Renormalizability and the Scalar Field

TL;DR

This paper explores a finite quantum field theory of scalar fields with a $rac{rac{}{6!}}$ interaction in four dimensions, maintaining key physical principles through fuzzy particles instead of pointlike ones.

Contribution

It introduces a renormalizable scalar field theory with fuzzy particles that preserves causality, Lorentz invariance, and unitarity, extending traditional quantum mechanics of extended objects.

Findings

The theory is finite with fuzzy particles.

Causality, Lorentz invariance, and unitarity are preserved.

The Kallen-Lehmann spectral representation is discussed.

Abstract

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.