THE FINITE MULTIVERSE
At the bottom of the page on the Origin of Quantum Probability I claimed that the number of universes in the multiverse is not infinite. Hugh Everett was effectively asked the same question at a closed meeting at Xavier University Physics Department, Cincinnati in 1962 (see reference [15] in my paper on the structure of the multiverse). Attendance was limited to a distinguished group – 23 of the leading physicists in quantum theory – who were trying to understand the more fundamental aspects of quantum mechanics like whether the wave function really collapsed or not. Everett was also present, although not as a formal participant. It was chaired by Boris Podolsky (he of the EPR paper – see the Quantum Entanglement page) and Everett had earlier sent him, on request, a copy of his PhD on the Many Worlds interpretation. So Podolsky had invited Everett to the meeting and Everett was free to participate in the discussions.
At one point in the meeting, Nathan Rosen (the “R” in the EPR paper) refers to Everett’s work as an alternative to wave function collapse, and invites Everett into the discussion. Everett explains his interpretation to the select audience and then Podolsky (presumably Boris, and not his son – the transcript uses only last names) says: “It looks like we would have a non-denumerable infinity of worlds” (meaning “an uncountable infinity of worlds” – see page on Playing with Infinity), to which quite profound statement Everett simply utters a laconical “Yes” (!)
Some have claimed that it is odd of Everett to have accepted that the number of parallel worlds is uncountably infinite, because, mathematically, the quantum-mechanical states representing these worlds should be countable, even if infinite in number (see Playing with Infinity for a definition of “countable”). I think there is a more concrete argument for saying that the number of parallel worlds should be countable, but before I tell you what it is, what about the puzzle of why Everett said “yes”? I believe that it was simply an impromptu response on the part of Everett, still only 32 years old, having been so recently rebuffed by Bohr himself and hating speaking in public, finding himself facing the core establishment of quantum physics and being asked a question that came rather out of the blue.
You might equally ask: why did Podolsky make the suggestion in the first place, that the number of parallel worlds is uncountably infinite? Perhaps he had in mind the fact that Schrödinger’s equation is continuous: it uses real numbers rather than natural numbers (strictly, it uses so-called complex numbers, but the components of complex numbers are real numbers) and so he may have reasoned that it would require an uncountably infinite number of parallel branches to capture all of the possible outcomes predicted by Schrödinger’s equation for a given experiment. I shall come back to this point about Schrödinger’s equation later, but first I want to tell you why I think that the number of parallel universes in the multiverse is in fact not only countable but finite.
Figure 1 is taken from Figure 6 of Tree Structure of the Many Worlds Interpretation. We saw there that the numbers of parallel universes in the branch kernels for the outcomes of a given experiment are in the same proportions to each other as those of the relevant probabilities. (So, for instance, KC1 : KC2 : KC3 → γ1 : γ2 : γ3 , where γ1 , γ2 and γ3 are the probabilities that the results of experiment C are C1, C2 and C3 respectively). So none of the branch kernels can be infinite, because ratios of infinite quantities are not properly defined (remember the discussion on Figure 5 of Playing with Infinity). After all, we can confidently verify the consistency of such ratios from the outcomes of multiply repeated C experiments (to within experimental error – see Figure 5 of The Origin of Quantum Probability). This means that the complete kernel for any quantum event in the multiverse, being the sum of its finite branch kernels, cannot be infinite. Therefore N, which we defined in the Tree Structure of the Many Worlds Interpretation as the smallest number of universes in the multiverse, and which was the product of all of the kernels of all of the quantum events in the multiverse, cannot be infinite either. So the number of universes in the multiverse is finite.
This conclusion, that the number of universes in the multiverse is finite, is unchanged even if that number is not N, but a multiple of N. It cannot be an infinite multiple, because that would mean, once again, expressing probabilities of infinite quantities as improperly defined ratios.
You may recall the Parallel Universe Probability Rule that I defined in The Origin of Quantum Probability:
Parallel Universe Probability Rule
The probability of an event happening in a universe is the same as the proportion of times that a universe with that event appears in the kernel of possible universes.
Now, quantum mechanical probabilities calculated from the Schrödinger equation are generally irrational: you can’t express them in the convenient ratios implied by this rule, such as the fractions (1/3), (1/5) and (7/15) that we used in the Tree Structure of the Many Worlds Interpretation. Indeed, as we saw in Playing with Infinity, an irrational quantity cannot be expressed as the ratio of two finite numbers. Equally though, the ratio of two infinite numbers is meaningless. Since the probabilities calculated under the Parallel Universe Probability Rule must be rational (since the components of the kernel are finite), we are forced to conclude that the (irrational) Schrödinger-generated quantum probabilities cannot generally emerge from a multiverse of parallel block universes.
So, to summarize where we are up to now:
1 The Many Worlds Interpretation of quantum mechanics circumvents the problem of wave-function collapse (see Many Worlds).
2 However, in turn, MWI introduces the difficulty that its topology is incompatible with that of the Minkowski block universe (see Tree Structure of the Many Worlds Interpretation).
3 The resolution is to populate the MWI tree with discrete parallel block universes in proportion to the weightings of the branches (see Tree Structure of the Many Worlds Interpretation).
4 However, that constrains calculated quantum probabilities to rational values only, through the Parallel Universe Probability Rule, and it also means that the number of parallel block universes in the multiverse is finite, albeit unimaginably vast.
I shall come back to the point about quantum probabilities being restricted to rational fractions in a moment. In the meantime, I want to tell you about an immediate consequence of the hypothesis that the number of parallel block universes in the multiverse is finite: all of these universes must be finite in the time and spatial dimensions. If any single universe were infinite in either time or space, then the Schrödinger equation, along with other quantum equations like those governing fields, would keep on generating, without limit, new quantum events within that universe. So that universe would contain an infinite number of quantum events. However, that means that it would consist of an infinite number of kernels. Remembering that the number of universes in any branch of the multiverse contains components of kernels from every branch of the multiverse, this would mean that the number of parallel block universes in the multiverse would be infinite, which we have ruled out. So the premise that we started out with is wrong: instead of suggesting a universe infinite in either time or space, we have to conclude that all of the parallel universes in the multiverse are finite both in time and in space.
By the way, when I talk about a “universe” here, I mean a block universe. That is different, and much greater, than the observable universe that people generally mean when they refer to the universe.
If our universe were not expanding, the most distant object from which we can receive photons would be 13.7 billion light years from us, because the Big Bang was 13.7 billion years ago. However, during all that time in which the photons have been travelling to us from the object, space has been swelling – expanding – so that the object is now much further from us: around 45 billion light years, in fact. So this is the radius of our observable universe.
According to virtually all modern cosmology theories, just as we cannot detect anything of the space beyond the radius of our universe, observers living in that distant space cannot detect us. Nevertheless, they, too, will ascribe a volume to their observable universe just as we do to ours. If they are not far beyond the radius of our universe, they will be able to detect the parts of our own universe lying closest to them: our two universes share some space – they overlap. If they are far enough from our universe, our two universes will be completely separate and expansion of space will ensure that the boundaries, while expanding themselves, never overlap.
Clearly, the observable universe (what people normally think of as our universe) is finite, but the stronger conclusion here is that the whole block universe is also finite.
Another consequence of a finite multiverse is that not every possible universe exists. A common theme in science fiction is to imagine an alternative scenario to our own universe – one in which Hitler won the Second World War, for example. Indeed, those physicists who believe that the universe beyond our observable universe extends to infinity also subscribe to the view that every possible scenario exists. (They might be correct, of course, if the universe were, indeed, infinite, but the above argument suggests that it is not.) I shall discuss this more closely in a later page.
I’d like to expand a little on what I said above about quantum probabilities being rational.
Go back to the page on the Origin of Quantum Probability and look at Figure 6 or Figure 9. In these figures, you see that the probability of a version of you being in a universe where the measured electron spin is “up” is ½ and ¾ respectively, because, in Figure 6, there is only one universe out of the two in the kernel where the spin is “up” and, in Figure 9, the spin is “up” in three of the four universes in the kernel. So, as we saw in that page, if you repeated the experiments many times, you would find these probabilities.
Figure 9 showed the case where an electron with an up-spin is passed through a filter inclined at 60°. We know that the probability that it passes through the filter is ¾ because quantum mechanics says that the probability of it passing through a filter inclined at an angle θ is cos2 (θ/2), or equivalently, (1 + cos θ)/2. When θ is 60°, this is ¾. But suppose that the angle θ is, say, 45°? Then quantum mechanics says that the probability is (1 + 1/√2)/2 = ½ + (√2)/4 (see Figure 2 in this page). Written as a decimal, the first eight digits are 0.85355339… This number is clearly irrational, because, as we saw in the Playing with Infinity page, the square root of 2 is irrational.
So, how many universes are in the kernel for the event where an electron passes through a 45° filter? If the probability were exactly 0.85355339, then the total number could be 100,000,000 in which the electron emerges in 85,355,339 of them. However, because the probability is irrational, it would take a countably infinite number of decimals to express it and there would be a correspondingly countably infinite number of universes in the kernel (see Figure 2).
So, in the case of the 45°-filter experiment, it would appear that the probability for the electron to pass through the filter is the ratio of a countably infinite number of universes where it passes through and a countably infinite total number of universes in the kernel. But we saw in the Playing with Infinity page that the ratio of two countably infinite quantities is not defined: it is effectively nonsense.
But now we have a contradiction: quantum mechanics says that the probability of an electron passing through the 45°-filter experiment is an irrational number. On the other hand, an irrational number cannot be expressed as the ratio of the number of parallel universes where it passes through and the total number of parallel universes in the kernel. So we have to conclude that the irrational number given by quantum mechanics for the probability is an approximation! In other words, rather than being a continuous theory, quantum mechanics should be re-cast as a digital theory that does not generate irrational numbers as probabilities. I shall return to this in a moment.
Meanwhile, let me go back to my speculation about what was on Podolsky’s mind when he asked Everett his question. I suggested that he might have been thinking that it would take an uncountably infinite number of parallel branches to capture every possible outcome of an experiment as predicted by Schrödinger’s equation. However, we have just seen that no event in the multiverse can have even a countably infinite number, let alone an uncountably infinite number of parallel universes in its kernel.
To see the implication of this, let us take a practical example: detection of a particle (such as a photon or electron) that appears to pass through a narrow slit onto a screen (see Figure 3).
The distribution of the particle incidence on the screen is traditionally calculated using optical formulae, but Schrödinger’s equation gives the same result, of course, which I have outlined in Figure 4. I have drawn a curve on the screen to show the probability that a particle will land there, peaked on the centreline. Strictly, this curve is the probability that a particle will land within a narrow strip, so it is really the probability per unit length of screen. The value given by Schrödinger’s equation for this probability is, in general, a real number (or, to be more specific, an irrational number) rather than a rational number (see page on Playing with Infinity for the definitions of these).
So how should we think of this experiment in terms of parallel universes? We can start by imagining that event C in Figure 1, has 10 branches instead of 3. Now let us split the observation screen into five slots (see Figure 5). Within each of these five slots, I have indicated the probability of a particle being detected in it. On the table in the right of the figure, I have drawn 10 separate universes, each represented by their particular version of the observation screen in which one particle has landed. These 10 universes comprise the kernel for the detection event.
In the first universe, shown at the top of the table, a particle has landed in the left-most slot of the observation screen. In the next two universes, it has landed in the second-left slot, and so on. So you see that the percentage probabilities shown in the observation screen in the left of the picture arise from the proportions in the kernel. For example, there are four universes out of the 10 in the kernel where the particle falls in the central slot, and so the probability that the particle will fall in the central slot is 40%.
So this is how the probability distribution for the single-slit experiment is represented in the parallel-block universe model. Of course, there will be many more branches than 10, but there will not be an infinite number. In this way, the probabilities that arise will always be rational. However, that means, as I said above, that Schrödinger’s equation can only be considered as an approximation, because it gives values for probabilities that are generally irrational. Does it make sense to say that Schrödinger’s equation could be rewritten as a discrete, digital theory?
On a trivial level, we already know that the equations of quantum theory and general relativity can be expressed in terms of natural numbers. After all, every time you use a digital computer to simulate an experiment in quantum mechanics or relativity, you are, of necessity, using integer approximations that are only as good as the precision – the number of bits, 1s and 0s – of the numbers in your computer. I, personally, have modelled interactions using the Schrödinger equation where the slopes of the plots are only approximations, the accuracy of which depends upon the sizes of the steps I use to model small changes in distance and time.
In 1954, shortly before Einstein’s friend, Michele Besso, died, Einstein wrote in a letter to him:
“I consider it quite possible that physics cannot be based on the field concept, i.e., on continuous structures. In that case, nothing remains of my entire castle in the air, gravitation theory included…”
Einstein was surely being gravely pessimistic about his theory of general relativity but the point is that he was questioning the basis of real numbers upon which the edifice of physics is based.
In the concluding paragraph of a report of the First Conference on Physics and Computation held at Massachusetts Institute of Technology in 1981, Richard Feyman noted:
“All these things suggest that it’s really true, somehow, that the physical world is representable in a discretized way…”
In 2014, Gerard ’t Hooft went so far as to say:
“… one may suspect that real numbers are nothing but a human invention.”
In that same paper, ’t Hooft goes on to outline an approach to devising a quantum theory where, instead of using real numbers for fundamental quantities like position and momentum, integers are used instead. Similar work has been reported by other authors: Ramin Zahedi has recast the field equations of the strong, electromagnetic and gravitational forces (and other equations from quantum theory) in terms of integers and matrices, and he prefaces his paper with a nice review of work in the field [Ramin Zahedi, arXiv:1501.01373v15].
In this page, we have come to some very controversial conclusions:
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The number of parallel universes in the multiverse is finite;
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No universe within the multiverse is infinite;
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Not every possible universe exists;
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Quantum probabilities are never irrational;
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Quantum mechanics is only an approximation to an equivalent theory which must be discrete (this must apply to general relativity as well).
I shall discuss how we might test these claims in a later page.
Click here to go to Self-Awareness in a Toy Universe [16]