Quantization of Big Bang in Crypto-Hermitian Heisenberg Picture

Abstract

A background-independent quantization of Universe near its Big Bang singularity is considered. Several conceptual issues are addressed in Heisenberg picture. (1) The observable spatial-geometry non-covariant characteristics of an empty-space expanding Universe are sampled by (quantized) distances $Q=Q(t)$ between space-attached observers. (2) In Q(t) one of the Kato’s exceptional-point times $t=\tau _{(EP)}$ is postulated real-valued. At such an instant the widely accepted “Big Bounce” regularization of the Big Bang singularity gets replaced by the full-fledged quantum degeneracy. Operators $Q(\tau _{(EP)})$ acquire a non-diagonalizable Jordan-block structure. (3) During our “Eon” (i.e., at all $t>\tau _{(EP)}$) the observability status of operators Q(t) is guaranteed by their self-adjoint nature with respect to an ad hoc Hilbert-space metric $\varTheta (t) \ne I$. (4) In adiabatic approximation the passage of the Universe through its $t=\tau _{(EP)}$ singularity is interpreted as a quantum phase transition between the preceding and the present Eon.

Appendix: Auxiliary Spaces and \({\mathscr {P}}{\mathscr {T}}\)-Symmetries

A few years after the publication of review [6], a series of rediscoveries and an enormous growth of popularity of the pattern followed the publication of pioneering letter [29] in which Bender with his student inverted the flowchart. They choose a nice illustrative example to show that the manifestly non-Hermitian F-space Hamiltonian H with real spectrum may be interpreted as a hypothetical input information about the dynamics (cf. also review [8] for more details).

Graphically, the flowchart of ${\mathscr {P}}{\mathscr {T}}$-symmetric quantum theory is schematically depicted in Fig. 9. For completeness let us add that the Bender’s and Boettcher’s construction was based on the assumption of ${\mathscr {P}}{\mathscr {T}}$-symmetry $H {\mathscr {P}}{\mathscr {T}} = {\mathscr {P}}{\mathscr {T}}H$ of their dynamical-input Hamiltonians where the most common phenomenological parity ${\mathscr {P}}$ and time reversal ${\mathscr {T}}$ entered the game. Mostafazadeh (cf. his review [9]) emphasized that their theory may be generalized while working with more general ${\mathscr {T}}$s (typically, any antilinear operator) and ${\mathscr {P}}$s (basically, any indefinite, invertible operator).

Fig. 9

THS interpretation of ${\mathscr {P}}{\mathscr {T}}$-symmetric Hamiltonians H

Several mathematical amendments of the theory were developed in the related literature, with the main purpose of making the constructions feasible. Let us only mention here that the useful heuristic role of operator ${\mathscr {P}}$ was successfully transferred to the Krein-space metrics $\eta$ (cf. [30] for a comprehensive review). In comment [31] we explained that in principle, the role of ${\mathscr {P}}$ could even be transferred to some positive-definite, simplified and redundant auxiliary-Hilbert-space metrics $\tilde{\mathscr {P}} = \varTheta _A \ne \varTheta _S$. Such a recipe proved encouragingly efficient [32]. Its flowchart may be summarized in the following diagram:

Besides the right-side flow of mapping we see here the auxiliary, unphysical left-side flow where, typically, the non-Dirac metric $\varTheta _A$ need not carry any physical contents. In some models such an auxiliary metric proved even obtainable in a trivial diagonal-matrix form [28].