Abstract:
This letter analyzes the limits that quantum mechanics imposes on the accuracy to which spacetime geometry can be measured. By applying the physics of computation to ensembles of clocks, as in GPS, we present a covariant version of the quantum geometric limit, which states that the total number of ticks of clocks and clicks of detectors that can be contained in a four volume of spacetime of radius r and temporal extent t is less than or equal to rt/pi x_P t_P, where x_P, t_P are the Planck length and time. The quantum geometric bound limits the number of events or `ops' that can take place in a four-volume of spacetime and is consistent with and complementary to the holographic bound which limits the number of bits that can exist within a three-volume of spacetime.

Abstract:
Spacetime geometry is supposed to be measured by identifying the trajectories of free test particles with geodesics. In practice, this cannot be done because, being described by Quantum Mechanics, particles do not follow trajectories. As a first step to study how it is possible to read spacetime geometry with quantum particles, we model these particles with classical extended objects. We propose to represent such extended objects by its covariant center of mass, which generically does not follow a geodesic of the background metric. We present a scheme that allows to extract some of components of an "effective" connection, namely, the connection that would be obtained if the locus of the center of mass is regarded as a geodesic. We discuss some issues that arise when trying to obtain all the components of the effective connection and its possible implications.

Abstract:
We argue that the conventional quantum field theory in curved spacetime has a grave drawback: The canonical commutation relations for quantum fields and conjugate momenta do not hold. Thus the conventional theory should be denounced and the related results revised. A Hamiltonian version of the canonical formalism for a free scalar quantum field is advanced, and the fundamentals of an appropriate theory are constructed. The principal characteristic feature of the theory is quantum-gravitational nonlocality: The Schroedinger field operator at time t depends on the metric at t in the whole 3-space. Applications to cosmology and black holes are given, the results being in complete agreement with those of general relativity for particles in curved spacetime. A model of the universe is advanced, which is an extension of the Friedmann universe; it lifts the problem of missing dark matter. A fundamental and shocking result is the following: There is no particle creation in the case of a free quantum field in curved spacetime; in particular, neither the expanding universe nor black holes create particles.

Abstract:
Employing standard results from spectral geometry, we provide strong evidence that in the classical limit the ground state of three-dimensional causal dynamical triangulations is de Sitter spacetime. This result is obtained by measuring the expectation value of the spectral dimension on the ensemble of geometries defined by these models, and comparing its large scale behaviour to that of a sphere (Euclidean de Sitter). From the same measurement we are also able to confirm the phenomenon of dynamical dimensional reduction observed in this and other approaches to quantum gravity -- the first time this has been done for three-dimensional causal dynamical triangulations. In this case, the value for the short-scale limit of the spectral dimension that we find is approximately 2. We comment on the relevance of these results for the comparison to asymptotic safety and Horava-Lifshitz gravity, among other approaches to quantum gravity.

Abstract:
Following our discussion [Physica A, 375 (2007) 123] to associate an analogous probabilistic description with spacetime geometry in the Schwarzschild metric from the macro- to the micro-domain, we argue that there is a possible connection among normalized probabilities, spacetime geometry (in the form of Schwarzschild radii) and quantum mechanics (in the form of complex wave functions). We show how this association along different (n)-nested surfaces --representing curve space due to an inhomogeneous density of matter-- preserves the postulates of quantum mechanics at different geometrical scales.

Abstract:
The rules of quantum mechanics require a time coordinate for their formulation. However, a notion of time is in general possible only when a classical spacetime geometry exists. Such a geometry is itself produced by classical matter sources. Thus it could be said that the currently known formulation of quantum mechanics pre-assumes the presence of classical matter fields. A more fundamental formulation of quantum mechanics should exist, which avoids having to use a notion of time. In this paper we discuss as to how such a fundamental formulation could be constructed for single particle, non-relativistic quantum mechanics. We argue that there is an underlying non-linear theory of quantum gravity, to which both standard quantum mechanics and classical general relativity are approximations. The timeless formulation of quantum mechanics follows from the underlying theory when the mass of the particle is much smaller than Planck mass. On the other hand, when the particle's mass is much larger than Planck mass, spacetime emerges and the underlying theory should reduce to classical mechanics and general relativity. We also suggest that noncommutative differential geometry is a possible candidate for describing this underlying theory.

Abstract:
Quantum mechanics in its presently known formulation requires an external classical time for its description. A classical spacetime manifold and a classical spacetime metric are produced by classical matter fields. In the absence of such classical matter fields, quantum mechanics should be formulated without reference to a classical time. If such a new formulation exists, it follows as a consequence that standard linear quantum mechanics is a limiting case of an underlying non-linear quantum theory. A possible approach to the new formulation is through the use of noncommuting spacetime coordinates in noncommutative differential geometry. Here, the non-linear theory is described by a non-linear Schrodinger equation which belongs to the Doebner-Goldin class of equations, discovered some years ago. This mass-dependent non-linearity is significant when particle masses are comparable to Planck mass, and negligible otherwise. Such a non-linearity is in principle detectable through experimental tests of quantum mechanics for mesoscopic systems, and is a valuable empirical probe of theories of quantum gravity. We also briefly remark on the possible connection our approach could have with loop quantum gravity and string theory.

Abstract:
We compute the intrinsic Hausdorff dimension of spacetime at the infrared fixed point of the quantum conformal factor in 4D gravity. The fractal dimension is defined by the appropriate covariant diffusion equation in four dimensions and is determined by the coefficient of the Gauss-Bonnet term in the trace anomaly to be generally greater than 4. In addition to being testable in simplicial simulations, this scaling behavior suggests a physical mechanism for the screening of the effective cosmological `constant' and inverse Newtonian coupling at very large distance scales, which has implications for the dark matter content and large scale structure of the universe.

Abstract:
We develop the first steps towards an analysis of geometry on the quantum spacetime proposed in [1]. The homogeneous elements of the universal differential algebra are naturally identified with operators living in tensor powers of Quantum Spacetime; this allows us to compute their spectra. In particular, we consider operators that can be interpreted as distances, areas, 3- and 4-volumes. The Minkowski distance operator between two independent events is shown to have pure Lebesgue spectrum with infinite multiplicity. The Euclidean distance operator is shown to have spectrum bounded below by a constant of the order of the Planck length. The corresponding statement is proved also for both the space-space and space-time area operators, as well as for the Euclidean length of the vector representing the 3-volume operators. However, the space 3-volume operator (the time component of that vector) is shown to have spectrum equal to the whole complex plane. All these operators are normal, while the distance operators are also selfadjoint. The Lorentz invariant spacetime volume operator, representing the 4- volume spanned by five independent events, is shown to be normal. Its spectrum is pure point with a finite distance (of the order of the fourth power of the Planck length) away from the origin. The mathematical formalism apt to these problems is developed and its relation to a general formulation of Gauge Theories on Quantum Spaces is outlined. As a byproduct, a Hodge Duality between the absolute dif- ferential and the Hochschild boundary is pointed out.

Abstract:
In a recent paper we have suggested that a formulation of quantum mechanics should exist, which does not require the concept of time, and that the appropriate mathematical language for such a formulation is noncommutative differential geometry. In the present paper we discuss this formulation for the free point particle, by introducing a commutation relation for a set of noncommuting coordinates. The sought for background independent quantum mechanics is derived from this commutation relation for the coordinates. We propose that the basic equations are invariant under automorphisms which map one set of coordinates to another- this is a natural generalization of diffeomorphism invariance when one makes a transition to noncommutative geometry. The background independent description becomes equivalent to standard quantum mechanics if a spacetime manifold exists, because of the proposed automorphism invariance. The suggested basic equations also give a quantum gravitational description of the free particle.