On the instability of the vacuum in multidimensional scalar theories

The 1-loop effective potential in a scalar theory with quartic interaction on the space $M^{4} \times T^{n}$ for $n=2$ is calculated and is shown to be unbounded from below. This is an indication of a possible instability of the vacuum of the $\lambda \phi^{4}$ theory on $M^{4}$, when it is regarded as a result of the dimensional reduction of the original six-dimensional model. The issue of stability for other values of the number $n$ of extra dimensions is also discussed.