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Authors: Alex Vilenkin
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Not only is the number of possible states of an O-region finite, the number of its possible histories is finite as well.
A
history
is described by a sequence of states at successive moments of
time. The notions of which histories are possible are very different in quantum and classical physics. In the quantum world, the future is not uniquely determined by the past. The same initial state can lead to a multitude of different outcomes, and we can only calculate their probabilities. As a result, the range of possibility is greatly enlarged. But once again, the quantum uncertainty does not allow us to distinguish between histories that are too close to one another.
history
is described by a sequence of states at successive moments of
time. The notions of which histories are possible are very different in quantum and classical physics. In the quantum world, the future is not uniquely determined by the past. The same initial state can lead to a multitude of different outcomes, and we can only calculate their probabilities. As a result, the range of possibility is greatly enlarged. But once again, the quantum uncertainty does not allow us to distinguish between histories that are too close to one another.
A quantum particle does not generally have a well-defined history. This is not surprising, since, as we know, it does not have a definite position. But the uncertainty does not amount to simply not knowing which path the particle followed between its emission and its subsequent detection. The situation is much weirder: the particle appears to follow many different paths at once, and all of them contribute to the final outcome.
Figure 11.4
.
The double-slit experiment.
.
The double-slit experiment.
This schizophrenic behavior is best illustrated in the famous double-slit experiment (
Figure 11.4
). The setup consists of a light source and a photographic plate, which is blocked by an opaque screen with two narrow parallel slits in it. The light that gets through the slits creates an image on the plate. The experiment was first performed in the early 1800s by the English physicist Thomas Young. He found that the image displays a pattern of alternating
bright and dark fringes. All points on the photographic plate receive light from both slits. But in some places the light waves arrive “in phase” (the crests and troughs of the two waves coincide) and reinforce one another, while in other places they are out of phase (the crest of one wave coincides with the trough of the other) and mutually cancel. Thus the pattern of fringes is explained by the wavelike nature of light.
Figure 11.4
). The setup consists of a light source and a photographic plate, which is blocked by an opaque screen with two narrow parallel slits in it. The light that gets through the slits creates an image on the plate. The experiment was first performed in the early 1800s by the English physicist Thomas Young. He found that the image displays a pattern of alternating
bright and dark fringes. All points on the photographic plate receive light from both slits. But in some places the light waves arrive “in phase” (the crests and troughs of the two waves coincide) and reinforce one another, while in other places they are out of phase (the crest of one wave coincides with the trough of the other) and mutually cancel. Thus the pattern of fringes is explained by the wavelike nature of light.
The weird part begins when we reduce the intensity of the source to the point that it emits individual photons one by one. Each photon makes a small spot on the photographic plate. Initially the spots seem to appear at random, but remarkably, after a while, a pattern begins to build up that exactly coincides with the pattern of fringes we had before. The photons arrive at the screen separately, so photons that passed through one of the slits have no way of interacting with photons that passed through the other. How, then, do they manage to “reinforce” or “cancel” one another?
To probe this strange behavior a bit further, we can try to see what happens if we force photons to go through one slit or the other. Suppose we run the experiment first with only one slit open and then with the other open for equal amounts of time and without changing the photographic plate in between. Since photons pass through the apparatus individually, this should make no difference, and we should get the same pattern. Right? Wrong. In this modified version of the experiment, no fringes are observed and the photograph shows only the outlines of the two slits.
It follows that the picture of a photon that goes through one of the slits and does not care whether or not the other one is open cannot be right. When both slits are open, the photon somehow “feels” the two possible histories that it may follow. They jointly determine the probability for the photon to hit a particular spot on the plate. This phenomenon is called
quantum interference
between the histories.
quantum interference
between the histories.
Quantum interference is rarely as apparent as in the double-slit experiment, but it affects the behavior of every particle in the universe. As they move from one place to another, particles “sniff” many different routes, so instead of a well-defined past, we have a tangled web of interfering histories.
How, then, can we be sure that some event has actually happened? How can we make any sense of the concept of history? The answer, once again, lies in the coarse-grained description.
As before, we divide space into little cells and define coarse-grained
states of the system (O-region in our case) by specifying the cell “addresses” for all particles. A coarse-grained history is given by a sequence of such states at regular time intervals, say, every 2 seconds. Now, the key point is that the effect of interference is usually strong only for histories that are very close to one another. If we increase the cell sizes and time intervals, different coarse-grained histories become more and more distinct from one another, and at some point their interference becomes completely negligible. We can then meaningfully talk about alternative histories of the system.
states of the system (O-region in our case) by specifying the cell “addresses” for all particles. A coarse-grained history is given by a sequence of such states at regular time intervals, say, every 2 seconds. Now, the key point is that the effect of interference is usually strong only for histories that are very close to one another. If we increase the cell sizes and time intervals, different coarse-grained histories become more and more distinct from one another, and at some point their interference becomes completely negligible. We can then meaningfully talk about alternative histories of the system.
The formulation of quantum mechanics in terms of alternative coarse-grained histories was developed relatively recently, in the 1990s, by Robert Griffiths, Roland Omnes, James Hartle, and Murray Gell-Mann. They found, in particular, that the minimum size of cells that is still consistent with a definite history is typically microscopic and that the minimum time interval is a tiny fraction of a second. Not surprisingly, history is a well-defined concept in the macroscopic world of human experience.
A coarse-grained history proceeds in finite time steps, and a history of any finite duration must consist of a finite number of moments. At each moment, the system can only be in a finite number of states, and it follows that the number of distinct histories of the system must also be finite.
Jaume and I did a quick, back-of-the-envelope estimate of the number of possible histories that can occur in an O-region from the big bang till the present. As one might expect, we got yet another “googolplexic”
ac
number: 10 to the power 10
150
. The actual numbers of quantum states and of histories in an O-region are not particularly important, but the finiteness of these numbers has profound implications, as we shall now discuss.
ac
number: 10 to the power 10
150
. The actual numbers of quantum states and of histories in an O-region are not particularly important, but the finiteness of these numbers has profound implications, as we shall now discuss.
Let us now take stock of the situation. It follows from the theory of inflation that island universes are internally infinite, so each of them contains an infinity of O-regions. And it follows from quantum mechanics that there is only a finite number of histories that can unfold in any O-region. Putting these two statements together, we arrive at the inevitable conclusion that every single history should be repeated an infinite number of times. According
to quantum mechanics, anything that is not strictly forbidden by conservation laws has a nonzero probability of happening. And any history that has a nonzero probability will happen—or rather has happened—in an infinite number of O-regions!
to quantum mechanics, anything that is not strictly forbidden by conservation laws has a nonzero probability of happening. And any history that has a nonzero probability will happen—or rather has happened—in an infinite number of O-regions!
Included among these infinitely replayed scripts are some very bizarre histories. For example, a planet similar to our Earth can suddenly collapse to form a black hole. Or it can emit a huge pulse of radiation and switch to another orbit, much closer to the central star. Such occurrences are extremely unlikely, but this only means that one will have to survey a very large number of O-regions before encountering one of them.
A striking consequence of the new picture of the world is that there should be an infinity of regions with histories absolutely identical to ours. Yes, dear reader, scores of your duplicates are now holding copies of this book. They live on planets exactly like our Earth, with all its mountains, cities, trees, and butterflies. The earths revolve around perfect copies of our Sun, and each sun belongs to a grand spiral galaxy—an exact replica of our Milky Way.
How far away are these earths populated by our duplicates? We know that matter contained in our O-region can be in about 10 to the 10
90
different states. A box containing, say, a googleplex (10 to the 10
100
) O-regions should exhaust all these possibilities, with a large margin. Such a box should be, roughly, a googolplex light-years across. At larger distances, O-regions, including ours, should recur.
90
different states. A box containing, say, a googleplex (10 to the 10
100
) O-regions should exhaust all these possibilities, with a large margin. Such a box should be, roughly, a googolplex light-years across. At larger distances, O-regions, including ours, should recur.
There should also be regions where histories are somewhat different from ours, with all possible variations. When Julius Caesar stood with his legions on the bank of the river Rubicon, he knew he was about to make a momentous decision. Crossing the river would amount to high treason, and there would be no way back. With the words
“Iacta alea est!”
—“The die is cast!”—he ordered the troops to advance. And the die was cast indeed: on some earths Caesar went on to become the dictator of Rome, while on others he was defeated, tried, and executed as an enemy of the state. Of course, on most earths there has never been a person by the name Caesar, and most places in the universe are nothing like our Earth—since there are many more ways for things to be different than for them to be the same.
“Iacta alea est!”
—“The die is cast!”—he ordered the troops to advance. And the die was cast indeed: on some earths Caesar went on to become the dictator of Rome, while on others he was defeated, tried, and executed as an enemy of the state. Of course, on most earths there has never been a person by the name Caesar, and most places in the universe are nothing like our Earth—since there are many more ways for things to be different than for them to be the same.
It may be fitting that this surreal picture of the world originated in the
town haunted by the spirit of Salvador Dalí. Like Dalí’s paintings, it blends weird, nightmarish features with recognizable reality. It is, however, a direct consequence of the inflationary cosmology. Jaume and I wrote a paper describing the new worldview and submitted it to the
Physical Review
, the leading physics journal. We ran the risk that the paper could be rejected for being “too philosophical,” but it was accepted without a glitch. In the discussion section at the end of the paper we wrote:
town haunted by the spirit of Salvador Dalí. Like Dalí’s paintings, it blends weird, nightmarish features with recognizable reality. It is, however, a direct consequence of the inflationary cosmology. Jaume and I wrote a paper describing the new worldview and submitted it to the
Physical Review
, the leading physics journal. We ran the risk that the paper could be rejected for being “too philosophical,” but it was accepted without a glitch. In the discussion section at the end of the paper we wrote:
The existence of O-regions with all possible histories, some of them identical or nearly identical to ours, has some potentially troubling implications. Whenever a thought crosses your mind that some terrible calamity might have happened, you can be assured that it
has
happened in some of the O-regions. If you narrowly escaped an accident, then you were not so lucky in some of the regions with the same prior history … . On the positive side, … some readers will be pleased to know that there are infinitely many O-regions where Al Gore is President
ad
and—yes!—Elvis is still alive.
1
has
happened in some of the O-regions. If you narrowly escaped an accident, then you were not so lucky in some of the regions with the same prior history … . On the positive side, … some readers will be pleased to know that there are infinitely many O-regions where Al Gore is President
ad
and—yes!—Elvis is still alive.
1
The press responded instantly—as Jaume had anticipated. The next month’s issue of the British magazine
New Scientist
published a review of our paper under the headline “The King Lives!”
New Scientist
published a review of our paper under the headline “The King Lives!”
We later learned that the picture of multiple clones of ourselves scattered throughout the universe had some lineage. The famous Russian physicist Andrei Sakharov expressed a similar idea in his 1975 Nobel Peace Prize lecture. He said, “In infinite space many civilizations are bound to exist, among them societies that may be wiser and more ‘successful’ than ours. I support the cosmological hypothesis which states that the development of the universe is repeated in its basic characteristics an infinite number of times.”
2
2
Some people even argued that it was self-evident that absolutely everything
must happen in an infinite universe. This claim, however, is false. Consider, for example, the sequence of odd numbers 1,3,5,7, … The sequence is infinite, but you cannot conclude that it contains all possible numbers. In fact, all even numbers are missing from the sequence. Similarly, infinity of space does not, by itself, guarantee that all possibilities are realized somewhere in the universe. We could, for example, have the same galaxy endlessly repeated in the infinite space.
must happen in an infinite universe. This claim, however, is false. Consider, for example, the sequence of odd numbers 1,3,5,7, … The sequence is infinite, but you cannot conclude that it contains all possible numbers. In fact, all even numbers are missing from the sequence. Similarly, infinity of space does not, by itself, guarantee that all possibilities are realized somewhere in the universe. We could, for example, have the same galaxy endlessly repeated in the infinite space.
This point was recognized by the South African physicists George Ellis and G. Brundrit.
3
They assumed that the universe is infinite and argued that it should contain an infinity of places very similar to our Earth. (Their analysis was based on classical physics, so they could only argue that other earths were similar, but not identical, to ours.) They had to assume in addition that the initial state of the universe varied randomly from one O-region to another, so that all possible initial states were exhausted in the infinite volume. Thus, the existence of our clones is not certain, but hinges on the assumptions of spatial infinity and the “exhaustive randomness” of the universe.
3
They assumed that the universe is infinite and argued that it should contain an infinity of places very similar to our Earth. (Their analysis was based on classical physics, so they could only argue that other earths were similar, but not identical, to ours.) They had to assume in addition that the initial state of the universe varied randomly from one O-region to another, so that all possible initial states were exhausted in the infinite volume. Thus, the existence of our clones is not certain, but hinges on the assumptions of spatial infinity and the “exhaustive randomness” of the universe.
In contrast, in eternal inflation these features do not have to be introduced as independent assumptions. It follows from the theory that island universes are infinite and that the initial conditions at the big bang are set by random quantum processes during inflation. The existence of clones is therefore an inevitable consequence of the theory.
It depends on what the meaning of the word “is” is.
—BILL CLINTON
The idea of many worlds or “parallel” universes has also been discussed in a totally different context. You might have heard of the many-worlds interpretation of quantum mechanics, which asserts that the universe is constantly splitting into multiple copies of itself, with all possible outcomes of every quantum process being realized in different copies. This may sound similar to eternal inflation, but the two theories are in fact completely different. To make sure they do not get confused, let us now make a brief detour into the world of many worlds.
Quantum mechanics is a phenomenally successful theory. It explains the structure of atoms, the electric and thermal properties of solids, nuclear reactions, and superconductivity. Physicists rely on it with complete confidence—and yet, the foundations of this theory are notoriously obscure, and debate about its interpretation is still ongoing.
The most contentious issue is the nature of quantum-mechanical probabilities. The
Copenhagen interpretation
, developed by Niels Bohr and his followers, holds that the quantum world is inherently unpredictable. According to Bohr, it is meaningless to ask where a quantum particle is, unless you perform a measurement to find this out. The probabilities for all possible outcomes of the measurement can be calculated using the rules of quantum mechanics. It appears that the particle “makes up its mind” and jumps to a certain position at the last moment, when the measurement is performed.
Copenhagen interpretation
, developed by Niels Bohr and his followers, holds that the quantum world is inherently unpredictable. According to Bohr, it is meaningless to ask where a quantum particle is, unless you perform a measurement to find this out. The probabilities for all possible outcomes of the measurement can be calculated using the rules of quantum mechanics. It appears that the particle “makes up its mind” and jumps to a certain position at the last moment, when the measurement is performed.
An alternative interpretation was proposed by Hugh Everett III in his Princeton doctoral thesis in the 1950s. He argued that each possible outcome of every quantum process is actually realized, but they all occur in different, “parallel” universes. With every measurement of a particle’s position, the universe branches into myriads of copies of itself, where the particle is found to be in all possible places. The branching process is fully deterministic, but we don’t know which of the branches is going to be the branch of
our
experience. Thus, the outcome of
our
measurement is still subject to the law of probability, and Everett showed that all the probabilities come out exactly the same as one finds using the Copenhagen interpretation.
4
our
experience. Thus, the outcome of
our
measurement is still subject to the law of probability, and Everett showed that all the probabilities come out exactly the same as one finds using the Copenhagen interpretation.
4
Since the choice of interpretation does not affect any results or predictions of the theory, most practicing physicists take an agnostic attitude toward the foundations of quantum mechanics and spend little time worrying about such issues. In the words of the particle physicist Isidor Rabi, “Quantum mechanics is just an algorithm. Use it. It works, don’t worry.”
5
This “shut up and calculate”
6
attitude works fine, except in quantum cosmology, where quantum mechanics is applied to the entire universe. The “orthodox” Copenhagen interpretation, which requires an external observer to perform measurements on the system, cannot even be formulated in this case: there are no observers external to the universe. Cosmologists, therefore, tend to favor the many-worlds picture.
5
This “shut up and calculate”
6
attitude works fine, except in quantum cosmology, where quantum mechanics is applied to the entire universe. The “orthodox” Copenhagen interpretation, which requires an external observer to perform measurements on the system, cannot even be formulated in this case: there are no observers external to the universe. Cosmologists, therefore, tend to favor the many-worlds picture.
Everett and some of his followers insist that parallel worlds are all
equally real, while others believe that they are just
possible
worlds and only one universe is real.
ae
The dispute may be purely semantic: When one says there is another, parallel universe, independent of ours, what exactly does this statement mean? As President Clinton said on a different subject, “It depends on what the meaning of the word ‘is’ is.”
7
Parallel universes are like parallel lines: they do not have any points in common. Each of them evolves in its own, separate space and time, which cannot be penetrated from anywhere in our universe. How, then, can we tell whether they are real or merely possible?
af
equally real, while others believe that they are just
possible
worlds and only one universe is real.
ae
The dispute may be purely semantic: When one says there is another, parallel universe, independent of ours, what exactly does this statement mean? As President Clinton said on a different subject, “It depends on what the meaning of the word ‘is’ is.”
7
Parallel universes are like parallel lines: they do not have any points in common. Each of them evolves in its own, separate space and time, which cannot be penetrated from anywhere in our universe. How, then, can we tell whether they are real or merely possible?
af
I should emphasize that none of this affects the worldview of eternal inflation that I described earlier in this chapter. If the many-worlds interpretation is adopted, then there is an ensemble of “parallel,” eternally inflating universes, each having an infinite number of O-regions. The new worldview applies to each of the universes in the ensemble.
Moreover, in contrast to parallel worlds, other O-regions are undeniably real. They all belong to the same spacetime, and given enough time, we may even be able to travel to other O-regions and to compare their histories with ours.
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