Temperature effects in five-dimensional Kaluza-Klein theory

We examine the combined effectsof thermal and quantum-mechanical fluctuations in a Kaluza-Klein universe with a single compact spatial dimension of length ${\overline{L}}_5$. From the one-loop effective potential for ${\overline{L}}_5$ two types of instability are evident. If initially ${\overline{L}}_5$ is less than a temperature-dependent critical length, ${\overline{L}}_5$ will shrink at least down to a length comparable to the Planck length. This instability has been noted by Appelquist and Chodos in the zero-temperature limit. If, on the other hand, ${\overline{L}}_5$ starts out larger than the critical length, it will tend to increase. This result suggests that temperature effects may play an important role in Kaluza-Klein cosmological models.