Dirac Sea and Hole Theory for Bosons II -- Renormalization Approach --
  In bosonic formulation of the negative energy sea, so called Dirac sea presented in the preceding paper [arXiv:hep-th/0603242], one of the crucial points is how to construct a positive definite inner product in the negative energy states, since naive attempts would lead to non-positive definite ones. In the preceding paper the non-local method is used to define the positive definite inner product. In the present article we make use of a kind of $\epsilon$-regularization and renormalization method which may clarify transparently the analytical properties of our formulation. 
Yoshinobu Habara, Yukinori Nagatani, Holger B. Nielsen and Masao   Ninomiya