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We examine the possibility of constraining the dimensions
M + 1 and
N of the physical space-time and the internal space, respectively, of the
D(=
M +
N + 1)-dimensional theory
L=√−
g
̂
(
1
2
K
̂
−2
R
̂
+
1
2
Φ;
AΦ;
B
g
̂
AB)
, which is the field-theory limit of both the bosonic string and the superstring when higher-order terms
K
̂
2φ
−4
(D−2)
R
̂
2
and
K
̂
2φ
−8
(D−2)
H
̂
ABC
2
can be ignored, where
φ
exp(
1
2
√(D−2)
K
̂
Φ)
is the dilation. The Wheeler-De Witt equation for the wave function of the Universe ψ is derived for the topology R × S
M
× R
N
, where the space R
N
is to approximate a (Ricci-flat) Kähler space. By requiring that ψ remain finite at the origin (zero (
D − 1)-volume) and assuming that the eigenvalue of the dilaton energy-momentum tensor vanishes, we show that the metric initially possesses the “no-scale” form d
s
2 = −
n
2 d
t
2 +
a
2(
t)d
x
2 +
a
−2(
t)d
y
2, provided that (
M −
N)
2 =
M +
N. This condition includes not only the cases
D = 26 (
M = 10,
N = 15) and
D = 10 (
M = 3,
N = 6), for which the dilaton beta function
β
φ
vanishes (to lowest order in α′) in the bosonic string and the superstring, respectively, but also the case
D = 5 (
M = 3,
N = 1) corresponding to the original Kaluza-Klein theory. Since
M = 3 for the physical space, then the superstring is allowed, while the bosonic string appears to be ruled out.
The analysis was motivated by a preprint of Enqvist, Mohanty and Nanopoulos.