Quantum averages from Gaussian random fields at the Plancklength scale

摘要

We show that the mathematical formalism of quantum mechanics can be interpreted as a method for approxima-tion of classical (measure-theoretic) averages of functions f : L_2(R~3) →R. These are classical physical variablesin our model with hidden variables - Prequantum Classical Statistical Field Theory (PCSFT). In this paper weprovide a simple stochastic picture of such a quantum approximation procedure. In the probabilistic terms this isnothing else than the approximative method for computation of averages for functions of random variables. Sincein PCSFT the space of hidden variables is L_2(R~3), the role of a classical random variable is played by randomfield. In PCSFT we consider Gaussian random fields representing random fluctuations at the prequantum lengthscale. Quantum mechanical expression for the average (given by the von Neumann trace formula) is obtainedthrough moving from the prequantum length scale to the quantum one (the scale at that we are able to performmeasurements).