Decoherent histories analysis of minisuperspace quantum cosmology

Abstract

Abstract. Recent results on the decoherent histories quantization of simple cosmological models (minisuperspace models) are described. The most important issue is the construction, from the wave function, of a probability distribution answering various questions of physical interest, such as the probability of the system entering a given region of configuration space at any stage in its entire history. A standard but heuristic procedure is to use the flux of (components of) the wave function in a WKB approximation as the probability. This gives sensible semiclassical results but lacks an underlying operator formalism. Here, we supply the underlying formalism by deriving probability distributions linked to the Wheeler-DeWitt equation using the decoherent histories approach to quantum theory, building on the generalized quantum mechanics formalism developed by Hartle. The key step is the construction of class operators characterizing questions of physical interest. Taking advantage of a recent decoherent histories analysis of the arrival time problem in non-relativistic quantum mechanics, we show that the appropriate class operators in quantum cosmology are readily constructed using a complex potential. The class operator for not entering a region of configuration space is given by the S-matrix for scattering off a complex potential localized in that region. We thus derive the class operators for entering one or more regions in configuration space. The class operators commute with the Hamiltonian, have a sensible classical limit and are closely related to an intersection number operator. The corresponding probabilities coincide, in a semiclassical approximation, with standard heuristic procedures.

Machian classical and semiclassical emergent time

“…t em(WKB) aligns with t em(JBB) at least to first approximation. As outlined in the Conclusion (see [50,51,52,53,6,54] for more details), this justifies the WKB ansatz leads one to a Machian Semiclassical Histories Timeless Records combined scheme: Level 3 of the current program. The present article covers Levels 1 and 2.…”

“…The main idea is then to consider (52) and (50) as a pair of equations for the unknowns t em(rec) and |χ . Regime 1) One might argue for the interaction term J to be quantitatively negligible as regards the observed l-physics, as both a small interaction term and an averaged quantity.…”

Quantum cosmology from the de Broglie–Bohm perspective

“…A promising alternative approach to quantum cosmology using the consistent histories quantization is to consider the full Klein-Gordon equation. In this approach both energy sectors, positive and negative, are simultaneously taken into account but the Hilbert space is defined with a different inner product (see [37] and references therein). Following closely Ref.…”

“…One possibility is to define a new inner product, as it was done in Ref. [37]. This is something yet to be explored in full superspace.…”

“…[19] with care, and contextualize it in the framework of the de Broglie-Bohm theory, and other quantization techniques, as the two frequencies (Klein-Gordon) approach of Ref. [36,37], in order to understand this discrepancy of results.…”