Reducible representations of CAR and CCR with possible applications to field quantization

Reducible representations of CAR and CCR are applied to second quantization of Dirac and Maxwell fields. The resulting field operators are indeed operators and not operator-valued distributions. Examples show that the formalism may lead to a finite quantum field theory. 1 Reducible representation of CAR The main objective of the paper is to discuss a reducible representation of canonical anti-commutation relations (CAR) which generalizes to Dirac electrons the construction pre-viously employed in [1, 2, 3] to electromagnetic fields. The case of canonical commutation relations (CCR) is briefly discussed in the last section. Beginning with the CAR operators b±, d ± of an irreducible representation we introduce the following four operators b(~p,±) = |~p〉〈~p | ⊗ b ± = c1(~p,±), d(~p,±) = |~p〉〈~p | ⊗ d ± = c2(~p,±) (1.1) satisfying a reducible representation of CAR. The momentum eigenvectors are normalized by 〈~p|~p ′ 〉 = δΓm(~p, ~p ′) = (2π)32 ~p2 +m2δ(3)(~p−~p ′).The reducible representation of CAR can be written in a compact form as cj(~p, s), cj′(~p ′, s′)† = δjj′δss′δΓm(~p, ~