Quantization of Big Bang in crypto-Hermitian Heisenberg picture

A background-independent quantization of the Universe near its Big Bang singularity is considered using a drastically simplified toy model. Several conceptual issues are addressed. (1) The observable spatial-geometry characteristics of our empty-space expanding Universe is sampled by the time-dependent operator Q=Q(t) of the distance between two space-attached observers ("Alice and Bob"). (2) For any pre-selected guess of the simple, non-covariant time-dependent observable Q(t) one of the Kato's exceptional points viz., $t=τ_{(EP}) is postulated {em real-valued}. This enables us to treat it as the time of Big Bang. (3) During our "Eon" \(i.e., at allt>τ_{(EP\)}) the observability status of operatorQ(t)is mathematically guaranteed by its self-adjoint nature with respect to an {em ad hoc} Hilbert-space metricΘ(t) \neq I. (4) In adiabatic approximation (i.e., in Heisenberg picture) the passage of the Universe through itst=τ_{(EP)}$ singularity is interpreted as a quantum phase transition between the preceding and the present Eon. It is worth adding that in our model the widely accepted "Big Bounce" regularization of the classical Big Bang singularity after quantization gets replaced by the full-fledged quantum degeneracy. At the quantum-phase-transition singularity, operator Q\(τ_(EP\)) becomes unobservable and acquires a non-diagonalizable Jordan-block structure.