Ghanashyam Date

A discrete time presentation of quantum dynamics

Inspired by the discrete evolution implied by the recent work on loop quantum cosmology, we obtain a discrete time description of usual quantum mechanics viewing it as a constrained system. This description, obtained without any approximation or explicit discretization, mimics features of the discrete time evolution of loop quantum cosmology. We discuss the continuum limit, physical inner product and matrix elements of physical observables to bring out various issues regarding the viability of a discrete evolution. We also point out how a continuous time could emerge without appealing to any continuum limit.

Homogeneous loop quantum cosmology: the role of the spin connection

Homogeneous cosmological models with non-vanishing intrinsic curvature require a special treatment when they are quantized with loop quantum cosmological methods. Guidance from the full theory which is lost in this context can be replaced by two criteria for an acceptable quantization, admissibility of a continuum approximation and local stability. A quantization of the corresponding Hamiltonian constraints is presented and shown to lead to a locally stable, non-singular evolution compatible with almost classical behaviour at large volume. As an application, the Bianchi IX model and its modified behaviour close to its classical singularity is explored.

Genericness of a big bounce in isotropic loop quantum cosmology.

The absence of isotropic singularity in loop quantum cosmology can be understood in an effective classical description as the Universe exhibiting a big bounce. We show that with a scalar matter field, the big bounce is generic in the sense that it is independent of quantization ambiguities and the details of scalar field dynamics. The volume of the Universe at the bounce point is parametrized by a single parameter. It provides a minimum length scale which serves as a cutoff for computations of density perturbations thereby influencing their amplitudes.

Quantum Suppression of the Generic Chaotic Behavior Close to Cosmological Singularities

In classical general relativity, the generic approach to the initial singularity is very complicated as exemplified by the chaos of the Bianchi IX model which displays the generic local evolution close to a singularity. Quantum gravity effects can potentially change the behavior and lead to a simpler initial state. This is verified here in the context of loop quantum gravity, using methods of loop quantum cosmology: The chaotic behavior stops once quantum effects become important. This is consistent with the discrete structure of space predicted by loop quantum gravity.

The Bianchi IX model in loop quantum cosmology

The Bianchi IX model has been used often to investigate the structure close to singularities of general relativity. Its classical chaos is expected to have, via the BKL scenario, implications even for the approach to general inhomogeneous singularities. Thus, it is a popular model to test consequences of modifications to general relativity suggested by quantum theories of gravity. This paper presents a detailed proof that modifications coming from loop quantum gravity lead to a non-chaotic effective behaviour. The way this is realized, independently of quantization ambiguities, suggests a new look at initial and final singularities.

Genericness of inflation in isotropic loop quantum cosmology

Nonperturbative corrections from loop quantum cosmology (LQC) to the scalar matter sector are already known to imply inflation. We prove that the LQC modified scalar field generates exponential inflation in the small scale factor regime, for all positive definite potentials, independent of initial conditions and independent of ambiguity parameters. For positive semidefinite potentials it is always possible to choose, without fine-tuning, a value of one of the ambiguity parameters such that exponential inflation results, provided zeros of the potential are approached at most as a power law in the scale factor. In conjunction with the generic occurrence of bounce at small volumes, particle horizon is absent, thus eliminating the horizon problem of the standard big bang model.

Consistency conditions for fundamentally discrete theories

The dynamics of physical theories is usually described by differential equations. Difference equations then appear mainly as an approximation which can be used for a numerical analysis. As such, they have to fulfil certain conditions to ensure that the numerical solutions can reliably be used as approximations to solutions of the differential equation. There are, however, also systems where a difference equation is deemed to be fundamental, mainly in the context of quantum gravity. Since difference equations in general are harder to solve analytically than differential equations, it can be helpful to introduce an approximating differential equation as a continuum approximation. In this paper implications of this change in viewpoint are analysed to derive the conditions that the difference equation should satisfy. The difference equation in such a situation cannot be chosen freely but must be derived from a fundamental theory. Thus, the conditions for a discrete formulation can be translated into conditions for acceptable quantizations. In the main example, loop quantum cosmology, we show that the conditions are restrictive and serve as a selection criterion among possible quantization choices.

Topological Interpretation of Barbero-Immirzi Parameter

We set up a canonical Hamiltonian formulation for a theory of gravity based on a Lagrangian density made up of the Hilbert-Palatini term and, instead of the Holst term, the Nieh-Yan topological density. The resulting set of constraints in the time gauge are shown to lead to a theory in terms of a real SU(2) connection which is exactly the same as that of Barbero and Immirzi with the coefficient of the Nieh-Yan term identified as the inverse of the Barbero-Immirzi parameter. This provides a topological interpretation for this parameter. Matter coupling can then be introduced in the usual manner, without changing the universal topological Nieh-Yan term.

A minimal covariant action for the free open spinning string field theory

Abstract The BRST transformations of the string fields of the free open spinning string is analyzed. After expanding the string fields into ghost zero modes the BRST transformations are diagonalized and it is found that in the Ramond sector the physical fields are contained in only two string fields. A minimal covariant classical action for both sectors is constructed. The component expansion of the string fields yields the correct set of physical and Stueckelberg fields which are needed to construct a covariant local action for the corresponding massless and massive states. This is explicitly verified for the first three mass-levels.

Effective actions from loop quantum cosmology: correspondence with higher curvature gravity

Quantum corrections of certain types and relevant in certain regimes can be summarized in terms of an effective action calculable, in principle, from the underlying theory. The demands of symmetries, local form of terms and dimensional considerations limit the form of the effective action to a great extent leaving only the numerical coefficients to distinguish different underlying theories. The effective action can be restricted to particular symmetry sectors to obtain the corresponding, reduced effective action. Alternatively, one can also quantize a classically (symmetry) reduced theory and obtain the corresponding effective action. These two effective actions can be compared. As an example, we compare the effective action(s) known in isotropic loop quantum cosmology with the Lovelock actions, as well as with more general actions, specialized to homogeneous isotropic spacetimes and find that the -scheme is singled out.