  Bell inequalities, understood as constraints between classical conditional
probabilities, can be derived from a set of assumptions representing a common
causal explanation of classical correlations. A similar derivation, however, is
not known for Bell inequalities in algebraic quantum field theories
establishing constraints for the expectation of specific linear combinations of
projections in a quantum state. In the paper we address the question as to
whether a 'common causal justification' of these non-classical Bell
inequalities is possible. We will show that although the classical notion of
common causal explanation can readily be generalized for the non-classical
case, the Bell inequalities used in quantum theories cannot be derived from
these non-classical common causes. Just the opposite is true: for a set of
correlations there can be given a non-classical common causal explanation even
if they violate the Bell inequalities. This shows that the range of common
causal explanations in the non-classical case is wider than that restricted by
the Bell inequalities.
