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Assuming the space dimension is not constant, but varies with the expansion
of the universe, a Lagrangian formulation of a toy universe model is given.
After a critical review of previous works, the field equations are derived
and discussed. It is shown that generalization of the FRW cosmology
is not unique. There is a free parameter in the theory, $C$, with
which we can fix the dimension of space, say, at the Planck time.
Different possibilities for this dimension are discussed. The standard
FRW model corresponds to the limiting case $C\rightarrow +\infty$.
Depending on the free parameter of the theory, $C$, the expansion of the
model can behave differentl from the standard cosmological models with
constant dimension. This is explicitly studied in the framework of quantum
cosmology. The Wheeler-DeWitt equation is written down. It turns out that
in our model universe, the potential of the Wheeler-DeWitt equation has
different characteristics relative to the potential of the de Sitter
minisuperspace. Using the appropriate boundary conditions and the
semiclassical approximation, we calculate the wave function of our
model universe. In the limit of $C\rightarrow +\infty$, corresponding
to the case of constant space dimension, our wave function does not
have a unique behavior. It can either lead to the Hartle-Hawking wave
function or to a modified Linde wave function, or to a more general
one, but not to that of Vilenkin. We also calculate the probability
density in our model universe. It is always more than the probability
density of the de Sitter minisuperspace in three-space as suggested by
Vilenkin, Linde, and other. In the limit of constant space dimension, the
probability density of our model universe approaches that of the Vilenkin
and Linde probability density, being $\exp(-2|S_E|)$, where $S_E$ is the
Euclidean action. Our model universe indicates therefore that the
Vilenkin wave function is not stable with respect to the variation of
space dimension.
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