By treating modulus and phase on equal footing, as prescribed by Dirac, local scale invariance can consistently accompany any Brans-Dicke ω-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 ∼ ωg 2 GU T /4π. Critical local Weyl invarianceThe Brans-Dicke theory [1] is described by the action(1)

At the classical level, a linear dilaton action offers an eternal inflation evolution governed by the unified (cosmological constant plus radiation) equation of state ρ − 3P = 4Λ. At the quantum mechanical mini superspace level, a 'two-particle' variant of the no-boundary proposal, notably 'oneparticle' energy dependent, is encountered. While a (gravity-anti-gravity) 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 maximal values for the cosmological scale factor and the dilaton form a grid {a 2 , aφ} ∼ √ 4n1 + 1 ± √ 4n2 + 1.

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 is 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.

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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 and the same time-independent singlevariable Hartle-Hawking wave function. It is only that its original argument, the cosmic scale factor a, is replaced by aφ (φ 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.