Quantum vs stochastic processes and the role of complex numbers

Abstract

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).