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Quantum mechanical operators and quantum fields are interpreted asrealizations of timespace manifolds. Such causal manifolds are parametrized bythe classes of the positive unitary operations in all complex operations, i.e.by the homogenous spaces $\D(n)=\GL(\C^n_\R)/\U(n)$ with $n=1$ for mechanicsand $n=2$ for relativistic fields. The rank $n$ gives the number of both thediscrete and continuous invariants used in the harmonic analysis, i.e. twocharacteristic masses in the relativistic case. 'Canonical' field theories withthe familiar divergencies are inappropriate realizations of the real4-dimensional causal manifold $\D(2)$. Faithful timespace realizations do notlead to divergencies. In general they are reducible, but nondecomposable - inaddition to representations with eigenvectors (states, particle) theyincorporate principal vectors without a particle (eigenvector) basis asexemplified by the Coulomb field.Comment: 36 pages, latex, macros include

Realizations of causal manifolds by quantum fields