Minimal Lagrangian surfaces in the tangent bundle of a Riemannian surface

Given an oriented Riemannian surface $(\Sigma, g)$, its tangent bundle $T\Sigma$ enjoys a natural pseudo-K\"{a}hler structure, that is the combination of a complex structure \J, a pseudo-metric \G with neutral signature and a symplectic structure \Om. We give a local classification of those surfaces of $T\Sigma$ which are both Lagrangian with respect to \Om and minimal with respect to \G. We first show that if $g$ is non-flat, the only such surfaces are affine normal bundles over geodesics. In the flat case there is, in contrast, a large set of Lagrangian minimal surfaces, which is described explicitly. As an application, we show that motions of surfaces in $\R^3$ or $\R^3_1$ induce Hamiltonian motions of their normal congruences, which are Lagrangian surfaces in $T\S^2$ or T \H^2 respectively. We relate the area of the congruence to a second-order functional $\mathcal{F}=\int \sqrt{H^2-K} dA$ on the original surface.Comment: 22 pages, typos corrected, results streamline

Generalized Helical Hypersurfaces Having Time-like Axis in Minkowski Spacetime

In this paper, the generalized helical hypersurfaces x=x(u,v,w) with a time-like axis in Minkowski spacetime E14 are considered. The first and the second fundamental form matrices, the Gauss map, and the shape operator matrix of x are calculated. Moreover, the curvatures of the generalized helical hypersurface x are obtained by using the Cayley–Hamilton theorem. The umbilical conditions for the curvatures of x are given. Finally, the Laplace–Beltrami operator of the generalized helical hypersurface with a time-like axis is presented in E14

Introduction to Loop Quantum Gravity

This article is based on the opening lecture at the third quantum geometry and quantum gravity school sponsored by the European Science Foundation and held at Zakopane, Poland in March 2011. The goal of the lecture was to present a broad perspective on loop quantum gravity for young researchers. The first part is addressed to beginning students and the second to young researchers who are already working in quantum gravity.Comment: 30 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:gr-qc/041005

Functional Evolution of Free Quantum Fields

We consider the problem of evolving a quantum field between any two (in general, curved) Cauchy surfaces. Classically, this dynamical evolution is represented by a canonical transformation on the phase space for the field theory. We show that this canonical transformation cannot, in general, be unitarily implemented on the Fock space for free quantum fields on flat spacetimes of dimension greater than 2. We do this by considering time evolution of a free Klein-Gordon field on a flat spacetime (with toroidal Cauchy surfaces) starting from a flat initial surface and ending on a generic final surface. The associated Bogolubov transformation is computed; it does not correspond to a unitary transformation on the Fock space. This means that functional evolution of the quantum state as originally envisioned by Tomonaga, Schwinger, and Dirac is not a viable concept. Nevertheless, we demonstrate that functional evolution of the quantum state can be satisfactorily described using the formalism of algebraic quantum field theory. We discuss possible implications of our results for canonical quantum gravity.Comment: 21 pages, RevTeX, minor improvements in exposition, to appear in Classical and Quantum Gravit