On the Interpretation of Quantum Cosmology
Franz Embacher
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Abstract
We formulate a "minimal" interpretational scheme for fairly general (minisuperspace) quantum cosmological models. Admitting as few exact mathematical structure as is reasonably possible at the fundamental level, we apply approximate WKB-techniques locally in minisuperspace in order to make contact with the realm of predictions, and propose how to deal with the problems of mode decomposition and almost-classicality without introducing further principles. In order to emphasize the general nature of approximate local quantum mechanical structures, we modify the standard WKB-expansion method so as to rely on exact congruences of classical paths, rather than a division of variables into classical and quantum. The only exact mathematical structures our interpretation needs are the space of solutions of the Wheeler-DeWitt equation and the Klein-Gordon type indefinite scalar product. The latter boils down to plus or minus the ordinary Work supported by the Austrian Academy of Scie...
Key takeaways
- For any local quantum structure we define that true local quantum mechanics is recovered if χ admits a probabilistic interpretation without severe problems.
- In analogy to, say, the momentum eigenstates exp(i k x) in ordinary quantum mechanics, it will also be necessary to consider wave functions ψ of distributional character, such that Q(Ξ, ψ) exists for all Ξ ∈ IH.
- Any decomposition operator K gives rise to a positive definite scalar product, and for any K one may formally apply the standard quantum mechanical assignment of probabilities and give IH (possibly redefined so as to contain all states with Q K (Ξ, Ξ) < ∞) the role of a quantum mechanical Hilbert space.
- Therefore we assume some wave function ψ to represent the actual quantum state of the universe.
- The expression P = |Q(Ξ, ψ)| 2 is just the standard quantum mechanical probability for finding φ 0 in the given interval, when measured at time t 0 .
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