A survey on quantum cosmology

Using general relativity to extrapolate back in time, the universe emerged from a single, unbelievably, small, dense, hot region . This is the hot big bang model. The events that have unfolded since appear to be adequately described by conventional cosmology. However, they fail to explain or describe the ultimate origin of the universe. The size of the universe tends to zero, and the strength of the gravitational field and the energy density of matter tend to infinity. The universe appears to have emerged from a singularity, a region of infinite curvature and energy density at which the known laws of physics break down. "Singularity theorems" by Hawking and Penrose showed that under reasonable assumptions any model of the expanding universe extrapolated backward in time will encounter an initial singularity. The theory predicting them - classical general relativity - breaks down at high curvatures to very small dimensions. In fact, the universe is reduced down to a size where one must incorporate quantum theory.

Quantum cosmology is precisely the theory where quantum mechanics is applied to the entire universe. Important processes occur as early as the so-called Planck time, 0.000...(43 zeros)...05 sec. The radius of the universe is of order of the Planck Length, 0.000...(32 zeros)...016 cm. On these scales the behaviour of the universe is governed by the laws of quantum mechanics.

At the very center of quantum cosmology, one must put forward quite definite laws of initial conditions, that is, conditions that must have existed at the very moment of creation. When adjoined with suitable laws governing the evolution of the universe, such proposals could conceivably lead to a complete explanation of all cosmological observations.

It is usually assumed that inflation improves dramatically on the hot big bang model in that it allows for the currently observed state of the universe to have arisen from a much broader, far more plausible set of initial conditions. Nevertheless, inflation does not relieve the observed state of the universe of all dependence on assumptions about initial conditions. Where do these assumptions then might come? " What happened before inflation? How did the universe actually begin?"

Like quantum mechanics, quantum cosmology attempts to describe a system fundamentally in terms of its wave function. One cannot know the gravitational field exactly at a given instant; one can only known the probability for a range of gravitational fields. Such wave function of the universe can be obtained by solving an equation called the Wheeler- DeWitt equation, which is the cosmological analogue of the Schrodinger equation. In the simplest cases, the spatial size of the universe is the analogue of position, and the rate of the universe's expansion represents the momentum.

A serious difficulty in this scheme is the lack of a complete quantum theory of gravity. Another question is the applicability of quantum mechanics to the entire universe. This is the fundamental assertion of quantum cosmology. It concerns the interpretation of quantum measurement theory. The world may be divided into two parts: microscopic systems and external macroscopic systems (such as observers). A measurement is an interaction between the observer and the microscopic system that leads to a permanent recording of the event. This scheme, known as the Copenhagen interpretation, meets with acute difficulties in a theory of the universe of which the observer is a part, there should be no fundamental division between observer and observed.

Another issue is the following. The question of classical initial conditions becomes one of quantum initial conditions: Of the many wave functions possible (solutions to the Wheeler - DeWitt equation), how is just one singled out?

In quantum cosmology one propose laws of initial or boundary conditions for the universe. Hartle and Hawking, Linde, and Vilenkin have made quite definite proposals.

Hartle and Hawking's proposal defines a particular wave function of the universe using the path integral or sum-over-histories method. The wave function of the universe may be calculated by summing over some class of histories for the universe. The technique is equivalent to solving the Wheeler - DeWitt equation. The precise solution obtained depends on how the class of histories summed over is chosen. To understand the choice made by Hartle and Hawking is to imagine the spatial extent of the universe at a particular time as a closed loop in the horizontal plane. The vertical axis represents time, then the loop changes in size as time passes. Various possible histories of the universe therefore appear as tubes swept out by the loop as it evolves in time. Hartle and Hawking proposed that one should consider only tube whose initial end shrinks to zero in a smooth, regular way. One therefore sums over geometries that have no boundary (except for the final end, which is open and corresponds to the present universe). Hence, the idea is called the no-boundary proposal.

The smooth closing off may be regarded as taking place in imaginary time and as such is distinctly nonclassical. The appearance of imaginary time is characteristic of tunnelling processes in quantum theory. Perhaps, then, the universe has tunnelled from "nothing". The evolution described by inflation and the big bang would have subsequently occurred after the tunnelling. Partly for this reason, Linde and Vilenkin independently put forward a "tunnelling" proposal.

The no - boundary and tunnelling proposals both, indicate that space-time behaves according to classical cosmology when the universe is larger. When the universe is smaller, however, the wave function indicates that classical space-time does not exist. After quantum creation, the wave function assigns probabilities to different evolutionary paths, one of which includes the inflation process. Both the no-boundary and tunnelling proposals seem to predict the conditions necessary for inflation, thereby eliminating the need for assumptions about the scalar-field matter that drove the rapid expansion. The no-boundary and tunnelling proposals also eliminate assumptions about the density perturbations.

Quantum cosmology may also become an observational subject. Recent astronomical observations have measured the small fluctuations in the temperature of an early time in the evolution of the universe. The features of these fluctuations can be worked out using quantum cosmology. Consequently, a simple quantum state like the no - boundary state may be put to a obsevational test.

Back to my homepage! When supergravity is formulated in canonical framework, one finds supersymmetry constraint. These constraints are indeed analogous to Dirac equations, and their anti-commutator gives back the 'Klein-Gordon' constraints corresponding to space-time co-ordinate transformations.

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