Two-particle wave function as an integral operator and the random field approach to quantum correlations

We propose a new interpretation of the wave function $\Psi(x,y)$ of a two-particle quantum system, interpreting it not as an element of the functional space $L_2$ of square-integrable functions, i.e., as a vector, but as the kernel of an integral (Hilbert–Schmidt) operator. The first part of the paper is devoted to expressing quantum averages including the correlations in two-particle systems using the wave-function operator. This is a new mathematical representation in the framework of conventional quantum mechanics. But the new interpretation of the wave function not only generates a new mathematical formalism for quantum mechanics but also allows going beyond quantum mechanics, i.e., representing quantum correlations (including those in entangled systems) as correlations of (Gaussian) random fields.