On the Role of Fiducial Structures in Minisuperspace Reduction and Quantum Fluctuations in LQC

On the Role of Fiducial Structures in Minisuperspace Reduction and Quantum Fluctuations in LQC

Fabio M. Mele, Johannes Münch
Okinawa Institute of Science and Technology & The Centre de Physique Théorique – UMR 7332 – CNRS and Aix-Marseille Université and Université de Toulon

Preprint

ABSTRACT

We study the homogeneous minisuperspace reduction within the canonical framework for a scalar field theory and gravity. Symmetry reduction is implemented via second class constraints for the field modes over a partitioning of the non-compact spatial slice $Sigma$ into disjoint cells. The canonical structure of the resulting homogeneous theories is obtained via the associated Dirac bracket which can only be defined on a finite number of cells homogeneously patched together and agrees with the full theory Poisson bracket for the averaged fields. This identifies a finite region $V_o$, the fiducial cell, whose size $L$ sets the physical scale over which homogeneity is imposed, namely a wavelength cutoff. The reduced theory results from 1) selecting a subset of $V_o$-averaged observables of the full theory; 2) neglecting inhomogeneous $vec kneqmathbf0$ modes with wavelengths $lambdageq L$ and $lambda< L$; 3) neglecting boundary terms encoding interactions between neighbouring cells. The error made is of order $mathcal O(1/kL)$. As a result, the off-shell structures of the reduced theory depend on the size of $V_o$ and different $V_o$ identify canonically inequivalent theories whose dynamics though is $V_o$-independent. Their quantisation leads then to a family of $V_o$-labeled quantum representations and the quantum version of an active rescaling of $V_o$ is implemented via a suitable dynamics-preserving isomorphism between the different theories. We discuss the consequences for statistical moments, fluctuations, and semiclassical states in both a standard and polymer quantisation. For a scalar field of mass $m$, we also sketch the quantum reduction and identify a subsector of the QFT where the results of the"first reduced, then quantised" theories can be reproduced with good approximation as long as $mgg1/L$. Finally, a strategy to include inhomogeneities in cosmology is outlined.