Solvable quantum cosmological model and the importance of quantizing in a special canonical frame

Abstract

By analyzing the simple cosmological model consisting of a real massless Klein-Gordon field with vanishing spatial derivatives in the Friedmann universe, we conclude that this model can be successfully quantized only by using an extrinsic time. If one attempts to quantize using an intrinsic time, one is faced with the problem of either not having a point of maximum expansion, which violates the correspondence principle, or a necessity to devise a new interpretation for a zero-normed quantum mechanics (in addition to the particle-antiparticle interpretation). However, if one uses an extrinsic time, none of these difficulties occur. In analyzing the distinction between these two quantization procedures, we have noted that there are two distinct types of quantum-mechanical The first type is the usual quantum-mechanical which we call coordinate-space where the topology of the classical phase space is usually planar and the phase space has no classically forbidden regions, although for a fixed energy, there can exist certain regions of coordinate space that are classically forbidden. The second type occurs when the phase space has classically forbidden regions, and we call into these regions phase-space tunneling. In terms of these two types of tunneling, quantization with an intrinsic time allows phase-space tunneling to occur, and it is the presence of this type of that gives this solution its undesirable features. On the other hand, quantization with a particular choice of extrinsic time absolutely forbids the occurrence of phase-space and it is the lack of this type of that gives this model its desirable features. Thus, based on this model and other general arguments, we propose that although coordinate-space tunneling is quantum-mechanically allowed, the distinctly different process, phase-space is not only classically forbidden, but also must be considered to be quantum-mechanically forbidden as well.