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The Wheeler-DeWitt Quantization Can Solve the Singularity Problem

We study the Wheeler-DeWitt quantum cosmology of a spatially flat Friedmann cosmological model with a massless free scalar field. We compare the consistent histories approach with the de Broglie-Bohm theory when applied to this simple model under two different quantization schemes: the Schrödinger-like quantization, which essentially takes the square-root of the resulting Klein-Gordon equation through the restriction to positive frequencies and their associated Newton-Wigner states, or the induced Klein-Gordon quantization, that allows both positive and negative frequencies together. We show that the consistent histories approach can give a precise answer to the question concerning the existence of a quantum bounce if and only if one takes the single frequency approach and within a single family of histories, namely, a family containing histories concerning properties of the quantum system at only two specific moments of time: the infinity past and the infinity future. In that case, as shown by Craig and Singh \cite{CS}, there is no quantum bounce. In any other situation, the question concerning the existence of a quantum bounce has no meaning in the consistent histories approach. On the contrary, we show that if one considers the de Broglie-Bohm theory, there are always states where quantum bounces occur in both quantization schemes. Hence the assertion that the Wheeler-DeWitt quantization does not solve the singularity problem in cosmology is not precise. To address this question, one must specify not only the quantum interpretation adopted but also the quantization scheme chosen.