Addressing the issue of nonunitarity in anisotropic quantum cosmology

Abstract

In the present work we show that the widely believed pathology of the non-unitarity of anisotropic quantum cosmological models cannot be a generic problem. We exhibit a non trivial example, a Bianchi-I model with an ultrarelativistic fluid, that has a well behaved time independent norm. We also show that a suitable operator ordering should produce time independent norms for the wave packets in the case of other more realistic fluids as well.In the absence of a generally accepted quantum theory of general relativity, quantum mechanical principles are applied to many individual gravitational systems. Cosmological models are certainly amongst the fields where this kind of quantization finds an application. The universe at its early stage of evolution, at an energy scale where classical general relativity loses its viability, requires a quantum description. For some excellent reviews, we refer to [1,2]. Quantum Cosmology has, however, many issues yet to be resolved. As time is a coordinate in a relativistic theory of gravity, there is a problem of the identification of a suitable time parameter against which the evolution of the universe would be described [3][4][5][6]. Moreover, the interpretation of the wave function faces a challenge in quantum cosmology. The Copenhagen interpretation fails as there is no exterior observer for the system. There are attempts in this directi on with a many world interpretation and with Bohmian trajectories [7]. Problems regarding the imposition of proper boundary conditions are there as well [1].The present work deals with another widely known problem, the alleged non-unitarity of the anisotropic quantum cosmological models.The corresponding Hamiltonian, although hermitian, is not self-adjoint. The norm of the wave packet is thus time dependent and hence there is a non-conservation of probability. It may be argued that the observed universe is isotropic but one is not really certain whether the very early universe, beyond the Planck scale, is actually isotropic or not! Furthermore, this feature definitely makes the scheme of quantization unreliable. For a very recent review, we refer to the work of Pinto-Neto and Fabris [7].There is a scheme of quantization of a cosmological model with a matter field, namely a perfect fluid. Following Schutz's formalism, where the fluid variables are given dynamical degrees of freedom [8,9], the relevant action can be written in terms of the metric tensor components representing the gravity sector and some *

Anisotropic models are unitary: A rejuvenation of standard quantum cosmology

“…However, specifcally commenting on how a particular extension effects a particular model requires solving Wheeler-deWitt equation, which can be tricky for many cases. For Bianchi-I, the extension can be done explicitly for a particular operator ordering [15,18]. In case of Binachi-V, IX, Kantowoski Sachs (KS) model, it is possible to do it explicitly only for some choices of the perfect fluid, for a particular operator ordering [16,17].…”

“…Although an example of Bianchi III is given here, and Bianchi I, V, IX and Kantowski-Sachs models were mentioned in [15,16,17], this can explicitly be worked out for all the anisotropic homogeneous models, which in fact come either under Bianchi classification or is a Kantowoski Sachs Model. In the desired co-ordinate system, upon quantization, we have…”

“…However, it was shown that for Bianchi V model, the alleged non-unitarity can possibly be "alleviated" in the sense that the probability becomes a constant at later stages of the evolution [14]. Very recently, it has been categorically shown with explicit examples that Bianchi I [15], Bianchi V and IX [16] models as well as a Kantowoski-Sachs model [17] can all have unitary evolution for a proper choice of operator ordering. Even when an explicit solution is not found, the existence of a self-adjoint extension for the Hamiltonian can be ascertained [15,16,17].…”

“…Very recently, it has been categorically shown with explicit examples that Bianchi I [15], Bianchi V and IX [16] models as well as a Kantowoski-Sachs model [17] can all have unitary evolution for a proper choice of operator ordering. Even when an explicit solution is not found, the existence of a self-adjoint extension for the Hamiltonian can be ascertained [15,16,17]. Furthermore, this is acheieved not at the cost of the anisotropy itself [18].…”

Scalar-metric quantum cosmology with Chaplygin gas and perfect fluid

“…It was found to show two distinct peaks. Similar studies has also been carried out in various other cosmological models [14][15][16].…”