Decoherence without einselection

Decoherence in a quantum measurement is typically explained as an interaction with the environment that destroys coherence between the system's eigenstates, a phenomenon known as environment-induced superselection (einselection). In this work, we demonstrate that einselection and the associated envariance are actually artifacts resulting from neglecting the non-equilibrium dynamics of the apparatus. We propose a new formalism of operator dressing, which we call the stochastic matrix integral (SMI), inspired by recent developments in quantum gravity algebras. This approach naturally arises from a modified Page-Wootters (PW) formula and describes decoherence as an interacting non-equilibrium process. It leads to the reduction of the Hilbert space and the emergence of an intrinsic non-unitary process as well as Born's rule. These outcomes are achieved without relying on the assumptions of einselection and pointer basis.