Minisuperspace Quantum Cosmology Research Articles

Schrödinger from Wheeler–DeWitt: The issues of time and inner product in canonical quantum gravity

The wave-function in quantum gravity is supposed to obey the Wheeler–DeWitt (WDW) equation, however there is neither a satisfactory probability interpretation nor a successful solution to the problem of time in the WDW framework. To gain some insight on these issues we compare quantization of ordinary systems, first in the usual way having the Schrödinger equation and second by promoting them as parametrized theories by introducing embedding coordinate fields, which yields first class constraints and the WDW equation. We observe that the time evolution in the WDW framework can be described with respect to the embedding coordinates, where the WDW equation becomes Schrödinger like, i.e. it involves first order timelike functional derivatives. Moreover, the equivalence with the ordinary quantization procedure determines a suitable Hilbert space with a viable probability interpretation. We then apply the same construction to general relativity by adding embedding fields without any prior coordinate choice. The reparametrized general relativity has two different types of diffeomorphism invariance, which arises from world-volume and target-space reparametrizations. As in the case of ordinary systems, the time evolution can be described with respect to the embedding fields and the WDW equation becomes Schrödinger like; the construction is almost identical to an ordinary parametrized field theory in terms of time evolution and Hilbert space structure. However, this time, the constraint algebra enforces the wave-function to be in a subspace of states annihilated by an operator that can be identified as the Hamiltonian. The implications of these results for the canonical quantization program, and in particular for the minisuperspace quantum cosmology, are discussed.

No smooth spacetime in Lorentzian quantum cosmology and trans-Planckian physics

In minisuperspace quantum cosmology, the Lorentzian path integral formulations of the no-boundary and tunneling proposals have recently been analyzed. But it has been pointed out that the wave function of linearized perturbations around a homogeneous and isotropic background is of an inverse Gaussian form and thus that their correlation functions are divergent. In this paper, we revisit this issue and consider the problem of perturbations in Lorentzian quantum cosmology by modifying the dispersion relation based on trans-Planckian physics. We consider two modified dispersion relations, the generalized Corley-Jacobson dispersion relation with higher momentum terms and the Unruh dispersion relation with a trans-Planckian mode cut-off, as examples. We show that the inverse Gaussian problem of perturbations in Lorentzian quantum cosmology is hard to overcome with the trans-Planckian physics modifying the dispersion relation at short distances.

Minisuperspace Quantum Cosmology in Metric and Affine Theories of Gravity

Minisuperspace Quantum Cosmology is an approach by which it is possible to infer initial conditions for dynamical systems which can suitably represent observable and non-observable universes. Here we discuss theories of gravity which, from various points of view, extend Einstein’s General Relativity. Specifically, the Hamiltonian formalism for f(R), f(T), and f(G) gravity, with R, T, and G being the curvature, torsion and Gauss–Bonnet scalars, respectively, is developed starting from the Arnowitt–Deser–Misner approach. The Minisuperspace Quantum Cosmology is derived for all these models and cosmological solutions are obtained thanks to the existence of Noether symmetries. The Hartle criterion allows the interpretation of solutions in view of observable universes.

Noether symmetries in extended gravity quantum cosmology

We summarize the use of Noether symmetries in Minisuperspace Quantum Cosmology. In particular, we consider minisuperspace models, showing that the existence of conserved quantities gives selection rules that allow to recover classical behaviors in cosmic evolution according to the so-called Hartle criterion. Such a criterion selects correlated regions in the configuration space of dynamical variables whose meaning is related to the emergence of classical observable universes. Some minisuperspace models are worked out starting from Extended Gravity, in particular coming from scalar-tensor, f(R) and f(T) theories. Exact cosmological solutions are derived.

Strings in compact cosmological spaces

We confront the problem of giving a fundamental definition to perturbative string theory in spacetimes with totally compact space (taken to be a torus for simplicity, though the nature of the problem is very general) and non-compact time. Due to backreaction induced by the presence of even a single string quantum, the usual formulation of perturbative string theory in a fixed classical background is infrared-divergent at all subleading orders in the string coupling, and needs to be amended. The problem can be seen as a closed string analogue of D0-brane recoil under an impact by closed strings (a situation displaying extremely similar infrared divergences). Inspired by the collective coordinate treatment of the D0-brane recoil, whereby the translational modes of the D0-brane are introduced as explicit dynamical variables in the path integral, we construct a similar formalism for the case of string-induced gravitational backreaction, in which the spatially uniform modes of the background fields on the compact space are quantized explicitly. The formalism can equally well be seen as an ultraviolet completion of a minisuperspace quantum cosmology with string degrees of freedom. We consider the amplitudes for the universe to have two cross-sections with specified spatial properties and string contents, and show (at the first non-trivial order) that they are finite within our formalism.

Tomographic representation of minisuperspace quantum cosmology and noether symmetries

The probability representation, in which cosmological quantum states are described by a standard positive probability distribution, is constructed for minisuperspace models selected by Noether symmetries. In such a case, the tomographic probability distribution provides the classical evolution for the models and can be considered an approach to select “observable” universes. Some specific examples, derived from Extended Theories of Gravity, are worked out. We discuss also how to connect tomograms, symmetries and cosmological parameters.

Quantum cosmology and tachyons

AbstractWe discuss the relevance of the classical and quantum rolling tachyons inflation in the frame of the standard, p‐adic and adelic minisuperspace quantum cosmology. The field theory of tachyon matter proposed by Sen in a zero‐dimensional version suggested by Kar leads to a model of a particle moving in a constant external field with quadratic damping. We calculate the exact quantum propagator of the model, as well as, the vacuum states and conditions necessary to construct an adelic generalization.

QUANTUM COSMOLOGY FROM INDUCED SCALAR–TENSOR THEORY OF GRAVITY AND THE ROLE OF CONSERVATION LAWS

In this paper, it is shown that the so-called Hartle criterion to select correlated regions in configuration space of dynamical variables, is directly connected to the presence of conservation laws, that is the existence of conservation laws in minisuperspace quantum cosmology constitutes a selection rule to recover observable universes. In this context, we develop the quantum cosmology for a scalar–tensor theory of gravity, induced from the reduction procedure of a 5D-unification theory. A class of exact solutions for Wheeler–DeWitt equation is derived and their connection to classical observable universes is discussed.

P-ADIC AND ADELIC MINISUPERSPACE QUANTUM COSMOLOGY

We consider the formulation and some elaboration of p-adic and adelic quantum cosmology. The adelic generalization of the Hartle–Hawking proposal does not work in models with matter fields. p-adic and adelic minisuperspace quantum cosmology is well defined as an ordinary application of p-adic and adelic quantum mechanics. It is illustrated by a few cosmological models in one, two and three minisuperspace dimensions. As a result of p-adic quantum effects and the adelic approach, these models exhibit some discreteness of the minisuperspace and cosmological constant. In particular, discreteness of the de Sitter space and its cosmological constant is emphasized.

Selection Rules in Minisuperspace Quantum Cosmology

The existence of a Noether symmetry for a given minisuperspace cosmological model is a sort of selection rule to recover classical behaviours in cosmic evolution since oscillatory regimes for the wave function of the universe come out. The so called Hartle criterion to select correlated regions in the configuration space of dynamical variables can be directly connected to the presence of a Noether symmetry and we show that such a statement works for generic extended theories of gravity in the framework of minisuperspace approximation. Examples and exact cosmological solutions are given for nonminimally coupled and higher--order theories.

Axionic wormholes: More on their classical and quantum aspects

As a system which is known to admit classical wormhole instanton solutions, Einstein-Kalb-Ramond (KR) antisymmetric tensor theory is revisited. As an untouched issue, the existence of fermionic zero-modes in the background of classical axionic wormhole space-time and its physical implications is addressed. In particular, in the context of a minisuperspace quantum cosmology model based on this Einstein-KR antisymmetric tensor theory, “quantum wormhole”, defined as a state represented by a solution to the Wheeler-DeWitt equation satisfying an appropriate wormhole boundary condition, is discussed. An exact, analytic wave function for quantum wormholes is actually found. Finally, it is proposed that the minisuperspace model based on this theory in the presence of the cosmological constant may serve as an interesting simple system displaying an overall picture of entire universe's history from the deep quantum domain all the way to the classical domain.

Comparing formulations of generalized quantum mechanics for reparametrization-invariant systems

A class of decoherence schemes is described for implementing the principles of generalized quantum theory in reparametrization-invariant `hyperbolic' models such as minisuperspace quantum cosmology. The connection with sum-over-histories constructions is exhibited and the physical equivalence or inequivalence of different such schemes is analyzed. The discussion focuses on comparing constructions based on the Klein-Gordon product with those based on the induced (a.k.a. Rieffel, Refined Algebraic, Group Averaging, or Spectral Analysis) inner product. It is shown that the Klein-Gordon and induced products can be simply related for the models of interest. This fact is then used to establish isomorphisms between certain decoherence schemes based on these products.

Mode decomposition and unitarity in quantum cosmology

Contrary to common belief there are perspectives for generalizing the notion of positive and negative frequency in minisuperspace quantum cosmology, even when the wave equation does not admit symmetries. We outline a strategy for doing so when the potential is positive. An underlying unitarity structure also emerges. Starting from the framework of the Klein-Gordon type quantization, I am led to a result that relies on global features of the model, and that is possibly related to structures encountered in the refined algebraic quantization scheme.

Observer dependence in quantum cosmology.

The main thesis of this paper is that minisuperspace quantum cosmology reproduces the observer dependence of the vacuum found in quantum field theory in curved spacetime. We show that the vacua picked up by the Hartle-Hawking no boundary proposal in two different minisuperspaces sharing the same classical limit ($k=0$ and $k=1$ de Sitter minisuperspaces) are nonequivalent. A closer inspection suggests that the observer dependence is introduced through the choice of the Wick rotation. This hypothesis is supported by the analysis of the static de Sitter minisuperspace, where two different Euclidean sections are available, and where again two inequivalent vacua are obtained. These examples are not conclusive, but point to the possibility of a similar ambiguity even in full quantum cosmology.

Conditional probabilities and entropy in (minisuperspace) quantum cosmology

This paper exploits an analogy with a time-independent Schrodinger equation to propose a concrete interpretation of a 'wavefunction of the universe' psi which solves the Wheeler-DeWitt equation for a minisuperspace quantum cosmology, and then uses this interpretation to define information theoretic notions of entropy. Given a wavefunction psi depending, in a configuration space representation, on some set of variables q0, ., qN, one is instructed to pick one of the qA, say q0 identical to a, as a 'time' and then view psi (Q; a) as an unnormalised conditional probability amplitude for finding the values Q identical to (q1, ., qN) at 'time' a. By identifying the 'natural' measure dQ on an a=constant hypersurface in minisuperspace, rho (Q, Q'; a)dQ varies as psi *(Q; a) psi (Q'; a)dQ is interpreted as representing a density matrix in the Q-representation; and, by viewing the Q as abstract operators for fixed a, rho is interpreted as an abstract density matrix. This interpretation admits no a priori notion of unitary evolution with respect to a 'translation' da, unitarity being viewed instead as a derived concept arising only in a semiclassical limit. The object then is to use rho to define reasonable measures of 'coarse-grained entropy' in accord with physical intuition which could serve as a diagnostic for an 'arrow of time'. Thus, e.g., by interpreting the diagonal components rho ( lambda i, lambda i; a) in some Oa-representation as conditional probabilities pi for measuring the eigenvalues lambda i of some operator Oa, one is led to consider an observable dependent entropy S0(a)=- Sigma ipi log pi. And similarly, at least when the a=constant hypersurfaces are flat, one can map psi into a 'Wigner function' fw and then use that fw to define a 'Wigner function entropy' Sw.