  One attractive interpretation of quantum mechanics is the ensemble
interpretation, where Quantum Mechanics merely describes a statistical ensemble
of objects and not individual objects. But this interpretation does not address
why the wave-function plays a central role in the calculations of
probabilities, unlike most other interpretations of quantum mechanics. On the
other hand, Classical Statistical Field Theory suffers from severe mathematical
inconsistencies (specially for Hamiltonians which are non-polynomial in the
fields, e.g. General relativistic statistical field theory). We claim that both
problems are related to each other and we propose a solution to both. We prove:
1) the wave-function is a parametrization of any probability distribution of a
statistical ensemble: there is a surjective map from an hypersphere to the set
of all probability distributions; 2) for a quantum system defined in a
2-dimensional real Hilbert space, the role of the (2-dimensional real)
wave-function is identical to the role of the Euler's formula in engineering,
while the collapse of the wave-function is identical to selecting the real part
of a complex number; 3) the collapse of the wave-function of any quantum system
is a recursion of collapses of 2-dimensional real wave-functions; 4) the
wave-function parametrization is key in the mathematical definition of
Classical Statistical Field Theory we propose here. The same formalism is
applied to Quantum Yang-Mills theory and Quantum Gravity in another article.
