Quantum Cosmology by Martin Bojowald

Author:Martin Bojowald
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

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for the volume, which must factorize into three independent equations

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for the extensions of the three directions, where The product as the total number of patches can then be parameterized to be proportional to a power of volume, but this does not fix the individual dependences of If dependences are such that depends only on the geometry of its own direction, for instance or more generally, the refinement resembles what we have in isotropic models, just for three directions independently. In this case, while step sizes may not immediately be equidistant, variables can be redefined to make them so. But in anisotropic models the generic behavior would be one where each depends on all the triad variables, which in general results in a difference equation whose steps cannot all be made uniform. In fact, it is rather natural to expect that depends on the extension in direction given by the co-triad component (This non-trivial case is required for consistent dynamics in the sense of stability [15]. stabilityAt least one of the variables can be made uniform, corresponding to the total volume as the isotropic average of the degrees of freedom. Also at the quantum level there thus seems to be an advantage to using Misner-type variables Misner variablessplitting the scale factors into the volume and anisotropy parameters.) One thus has to deal with difference equations on a genuinely non-uniform lattice, which poses new mathematical and numerical problems. Methods of handling non-equidistant difference difference equation!non-uniform equations of this form are being developed [16, 17].

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