Boundary Quantum Mechanics

A reformulation of a physical theory in which measurements at the initial and final moments of time are treated independently is discussed, both on the classical and quantum levels. Methods of the standard quantum mechanics are used to quantize boundary phase space to obtain boundary quantum mechanics -- a theory that does not depend on the distinction between the initial and final moments of time, a theory that can be formulated without reference to the causal structure. As a supplementary material, the geometrical description of quantization of a general (e.g. curved) configuration space is presented.Comment: 35 pages, in Gravitation: Following the Prague Inspiration (To celebrate the 60th birthday of Jiri Bicak), O.Semerak, J.Podolsky, M.Zofka (eds.), World Scientific, Singapore, 2002, pp. 289--32

Minimal surfaces and entanglement entropy in anti-de Sitter space

According to Ryu and Takayanagi, the entanglement entropy in conformal field theory (CFT) is related through the AdS/CFT correspondence to the area of a minimal surface in the bulk. We study this holographic geometrical method of calculating the entanglement entropy in the vacuum case of a CFT which is holographically dual to empty anti-de Sitter (AdS) spacetime. Namely, we investigate the minimal surfaces spanned on boundaries of spherical domains at infinity of hyperbolic space, which represents a time-slice of AdS spacetime. We consider a generic position of two spherical domains: two disjoint domains, overlapping domains, and touching domains. In all these cases we find the explicit expressions for the minimal surfaces and the renormalized expression for the area. We study also the embedding of the minimal surfaces into full AdS spacetime and we find that for a proper choice of the static Killing vector we can model a dynamical situation of "tearing" of the minimal surface when the domains on which it is spanned are moved away from each other.Comment: 36 pages, 21 figures, for version with high-resolution figures see http://utf.mff.cuni.cz/~krtous/papers

Billiard in the space with a time machine

We study a system of an elastic ball moving in the non-relativistic spacetime with a nontrivial causal structure produced by a wormhole-based time machine. For such a system it is possible to formulate a simple model of the so-called `grandfather paradox': for certain `paradoxical' initial conditions the standard straight trajectory of the ball would self-collide inconsistently. We analyze globally consistent solutions of local equations of motion, namely, we find all trajectories with one self-collision. It is demonstrated that all standard initial conditions have a consistent evolution, including those `paradoxical' ones, for which the inconsistent collision-free trajectory is superseded by a special consistent self-colliding trajectory. Moreover, it is shown that for a wide class of initial conditions more than one globally consistent evolution exist. The nontrivial causal structure thus breaks the uniqueness of the classical theory even for locally deterministic physical laws.Comment: 13 pages, 16 figure

Charged particle in higher dimensional weakly charged rotating black hole spacetime

We study charged particle motion in weakly charged higher dimensional black holes. To describe the electromagnetic field we use a test field approximation and use the higher dimensional Kerr-NUT-(A)dS metric as a background geometry. It is shown that for a special configuration of the electromagnetic field the equations of motion of charged particles are completely integrable. The vector potential of such a field is proportional to one of the Killing vectors (called primary Killing vector) from the `Killing tower' of symmetry generating objects which exists in the background geometry. A free constant in the definition of the adopted electromagnetic potential is proportional to the electric charge of the higher dimensional black hole. The full set of independent conserved quantities in involution is found. It is demonstrated, that Hamilton-Jacobi equations are separable, as well as the corresponding Klein-Gordon equation and its symmetry operators.Comment: 9 pages, no figure