  In the operational approach to general probabilistic theories one
distinguishes two spaces, the state space of the "elementary systems" and the
physical space in which "laboratory devices" are embedded. Each of those spaces
has its own dimension- the minimal number of real parameters (coordinates)
needed to specify the state of system or a point within the physical space.
Within an operational framework to a physical theory, the two dimensions
coincide in a natural way under the following "closeness" requirement: the
dynamics of a single elementary system can be generated by the invariant
interaction between the system and the "macroscopic transformation device" that
itself is described from within the theory in the macroscopic (classical)
limit. Quantum mechanics fulfils this requirement since an arbitrary unitary
transformation of an elementary system (spin-1/2 or qubit) can be generated by
the pairwise invariant interaction between the spin and the constituents of a
large coherent state ("classical magnetic field"). Both the spin state space
and the "classical field" are then embedded in the Euclidean three-dimensional
space. Can we have a general probabilistic theory, other than quantum theory,
in which the elementary system ("generalized spin") and the "classical fields"
generating its dynamics are embedded in a higher-dimensional physical space? We
show that as long as the interaction is pairwise, this is impossible, and
quantum mechanics and the three-dimensional space remain the only solution.
However, having multi-particle interactions and a generalized notion of
"classical field" may open up such a possibility.
