10,319 research outputs found

    Non-Commutative Geometry, Multiscalars, and the Symbol Map

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    Starting from the concept of the universal exterior algebra in non-commutative differential geometry we construct differential forms on the quantum phase-space of an arbitrary system. They bear the same natural relationship to quantum dynamics which ordinary tensor fields have with respect to classical hamiltonian dynamics.Comment: 8 pages, late

    Renormalization Group Flow of Quantum Gravity in the Einstein-Hilbert Truncation

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    The exact renormalization group equation for pure quantum gravity is used to derive the non-perturbative \Fbeta-functions for the dimensionless Newton constant and cosmological constant on the theory space spanned by the Einstein-Hilbert truncation. The resulting coupled differential equations are evaluated for a sharp cutoff function. The features of these flow equations are compared to those found when using a smooth cutoff. The system of equations with sharp cutoff is then solved numerically, deriving the complete renormalization group flow of the Einstein-Hilbert truncation in d=4d=4. The resulting renormalization group trajectories are classified and their physical relevance is discussed. The non-trivial fixed point which, if present in the exact theory, might render Quantum Einstein Gravity nonperturbatively renormalizable is investigated for various spacetime dimensionalities.Comment: 58 pages, latex, 24 figure

    On the Possibility of Quantum Gravity Effects at Astrophysical Scales

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    The nonperturbative renormalization group flow of Quantum Einstein Gravity (QEG) is reviewed. It is argued that at large distances there could be strong renormalization effects, including a scale dependence of Newton's constant, which mimic the presence of dark matter at galactic and cosmological scales.Comment: LaTeX, 18 pages, 4 figures. Invited contribution to the Int. J. Mod. Phys. D special issue on dark matter and dark energ

    A Proposal for a Differential Calculus in Quantum Mechanics

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    In this paper, using the Weyl-Wigner-Moyal formalism for quantum mechanics, we develop a {\it quantum-deformed} exterior calculus on the phase-space of an arbitrary hamiltonian system. Introducing additional bosonic and fermionic coordinates we construct a super-manifold which is closely related to the tangent and cotangent bundle over phase-space. Scalar functions on the super-manifold become equivalent to differential forms on the standard phase-space. The algebra of these functions is equipped with a Moyal super-star product which deforms the pointwise product of the classical tensor calculus. We use the Moyal bracket algebra in order to derive a set of quantum-deformed rules for the exterior derivative, Lie derivative, contraction, and similar operations of the Cartan calculus.Comment: TeX file with phyzzx macro, 43 pages, no figure
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