  We argue that the complex numbers are an irreducible object of quantum
probability. This can be seen in the measurements of geometric phases that have
no classical probabilistic analogue. Having complex phases as primitive
ingredient implies that we need to accept non-additive probabilities. This has
the desirable consequence of removing constraints of standard theorems about
the possibility of describing quantum theory with commutative variables.
Motivated by the formalism of consistent histories and keeping an analogy with
the theory of stochastic processes, we develop a (statistical) theory of
quantum processes. They are characterised by the introduction of a "density
matrix" on phase space paths -thus including phase information- and fully
reproduce quantum mechanical predictions. In this framework wecan write quantum
differential equations, that could be interpreted as referring to a single
system (in analogy to Langevin's equation). We describe a reconstruction
theorem by which a quantum process can yield the standard Hilbert space
structure if the Markov property is imposed. Finally, we discuss the relevance
of our iresults for the interpretation of quantum theory (a sample space if
possible if probabilities are non-additive) and quantum gravity (the Hilbert
space arises after the consideration of a background causal structure).
