Abstract
We derive ``Bell inequalities\'\' in four dimensional phase space and prove the following ``three marginal theorem\'\' for phase space densities $\\rho(\\overrightarrow{q},\\overrightarrow{p})$, thus settling a long standing conjecture : ``there exist quantum states for which more than three of the quantum probability distributions for $(q_1,q_2)$, $(p_1,p_2)$, $(q_1,p_2)$ and $(p_1,q_2)$ cannot be reproduced as marginals of a positive $\\rho(\\overrightarrow{q},\\overrightarrow{p})$\'\'. We also construct the most general positive $\\rho(\\overrightarrow{q},\\overrightarrow{p})$ which reproduces any three of the above quantum probability densities for arbitrary quantum states. This is crucial for the construction of a maximally realistic quantum theory.