A random map model for quantum interference

Quantum interference of particle systems results from the wave properties of the particles and are predicted theoretically from the superposition of the wave functions. In place of wave functions we use deterministic chaotic maps as the underlying mechanism that produces the observed probability density functions. Let ψ i ( x , t ) , i = 1 , 2 be two wave functions of a quantum mechanical particle system. For each ψ i ( x , t ) we define deterministic nonlinear point transformations τ i ( x ) whose unique probability density function is the observed density ρ i ( x , t ) = ψ i ∗ ( x , t ) ψ i ( x , t ) . We consider the wave function ψ ( x , t ) = a ψ 1 ( x , t ) + b ψ 2 ( x , t ) and show that we can associate with ψ ( x , t ) , a random chaotic map that switches (probabilistically between) τ 1 ( x ) , τ 2 ( x ) and the identity map I ( x ) and whose probability density function f t ( x ) equals ψ ∗ ( x , t ) ψ ( x , t ) , where t denotes time. This description of quantum interference of particle systems allows a more insightful interpretation than wave mechanics.