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 the complex phases as primitive ingredient implies that we need to accept nonadditive 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 characterized by the introduction of a “density matrix” on phase space paths (it thus includes phase information) and fully reproduces quantum mechanical predictions. We can write quantum differential equations (in analogy to Langevin equation) that could be interpreted as referring to individual quantum systems. We describe the reconstruction theorem by which a quantum process can yield the standard Hilbert space structure if the Markov property is imposed. We discuss the relevance of our results for the interpretation of quantum theory (a sample space is possible if probabilities are nonadditive) and quantum gravity (the Hilbert space arises here after the consideration of a background causal structure).
A model for quantum dynamics is presented in terms of a generalization of Markov chains. We first consider measurements and events on an amplitude space. Markov amplitude chains on an amplitude space are then studied. Quantum chains are defined and characterized by a Markov and weak stationarity property. We then consider random phase transformations and study the changes that result in an amplitude space and in the quantum chains due to such transformations. A perturbation expansion involving a potential perturbing the ‘‘free’’ motion is proved. Quantum processes are defined and their relation to quantum chains is discussed. The existence of an arbitrary quantum chain is proved using path chains.