A closed Friedmann-Lemaitre-Robertson-Walker Wheeler-DeWitt model is formulated as an exact reflection problem. The derivative of its reflection phase defines a relational crossing duration whose first two clock moments follow from a covariant scalar-clock observable. An analytic expression is obtained for this duration, its classical recollapse limit, and the leading quantum correction. A finite spectral packet also gives an operational, equal-prior minimum error probability for distinguishing the two orientations of recollapse with a geometric reading. The construction is extended to a bounded finite quantum detector. Its multichannel reflection matrix yields probe transitions, spectral-probe correlations, and a matrix duration. A direct weak-coupling calculation verifies the predicted transition and duration scalings. Scalar-clock conditioning and oriented geometric sections are shown to be local representations of the same positive-frequency Dirac sector. These results define duration and transition observables entirely through correlations and scattering records of a stationary constrained state, without introducing a background clock.
A closed Friedmann-Lemaitre-Robertson-Walker Wheeler-DeWitt model is formulated as an exact reflection problem. The derivative of its reflection phase defines a relational crossing duration whose first two clock moments follow from a covariant scalar-clock observable. An analytic expression is obtained for this duration, its classical recollapse limit, and the leading quantum correction. A finite spectral packet also gives an operational, equal-prior minimum error probability for distinguishing the two orientations of recollapse with a geometric reading. The construction is extended to a bounded finite quantum detector. Its multichannel reflection matrix yields probe transitions, spectral-probe correlations, and a matrix duration. A direct weak-coupling calculation verifies the predicted transition and duration scalings. Scalar-clock conditioning and oriented geometric sections are shown to be local representations of the same positive-frequency Dirac sector. These results define duration and transition observables entirely through correlations and scattering records of a stationary constrained state, without introducing a background clock.
We give an explicit, rigorous framework for calculating quantum probabilities in a model theory of quantum gravity. Specifically, we construct the decoherence functional for the Wheeler-DeWitt quantization of a flat Friedmann-Robertson-Walker cosmology with a free, massless, minimally coupled scalar field, thus providing a complete decoherent histories formulation for this quantum cosmological model. The decoherence functional is applied to study predictions concerning the model's Dirac (relational) observables; the behavior of semiclassical states and superpositions of such states; and to study the singular behavior of quantum Wheeler-DeWitt universes. Within this framework, rigorous formulas are given for calculating the corresponding probabilities from the wave function when those probabilities may be consistently defined, thus replacing earlier heuristics for interpreting the wave function of the universe with explicit constructions. It is shown according to a rigorously formulated standard, and in a quantum-mechanically consistent way, that in this quantization these models are generically singular. Independent of the choice of state we show that the probability for these Wheeler-DeWitt quantum universes to ever encounter a singularity is unity. In addition, the relation between histories formulations of quantum theory and relational Dirac observables is clarified.
This note applies the frame-deficit dictionary to a genuinely dynamical quantum geometry: the exactly solvable flat Friedmann–Robertson–Walker universe with a massless scalar field, where the Wheeler–DeWitt equation becomes a 1+1 wave equation. The model is studied in a minisuperspace lattice, with expanding and contracting cosmological branches and the scalar field serving as an internal clock. Unlike previous toy models where geometry was treated as an external label, the geometry here is a quantum degree of freedom with its own conjugate momentum. The paper establishes four exact results. First, the timeless Wheeler–DeWitt kernel admits relational evolution: conditioning on the scalar clock recovers unitary evolution with no dispersion. Second, an observer without access to the clock carries an exact information deficit determined by the clock-momentum distribution. Third, a superposition of expanding and contracting universes decoheres through matter records, with the interference of cosmological histories controlled exactly by record overlap. Fourth, an entropic trade-off is established between the sharpness of the universe's history and the information hidden behind the clock: a more classical size history requires a larger information deficit. The result extends the frame-deficit framework from quantum reference frames to quantum cosmology, showing how missing reference information, decoherence, and classicality costs arise in a model with genuinely quantized geometry. A bibliographic correction concerning the attribution of previous quantum cosmology decoherence work is recorded in the paper. The scope is limited to the exactly solvable minisuperspace model and its lattice realization. No claim is made for full Einstein dynamics or a complete theory of quantum gravity. All results are based on pre-registered predictions, symbolic derivations, and computational verification. Reproducibility materials are included.
We solve cyclic finite-dimensional quantum histories for arbitrary time-dependent unitary steps, without assuming that one step has finite order. The propagation Hamiltonian is a unitary connection Laplacian on a cycle; its complete gauge invariant is the monodromy M=UL−1⋯U0. Its spectrum is λa,k=1−cos((2πk−θa)/L), where eiθa∈spec(M). Thus the exact history sector is isomorphic to Fix(M), while frustration, the gap above a nonempty zero-energy sector, the determinant, and the finite-temperature trace are obtained in closed form. Ordinary spectral data recover the multiset of monodromy phase cosines but not phase orientation; low energy certifies proximity to an exact relational history. We then define the predictive quotient of a sharp finite clock relative to an accessible operator system as the unique coarsest event alphabet preserving all conditional statistics on a history sector. A finite-error theorem shows that threshold clustering recovers this quotient when the minimum diamond separation of inequivalent event channels exceeds four times the estimation error, and proves an optimal record-count bound. With full matrix access and homogeneous step U, the minimal number of clock events is the projective order of U. We distinguish the normalizer of the clock algebra from transformations preserving the coherent history code and classify oriented exact sharp clock changes by U(r)×ZL on a rank-r history sector; without orientation the cyclic factor becomes dihedral. Reversible changes of full-information clock fibers are necessarily unitary, so irreversible coarse-graining is not exact clock covariance. Minimal realizations of a complete history Gram kernel are uniquely unitarily equivalent, with a finite-data Procrustes bound. Independent finite-matrix code verifies the main results.
We relate positive observable quotients to Page--Wootters (PW) clockdescriptions. Physicalization identifies a quotient by its kernel, andjointly faithful clock reductions recover the same exactness relation.Admissible PW data give a faithful functor into an independently definedquotient-clock category after representative lifts are identified modulophysical null directions. With algebraic lifts, it is full on its essentialimage at the level of retained state-and-clock data. A restricted converse characterizes strictly periodic, globallydeparametrizable realizations with unitary one-slice conditioning:one reference reduction must be unitary and the relative reductions mustform a strongly continuous periodic unitary group. The construction includesthe closed-operator domains required for unbounded generators. Retaining clock-event probabilities gives a separate strictness theorem.Normalizable stationary models with additive constraints and covariantperiodic clocks have state-independent clock marginals. A positivetwo-qubit constraint quotient violates this invariant despite invertiblereductions. This separates it from the additive class; its realization isitself interacting. We then define a broader stationary-conditioning class allowing generalizedcovariant clocks, unrestricted self-adjoint interactions, and separableenvironmental dilations. Its normal quantum-instrument data descendisometrically to positive quotients. Conversely, every normalized amplitudequotient-clock datum has an exact realization: a positive projectorconstraint selects the image of its isometric history map. Standardinstrument dilation extends the result to absolutely continuous normalcompletely positive instruments. For finite-dimensional physical and outputspaces with Sobolev-regular histories, a bounded finite-rank interactionsuffices by standard Hermitian completion and spectral isolation. Hencethis normalized operational sector has no obstruction to realization inthe broad class. For a fixed canonical periodic clock, we then bound the interactionremaining after arbitrary bounded clock-local Hermitian terms are allowed.Translation Fisher information gives a lower bound valid under separableenvironmental dilations. For instruments with orthogonal amplitude columns,a finite-rank construction attains the exact minimum while selecting theprescribed physical kernel. The modulated qubit family yields an explicitstrength--frequency threshold and a finite-event approximation-error bound. An exact Gaussian differential-image quotient coupled to a full oscillatorprovides an additive realization with matching physical metrics,conditioning, reconstruction, and relational probabilities, including finitereading intervals. These results distinguish restricted strictness frombroad representability; they neither identify all microscopic realizationsnor establish equivalence or strict superiority across every QRT and PWformulation.
Duration can be reconstructed from finite physical records through the dynamics and calibration of internal clocks. Finite classical, relativistic, and stationary quantum models show how the available records and clock interactions affect this reconstruction. For a prepared oscillator, exact optimal error bounds identify when adding a record combining position and momentum improves the duration estimate, while a coupled pointer provides a consistency test for sampled readings. Separatrix passage, Sagnac returns, and quantum backflow illustrate the construction across these settings. The calibrated predictions agree with conventional descriptions without an independently supplied time interval.