Path-integral and operator formalism in quantum gravity

The meaning of the conformal rotation in the path-integral formulation of quantum gravity is investigated. We show that adopting the Euclidean path integral with the path of the conformal mode taken along the imaginary axis is equivalent to regarding the mode as an indefinite-metric one in the operator formalism based on canonical quantization. In order to quantize the mode as an indefinite-metric one, the integration path of the inner product in the Schrödinger representation should be taken along the imaginary axis. In this sense the path of the conformal mode is not rotated but is along the imaginary axis from the beginning in the path-integral representation as well as in the operator formulation. This is demonstrated explicitly in the case of linearized gravity and in the case of the minisuperspace model for the closed universe. For the latter case the resultant quantization with indefinite metric significantly deviates from the usual treatment where a positive-definite metric of the wave-function space is a priori assumed.