Nonlinear Wightman fields

A nonlinear Wightman field is taken to be a nonlinear map from a linear space of test functions to a linear space of Hilbert space operators, with inessential modifications to other axioms only to the extent dictated by the introduction of nonlinearity. Two approaches to nonlinear quantum fields are constructed and discussed, the first of which, starting from Lagrangian QFT, offers a fresh perspective on renormalization, while the second, starting from linear Wightman fields, provides an extensive range of well-defined nonlinear theories.

💡 Research Summary

The paper introduces the notion of a “non‑linear Wightman field” as a systematic extension of the traditional Wightman axioms. In the standard framework a test function (f) is mapped linearly to a field operator (\phi(f)) acting on a Hilbert space, and the axioms (locality, spectrum condition, positivity, etc.) are built around this linearity. The author relaxes only the linearity requirement, allowing a non‑linear map (\Phi:\mathcal{S}(\mathbb{R}^d)\rightarrow\mathcal{B}(\mathcal{H})) while preserving the other axioms as far as possible. Two concrete constructions are presented.

The first construction starts from a Lagrangian quantum field theory. The interaction Lagrangian (\mathcal{L}_{\text{int}}(\phi)) is re‑interpreted as a non‑linear functional (\mathcal{F}