Quantization of Big Bang in crypto-Hermitian Heisenberg picture

Abstract

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., $t={\tau }_{(EP)}$) is postulated {\em real-valued}. This enables us to treat it as the time of Big Bang. (3) During our ``Eon'' (i.e., at all $t>{\tau }_{(EP)}$) the observability status of operator $Q(t)$ is mathematically guaranteed by its self-adjoint nature with respect to an {\em ad hoc} Hilbert-space metric $\Theta (t)\ne I$. (4) In adiabatic approximation (i.e., in Heisenberg picture) 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. 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 $Q({\tau }_{(EP)})$ becomes unobservable and acquires a non-diagonalizable Jordan-block structure.

16 pp., 9 figures, written version of the talk presented to the PHHQP XIII conference in Jerusalem (see http://www.as.huji.ac.il/isf/phhqp13). The videorecorded form of the talk is available, under number 18, here: http://www.as.huji.ac.il/node/877

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• Making Sense of Non-Hermitian Hamiltonians
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• Time-dependent version of cryptohermitian quantum theory
• Maximal couplings in PT-symmetric chain-models with the real spectrum of energies
• Cryptogauge symmetry and cryptoghosts for crypto-Hermitian Hamiltonians
• Non-Hermitian Heisenberg representation
• Multiply Degenerate Exceptional Points and Quantum Phase Transitions
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Cited by in corpus (2)

• Non-Hermitian interaction representation and its use in relativistic quantum mechanics
• Wheeler-DeWitt equation and the applicability of crypto-Hermitian interaction representation in quantum cosmology