Repeatedly over the years I've discussed quantum mechanics in this blog and the main aim of this particular essay is to present, in simple terms, my interpretation of it. I've thought about quantum physics sometimes over the last six months when not encountering supernatural beings late at night in the Auckland CBD or thinking about God and now believe I can set out my understanding quite briefly and clearly. It is a conceptual rather than a mathematical interpretation; in fact, it is so simple and common sensical that it may disappoint some of my readers. I want then to talk a little about Bayesian probability and, in the last part, I want to talk about myself in the hope that the person I originally started writing this blog for still reads it.
I am not a fully trained physicist but you don't need a doctorate in physic to either invent or understand the interpretation I am going to present. Despite the simplicity of this interpretation, I have gathered the impression that it is unusual; I cannot remember learning about anyone else having espoused this particular interpretation. The reason I approach quantum mechanics from a different perspective than others who have studied it in more depth and in institutions other than the University of Auckland is that the textbooks I read explained it from a historical perspective whereas at other universities, I believe, they attempt to explain it ahistorically and from the ground up. At other universities, they teach quantum mechanics using Dirac notation whereas the textbooks I read came at it from a wave perspective. It is the clarity and pedagogical effectiveness of these textbooks, Physics for Scientists and Engineers by Serway and Jenett, that has enabled me to come up with a new interpretation of quantum mechanics. Of course, it might seem that I am being arrogant when I say that it is new – perhaps others have come up with the same interpretation or similar interpretations to it in the past but, if so, I have never heard about it or about any of them.
The easiest way to approach quantum mechanics is to think of a one-dimensional system. We can imagine a curve or sine wave, a diagram in which the horizontal axis relates to space and the vertical axis relates to something called a probability density or probability amplitude. The probability of finding a particle between two points, a and b, is the area under the curve, squared, between these two points. Because we are assuming that one and only one particle definitely exists somewhere along this single dimension the absolute square of this area must, in total, add up to unity, one. Sometimes a wave function can have complex values, both a real and an imaginary part, and so, even when considering a one dimensional system, we need two other axes, one for the real and one for the imaginary part of the probability amplitude. In reality, when wanting to describe a particle in the real world, we need three spatial dimensions, one temporal dimension, and two other dimensions related to the real and imaginary parts of the probability amplitude. Thus the wave function associated with a single particle actually involves six dimensions and if we want to conceptualise or hypothesise a wave function involving two particles, I think we need twelve dimensions.
In order to calculate the wave function we need a few measurements. One then works out what wave function or wave functions will fit these measurements, a wave-function which, when squared, because we are making the reasonable assumption that one and only particle is being described by it, at every moment in time has under it an area the integrates to equal exactly unity, one. We also need to ensure that the wave function satisfies the appropriate equation, the Schrodinger equation or, if we want to be more precise, the Dirac equation. Having calculated the wave functions that conform to the measurements (there can be more than one), we can use the wave functions so calculated to surmise the probabilities associated with other measurements that we have not yet made, measurements concerning where the particle will be in the future or where it must have been sometime in the past.
The wave function is a kind of hypothetical or ideal structure, a structure always associated with some set of measurements. Suppose we use the wave function associated with a particle to work out where it will be at some future moment, which we call t, and then, at time t, perform a measurement to see whether or not it is at the location or not. We can either stick with our original wave function and pretend that the later measurement never occurred. Or we can revise the wave function we calculated by incorporating this later measurement into the data set, an incorporation that will force us to calculate a different wave function to associate with that particle.
Thus, in my interpretation, measurements are real and the wave function is unreal. The wave function is something hypothetical, abstract, idealised. It is a kind of model associated with some set of measurements. There is at least one wave function associated with every set of measurements or observations and these wave functions can be used to make probabilistic estimates concerning other potential measurements but, when these measurements are actually performed, either these later measurements must be ignored or, if not ignored, included in the set of measurements. And if they are included in the set of measurements, a new wave function must be calculated that incorporates these later measurements. My interpretation is that simple.
A long long time ago, in this blog, when first bringing up the topic of quantum mechanics, I discussed the Schrodinger's Cat paradox. I'm not going to describe this thought experiment again in detail because the reader may be able to find those original posts or, alternatively, just look up the Schrodingers' cat paradox on Wikipedia. According to my interpretation of quantum mechanics, the apparent paradox can be easily dissolved. Wigner is looking a box containing a cat, a gun, and piece of uranium with a fifty-fifty chance of decaying in a certain period of time, a piece of uranium which will cause the gun to kill the cat if it decays. According to traditional interpretations of quantum mechanics, the cat is both alive and dead at the same time but in my interpretation the cat is both alive and dead only with respect to the set of measurements, observations, phenomenological experience possessed by or accessible to Wigner before he opens the box and makes the necessary observation, performs the measurement. The cat, though, because it can access or has available to it a different set of measurements, a different set of observations, a different phenomenological world, is, if it is alive and thus a conscious observing being, definitely alive. If Wigner, rather than opening the box himself, asks an assistant to do it, then the assistant, until such time as he tells Wigner whether the cat is alive or dead, has accessible to him or is possessed of or can register a different set of measurements than Wigner. The assistant has as part of the set of measurements available to him, in the period between opening the box and telling Wigner what he saw, information concerning whether the cat is alive of dead, that Wigner does not have until he himself is told. Because more than one set of measurements or observations can be associated with a particle, more than one wave function can legitimately be associated with it – if two different physicists have made two different sets of measurements of a particular quantum system, they will come up with two different wave functions for it and thus calculate two different estimates of the probability related to future measurements of the system in question. I'm going to repeat myself here because the idea is so simple and so obvious – a wave function is an abstract idealised conception always associated with some set of measurements. If two different people have made, collated, two different sets of measurements associated with a single system, they will arrive at two different wave functions to associate with it even though both sets ostensibly concern a single system.
I'll give another example. When you fire an electron through a slit and then observe where it lands on a screen some distance from the slit, you can make a probabilistic estimate of where on the screen it will land. The wave function that enables you to do this is calculated through some set of measurements – the width of the slit, the momentum of the electron, and the distance to the screen. However if we observe where it actually lands, one can revise one's understanding of the wave function it must have had when going through the slit and on transit to the screen. If we fire thousands of electrons through the slit, we can either ignore where they land and base our calculation of the wave function solely on the width of the slit, the distance to the screen and the momenta of individual electrons. Or we can calculate a different wave function for every electron basing the calculation on a set of measurements that includes where each lands. Or we can somehow calculate a kind of aggregate wave function that includes not only the width the slit, the distance to the screen and the momenta of the particles, but also the information given by where the electrons tend on average to arrive. The choice depends on the experiment and experimenters.
Much of the confusion surrounding quantum mechanics arises from a confusion about the nature of probability itself. Physicists themselves are often reluctant to admit how central a role probability plays in quantum mechanics, even though this was first suggested by Max Born in 1928, because the nature of probability is itself so misunderstood. Probability estimates or calculations arise from the uncertainty of particular agents. Suppose Bob knows that Alice drew a card from a deck yesterday; he may estimate that the probability of her having drawn the ace of spades as being one in fifty-two. However Alice herself, together with everyone she has told, knows for sure that she either drew the ace of spades or did not. The probability from her point of view, and from the point of view of others who have more information than Bob has, is either one or zero. Because I believe that the world is deterministic, the same type of argument applies when trying to calculate whether or not Alice will draw at the ace of spades tomorrow. From the perspective of omniscience events are always either absolutely certain or else impossible. It is because individuals have limited knowledge that we have any reason to believe in chance at all; probability calculations arise from individuals' uncertainty about the outcomes associated with particular scenarios. This may lead us to want to think of probability in a Bayesian way. Thomas Bayes's theory of probability is explicitly subjective, explicitly concerned with the knowledge of individuals, explicitly concerned with the relationship between hypotheses and evidence. But there is a problem with Bayes' Theorem as I want now to discuss.
Bayes's Theorem supposedly allows us to calculate the probability of probabilities. However, in reality, Bayes Theorem only enables us to say which of two competing hypotheses is more likely – it provides us with a rough guide as to which of two probability estimates is better. Suppose, for instance, we hypothesise that a fair die will come up 6 one time out of every six throws. We then roll a die six times and it comes up 6 once. The probability of our hypothesis being true given this evidence is the probability of this evidence given that they hypothesis is true multiplied by the probability that the hypothesis is true divided by the probability of the evidence. However in the absence of a more general hypothesis concerning the probability of the evidence in general, we cannot perform this calculation. Mathematically, however, the probability of the evidence equals the probability of the evidence given the hypothesis being true plus the probability of the evidence given the hypothesis being untrue. However we still cannot perform the calculation because assuming the hypothesis to be false gives us no clue as to how to calculate the probability of the evidence (given that they hypothesis is false.) We need a second competing hypothesis to perform the calculation.
We could approach this problem in the following way. Our main hypothesis is that a die will come up 6 on average once in every six throws. The probability of the evidence given this hypothesis can be worked out to be 5/6 to the power of five. We can pick, although for no particularly good reason, the following competing hypothesis: a die will come up 6 once in every two rolls. The probability of the evidence given this second competing hypothesis is 6 times 1/2 to the power of 6. To make the calculation easier, we shall assume that the probability of both hypotheses, before any evidence is taken into account, is equal. If we also pretend that these two hypotheses, the two priors, are mutually exclusive and mutually exhaustive, then we can assess the probability of the hypothesis given the evidence as 5/6 to the power of five divided by the sum of 5/6 to the power of five and 6 times 1/2 to the power of 6. Of course, these two hypotheses are not mutually exhaustive but we can weigh the first hypothesis against the second by saying that the first hypothesis is 2 times 5/3 to the power of four weightier or more likely than the second hypothesis. The two values of the hypotheses can then, if we want, be substituted in as the priors of another calculation if we then roll the die six more times or however many times we want; hopefully the more often the die is rolled and the evidence tabulated, the closer we will come to a hypothesis that absolutely fits the evidence. The mathematics in this paragraph is complicated but I hope I have made no mistakes and the more mathematically minded of my readers can check my maths to see if I have it completely right.
We might also imagine rolling a die six thousand times and it coming up a 6 a thousand times. This is certainly what we would expect but the statisticians among my readers will of course know that for a 6 to come up exactly a thousand times our of six thousand rolls is still extremely unlikely. Yet hopefully you'll also understand that this is still the most likely outcome. The probability of the hypothesis, that the probability of rolling a 6 is 1/6, is vastly more weighty or more likely than a competing hypothesis such as the probability of rolling a 6 being 1/2. The hypotheses that come closest to being the best are the closest to 1/6 such as, for instance, a probability of 999/6000.
Although there are cheerleaders for Bayes' Theorem among the mathematical community, it is still not the perfect way of approaching probabilistic problems, still not the best way to understand what chance and randomness actually are. Its main advantage is that it seems to be explicitly subjective and begins with hypotheses, with rational problem solvers trying to make the best possible guesses. The hurdle all the Bayesian advocates admit to be problematic, the problem they all recognise when attempting to explain Bayes Theorem to the laity, is that, at first, the probabilities assigned to the hypotheses, the priors, are arbitrary. However, as they point out, the more evidence accumulates, the more the posterior probabilities should get closer and closer to the truth. So evidence increasingly should lead to a correct assignment of probability to a generic situation without any of the messy assumptions involved in older and more 'traditional' ways of understanding the nature of chance. The thing though that these cheerleaders or advocates forget to mention is that in order to perform a Bayesian calculation we need two competing hypotheses, not just one. (I have written about this before, last year, and if the reader is still unsure, he or she might be able to find and read this post.)
Even though Bayes Theorem is imperfect, it foregrounds the subjective nature of probability estimates. It fits with what I said earlier about how both the past and future may well be deterministic and that therefore probability must be considered a kind of estimate of how likely something will be from the perspective of an observer with limited information; probability estimates cannot be disentangled from the uncertainty of conscious agents or, I think to put it even better, an uncertainty related to some kind of schema associated with some set of measurements, a schema that enables someone who can work through all the complicated ever ramifying mathematics to make approximate predictions about future measurements. But these estimates are, like I say, approximate, uncertain.
The interpretation of quantum mechanics I have presented in this essay is actually very very simple. It is far better than, say, the Many Worlds Interpretation of quantum mechanics invented by Hugh Everett. Everett thought that because the wave function only gives probabilities of future events and permutations, every possible event and permutation occurs, each in its own world. But this is really fucking stupid. I know people, especially Hollywood screen writers making up superhero stories, love the idea of a multiverse but the reason it is stupid is that many Many Worlds enthusiasts say that measurements never really occur, that the wave function is real and measurements are not, even though we cannot have a wave function at all without at least a couple of measurements to establish what it is or might be. It seems really stupid to me as well that Everett and his followers seem to think that even though there is an infinite number of futures there is only a single past. For the sake of consistency, they should either argue that there is only one past and one future (that the universe is deterministic) or that if there are indeed an infinite number of futures, as they want to claim, there must be an infinite number of pasts (and presumably an infinite number of presents as well). If this second paradigm seems to them the best, then there can indeed be no real measurements because all possible measurements happen, and if all possible measurements happen I suppose we might still be able to have wave functions but these wave functions are things that we can know nothing about because to know anything at all about them we actually do need at least a couple of measurements.
My interpretation, which is so simple (although it follows directly from other essays I've written over many years in this blog), simply seems the best and most obvious way to make sense of quantum mechanics. The only conceptual pill to be swallowed is the idea that probabilities are subjective estimates resulting from complex mathematics or worked out by ideal totally rational minds. Why then have so few people, if anyone at all apart from me, posed this as the best possible interpretation? Partly the physicists screwed it up through their choice of terminology. Almost from the beginning they talked about 'wave function collapse'. The wave function was something diffuse, distributed over a volume, but particles, when observed, when measured, seem to collapse to points. What the physicists who spoke about 'wave function collapse' missed is that if, when measured, a particle is observed to have a distinct localised location, its momentum must become quite a lot more inexact, as per the Heisenberg uncertainty principle. The wave function doesn't collapse; rather another measurement has occurred resulting in a change to the wave function, in fact meaning that we must posit a whole new wave function that includes the new measurement as well as the prior ones as a data point in the total set of measurements.
This essay has been concerned with quantum mechanics again and a little with Bayes' Theorem. I feel that it is still unclear. If I am allowed to say this, it feels to me as though the world has changed around me, a change which makes it more difficult to express my ideas than it used to be; perhaps this has something to do with Trump's election. A lot has actually happened to me this year and the person I have spoken to almost all the time this year, albeit telepathically, may be surprised to learn that I am still obsessed by quantum mechanics despite everything else that has happened. I actually want to write about God, about Divine Hiddenness, and the Problem of Evil, even more than about quantum mechanics, but I am not far along the road enough yet to know what precisely I should say about all of this. One small thing I would like to do is to name myself: the author of the Silverfish blog is an Aucklander called Andrew Judd. If the person who I still cannot be sure absolutely exists and who I have been talking to indeed does have a friend called Emily Bronte (or something like that – I am being deliberately disingenuous to conceal people's true identities), a friend who has an interest in physics, perhaps one of you two can write to me and tell me if the theory I have proposed has legs at all. And, remember, if you ever get emotionally distressed, the best thing to do is to "Stomp! Stomp! Stomp!" If you ever do want to contact me, of course, you know my email address. If you don't contact me soon I may be forced to think that despite you seeming about as real as a disembodied person can be, although you seem the most plausible of impossible beings, you are indeed impossible and I will have to figure out another explanation for why God gave me the experiences he gave me this year. Here's hoping for a change in the weather.
