We compute the quantum fluctuations of a non-minimally coupled self-interacting scalar field around a classical self-consistent inflationary solution; this system is free of the instability usually associated with negative curvature couplings. By conformally transforming the original system it is possible to eliminate the direct coupling of the field to the curvature. Investigating the quantum fluctuations in this "conformal frame," where it is possible to do a more complete calculation, and using the same criteria for selecting the quantum state in both frames, we find the results in the two frames to be incompatible with each other. This illustrates, in a specific example, the inherent ambiguity in relating the quantum behavior of two systems that are connected by a field-dependent conformal transformation at the classical level if one follows the usual naive approach. for estimating density perturbations in inflation. Possible ways to get around this problem are discussed.
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Fakir et al. (1993) studied this question.