space-time is just a figment of our imagination.When we apply Feynman’s sum over histories to the universe, the analogue ofthe history of a particle is now a complete curved space-time which representsthe history of the whole universe. For the technical reasons mentioned above,these curved space-times must be taken to be Euclidean. That is, time isimaginary and is indistinguishable from directions in space. To calculate theprobability of finding a real space-time with some certain property, one addsup the waves associated with all the histories in imaginary time that have thatproperty. One can then work out what the probable history of the universewould be in real time.
THE NO BOUNDARY CONDITION
In the classical theory of gravity, which is based on real space-time, there areonly two possible ways the universe can behave. Either it has existed for an infi-nite time, or else it had a beginning at a singularity at some finite time in thepast. In fact, the singularity theorems show it must be the second possibility. Inthe quantum theory of gravity, on the other hand, a third possibility arises.Because one is using Euclidean space-times, in which the time direction is onthe same footing as directions in space, it is possible for space-time to be finitein extent and yet to have no singularities that formed a boundary or edge.Space-time would be like the surface of the Earth, only with two more dimen-sions. The surface of the Earth is finite in extent but it doesn’t have a boundaryor edge. If you sail off into the sunset, you don’t fall off the edge or run into asingularity. I know, because I have been around the world.
If Euclidean space-times direct back to infinite imaginary time or else startedat a singularity, we would have the same problem as in the classical theory ofspecifying the initial state of the universe. God may know how the universebegan, but we cannot give any particular reason for thinking it began one wayrather than another. On the other hand, the quantum theory of gravity hasopened up a new possibility. In this, there would be no boundary tospace-time. Thus, there would be no need to specify the behavior at theboundary. There would be no singularities at which the laws of science brokedown and no edge of space-time at which one would have to appeal to God orsome new law to set the boundary conditions for space-time. One could say:”The boundary condition of the universe is that it has no boundary.” The uni-verse would be completely self-contained and not affected by anything outsideitself. It would be neither created nor destroyed. It would just be.
It was at the conference in the Vatican that I first put forward the suggestionthat maybe time and space together formed a surface that was finite in size butdid not have any boundary or edge. My paper was rather mathematical, how-ever, so its implications for the role of God in the creation of the universe werenot noticed at the time-just as well for me. At the time of the Vatican confer-ence, I did not know how to use a no boundary idea to make predictions aboutthe universe. However, I spent the following summer at the University ofCalifornia, Santa Barbara. There, a friend and colleague of mine, Jim Hartle,worked out with me what conditions the universe must satisfy if space-timehad no boundary.
I should emphasize that this idea that time and space should be finite withoutboundary is just a proposal. It cannot be deduced from some other principle.Like any other scientific theory, it may initially be put forward for aesthetic ormetaphysical reasons, but the real test is whether it makes predictions thatagree with observation. This, however, is difficult to determine in the case ofquantum gravity, for two reasons. First, we are not yet sure exactly which the-ory successfully combines general relativity and quantum mechanics, thoughwe know quite a lot about the form such a theory must have. Second, anymodel that described the whole universe in detail would be