The existence of a minimal observable length has long been suggested in quantum gravity as well as in string theory. In this context a generalized uncertainty relation has been derived which quantum theoretically describes the minimal length as a minimal uncertainty in position measurements. Here we study in full detail the quantum mechanical structure which underlies this uncertainty relation. DAMTP/94-105, hep-th/9412167, and Phys.Rev.D52:1108 (1995) * supported by Studienstiftung des Deutschen Volkes, BASF-fellow, a

Treating the cosmological constant as a thermodynamic pressure and its conjugate quantity as a thermodynamic volume, we reconsider the critical behaviour of charged AdS black holes. We complete the analogy of this system with the liquid-gas system and study its critical point, which occurs at the point of divergence of specific heat at constant pressure. We calculate the critical exponents and show that they coincide with those of the Van der Waals system.

We investigate the critical behaviour of charged and rotating AdS black holes in d spacetime dimensions, including effects from non-linear electrodynamics via the Born-Infeld action, in an extended phase space in which the cosmological constant is interpreted as thermodynamic pressure. For Reissner-Nördstrom black holes we find that the analogy with the Van der Walls liquid-gas system holds in any dimension greater than three, and that the critical exponents coincide with those of the Van der Waals system. We find that neutral slowly rotating black holes in four spacetime dimensions also have the same qualitative behaviour. However charged and rotating black holes in three spacetime dimensions do not exhibit critical phenomena. For Born-Infeld black holes we define a new thermodynamic quantity B conjugate to the Born-Infeld parameter b that we call Born-Infeld vacuum polarization. We demonstrate that this quantity is required for consistency of both the first law of thermodynamics and the corresponding Smarr relation.

Two observers determine the entanglement between two free bosonic modes by each detecting one of the modes and observing the correlations between their measurements. We show that a state which is maximally entangled in an inertial frame becomes less entangled if the observers are relatively accelerated. This phenomenon, which is a consequence of the Unruh effect, shows that entanglement is an observer-dependent quantity in non-inertial frames. In the high acceleration limit, our results can be applied to a non-accelerated observer falling into a black hole while the accelerated one barely escapes. If the observer escapes with infinite acceleration, the state's distillable entanglement vanishes. Entanglement is a property of multipartite quantum states that arises from the tensor product structure of the Hilbert space and the superposition principle. It is considered to be a resource for quantum information tasks such as teleportation [1] and has applications in quantum control [2] and quantum simulations [3]. Non-relativistic bipartite entanglement can be quantified uniquely for pure states by the von Neumann entropy and for mixed states several measures have been proposed such as entanglement cost, distillable entanglement and logarithmic negativity [4]. Understanding entanglement in the relativistic framework is crucial from both fundamental and practical perspectives. Relativistic space-time presents naturally a more complete setting for theoretical considerations and many experimental set-ups require such a treatment. This program is therefore an important and topical one. It is only in this framework that we can understand quantum information tasks involving entanglement between moving observers. A central question in the field of relativistic quantum information is whether entanglement is observer-independent. So far, it has been shown that entanglement between inertial moving parties remains constant although the entanglement between some degrees of freedom can be transferred to others [5].In this letter we investigate the entanglement between two modes of a non-interacting massless scalar field when one of the observers describing the state is uniformly accelerated. We consider a maximally entangled pure state in an inertial frame and describe its entanglement from a non-inertial perspective. Our results imply that only inertial observers in flat spacetime agree on the degree of entanglement, whereas non-inertial observers see a degradation. While Minkowski coordinates (t, z) are the most suitable to describe the field from an inertial perspective, * Published before under name Fuentes-Guridi Rindler coordinates (τ, ξ) are appropriate for discussing the viewpoint of an observer moving with uniform acceleration. Two different sets of Rindler coordinates, which differ from each other by a sign change in the temporal coordinate, are necessary for covering Minkowski space. These sets of coordinates define two Rindler regions that are causally disconnected from each other. A particle undergoing uniform accele...