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Authors: Alex Vilenkin
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The universe emerging from quantum tunneling does not have to be perfectly spherical. It can have a variety of different shapes, and it can also be filled with different kinds of false vacuum. As usual in quantum theory, we cannot tell which of these possibilities has been realized, but can only calculate their probabilities. Could it be, then, that there is a multitude of other universes that started differently from our own?
This issue is closely related to the thorny question of how quantum probabilities are to be interpreted. As we discussed in Chapter 11, there are two major alternatives. According to the Copenhagen interpretation, quantum mechanics assigns probabilities to all possible outcomes of an experiment, but only one of these outcomes actually happens. The Everett interpretation, on the other hand, asserts that all possible outcomes are realized in disconnected, “parallel” universes.
If the Copenhagen interpretation is adopted, then the creation was a one-shot event, with a single universe popping out of nothing. This, however, leads to a problem. The most likely thing to pop out of nothing is a tiny Planck-sized universe, which would not tunnel, but would instantly recollapse and disappear. Tunneling to a larger size has a small probability and therefore requires a large number of trials. It appears to be consistent only with the Everett interpretation.
In the Everett picture, there is an ensemble of universes with all possible initial states. Most of them are Planck-sized “flicker” universes, which blink in and out of existence. But in addition, there are some universes that tunnel to a larger size and inflate. The crucial difference from the Copenhagen interpretation is that all these universes are not merely possible, but real.
9
Since observers cannot evolve in the “flickers,” only large universes will be observed.
9
Since observers cannot evolve in the “flickers,” only large universes will be observed.
All of the universes in the ensemble are completely disconnected from one another. Each has its own space and its own time. Calculations show that the most probable—and thus the most numerous—of the tunneling universes are the ones nucleating with the smallest initial radius and the highest energy density of the false vacuum. Our best guess, then, is that our own universe also nucleated in this way.
In scalar field models of inflation, the highest vacuum energy density is reached at the top of the energy hill, and thus most of the universes will nucleate having the scalar field in that vicinity. This is the most favorable starting point for inflation. Remember, I promised to explain how the field got to the top of the hill. In the tunneling-from-nothing scenario, this is where it was when the universe came into being.
The nucleation of the universe is basically a quantum fluctuation, and its probability decreases rapidly with the volume it encompasses. Universes nucleating with a larger initial radius have smaller probability, and in the limit of infinite radius, the probability vanishes. An infinite, open universe has a strictly zero probability of nucleating, and thus all universes in the ensemble must be closed.
In July 1983 several hundred physicists from all over the world gathered in the Italian city of Padova for the tenth International Conference on General Relativity and Gravitation. The conference site was the thirteenth-century Palazzo della Ragione, the old courthouse, located at the very heart of Padova. The ground floor of the Palazzo is taken by the famous food market, which spills outdoors into the adjacent piazza. The upper floor is occupied by a spacious hall, frescoed with signs of the zodiac around its perimeter. That’s where the lectures were held. The highlight of the program was the talk by Stephen Hawking, entitled “The Quantum State of the Universe.” The entrance to the lecture hall was through a long stairway, and it was a nontrivial task to carry Hawking in his wheelchair up the stairs. I was glad I arrived early, since by the time Hawking appeared on stage, the hall was completely packed.
In his talk Hawking unveiled a new vision for the quantum origin of the universe, based on the work he had done with James Hartle of the University of California at Santa Barbara.
10
Instead of focusing on the early moments of creation, he asked a more general question: How can we calculate the quantum probability for the universe to be in a certain state? The universe could follow a large number of possible histories before it got to that state, and the rules of quantum mechanics can be used to determine how much
each particular history contributes to the probability.
bk
The final result for the probability depends on what class of histories is included in the calculation. The proposal of Hartle and Hawking was to include only histories represented by spacetimes that have no boundaries in the past.
10
Instead of focusing on the early moments of creation, he asked a more general question: How can we calculate the quantum probability for the universe to be in a certain state? The universe could follow a large number of possible histories before it got to that state, and the rules of quantum mechanics can be used to determine how much
each particular history contributes to the probability.
bk
The final result for the probability depends on what class of histories is included in the calculation. The proposal of Hartle and Hawking was to include only histories represented by spacetimes that have no boundaries in the past.
A space without boundaries is easy to understand: it simply means a closed universe. But Hartle and Hawking required that the spacetime should also have no boundary, or edge, in the past direction of time. It should be closed in all four dimensions, except for the boundary corresponding to the present moment (see
Figure 17.4
).
Figure 17.4
).
Figure 17.4
.
A two-dimensional spacetime without a past boundary.
.
A two-dimensional spacetime without a past boundary.
A boundary in space would mean that there is something beyond the universe, so that things can come in and go out through the boundary. A boundary in time would correspond to the beginning of the universe, where some initial conditions would have to be specified. The proposal of Hartle and Hawking asserts that the universe has no such boundary; it is
“completely self-contained and not affected by anything outside itself.” That sounded like a very simple and attractive idea. The only problem was that spacetimes closed to the past, of the kind shown in
Figure 17.4
, do not exist. There should be three spacelike and one timelike direction at every spacetime point, but a closed spacetime necessarily has some pathological points with more than one timelike direction (see
Figure 17.5
).
“completely self-contained and not affected by anything outside itself.” That sounded like a very simple and attractive idea. The only problem was that spacetimes closed to the past, of the kind shown in
Figure 17.4
, do not exist. There should be three spacelike and one timelike direction at every spacetime point, but a closed spacetime necessarily has some pathological points with more than one timelike direction (see
Figure 17.5
).
To resolve this difficulty, Hartle and Hawking suggested that we switch from real time to Euclidean time. As we discussed earlier in this chapter, Euclidean time is no different from another spatial direction; so the spacetime simply becomes a four-dimensional space, and there is no problem making it closed. Thus, the proposal was that we calculate probabilities by adding up contributions from all Euclidean spacetimes without boundaries. Hawking emphasized that this was only a proposal. He had no proof that it was correct, and the only way to find out was to check whether or not it makes reasonable predictions.
Figure 17.5
.
Same as in
Figure 17.4
, with timelike and spacelike directions indicated by solid and dashed lines, respectively. The point
P
is pathological, since all directions are timelike at that point.
.
Same as in
Figure 17.4
, with timelike and spacelike directions indicated by solid and dashed lines, respectively. The point
P
is pathological, since all directions are timelike at that point.
The Hartle-Hawking proposal has a certain mathematical beauty about it, but I thought that after switching to Euclidean time it lost much of its intuitive
appeal. Instead of summing over possible histories of the universe, it instructs us to sum over histories that are certainly impossible, because we do not live in Euclidean time. So, after the scaffolding of the original motivation is dropped, we are left with a rather formal prescription for calculating probabilities.
11
appeal. Instead of summing over possible histories of the universe, it instructs us to sum over histories that are certainly impossible, because we do not live in Euclidean time. So, after the scaffolding of the original motivation is dropped, we are left with a rather formal prescription for calculating probabilities.
11
In the conclusion of his talk, Hawking discussed the implications of the new proposal for the inflationary universe. He argued that the main contribution to the sum over histories is given by the Euclidean spacetime having the form of a hemisphere—the same as appeared in my tunneling calculation—and that the following evolution is represented by the inflationary expansion in ordinary time. (Switching back to ordinary time from the Euclidean formalism was a tricky procedure, which I will not try to describe here.) The result was the same spacetime history as in my
Figure 17.3
, but obtained from a very different starting point.
Figure 17.3
, but obtained from a very different starting point.
I expected that Hawking would mention my work on quantum tunneling from nothing and was disappointed when he didn’t. But I was sure that with Hawking now in the field, the whole subject of quantum cosmology, and my work in particular, would receive a lot more attention than before.
An important difference between the “tunneling from nothing” and “no boundary” proposals is that they give very different, and in some sense opposite, predictions for the probabilities. The tunneling proposal favors nucleation with the highest vacuum energy and the smallest size of the universe. The no-boundary prescription, on the contrary, suggests that the most likely starting point is a universe of the smallest vacuum energy and largest possible size. The most probable thing to pop out of nothing is then an infinite, empty, flat space. I find this very hard to believe!
The conflict between the two approaches became apparent only after some initial confusion. The result in my 1982 paper was that larger universes had a
higher
probability of nucleating, so it looked as though the two proposals were in agreement. I kept returning to my calculation, because the result was so counter-intuitive. In 1984 I found an error, which reversed the probability trend. At the time, Hawking was visiting Harvard, and I rushed
to talk to him and share my new insight. But Stephen was unconvinced and thought that I had gotten it right the first time around.
bl
higher
probability of nucleating, so it looked as though the two proposals were in agreement. I kept returning to my calculation, because the result was so counter-intuitive. In 1984 I found an error, which reversed the probability trend. At the time, Hawking was visiting Harvard, and I rushed
to talk to him and share my new insight. But Stephen was unconvinced and thought that I had gotten it right the first time around.
bl
Hawking is a legend among physicists and far beyond. I admire both his science and his spirit and treasure the opportunities to talk to him. Since it takes him so much effort to communicate, people are often reluctant to approach him. It took me a while to realize that Stephen actually enjoys conversation and does not even mind some joking around. We have very different views on eternal inflation and on quantum cosmology, but this only makes the discussion more interesting.
In 1988 I took the battle to Hawking’s turf and gave a talk to his group at Cambridge University, highlighting the advantages of my approach. After the talk, Hawking rolled up to me in his wheelchair. I expected some critical remarks, but instead he invited me over for dinner. After a meal of duck with potatoes and a plum pie, cooked by Stephen’s mother, we talked about the use of wormholes—shortcut tunnels through spacetime—for intergalactic travel. This is the physicist’s notion of a light after-dinner conversation. As for the no-boundary proposal, Stephen did not change his mind.
The dispute between proponents of the two approaches is still going on. There was even an “official” debate at the Cosmo-98 conference in Monterey, California, with Hawking defending the no-boundary proposal and Andrei Linde and me arguing for the tunneling one.
bm
It was not actually much of a debate. It takes a long time for Hawking to compose sentences with his speech synthesizer, so we did not progress much beyond the prepared statements.
bm
It was not actually much of a debate. It takes a long time for Hawking to compose sentences with his speech synthesizer, so we did not progress much beyond the prepared statements.
We could resolve the dispute if we devised some observational test to distinguish between the two proposals. This, however, appears rather unlikely, and the reason is eternal inflation. Quantum cosmology makes predictions about the initial state of the universe, but in the course of eternal inflation any effect of the initial conditions is completely erased. Take, for example, the string theory landscape that we discussed before. We can start in one inflating vacuum or another, but inevitably bubbles of other vacua
will be formed, and the entire landscape will be explored. The properties of the resulting multiverse will be independent of how inflation started.
12
will be formed, and the entire landscape will be explored. The properties of the resulting multiverse will be independent of how inflation started.
12
Figure 17.6
.
Discussing quantum cosmology with Hawking. From left to right: the author, Bill Unruh of the University of British Columbia, and Stephen Hawking (drinking tea with the help of his nurse). (Courtesy of Anna Zytkow)
.
Discussing quantum cosmology with Hawking. From left to right: the author, Bill Unruh of the University of British Columbia, and Stephen Hawking (drinking tea with the help of his nurse). (Courtesy of Anna Zytkow)
Thus, quantum cosmology is not about to become an observational science. The dispute between different approaches will probably be resolved by theoretical considerations, not by observational data. For example, the quantum state of the universe may be determined by some new, yet to be discovered, principle of string theory. It may, of course, differ from either of the present proposals. This issue is not likely to be settled any time soon.
BOOK: Many Worlds in One: The Search for Other Universes
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