  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.
