We are now ready to put together all the previous sub-steps to show that our universe is absolutely indeterministic. This is the first step in our quest to glimpse the higher reality.

The hidden variables of the de Broglie-Bohm theory were not ruled out by the Bell Inequality (because the Bell Inequality only rules out local hidden variables). So, at first sight, it looks as though we cannot rule out the possibility that hidden variables underlie all interactions in the universe, which would mean that, in principle, the outcomes of these interactions are predictable. If we go back to the double-slit experiment – the one that Feynman regarded as encapsulating the very essence of quantum mechanics – we can imagine an electron following one of the detailed, convoluted paths described by the de Broglie-Bohm equations, starting at one of the double slits, and going all the way to its destination on the detecting screen. If that’s what happens, then the universe is deterministic rather than indeterministic, because every twist and turn of the journey, as well as the final destination, can be predicted.

However, what the de Broglie-Bohm theory doesn’t tell you, because it can’t, is exactly which slit the electron has come through and, in particular, the exact point where, within the breadth of that slit, its path begins. The detailed shape of each path from the slit to the detecting screen is described precisely by the theory, but on the question of which of the indefinitely large number of these paths the electron takes, the theory is silent! It is literally a matter of chance which particular path the electron takes. In this context, the de Broglie-Bohm theory is actually statistical. So, although the theory is grounded in the electron having a defined position and momentum at every moment along its path, which particular path itself is not determined by the theory. In other words, the de Broglie-Bohm theory does not, after all, scupper the claim that our universe is indeterministic.

In case you have any remaining doubt about ruling out the de Broglie-Bohm theory as an argument against indeterminism, there is a more fundamental reason for rejecting it. As we have seen, for entangled particles, the matching between Alice’s and Bob’s measurement outcomes can’t be accounted for by local hidden variables. So, quantum theory is nonlocal. However, that doesn’t mean that a message is somehow transmitted instantaneously – faster than the speed of light, which is forbidden by relativity – from Alice’s detector to Bob’s detector. That’s not how it works – there is nothing Alice can do to influence Bob’s result to make it match with hers. Quantum mechanics as developed by Heisenberg and Schrödinger, while remarkably successful, didn’t take account of special relativity (more of which later), and, in particular, the speed-of-light limitation of how fast information can get from place to place. Paul Dirac (who was the first to apply the mathematics of Hilbert spaces to quantum mechanics) later upgraded the theory to make it relativistic, and this was eventually developed by others into the modern quantum field theory that regards particles as excitations of fields (so the photon is an excitation of the quantum electromagnetic field, and the electron is an excitation of the quantum electron field, and some of these quantum fields interact with each other, which we see as particles interacting).

While the nonlocality of quantum mechanics didn’t prevent it from being upgraded to quantum field theory, the de Broglie-Bohm theory is even more nonlocal than the original quantum mechanics ever was. In de Broglie-Bohm theory, the guiding wave for a particle is shaped, from moment to moment, by the position of all of the particles in the universe, instantaneously, and any movement of any particle affects all the others reciprocally! This seems to be a show-stopper with regard to modifying the theory to comply with relativity. Furthermore, the theory resists all attempts to modify it to accommodate the creation and annihilation of particles, which, in contrast, quantum field theory takes in its stride. So, while the de Broglie-Bohm theory has served its purpose, with Bell’s help, in drawing attention to the nonlocality of the universe, we have nevertheless exhausted all the arguments for using it to make a case against the universe being indeterministic.

One loophole remains, however. Returning to the double-slit experiment, before we could rule out any possibility of the electron having both a definite position and momentum on its way to the detecting screen, we wondered whether the unpredictability of its final destination on the screen might be due to stray field excitations in its path, subjecting it to buffeting which could, in principle, be accounted for if we knew exactly where all these stray excitations were. Well, rather than thinking of the electron as a kind of miniscule cannon ball, a more accurate description of the electron would be an excitation of the quantum electron field: in other words, we might visualize the electron as a stable combination of waves in the field. Might these waves interact with the random, stray fields that we had originally pictured, giving the impression of randomness, whereas, if we only knew the exact configuration of these stray fields, the point on the screen where the electron is finally detected would be completely determinable?

Let’s make the question more quantitative: we can use the experiment where two entangled particles travel in opposite directions, and Alice and Bob incline their detectors at a mutual angle of 120º, noting how often their results of spin measurement match each other. According to quantum mechanics, and, as verified by experiments of the type conducted by Aspect and those who came after him, the proportion of matches to the total number of experiments is ¾. However, this doesn’t mean that, if Alice and Bob carry out a sequence containing 100 experiments, they will always find precisely 75 matches and 25 non-matches. In practice, 100 measurements might yield only 71 matches (with 29 non-matches). So, they try again: this time it’s 80 matches. Again: it’s 77 matches, and so it goes on. If they repeat the sequence of 100 measurements often enough, and plot the results on a histogram, they’ll find something like the next Figure, which shows the results of conducting 100,000 sequences of experiments, with each sequence containing 100 experiments (so this is showing the results of 10 million measurements, and is actually a computer simulation).

If a sequence contained only one experiment instead of 100, there would be only two possible outcomes: either there would be a match or there would be a no-match. If the sequence had two experiments, then there would be three possible outcomes: two matches, one match or no matches. This illustrates the fact that there is always one more possible outcome than the number of experiments – in 100 experiments, there are 101 possible outcomes. In the Figure, these are plotted along the x-axis, beginning at zero and ending at 100. As there happen to be 100 experiments, I have labelled the x-axis in percentages, but it could equally well have been called bins, or something else.

Plotted on the y-axis is the number of sequences, each containing 100 experiments, that Alice and Bob carry out. So, looking at the histogram, you can see from the peak (of around 9,200 sequences) that sequences containing 75% matches are the most common finding, which, from what we saw earlier, is what you would expect from spin-filters inclined at a mutual angle of 120º. However, there is still a substantial number of sequences (about 4,500) where only 70% of the 100 experiments yielded a match, and, similarly, where as many as 80% of the outcomes matched.