4,693 research outputs found

    On Pairs of Difference Operators Satisfying: [P,Q] = Id

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    Different finite difference replacements for the derivative are analyzed in the context of the Heisenberg commutation relation. The type of the finite difference operator is shown to be tied to whether one can naturally consider PP and XX to be self-adjoint and skew self-adjoint or whether they have to be viewed as creation and annihilation operators. The first class, generalizing the central difference scheme, is shown to give unitary equivalent representations. For the second case we construct a large class of examples, generalizing previously known difference operator realizations of [P,X]=Id[P,X]=Id.Comment: 32 pages, plain Te

    Gravitational Vacuum Condensate Stars

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    A new final state of gravitational collapse is proposed. By extending the concept of Bose-Einstein condensation to gravitational systems, a cold, dark, compact object with an interior de Sitter condensate pV=ρVp_{_V} = -\rho_{_V} and an exterior Schwarzschild geometry of arbitrary total mass MM is constructed. These are separated by a shell with a small but finite proper thickness \ell of fluid with equation of state p=+ρp=+\rho, replacing both the Schwarzschild and de Sitter classical horizons. The new solution has no singularities, no event horizons, and a global time. Its entropy is maximized under small fluctuations and is given by the standard hydrodynamic entropy of the thin shell, which is of order kBMc/k_{_B}\ell Mc/\hbar, instead of the Bekenstein-Hawking entropy formula, SBH=4πkBGM2/cS_{_{BH}}= 4\pi k_{_B} G M^2/\hbar c. Hence unlike black holes, the new solution is thermodynamically stable and has no information paradox.Comment: 17 pages, LaTeX fil
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