  We use a result of Hawking and Gilkey to define a Euclidean path integral of
gravity and matter which has the special property of being independent of the
choice of basis in the space of fields. This property allows the path integral
to describe also physical regimes that do not admit position bases. These
physical regimes are pre-geometric in the sense that they do not admit a
mathematical representation of the physical degrees of freedom in terms of
fields that live on a spacetime. In regimes in which a spacetime representation
does emerge, the geometric properties of the emergent spacetime, such as its
dimension and volume, depend on the balance of fermionic pressure and bosonic
and gravitational pull. That balance depends, at any given energy scale, on the
number of bosonic and fermionic species that contribute, which in turn depends
on their masses. This yields an explicit mechanism by which the effective
spacetime dimension can depend on the energy scale.
