We investigate how the quantum dynamics of gravitational models depends on the internal clock employed in quantisation. Our previous result on the quantum Friedmann–Lemaitre model (Małkiewicz 2015 What is dynamics in quantum gravity? (arXiv:1505.04730)) demonstrates that almost all physical predictions from quantum cosmological dynamics, e.g. the scale of the quantum bounce or the number of bounces, depend on the choice of internal clock. In the present paper we show that there exists an important prediction concerning the quantum dynamics which holds in all internal clocks. Namely, we prove that any quantum state asymptotically, i.e. away from a quantum interaction regime, admits a unique classical limit described by unique classical solutions irrespective of the internal clock used for the dynamical description of the given state. We demonstrate this property explicitly for the semiclassical Bianchi Type I model, which includes the semiclassical Friedmann–Lemaitre model of Małkiewicz (2015 What is dynamics in quantum gravity? (arXiv:1505.04730)) as a special case. Our methods include the theory of pseudo-canonical transformations and the phase space portraits based on coherent states. As a by-product of the pursuit of our main goal, we elaborate the semiclassical description of anisotropic singularity resolution.

We discuss the question of time in a Bianchi I quantum cosmology in the framework of singularity avoidance. We show that time parameters fall into two distinct classes, that are such that the time development of the wave function either always leads to the appearance of a singularity (fast-gauge time) or that always prevents it from occurring (slow-gauge time). Furthermore, we find that, in the latter case, there exists an asymptotic regime, independent of the clock choice. This may point to a possible solution of the clock issue in quantum cosmology if there exists a suitable class of clocks all yielding identical relevant physical consequences.

We apply the reduced phase space quantization to the Kasner universe. We construct the kinematical phase space, find solutions to the Hamilton equations of motion, identify Dirac observables and arrive at physical solutions in terms of Dirac observables and an internal clock. We obtain the physical Hilbert space, which is the carrier space of the self-adjoint representation of the Dirac observables. Then, we discuss the problem of time. We demonstrate that the inclusion of evolution in a gravitational system, at classical level as well as at quantum level, leads respectively to canonically and unitarily inequivalent theories. The example of Hubble operator in two different clock variables and with two distinct spectra is given.

We study the spectral properties of the anisotropic part of the Hamiltonian entering the quantum dynamics of the Mixmaster universe. We derive the explicit asymptotic expressions for the energy spectrum in the limit of large and small volumes of the universe. Then, we study the threshold condition between both regimes. Finally, we prove that the spectrum is purely discrete for any volume of the universe. Our results validate and improve the known approximations to the anisotropy potential. They should be useful for any approach to the quantization of the Mixmaster universe.

The Mixmaster solution to Einstein field equations was examined by C. Misner in an effort to better understand the dynamics of the early universe. We highlight the importance of the quantum version of this model for the early universe. This quantum version and its semi-classical portraits are yielded through affine and standard coherent state quantizations and more generally affine and Weyl–Heisenberg covariant integral quantizations. The adiabatic and vibronic approximations widely used in molecular physics can be employed to qualitatively study the dynamics of the model on both quantum and semi-classical levels. Moreover, the semi-classical approach with the exact anisotropy potential can be effective in the numerical integration of some solutions. Some promising physical features such as the singularity resolution, smooth bouncing, the excitation of anisotropic oscillations and a substantial amount of post-bounce inflation as the backreaction to the latter are pointed out. Finally, a realistic cosmological scenario based on the quantum mixmaster model, which includes the formation and evolution of local structures is outlined.

The basic tenet of the present work is the assumption of the lack of external and fixed time in the Universe. This assumption is best embodied by general relativity, which replaces the fixed space-time structure with the gravitational field, which is subject to dynamics. The lack of time does not imply the lack of evolution but rather brings to the forefront the role of internal clocks which are some largely arbitrary internal degrees of freedom with respect to which the evolution of timeless systems can be described. We take this idea seriously and try to understand what it implies for quantum mechanics when the fixed external time is replaced by an arbitrary internal clock. We put the issue in a solid, mathematically rigorous framework. We find that the dynamical interpretation of a quantum state of a timeless system depends on the employed internal clock. In particular, we find that the continuous spectra of well-known dynamical observables like the position of a free particle on the real line may turn discrete if measured in unusual clocks. We discuss the meaning of our result for attempts at quantization of global gravitational degrees of freedom.