A proper scalar product for tachyon representations in configuration space

TL;DR

This paper introduces a new positive-definite inner product for tachyonic scalar fields, enabling a consistent quantum field theory framework with unitary Poincaré transformations and a novel Fourier transform.

Contribution

It proposes a non-local inner product for tachyon solutions, ensuring positive definiteness and unitarity, and develops a new Fourier transform linking configuration and momentum spaces.

Findings

Inner product is positive definite for oscillatory solutions.

Poincaré transformations are unitarily implemented.

A new unitary Fourier transform is established.

Abstract

We propose a new inner product for scalar fields that are solutions of the Klein-Gordon equation with $m^2<0$. This inner product is non-local, bearing an integral kernel including Bessel functions of the second kind, and the associated norm proves to be positive definite in the subspace of oscillatory solutions, as opposed to the conventional one. Poincar\'e transformations are unitarily implemented on this subspace, which is the support of a unitary and irreducible representation of the proper orthochronous Poincar\'e group. We also provide a new Fourier Transform between configuration and momentum spaces which is unitary, and recover the projection onto the representation space. This new scenario suggests a revision of the corresponding quantum field theory.