Abstract
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 single-variable 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.
Highlights
The mini-superspace approximation [1], while being premeditatedly naive and simple by construction, is still one of the best available theoretical tools to probe the quantum cosmology
The prototype mini-superspace Hartle-Hawking model is economical in its ingredients
Is the exact ψHH (a) a unique fingerprint of the underlying theory of general relativity? We first prove that the answer to this question is negative
Summary
The mini-superspace approximation [1], while being premeditatedly naive and simple by construction, is still one of the best available theoretical tools to probe the quantum cosmology. The conclusion holds for an arbitrary Brans-Dicke parameter [10] This brings us to the more general topic of no-scale quantum cosmology, where the underlying gravitational theory exhibits local scale invariance. The latter local symmetry is translated into an additional constraint (on top of the Hamiltonian constraint), and may have a far reaching impact on the mini-superspace. One may invoke KaluzaKlein reduced higher dimensional local scale symmetric theories [12] This allows the Weyl vector to enter the game, and even govern the wave function behavior [13] near Big Bang. The corresponding Weyl-Dirac gravity is field theoretically formulated by the action
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