Renormalizability and the Scalar Field

The infinite dimensional generalization of the quantum mechanics of extended objects, namely, the quantum field theory of extended objects is presented. The paradigm example studied in this paper is the Euclidean scalar field with a $\lambda (\phi^{6})/6!$ interaction in four spacetime dimensions. The theory is found to be finite when the virtual particle intermediate states are characterized by fuzzy particles instead of ordinary pointlike particles. Causality, Lorentz invariance, and unitarity (verified up to fourth order in the coupling constant) are preserved in the theory. In addition, the Kallen-Lehmann spectral representation for the propagator is discussed.