Mental Workout
December 08, 2002

This is gonna get your mind hurting, but, as my trainer says, you'll thank me for it in the morning:

In practice, calculating probabilities in quantum cosmology using the full path integral is formidably difficult and an approximation has to be used. This is known as the semiclassical approximation because its validity lies somewhere between that of classical and quantum physics. In the semiclassical approximation one argues that most of the four dimensional geometries occuring in the path integral will give very small contributions to the path integral and hence these can be neglected. The path integral can be calculated by just considering a few geometries that give a particularly large contribution. These are known as instantons. Instantons don't exist for all choices of boundary three geometry; however those three geometries that do admit the existence of instantons are more probable than those that don't. Therefore attention is usually restricted to three geometries close to these.

Remember that the path integral is a sum over geometries with four spatial dimensions. Therefore an instanton has four spatial dimensions and a boundary that matches the three geometry whose probability we wish to compute. Typical instantons resemble (four dimensional) surfaces of spheres with the three geometry slicing the sphere in half. They can be used to calculate the quantum process of universe creation, which cannot be described using classical general relativity. They only usually exist for small three geometries, corresponding to the creation of a small universe. Note that the concept of time does not arise in this process. Universe creation is not something that takes place inside some bigger spacetime arena - the instanton describes the spontaneous appearance of a universe from literally nothing. Once the universe exists, quantum cosmology can be approximated by general relativity so time appears.

People have found different types of instantons that can provide the initial conditions for realistic universes. The first attempt to find an instanton that describes the creation of a universe within the context of the `no boundary' proposal was made by Stephen Hawking and Ian Moss. The Hawking-Moss instanton describes the creation of an eternally inflating universe with `closed' spatial three-geometries....

Posted by Jody at December 8, 2002 01:21 PM

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