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Ashtekar quantum cosmology paper new this month

  1. Jun 20, 2003 #1


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    What I suspect is a major paper by Ashtekar (and others) appeared in arXiv this month----on June 9.

    Mathematical structure of loop quantum cosmology

    Ashtekar, Bojowald, Lewandowski



    Here is the concluding paragraph:

    "We conclude with a general observation. The way in which the big-bang singularity is resolved has potentially deep implications on questions about the origin of the universe. For instance, the question of whether the universe had a beginning at a finite time is now ‘transcended’. At first, the answer seems to be ‘no’ in the sense that the quantum evolution does not stop at the big bang. However, since space-time geometry ‘dissolves’ near the big-bang, there is no longer a notion of time, or of ‘before’ or ‘after’ in the familiar sense. Therefore, strictly, the question is no longer meaningful. The paradigm has changed and meaningful questions must now be phrased differently, without using notions tied to classical space- times. A similar shift of paradigm occurred already with the advent of general relativity. Before Einstein, philosophers argued that the universe could not have a finite beginning because, if it did, one could ask what there was before. However, this question pre-supposes that space-time is an eternal, passive arena and matter simply evolves in it. With general relativity, we learned that space and time are ‘born with matter’, whence the question of ‘what was there before’ is no longer meaningful. Loop quantum cosmology brings about a
    further shift of paradigm, weeding out certain questions that seemed meaningful in classical general relativity and requiring that they be replaced by more refined questions, formulated in the context of quantum space-times."
    Last edited: Jun 20, 2003
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  3. Jun 20, 2003 #2


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    progress towards solving the problem of time-evolution

    Ashtekar et al appear to have made some progress is constructing a Hamiltonian operator with which to model time-evolution of the universe---in a way, moreover, naturally compatible with the idea of quantizing time in discrete steps.

    Here is a quote from page 23 of the paper that just came out this month---"Mathematical structure of loop quantum cosmology."

    "Therefore, to obtain a well-defined operator, we used loops enclosing an area ao, the smallest non-zero quantum of area in quantum geometry. The resulting operator can be regarded as a ‘good’ quantization of the classical constraint function because it has the correct classical limit. The resulting quantum constraint equation has novel and physically appealing properties. First, it is a difference –rather than differential— equation and thus provides a ‘discrete time evolution’. Second, the coefficients in this difference equation are such that the ‘evolution’ does not break down at the singularity; quantum physics does not stop at the big-bang! This occurs without fine tuning matter or making it violate energy conditions. Furthermore, while in consistent discrete models the singularity is often ‘avoided’ because discrete ‘time steps’ are such that one simply leaps over the point where the singularity is expected to occur, here, one can and does confront the singularity head on only to find that it has been resolved by the quantum ‘evolution’. Furthermore, these features are robust. However, near the big-bang, the state is ‘extremely quantum mechanical,’ with large fluctuations. Thus, the classical space-time ‘dissolves’ near the big-bang. In this regime, we can analyze the structure only in quantum mechanical terms; we can no longer use our classical intuition which is deeply rooted in space-times and small fluctuations around them."

    The quantum Hamiltonian contraint (e.g. equation 4.10 on page 17) necessarily contains a matter term.

    This form of the LQG Hamiltonian constraint is recognizable as a quantum version of the Einstein equation of standard GR and analogous to the earlier Wheeler-DeWitt "quantum Einstein equation" (e.g. equation 4.12 on page 18, provided for comparison.)
    Last edited: Jun 21, 2003
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