The nature of the scalar field responsible for the cosmological inflation is found to be rooted in the most fundamental concept of Weyl's differential geometry: the parallel displacement of vectors in curved space–time. Within this novel geometrical scenario, the standard electroweak theory of leptons based on the
SU
(2)
L
⊗
U
(1)
Y
as well as on the conformal groups of space–time Weyl's transformations is analysed within the framework of a general-relativistic, conformally covariant scalar-tensor theory that includes the electromagnetic and the Yang–Mills fields. A Higgs mechanism within a spontaneous symmetry breaking process is identified and this offers formal connections between some relevant properties of the elementary particles and the dark energy content of the Universe. An ‘effective cosmological potential’:
V
eff
is expressed in terms of the dark energy potential:
via the ‘mass reduction parameter’:
, a general property of the Universe. The mass of the Higgs boson, which is considered a ‘free parameter’ by the standard electroweak theory, by our theory is found to be proportional to the mass
which accounts for the measured cosmological constant, i.e. the measured content of vacuum-energy in the Universe. The non-integrable application of Weyl's geometry leads to a Proca equation accounting for the dynamics of a
ϕ
ρ
-particle, a vector-meson proposed as an an optimum candidate for dark matter. On the basis of previous cosmic microwave background results our theory leads, in the condition of cosmological ‘critical density’, to the assessment of the average energy content of the
ϕ
ρ
-excitation. The peculiar mathematical structure of
V
eff
offers a clue towards a very general resolution of a most intriguing puzzle of modern quantum field theory, the ‘Cosmological Constant Paradox’ (here referred to as the ‘
Λ
-Paradox’). Indeed, our ‘universal’ theory offers a resolution of the
Λ
-Paradox for all exponential inflationary potentials:
V
Λ
(
T
,
ϕ
)∝
e
−
nϕ
, and for all linear superpositions of these potentials, where
n
belongs to the mathematical set of the ‘real numbers’. An explicit solution of the
Λ
-Paradox is reported for
n
=2. The resolution of the
Λ
-Paradox cannot be achieved in the context of Riemann's differential geometry.
This article is part of the themed issue ‘Second quantum revolution: foundational questions’.
NONCOMMUTATIVE GRAVITY, A "NO STRINGS ATTACHED" QUANTUM–CLASSICAL DUALITY, AND THE COSMOLOGICAL CONSTANT PUZZLE
