The wave-function of the universe is another of cosmologist and physicist Stephen Hawking's unusual but insightful ideas. It involves starting with the idea of not just our universe but all possible universes, then calculating the relative likelihoods of each - much like calculating the wave-function of a particle.
The wave-function of a particle is essentially the function that determines its most likely location at any given time. Wave-functions are largest where that particle is observed to be, but also extend throughout the known universe in accordance with the sum-over-paths method. Everything has a wave-function - elementary particles possess wave-functions and make up all other matter in the universe, so this is a logical conclusion. Large and common objects such as rubber balls have wave-functions; for example, the wave-function of a ball sitting on a flat surface is largest where it is observed - say, a table - but also extends everywhere around us - beside your computer, on the moon, or even in another galaxy. However, the likelihood of the ball suddenly appearing in any of these locations is infinitesimal. The likelihood of such changes in location depends on Planck's constant.
Hawking and Hartle propose to calculate the wave-function of the universe using the sum-over-paths method, which begins with the assumption that the universe has all possible histories. Moreover, they would calculate this sum in imaginary time, not ordinary time. This is because imaginary time travels at right angles to ordinary time and "meets" with the three spatial dimensions to create a smooth surface similar to the surface of the earth. This eliminates the singularities (points of infinite curvature) present in ordinary time, allowing the history of the universe to be reliably calculated. Also unlike ordinary time, imaginary time has no beginning or end, so progression through it is determined entirely by physical laws.
For Hawking and Hartle's calculation, you must begin with a wave-function describing all possible universes - an infinite number. The wave-function is large near our own universe and infinitesimal near others in which life is impossible or the known laws of physics do not apply. Because of the wave-function's concentration in our own universe, it is the most likely of them all, but there is a chance - albeit vanishingly small - that an object from this universe would suddenly make a quantum leap into another one. Proving this conjecture mathematically is one of the primary goals of quantum cosmology, which applies quantum theory to the large structures of cosmology.
The Hawking/Hartle theory also postulates the existence of wormholes connecting the different universes, as in the first image below. According to them, the multitude of universes should be connected by wormholes, as in the second image below, although these wormholes are not an efficient or readily available means of transportation. You can see that some of these universes are very rich, and others are quite barren. Similarly, some are connected with many others, while others are isolated.
The wave-function of Hawking and Hartle raises two major controversies long debated among scientists. The first of these is an apparent return to the anthropic principle, which basically says that the universe is the way it is because we wouldn't be here if it was any other way. It has two basic forms - the "weak" and "strong" versions. The weak version states that the existence of intelligent life is experimental evidence to help us understand the universe's seemingly random physical constants. The strong version, much more controversial, states that these apparently random constants are not random but were instead chosen by some supreme being to make life in our universe possible. Hawking and Hartle's idea appeals to the weak anthropic principle, but seems to destroy the strong version - a supreme being is not necessary to explain the existence of a universe uniquely suited to intelligent life.
The second controversy relates to Schrödinger's cat and the many-worlds theories. The well-known Schrödinger's cat paradox states that, by the uncertainty principle, a particle is in a sum of all possible states until it is observed, a process called reducing the wave-function. Schrödinger postulated a paradox that arises by this theory - suppose a cat is placed in a box connected to a gun which is in turn connected to a Geiger counter measuring a uranium atom. If the unstable atom decays, the Geiger counter will register it, the gun will go off, and the cat will be killed. If it doesn't decay, the process will not be initiated and the cat will live. Before observing the cat, quantum theory states that it is in a superimposed condition of both dead and live states. Most physicists either assume the wave-function is always being reduced by some cosmic observer - a supreme being - or simply ignore the problem. A third way of dealing with the paradox is the many-worlds theory, suggested by Hugh Everett, which states that the universe is constantly splitting off new offshoots, so that in one universe, the cat is dead, and in another it is alive.
In the traditional many-worlds theory, contact between the worlds is mathematically impossible and therefore the idea cannot be tested. By the physics principle of Occam's razor, which states that the simpler of two competing explanations is generally correct, we should throw out the many-worlds theory because it is irrelevant to our universe and completely untestable. (Many scientists use Occam's razor against the existence of a supreme being as well.)
Hawking and Hartle's idea revives the many-worlds theory with one important twist - communication between the worlds in the form of wormholes discussed above is possible in their formulation. Thus, their idea is both testable and directly relevant to our universe.