Conservation of Quanta for Nonlocal Fields

Abstract

A definition of spatio and/or temporal integration of nonlocal expressions is set forth. The latter is applied to the computation of the momentum-energy vector and an invariant from nonlocal expressions corresponding to the stress-energy momentum tensor and a four-vector. The latter four-vector is interpreted to be the quantum current density, and the invariant surface integral of this four-vector over a space-like surface is associated with the operator for the number of quanta. Application to a massless boson field leads to the conclusion that only the energy is positive definite, while the operator for the number of quanta is not, even though the boson field is real (Hermitian). In the usual quantum theory of fields a four-vector current exists only for complex fields.