoing a local evolution move with its (time reversed) inverse are derived. Finally, the implications of these results for a Pachner move generated dynamics in simplicial quantum gravity models are commented on.

A temporally varying discretization often features in discrete gravitational systems and appears in lattice field theory models subject to a coarse graining or refining dynamics. To better understand such discretization changing dynamics in the quantum theory, an according formalism for constrained variational discrete systems is constructed. While this paper focuses on global evolution moves and, for simplicity, restricts to flat configuration spaces , a Paper II [P. A. Höhn, “Quantization of systems with temporally varying discretization. II. Local evolution moves,” J. Math. Phys., eprint arXiv:1401.7731 [grqc].] discusses local evolution moves. In order to link the covariant and canonical picture, the dynamics of the quantum states is generated by propagators which satisfy the canonical constraints and are constructed using the action and group averaging projectors. This projector formalism offers a systematic method for tracing and regularizing divergences in the resulting state sums. Nontrivial coarse graining evolution moves lead to nonunitary, and thus irreversible, projections of physical Hilbert spaces and Dirac observables such that these concepts become evolution move dependent on temporally varying discretizations. The formalism is illustrated in a toy model mimicking a “creation from nothing.” Subtleties arising when applying such a formalism to quantum gravity models are discussed.

We discuss the quantization of linearized gravity on globally hyperbolic, asymptotically flat, vacuum spacetimes, and the construction of distinguished states which are both of Hadamard form and invariant under the action of all bulk isometries. The procedure, we follow, consists of looking for a realization of the observables of the theory as a subalgebra of an auxiliary, nondynamical algebra constructed on future null infinity ℑ^{+}. The applicability of this scheme is tantamount to proving that a solution of the equations of motion for linearized gravity can be extended smoothly to ℑ^{+}. This has been claimed to be possible provided that a suitable gauge fixing condition, first written by Geroch and Xanthopoulos [“Asymptotic simplicity is stable,” J. Math. Phys.19, 714 (1978)], is imposed. We review its definition critically, showing that there exists a previously unnoticed obstruction in its implementation leading us to introducing the concept of radiative observables. These constitute an algebra for which a Hadamard state induced from null infinity and invariant under the action of all spacetime isometries exists and it is explicitly constructed.