We perform the 4-dimensional Kaluza-Klein (KK) reduction of the 5-dimensional locally scale invariant Weyl-Dirac gravity. While compactification unavoidably introduces an explicit length scale into the theory, it does it in such a way that the KK rad
ius can be integrated out from the low energy regime, leaving the KK vacuum to still enjoy local scale invariance at the classical level. Imitating a $U(1)timestilde{U}(1)$ gauge theory, the emerging 4D theory is characterized by a kinetic Maxwell-Weyl mixing whose diagonalization procedure is carried out in detail. In particular, we identify the unique linear combination which defines the 4D Weyl vector, and fully classify the 4D scalar sector. The later consists of (using Weyl language) a co-scalar and two in-scalars. The analysis is performed for a general KK $m$-ansatz, parametrized by the power $m$ of the scalar field which factorizes the 4D metric. The no-ghost requirement, for example, is met provided $-frac{1}{2}leq m leq 0$. An $m$-dependent dictionary is then established between the original 5D Brans-Dicke parameter $omega_5$ and the resulting 4D $omega_4$. The critical $omega_5=-frac{4}{3}$ is consistently mapped into critical $omega_4 = -frac{3}{2}$. The KK reduced Maxwell-Weyl kinetic mixing cannot be scaled away as it is mediated by a 4D in-scalar (residing within the 5D Weyl vector). The mixing is explicitly demonstrated within the Einstein frame for the special physically motivated choice of $m=-frac{1}{3}$. For instance, a super critical Brans-Dicke parameter induces a tiny positive contribution to the original (if introduced via the 5-dimensional scalar potential) cosmological constant. Finally, some no-scale quantum cosmological aspects are studied at the universal mini-superspace level.
We elevate the field theoretical similarities between Maxwell and Weyl vector fields into a full local scale/gauge invariant Weyl/Maxwell mutual sourcing theory. In its preliminary form, and exclusively in four dimensions, the associated Lagrangian i
s dynamical scalar field free, hosts no fermion matter fields, and Holdom kinetic mixing is switched off. The mutual sourcing term is then necessarily spacetime curvature (not just metric) dependent, and inevitably Ricci linear, suggesting that a non-vanishing spacetime curvature can in principle induce an electromagnetic current. In its mature form, however, the Weyl/Maxwell mutual sourcing idea serendipitously constitutes a novel variant of the gravitational Weyl-Dirac (incorporating Brans-Dicke) theory. Counter intuitively, and again exclusively in four dimensions, the optional quartic scalar potential gets consistently replaced by a Higgs-like potential, such that the co-divergence of the Maxwell vector field resembles a conformal vacuum expectation value.
We prove that, at the mini superspace level, and for an arbitrary Brans-Dicke parameter, one cannot tell traditional Einstein-Hilbert gravity from local scale invariant Weyl-Dirac gravity. Both quantum mechanical cosmologies are governed by the one a
nd the same time-independent single-variable Hartle-Hawking wave function. It is only that its original argument, the cosmic scale factor $a$, is replaced by $aphi$ ($phi$ being the dilaton field) to form a Dirac in-scalar. The Weyl vector enters quantum cosmology only in the presence of an extra dimension, where its fifth component, serving as a 4-dim Kaluza-Klein in-scalar, governs the near Big Bang behavior of the wave function. The case of a constant Kaluza-Klein in-radius is discussed in some detail.
We postulate a Planck scale horizon unit area, with no bits of information locally attached to it, connected but otherwise of free form, and let $n$ such geometric units compactly tile the black hole horizon. Associated with each topologically distin
ct tiling configuration is then a simple, connected, undirected, unlabeled, planar, chordal graph. The asymptotic enumeration of the corresponding integer sequence gives rise to the Bekenstein-Hawking area entropy formula, automatically accompanied by a proper logarithmic term, and fixes the size of the horizon unit area, thereby constituting a global realization of Wheelers it from bit phrase. Invoking Polyas theorem, an exact number theoretical entropy spectrum is offered for the 2+1 dimensional quantum black hole.
Unification of Randall-Sundrum and Regge-Teitelboim brane cosmologies gives birth to a serendipitous Higgs-deSitter interplay. A localized Dvali-Gabadadze-Porrati scalar field, governed by a particular (analytically derived) double-well quartic poten
tial, becomes a mandatory ingredient for supporting a deSitter brane universe. When upgraded to a general Higgs potential, the brane surface tension gets quantized, resembling a Hydrogen atom spectrum, with deSitter universe serving as the ground state. This reflects the local/global structure of the Euclidean manifold: From finite energy density no-boundary initial conditions, via a novel acceleration divide filter, to exact matching conditions at the exclusive nucleation point. Imaginary time periodicity comes as a bonus, with the associated Hawking temperature vanishing at the continuum limit. Upon spontaneous creation, while a finite number of levels describe universes dominated by a residual dark energy combined with damped matter oscillations, an infinite tower of excited levels undergo a Big Crunch.
By treating modulus and phase on equal footing, as prescribed by Dirac, local scale invariance can consistently accompany any Brans-Dicke $omega$-theory. We show that in the presence of a soft scale symmetry breaking term, the classical solution, if
it exists, cannot be anything else but general relativistic. The dilaton modulus gets frozen up by the Weyl-Proca vector field, thereby constituting a gravitational quasi-Higgs mechanism. Assigning all grand unified scalars as dilatons, they enjoy Weyl universality, and upon symmetry breaking, the Planck (mass)$^2$ becomes the sum of all their individual (VEV)$^2$s. The emerging GUT/Planck (mass)$^2$ ratio is thus $sim omega g_{GUT}^2/4pi$.
Pre-gauging the cosmological scale factor $a(t)$ does not introduce unphysical degrees of freedom into the exact FLRW classical solution. It seems to lead, however, to a non-dynamical mini superspace. The missing ingredient, a generalised momentum en
joying canonical Dirac (rather than Poisson) brackets with the lapse function $n(t)$, calls for measure scaling which can be realised by means of a scalar field. The latter is essential for establishing a geometrical connection with the 5-dimensional Kaluza-Klein Schwarzschild-deSitter black hole. Contrary to the Hartle-Hawking approach, (i) The $t$-independent wave function $psi(a)$ is traded for an explicit $t$-dependent $psi(n, t)$, (ii) The classical FLRW configuration does play a major role in the structure of the most classical cosmological wave packet, and (iii) The non-singular Euclid/Lorentz crossovers get quantum mechanically smeared.
A gravity-anti-gravity (GaG) odd linear dilaton action offers an eternal inflation evolution governed by the unified (cosmological constant plus radiation) equation of state $rho-3P=4Lambda$. At the mini superspace level, a two-particle variant of th
e no-boundary proposal, notably one-particle energy dependent, is encountered. While a GaG-odd wave function can only host a weak Big Bang boundary condition, albeit for any $k$, a strong Big Bang boundary condition requires a GaG-even entangled wave function, and singles out $k=0$ flat space. The locally most probable values for the cosmological scale factor and the dilaton field form a grid ${a^2,aphi}simsqrt{4n_1+1}pmsqrt{4n_2+1}$.
A quantum Schwarzschild black hole is described, at the mini super spacetime level, by a non-singular wave packet composed of plane wave eigenstates of the momentum Dirac-conjugate to the mass operator. The entropy of the mass spectrum acquires then
independent contributions from the average mass and the width. Hence, Bekensteins area entropy is formulated using the $langle text{mass}^2 rangle$ average, leaving the $langle text{mass} rangle$ average to set the Hawking temperature. The width function peaks at the Planck scale for an elementary (zero entropy, zero free energy) micro black hole of finite rms size, and decreases Doppler-like towards the classical limit.
We discuss the cosmological constant problem, at the minisuperspace level, within the framework of the so-called normalized general relativity (NGR). We prove that the Universe cannot be closed, and reassure that the accompanying cosmological constan
t $Lambda$ generically vanishes, at least classically. The theory does allow, however, for a special class of $Lambda ot=0$ solutions which are associated with static closed Einstein universe and with Eddington-Lema^{i}tre universe.
