Action Principle and Nonlocal Field Theories

Lagrangian densities are introduced for nonlocal field theories, which make the application of the action principle possible. The action principle is then applied to classical and quantum nonlocal field theories. General formulas for conserved densities are derived by use of a generalized variation method, under the assumption of $c$-number variations. These formulas are also applied to a particular model due to Kristensen and Møller. The charge and energy-momentum vector are, thereby, shown to be equal to the expressions given by Pauli, which he derived by other means. This model is also quantized by use of the Yang-Feldman method. But as this method leads to a noncanonical quantization, the above derived quantities are in general no longer conserved. This is due to the fact that the assumption of $c$-number variations essentially restricts the quantization to a canonical one. The action principle with $q$-number variations is therefore considered. Thus new integral conserved quantities are derivable. However, one also gets a general consistency condition for the above model as well as for all other similar models. The fulfilment of this condition is required by (i) stationarity of the total action and (ii) uniqueness of the integral conserved quantities. It is also a necessary condition for the existence of a unitary $S$ operator. The Kristensen-Møller model is shown to violate the above condition, and is therefore not consistent.