Reality in the Eye of the Beholder: Probability and Quantum Subjectivity
Quantum mechanics brings uncertainty into physical reality.
Author’s Note: This post is Part 2 of a two-part exploration of interpretations of probability and their implications for quantum mechanics. Part 1 is available here.
In the previous post, we took one understanding of probability – frequentism – and saw how it leads us to a quantum multiverse. Across this multiverse, all physically possible versions of history, and your life, unfold across innumerable parallel universes. The multiverse is awe-inspiring, but it is natural to also wonder whether it might be a bit too much. A fundamental principle of science is Occam’s Razor, which implores us to posit the fewest entities necessary to explain a phenomenon. The multiverse seems a maximalist solution that posits the existence of countless whole universes in order to explain probability and quantum mechanics. It is therefore prudent to wonder: is there no simpler solution?
In this post, we will seek such a solution by exploring the other standard interpretation of probability – Bayesianism. We will see that it indeed offers a simple notion of probability that eliminates the need for parallel universes, but only by introducing subjectivity into the heart of physics. Bayesianism can free us from the multiverse if we let it, but if we are not careful it may well also unmoor us from objective reality itself!
Bayesianism: Probability for the People
Let us imagine a game. I flip a coin. If it shows heads, we win one million dollars. If it shows tails, we win nothing. You are too nervous to watch, so you close your eyes. You hear the coin land, but cannot see which face shows. At this moment, what is the probability that the coin landed on heads? It depends on who you ask! Here is how we might each answer the question.
Me: I can see that the coin shows heads. Therefore, it is certain that the coin landed on heads. Therefore, the probability is 1.
You: The coin was equally likely to land on tails as heads. Since I cannot see the coin, I know nothing about how it landed. Therefore, it is equally likely that the coin shows tails as heads. Therefore, the probability is ½.
Who is wrong? Surely neither of us! We both reason correctly using valid principles of probability. However, we reach different answers because we have different information.
Bayesian probability is based on two principles that emerge from this example. First, probability is subjective. This means that two people with different information about a possibility can rationally assign different probabilities to it. Second, probability represents a statement of belief. Specifically, it represents how confidently the person assigning the probability believes the possibility to be true. A probability of zero means that you believe it to be impossible, a probability of one means you believe it is certain. Between zero and one, there is a spectrum where the closer the probability is to one, the more likely you believe it is. This interpretation is consistent with the above example. I am certain that the coin shows heads so I assign a probability of 1, while you are unsure so you assign a probability of ½. Bayesian probabilities are not objective properties of the world; they are formed in the minds of people as they interact with the world and learn information about it.
Bayesianism has two clear advantages over frequentism. First, because it does not depend on the frequency of an event, it allows us to assign probabilities to one-off events without requiring us to imagine parallel universes. Second, it explicitly deals with how probability is perceived by real people, rather than abstractly. This makes it ideally suited to examples like that above where people actively participate in a game or experiment rather than watching from a distance. Indeed, since frequentism insists probabilities are objective, it must bend over backwards to explain how you and I can even perceive different probabilities.1 The Bayesian explanation provided above is incomparably simpler because it treats the people involved as integral to any probabilistic situation. In Bayesianism, human experience is the point, while in frequentism it is a technical nuisance.
Bayesiansim in Quantum Mechanics
Bayesianism therefore seems ideally suited to quantum mechanics. In classical physics (that which came before quantum mechanics), performing experiments was fundamentally a passive process. As long as enough care was taken, it was assumed that we could always measure anything about a physical system without changing the system. This meant that we did not need to include the person performing the experiment in the experiment itself; the physical world we studied could be treated independently from human experience.
However, quantum mechanics is different. We discussed last time that one property of quantum mechanics is that it describes nature in terms of probabilities instead of certainties. Another fact about quantum mechanics is that measuring a quantum physical system changes it. This means that a person conducting a quantum experiment must necessarily be included as part of the system itself; they cannot be ignored because the system would behave differently if they were not there measuring it. This makes studying quantum mechanics similar to the example described in the previous section, where observations of people involved is relevant. Indeed, a famous thought experiment in quantum mechanics, referred to as Wigner’s Friend, is essentially the quantum mechanical version of that example. As we have seen, Bayesianism naturally accounts for the involvement of people in a probabilistic process and so it is a useful tool for understanding quantum mechanical processes.
Indeed, Bayesianism even offers a perspective on why quantum mechanics deals in probabilities. The Heisenberg Uncertainty Principle is a fundamental principle of quantum mechanics that says that it is impossible to determine all properties of a quantum system perfectly. It arises from the way that a quantum system is changed by being measured. Specifically, even if we have measured one property of a system (such as its position) perfectly we cannot necessarily measure another (such as its speed) without this second measurement changing the system in a way that invalidates the first measurement. The Heisenberg Uncertainty Principle makes it impossible to ever collect all conceivable information about a quantum system. According to Bayesianism, probabilities reflect incomplete knowledge; if we knew everything then we could say what would happen with certainty instead. So, we can see that probabilities in quantum mechanics arise because the Heisenberg Uncertainty Principle makes complete knowledge of a quantum system impossible, which naturally leaves Bayesian probabilities as an effective tool to study quantum systems.
A Subjective Universe
However, while practically Bayesianism is useful for quantum mechanics, it presents a significant conceptual problem. Quantum mechanics represents all physical entities by quantum states (or wavefunctions). The combination of all of these quantum states constitutes our best physical description of nature. These quantum states are intrinsically probabilistic. They do not describe what we will see happen with certainty; instead, they describe the probabilities of all different possibilities. However, if we accept the Bayesian interpretation of probability then these probabilities are not objective properties of the world; they are subjective values that exist in the minds of people. Since quantum states are probabilistic, this means that quantum states must themselves be subjective and exist in the minds of people. Since these quantum states form our best description of physical reality, this means that reality itself, as we understand it, is not objective.
Achieving absolute, objective truth is the driving motivation of physics. From Galileo’s determination to “measure what is measurable, and make measurable what is not so” through to Stephen Hawking’s desire to “know the mind of God”, physicists have always aimed to use mathematics and rigorous experiment to rise above subjectivity. By persistently seeking better data and better theories, they have striven to see beyond how things appear to find the hidden reality underneath. The logic of Bayesianism applied to quantum mechanics would seem to mark an end to this quest. It suggests that there is no objective truth hidden behind our impressions; reality itself is in the eye of the beholder.
The Nature of Reality
Our exploration of probability and quantum mechanics has therefore led us to a choice between two options. We can believe in frequentist probability which leads us to an incredible multiverse of innumerable parallel universes. Or we can believe in Bayesian probability which leads us to the disquieting conclusion that reality is intrinsically subjective. It is not clear that either provides a satisfactory basis for physics. The first interpretation forces us to refer constantly to an immense, unseeable reality just to make basic statements, while the second compels us to reject objective reality itself.
However, the two perspectives are not as incompatible as they may seem; they can be synthesised into a more compelling perspective. According to the first interpretation, an omniscient observer would see a singular, objective reality. This reality consists of the multiverse across which all possible versions of you experience all possible versions of your life. However, we experience only one of these versions of ourselves and our lives. This results in a subjective experience that includes only one of the many possible truths of what happens in our lives; that manifested in our own universe. This subjective reality is what is captured by the second interpretation.
Reality as we experience it is subjective, but we can imagine that our reality is only part of a larger objective whole. We can therefore use the Bayesian interpretation to understand and study probability and quantum mechanics, even while the frequentist interpretation provides a firm foundation of ultimate objectivity. The value of the Bayesian interpretation is that it represents probability, and quantum mechanics, as we experience them, instead of how they may be in the mind of a supernatural omniscience. Bayesian probability is ultimately about uncertainty; it is because we do not know everything that we experience an unpredictable, subjective reality. Perhaps this is why quantum mechanics inspires both such awe and such distaste – it places this cornerstone of human experience into the heart of nature. Viewed in this way, quantum mechanics brings uncertainty into physical reality.
On the surface, it seems that frequentism would imply that there must be one correct answer for the probability. Indeed, we may be inclined to conclude that I am right to say that the probability is one because the coin has already landed, and you are wrong as a result of not having all of the information. However, deferring to a person with all possible information begs the question of what the probability is because such a person will always perceive all possible outcomes to be either certain or impossible and so would have no need to refer to probabilities.
In fact, we can use frequentism to vindicate both assessments of the probability by the following, rather obscure, reasoning. Imagine many parallel universes in which the experiment took place. Since you cannot see the outcome of the coin flip, nothing in your experience is affected by it and so an identical version of you remains across all the universes. In half of these universes the coin shows heads, so you conclude that the probability of heads is ½. However, I see the outcome and am affected by it – I become very happy if I see heads (because we win) and very sad if I see tails (because we lose). Therefore, across the universes I split into two distinct versions of me – Happy Paul and Sad Paul – depending on the outcome of the coin flip. Every Happy Paul is in a universe where the coin shows heads, so for Happy Paul (who is present in the example) the probability of the coin showing heads is one. You and I disagree about the outcome because I have been split into two distinct people by seeing the coin and you are only asking one version of me, while you remain a single person across all of the universes.