TITLE:
Brief Note on a Scalar Quantum Field with Finite Lifetime in a Lorentz Invariant Non-Rectangular Euclidean Space
AUTHORS:
Carmen Tornow
KEYWORDS:
Lorentz Invariance, Scalar Field, Renormalisation, Lifetime
JOURNAL NAME:
Journal of Modern Physics,
Vol.5 No.14,
August
28,
2014
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.