Connection between Local Commutativity and Regularity of Wightman Functions

It is proved that a Wightman function (vacuum expectation value of a product of field operators) will be analytic and one valued at a real set of space-time points if and only if the fields possess a property of weak local commutativity at the same points. This statement assumes the validity of $\mathrm{TCP}$ invariance. A similar but more complicated statement is proved for theories without $\mathrm{TCP}$ invariance.