Switching internal times and a new perspective on the 'wave function of the universe'

TL;DR

The paper develops a framework for quantum general covariance using dynamical quantum reference frames and applies it to a simple flat FRW cosmology with a massless scalar field. It demonstrates how to perform both classical and quantum internal-time switches between the scale factor and the scalar field clocks by relating Dirac (neutral) quantization to reduced quantizations through trivialization maps. A complete relational quantum theory emerges as the conjunction of Dirac and reduced quantizations, enabling a perspective-neutral global state that encodes all internal perspectives and links to relational states used for predictions. The work also discusses extensions to loop quantum cosmology and offers a fresh interpretational view of the wave function of the universe as a unifying, clock-neutral structure that nevertheless yields operationally meaningful relative states.

Abstract

Despite its importance in general relativity, a quantum notion of general covariance has not yet been established in quantum gravity and cosmology, where, given the a priori absence of coordinates, it is necessary to replace classical frames with dynamical quantum reference systems. As such, quantum general covariance bears on the ability to consistently switch between the descriptions of the same physics relative to arbitrary choices of quantum reference system. Recently, a systematic approach for such switches has been developed (arXiv:1809.00556, 1809.05093, 1810.04153). It links the descriptions relative to different choices of quantum reference system, identified as the correspondingly reduced quantum theories, via the reference-system-neutral Dirac quantization, in analogy to coordinate changes on a manifold. In this work, we apply this method to a simple cosmological model to demonstrate how to consistently switch between different internal time choices in quantum cosmology. We substantiate the argument that the conjunction of Dirac and reduced quantized versions of the theory defines a complete relational quantum theory that not only admits a quantum general covariance, but, we argue, also suggests a new perspective on the 'wave function of the universe'. It assumes the role of a perspective-neutral global state, without immediate physical interpretation, that, however, encodes all the descriptions of the universe relative to all possible choices of reference system at once and constitutes the crucial link between these internal perspectives. While, for simplicity, we use the Wheeler-DeWitt formulation, the method and arguments might be also adaptable to loop quantum cosmology.

Figures (2)

• Figure 1: Diagrammatic overview of the relation between Dirac and the four reduced quantizations. In a nutshell, the physical Hilbert space is mapped to any of the four (positive or negative frequency) reduced Hilbert spaces by first trivializing the Hamiltonian constraint via $\mathcal{T}_\phi$ or $\mathcal{T}_\alpha$ to the corresponding choice of internal time variable and subsequently projecting onto the classical internal time gauge fixing condition. (Details in the main text.)
• Figure 2: Reduced probability distributions coming from the same physical state (defined through (\ref{['physstate']}, \ref{['n']}) and here $n=100$), but described relative to the choices of (a) $\phi$ and (b) $\alpha$ as internal times in the 'expanding-forward' (green) quadrant of fig. \ref{['fig:cs']}. Recall that in the reduced theory the usual modulus square of the wave function is the probability distribution, see (\ref{['redIP']}). Due to the symmetry of the model in $\alpha$ and $\phi$, reduced probability distributions will always behave symmetrically.