A time machine can be constructed by the relative motion of one mouth of a wormhole. The model has some remaining problems to be solved. Among the problems, the stability problem arises from the forming of the Cauchy horizon, where rays of early times will accumulate and diverge. This stability problem can be solved at the classical level. For quantum stability, Kim and Thorne recently tried to calculate the vacuum fluctuation of quantized fields by the point-splitting method. It was shown that the vacuum fluctuations produce a renormalized stress-energy tensor that diverges as one approaches the Cauchy horizon, which might be cut off by quantum gravity. However, there is a controversy. Hawking conjectures an observer-independent location for the breakdown in the semiclassical theory. In this paper, we deal with this quantum stability problem using another method: "particle production by an arbitrary gravitational field." When the wormhole forms in the infinite past, the result is finite, while it is divergent near the Cauchy horizon when the wormhole forms at a finite time. If we adopt the Kim-Thorne conjecture, then the divergence might be cut off by quantum gravity; therefore, the total energy cannot prevent the formation of the closed timelike curves when one is within a Planck length.
