The Nonlinear Field Space Theory

TL;DR

This paper introduces a nonlinear field space framework for scalar fields, revealing new physical effects like non-locality, modified uncertainty relations, and vacuum energy shifts, with implications for quantum gravity and condensed matter physics.

Contribution

It extends the concept of nonlinear geometry from particle phase spaces to field theories, providing a novel approach to address divergences and fundamental interactions.

Findings

Non-locality emerges from nonlinear field space.

Uncertainty relations are generalized in this framework.

Vacuum energy and propagation speeds are shifted.

Abstract

In recent years the idea that not only the configuration space of particles, i.e. spacetime, but also the corresponding momentum space may have nontrivial geometry has attracted significant attention, especially in the context of quantum gravity. The aim of this letter is to extend this concept to the domain of field theories, by introducing field spaces (i.e. phase spaces of field values) that are not affine spaces. After discussing the motivation and general aspects of our approach we present a detailed analysis of the prototype (quantum) Nonlinear Field Space Theory of a scalar field on the Minkowski background. We show that the nonlinear structure of a field space leads to numerous interesting predictions, including: non-locality, generalization of the uncertainty relations, algebra deformations, constraining of the maximal occupation number, shifting of the vacuum energy and renormalization of the charge and speed of propagation of field excitations. Furthermore, a compact field space is a natural way to implement the "Principle of Finiteness" of physical theories, which once motivated the Born-Infeld theory. Thus the presented framework has a variety of potential applications in the theories of fundamental interactions (e.g. quantum gravity), as well as in condensed matter physics (e.g. continuous spin chains), and can shed new light on the issue of divergences in quantum field theories.