  Relations between two definitions of (total) angular momentum operator, as a
generator of rotations and in the Lagrangian formalism, are explored in quantum
field theory. Generally, these definitions result in different angular momentum
operators, which are suitable for different purposes in the theory. From the
spin and orbital angular momentum operators (in the Lagrangian formalism) are
extracted additive terms which are conserved operators and whose sum is the
total angular momentum operator.
