Dr. Tonomura and Belsazar from Wikimedia Commons CC BY-SA 3.0
3.1 Patterns in an indeterministic universe
In order to reconcile the indeterminacy of our universe and the determinacy of its purely mathematical structure, it will be useful to have a standard experiment to refer to later on. We shall use the apparatus shown in the figure, where someone has prepared a particle in the vertically upward spin state (using a Stern-Gerlach filter not shown in the diagram) so that the particle is sent through a second Stern-Gerlach filter inclined at 60º to the vertical. The algebra of quantum mechanics gives a probability – the Born probability – of ¾ that Alice will detect a particle with spin-up in the vertical direction. We shall refer to this henceforth as “the 60º experiment”.
Alice’s 60º experiment.
The algebra of quantum mechanics says, if Alice performs a sequence of 100 of her 60º experiments, then there will be close to 75 results of spin-up, but it cannot predict exactly how many such results there will be in any given sequence. The histogram shows the expected distribution of results.
But what does that mean in a single experiment? If Alice detects the particle as spin-up in one experiment, that doesn’t tell her the transmission probability, any more than if she had detected the particle as spin-down. What she has to do is to perform a whole sequence of experiments, each sequence containing a large number of individual measurements, say 100. She might find 70 spin-up results in the sequence, which is close to, but not exactly the 75 predicted by quantum mechanics. If she performs a second sequence of 100 measurements, she might find 77 spin-up results. If she carries on like this, conducting, say, 100,000 sequences of 100 measurements each and plots the number of sequences containing a given number of spin-up results, she will end up with a plot like that in the histogram. In fact, I have just copied this figure from the one in Step 1.11 (Our Absolutely Indeterministic Universe), because I lazily chose the experiment for which the outcomes for the two quantum mechanical problems are identical. All I have done is to relabel the axes accordingly.
Now, think of these measurement events as they would appear in a Minkowski diagram of the block universe. If the measurements are being made, one after the other, on the same Stern-Gerlach filter, then they will appear like beads strung along the filter’s vertical world line. Although the outcome of each individual measurement can’t be predicted, you can see that, nevertheless, the sequences form a pattern peaked around the 75% mark when plotted as in the diagram. How does this pattern arise in an indeterministic universe? Each measurement is identical to every other, so what is it that determines that there are going to be three times as many spin-up outcomes as spin-down along the world line? To say that it is determined by the quantum-mechanical probability (or Born probability) misses the point – quantum mechanics is a description of how often to expect a particular outcome, not the cause of one particular outcome appearing rather than a different one.
Could it be that each measurement is influenced somehow by the outcomes of the previous measurements in the sequence, establishing a connection between them, leading to the pattern? Well, no, as we can easily show by carrying out each sequence on an array of 100 filters all at the same moment but spaced so far apart from each other that no outcome can influence any of the other measurements. This is in contrast to conducting each sequence of 100 measurements sequentially in time with the same Stern-Gerlach filter. Nevertheless, with the spaced-out measurements, we would still find that the plot of 100,000 such sequences is the same as the histogram in the figure.
We saw earlier that all outcomes embedded in the block universe are already fixed: they don’t wink in and out alternately red and green like fairy lights blinking on a Christmas tree. That would make no sense – if the outcomes were changing, then it would mean they were changing with time. But the time dimension is part of the block universe structure itself – and each outcome is embedded at a fixed point in space and time, and so it doesn’t change, which is just what you should expect of a mathematical structure. Otherwise, it would be like saying that one of the black squares in the structure of stacked planes showing the glider’s walk will suddenly turn white – that’s not going to happen!
Nevertheless, perhaps there is a way for randomness to be introduced into the structure while keeping the structure solidly fixed? Suppose that a random number generator is somehow embedded within the mathematical structure of the universe, determining the outcome of every quantum interaction or measurement. A random number generator produces streams of numbers, for instance, between 0 and 1, and so, for example, to determine the outcome in the case of Alice’s 60º experiment, any number it produces from 0.0 up to (but not including) 0.75 will be allocated a spin-up outcome and from 0.75 up to 1.0 it will be spin-down. Taken over a large number of measurements, this will produce the 75% of spin-up outcomes expected for the experiment. Our putative random number generator is part of the structure of the universe, and so no part of it “winks in and out” – it is fixed just as much as its random output is fixed.
“Middle-square” method of generating (pseudo)random numbers
However, that idea won’t work. Commonly, a random number generator is really a pseudo-random number generator, so-called because they use arithmetical formulae which are ultimately predictable. Since, as we have seen, the universe is fundamentally unpredictable, then we know that a pseudo-random number generator can’t be behind patterns like those shown by the histogram.
There is another kind of random number generator, though, called a true random number generator. Instead of using mathematical formulae, these generators use physical phenomena such as “noise” in electrical components, nuclear radiation emission, cosmic background radiation and so on. However, ultimately, all of these phenomena owe their randomness to quantum-mechanical uncertainty, which would mean that the randomness required to produce quantum outcomes through the random number generator would ultimately be generated by those very quantum outcomes! So, we are left without an explanation for a universe which is fundamentally indeterministic.
So, we still have to account for the histogram pattern that emerges from the sequences of quantum outcomes. Again, you can’t explain the pattern by saying that each outcome has a 75% probability of being spin-up: the probability is something you measure after you’ve got enough data points. As I said, quantum mechanics doesn’t determine which particular outcome you get for a given event; it merely tells you, for instance, the percentage of spin-up outcomes that you should expect to find in an indefinitely large number of measurements.
