The Nonlinear Field Space Theory

Jakub Mielczarek,Tomasz Trześniewski

doi:10.1016/j.physletb.2016.06.006

Jakub Mielczarek, Tomasz Trześniewski

Open Access

https://doi.org/10.1016/j.physletb.2016.06.006

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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.

Highlights

Depending on its type, the field theoretical description of Nature is assigning scalars, vectors, tensors or spinors to the points of space
The space of all possible values of a field, i.e. the field space, is a generalization of the particle phase space, with the number of degrees of freedom going to infinity
While nontrivial, curved phase spaces for particles and strings have been investigated in the context of quantum gravity [1,2,3,4,5] and string theory [6,7,8], the spaces of fields are typically assumed to be linear – flat and infinite

Summary

Introduction

The field theoretical description of Nature is assigning scalars, vectors, tensors or spinors to the points of space. An important advantage of such a nontrivial structure of the field space is a possibility of restrictions on field values. This is encouraging since one can expect that for physical systems only finite values of fields are allowed, whereas in standard field theories arbitrary large values are possible, leading to different kinds of divergences. Let us stress that NFST should not be confused with Field Theories on Curved Spaces [14] or the Group Field Theory [15] In the latter cases the field space is flat, while the background space(time), or momentum space, is curved. In the example of NFST studied we assume that the background is Minkowski spacetime

The scalar field

Spherical phase space

Dynamics

Quantum dynamics

Empirical consequences

Summary