On the Construction of Relativistic Representation Spaces in Functional Quantum Theory of the Nonlinear Spinor Field

Functional quantum theory is defined by an isomorphism of the state space H of a conventional quantum theory into an appropriate functional state space It is a constructive approach to quantum theory in those cases where the state spaces H of physical eigenstates cannot be calculated explicitly like in nonlinear spinor field quantum theory. For the foundation of functional quantum theory appropriate functional state spaces have to be constructed which have to be representation spaces of the corresponding invariance groups. In this paper, this problem is treated for the spinor field. Using anticommuting source operator, it is shown that the construction problem of these spaces is tightly connected with the construction of appropriate relativistic function spaces. This is discussed in detail and explicit representations of the function spaces are given. Imposing no artificial restrictions it follows that the resulting functional spaces are indefinite. Physically the indefiniteness results from the inclusion of tachyon states. It is reasonable to assume a tight connection of these tachyon states with the ghost states introduced by Heisenberg for the regularization of the nonrenormalizable spinor theory.