Analysing the measurement problem: stochastic trajectories of the Q function

We analyse the measurement problem and Schrödinger’s cat paradox using a reality model based on stochastic trajectories relating to the Q function. A field mode A prepared in a superposition of x eigenstates is coupled to a macroscopic meter, creating a two-mode cat state. A measurement x of the meter gives the outcome of a measurement x on system A. We model the measurement of x as a direct amplification. The time evolution of the Q function is solved using a Fokker-Planck equation. A stochastic equivalence for amplitudes p and x (of each mode) yields forward-backwards stochastic differential equations that are solved via boundary conditions in the past and future.



The results of our simulations show how the meter system can be regarded as being “in one or other” of two macroscopically distinct “states” before the final readout, since each amplitude x of the meter links in the simulation to a definite meter outcome. The retrocausal amplitudes of x associated with a definite meter outcome are traced, to define a restricted distribution Q_loop at the initial time. We show that the distribution Q_loop cannot correspond to a quantum state, since the predicted uncertainty for x and p measurements is too small. However, the distribution for x and p of system A, conditioned on the specific outcome for the meter, corresponds precisely to the associated eigenstate of A, consistent with the measurement postulate. The wave function collapse is explained as the loss of information about p of the meter to the observer in the final readout.