Brief Note on a Scalar Quantum Field with Finite Lifetime in a Lorentz Invariant Non-Rectangular Euclidean Space

Carmen Tornow

doi:10.4236/jmp.2014.514135

Abstract

A not necessary rectangular Euclidean space (NoNRES) is constructed, in which one obtains a generally Lorentz invariant scalar product for the low energy sector (LES). This sector is defined for energies below the Planckian limit. If the energy is zero, the NoNRES becomes rectangular and due to the Lorentz invariance, it is applicable for the complete LES of the theory. In contrast to the usual Minkowski space the metric of the NoNRES depends on the kinetic energy of the observed quantum particles. It is assumed that this metric may be useful to derive the scattering cross-section of the corresponding quantum field theory. This assumption is related to the occurrence of divergent loop momentum integrals caused by including the infinite energy range above the Planckian limit (high energy sector or HES). Due to its energy dependence, the metric in both energy sectors differs. In the HES, it depends on the effective dimension of the NoNRES. This dependency results from fluctuations of the space-time above the Planckian limit. Even if they are not part of the theory (as they would be in quantum gravity), these fluctuations should not be ignored. The effective dimension decreases if the energy of the considered particle increases. Since this is true for the HES only, the ultraviolet divergences of loop integrals seem to vanish without distorting the results of the LES. The mechanism is illustrated by calculating the tadpole integral occurring for a simple self-interacting scalar quantum field (with the Higgs mass as example). One obtains a finite contribution for the integral and consequently for the lifetime of the scalar particle.

Highlights

Ultraviolet (UV) divergences in quantum field theories (QFTs) without regularization and renormalization occurHow to cite this paper: Tornow, C. (2014) Brief Note on a Scalar Quantum Field with Finite Lifetime in a Lorentz Invariant Non-Rectangular Euclidean Space
The metric, gμν, in the not necessary rectangular Euclidean space (NoNRES), which depends on the particle energy, E, allows to derivate an effective dimension, dim(E), which is the key parameter of the integration measure ddim(E)k
In contrast to the NoNRES, the effective dimension of the Minkowski space (MS) always gives dimMS(E) = 4 for all values of E, i.e., it is independent of the particle energy

Summary

Introduction

Ultraviolet (UV) divergences in quantum field theories (QFTs) without regularization and renormalization occur. The divergences in loop-momentum integrals originate from the fact, that special QFTs, e.g. quantum electrodynamics, are low energy approximations of a complete theory of all possible quantum particle interactions, including gravity [3] In this study it is asked whether there is chance to simplify the regularization and renormalization procedure. Values, and between the energy sector of standard QFT and the HES an appropriate transition region might exist, since it is speculated that the effects related to the violation of Lorentz invariance (e.g. polarization independent refractivity [4]) can already be detected at energies much lower than EP = mPc2 The applicability of this integration method is discussed and further steps to establish this technique are outlined

Invariance of Lorentz Transformation in the Non-Rectangular Euclidean Space

The Effective Dimension of the NoNRES

The Interacting Scalar Particle in the NoNRES

Conclusions