# Mini-Superspace Universality and No-Scale Quantum Cosmology

## TL;DR

This paper demonstrates that at the mini-superspace level, Einstein-Hilbert and Weyl-Dirac gravity produce identical quantum cosmologies governed by the same wave function, with differences arising only in higher-dimensional contexts.

## Contribution

It proves the universality of mini-superspace quantum cosmology across different gravity theories and explores the role of extra dimensions and scalar fields.

## Key findings

- Einstein-Hilbert and Weyl-Dirac gravity share the same wave function in mini-superspace.
- The wave function's argument shifts from scale factor to a scalar combination involving the dilaton.
- The Weyl vector influences quantum cosmology only with extra dimensions, affecting near Big Bang behavior.

## 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 $a\phi$ ($\phi$ 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.