CHAPTER ELEVEN

The Mist

Science, even more than poetry, is the art of synecdoche. A scientific theory represents a whole with parts: elements, neurons, genes, whatever. Quantum mechanics is a fantastically successful synecdoche, even though wave functions, unlike elements, neurons and genes, aren’t part of nature; they’re not real. Or are they? Q Is for Quantum takes aim at this conundrum.

Q Is for Quantum (note the synecdochic title) is a synecdoche of a synecdoche, which reduces quantum mechanics to balls falling in and out of boxes. My forthcoming take on Q is a synecdoche of a synecdoche of a synecdoche. Yes, this is a warning, and an excuse. Before I riff on Q, here’s how I crossed paths with its author, Terry Rudolph. He reached out to me after reading “Is the Schrodinger Equation True?”, a column in which I express qualms about mathematical descriptions of the world. An edited version of Rudolph’s email:

Dear John,

I just saw your article on the “truth” (or otherwise) of the Schrodinger equation. It is a topic close to my heart (and my professional work)--not, as many presume, because he is my grandfather--in fact I didn’t know that when I became a physicist. Rather I find disconcerting all the options for the nature of reality forced upon us if we accept the main mathematical object in the equation—the quantum state itself—as somehow “real”.

I’m writing primarily to point out to you that it is possible to learn a large subset of quantum theory without using linear algebra or complex numbers. In fact, only basic arithmetic is necessary. I wrote up and self-published the method as a book, Q Is for Quantum. As you will rapidly appreciate, I am not skilled in science communication, but my book has helped school students learn this subset of quantum theory, which has been extremely gratifying.

Best, Terry Rudolph

Professor of Quantum Physics 
Imperial College London

Terry Rudolph, Google confirms, is indeed Schrodinger’s grandson. More impressively, Rudolph has contributed to quantum foundations, which attempts to resolve the theory’s paradoxes. Rudolph is the R of the PBR theorem, named after Matthew Pusey, Jonathan Barrett and Rudolph. The theorem, published in 2012, has been described as the most important insight into quantum mechanics since Bell published his theorem in 1964. Also, in 2015, Rudolph co-founded a firm to build quantum computers. This guy is an expert, a hands-on pro.

I confess to Rudolph that I have struggled to get through texts by David Griffiths and Leonard Susskind. Those are fine books, Rudolph says, but they teach “very much more than is required to quantitatively understand” quantum mechanics. So Susskind’s “theoretical minimum” isn’t truly minimal? Now you tell me! When physicists say you can’t get quantum mechanics without the math, they usually mean you need to study complex numbers, differential equations, matrices and all the rest.

But you don’t need all that fancy math to get quantum mechanics, according to Rudolph; you only need arithmetic and a little ordinary algebra. Rudolph says his system can calculate “anything the standard formalism can calculate with at worst some small overhead.” Rudolph doesn’t intend his system to replace the standard ones, which can solve many tough physics problems more efficiently; his method just provides an easier introduction to quantum mysteries.

Not that Rudolph’s slim little paperback is an easy read. To absorb its meaning, I must study it, write down what I think I’ve learned, study it again. Above all, I must do the math exercises. You can’t get quantum mechanics without the math, Rudolph says, any more than you can get Van Gogh’s “Starry Night” by looking at a black and white photo of the painting--a photo your dog has chewed. I fear my attempt to compress Rudolph’s already pithy book will produce the equivalent of that colorless, mangled photo, but here goes.

Balls and Boxes 

Rudolph explains how quantum mechanics works by explaining how quantum computers work. But first, he needs to explain how conventional computers work. He represents computation with balls falling in and out of boxes; each box has a hole in the top and bottom, for input and output. The balls come in two colors, white and black, which Rudolph represents with the letters W (for white) and B (for black).

The balls correspond to the strong and weak electrical pulses that serve as basic units of information in a digital computer, which in turn correspond to the ones and zeros of a binary code. The white and black balls, and ones and zeros, are bits, which you can think of as answers to yes or no questions. The boxes are the switches, also called transistors or gates, that alter electrical pulses.

The boxes alter the color of balls in accordance with simple, logical rules. A box called NOT changes a ball from white to black or vice versa. A box called SWAP takes in two balls and reverses their positions. Some boxes have a C appended to them, with C standing for control. A CNOT box performs a simple if/then operation. The box accepts two balls at the same time, the control ball and the target ball. If the control ball is black, then the target ball changes color. If the control ball is white, the target ball’s color does not change. A CSWAP box works in a similar fashion.

Everything that conventional computers do—adding more digits to pi, tracking storms, calculating how much interest we’ve earned, guiding missiles to targets—is based on simple, logical operations like these. Smartphones and supercomputers contain zillions of boxes that alter voltages of electrical currents, turning high-voltage pulses into low-voltage ones or vice versa.

Logical operations such as NOT, SWAP, CNOT and CSWAP underpin conventional computing and, Rudolph comments, much of human thought. “Almost everything we ever try to explain or discuss or argue about,” Rudolph says, “is built from applying these sorts of basic logical constructions to facts/assertions/propositions we take to be fundamentally (or self-evidently) true or false.”

So cogitation = computation? This, I must point out, is a debatable assertion. Roger Penrose, for one, objects to the idea that human thought—especially creative thought, especially the creative thoughts of Penrose himself--is reducible to conventional computation. Penrose thinks something unconventional is going on in brains, which might involve quantum computation. And that brings me to the next section of Rudolph’s book.

PETE Boxes

First, a quick clarification. Quantum mechanics often gets credit for catalyzing the computer revolution, because it has helped scientists discover materials suited for building microchips, which are packed with microscopic switches. But these chips aren’t quantum computers, because their computations are based on the old-fashioned logic described above. Quantum computing is based on a radically different logic.

Rudolph eases us into quantum computing by telling us about a box whose workings are “profoundly mysterious.” He calls the box PETE after his friend Pete Shadbolt, a physicist who has built and tested versions of the box. The PETE box is a quantum transistor, or switch, which manipulates not electrons but pulses of light. The PETE box alters balls in peculiar ways. If you drop a single white ball into a PETE box, a white ball or a black ball might pop out the bottom. If you keep dropping white balls in the PETE box, you eventually get an equal number of white and black balls. You might think, What’s the big deal? The box probably just has a randomizing mechanism, like an automatic coin flipper.

But if the PETE box worked that way, you would continue getting an equal mix of white and black balls if you dropped white balls through two PETE boxes stacked on top of each other. Right? But no, things aren’t that simple. If you drop a white ball through two PETE boxes, you get all white balls—but only if you don’t look at the balls as they pass from the first box to the second. Similarly, a black ball dropped through two PETE boxes produces all black balls if you don’t look between the boxes.

The second box seems to have eliminated the randomness produced by the first box. What gives? Rudolph proposes a peculiar model to explain the peculiar properties of PETE boxes. If you drop a single white ball into a PETE box, it enters what Rudolph calls a “misty” state. The white ball has been transformed into a white ball and a black ball, separated by a comma, inside a cartoon cloud. See drawing.

This drawing signifies that the white ball now has an equal chance of being either white or black when/if you look at it. Mistiness represents “a completely new possible state of physical being, and a completely new state of logical being,” Rudolph writes. Conventional physics, and logic, say a ball must be either black or white; it can’t be black and white at the same time. That is “as ridiculous as a cat which is both fat ‘and’ skinny,” Rudolph says.

Another odd fact to keep in mind: As soon as you look at misty states, the mist vanishes, and you see plain old white balls and black balls. Let’s go back for a moment to the experiment in which you drop a white ball into two stacked-up PETE boxes. If you look at the balls as they pass from the first box to the second, you’ll see them emerging as black and white in equal proportions. The second PETE box produces the same output. The two PETE boxes only produce all white balls if you don’t look at the balls as they pass from the first box to the second.

This is a weird sort of quasi-randomness. A single PETE box randomly turns white balls into white and black balls. You can’t know in advance the color of any individual ball emerging from the box, any more than you can know whether a flipped coin will end up as heads or tails. But two PETE boxes leave the white balls unchanged. The randomness seems to depend on whether and how you’re looking. This quasi-randomness recalls Feynman’s description of light bouncing off glass and Susskind’s discussion of spin measurements. Rudolph makes the quasi-randomness more clear and hence even weirder.

Negative Balls

I need to step back a moment to emphasize what a chasm we’ve crossed as we moved from conventional to quantum computing. In conventional computing, the balls falling in and out of ordinary boxes correspond to real things: electrical pulses that you can measure with a voltmeter. Misty states, in contrast, are postulated, inferred. You know what goes into a PETE box, and you know what comes out of it when you look. But you do not know what happens in between, you’re guessing. You’ve invented a model, a set of rules, to explain what happens. But you can’t be sure what happens, because as soon as you look at a misty state, it vanishes, and all you see are plain old white and black balls, which, again, Rudolph represents with cartoon balls or with Ws and Bs.

There is great potential for confusion here, because Rudolph shows the balls in the misty states, and he spells out specific mathematical rules governing their behavior. By adding, subtracting and multiplying misty balls, you calculate exactly what happens when you drop this combination of white and black balls into this combination of PETE boxes and ordinary boxes. After handling misty balls for a while, you might think they are as real as the balls in conventional computers, but they aren’t. A misty state is like the wave sloshing back and forth in the Wikipedia video of the particle in a box. You’re looking at the wave, right there in the video, but as soon as you open the box, the wave vanishes, and you just see the particle. Same with the misty state.

Now things get even odder. If you drop a single black ball into the PETE box, you get a misty state consisting of a white ball and a negative black ball, represented either by a black ball with a minus sign in front of it or with -B. Depending on your combination of balls and boxes, you can get a negative white ball, too, -W; and negative multiples of white and black balls, such as -WB or -WBB. When Rudolph first dropped that -B into his text, I thought, What the…??!! How can a ball be negative? What does that even mean?

Rudolph grants that a “negative ball” makes no sense; it does not correspond to anything that we can see, like a strong or weak electrical pulse in a computer. Negative balls exist only in misty states. A negative ball is what my PEP553 professor Ed Whittaker would call “mathematical skullduggery.” But if you assume there is such a thing as a negative ball, you can predict what happens to balls passing through PETE boxes.

In Rudolph’s system, the minus sign often leads to the cancellation of terms in the equations. A negative black ball, or -B, cancels a positive black ball, B. A negative white and black ball, -WB, cancels out a positive white and black ball, WB. The closest real-world analogue to this cancellation is wave interference; when the peak of one wave coincides with the trough of another, they cancel each other out. But as we know, quantum interference, or cancellation, has no real-world analogue.

Rudolph goes on to show what happens when you process various misty states through combinations of PETE boxes and conventional, non-PETE boxes. To make sure we grasp how his system works, he gives us exercises that require calculating the output of a particular set of boxes given a particular input. The calculations for the most part are ones I learned eons ago when studying algebra, not linear algebra but plain old algebra. For example:

W x (W - B) = WW - WB.

Or: (W + B) x (W – B) = WW - WB + WB - BB = WW - BB.

Note how the positive and negative WBs cancel out in the second example. This cancellation of terms within misty states is crucial to quantum computers; it gives them their peculiar power. The negative balls enable a quantum computer to perform more calculations and hence to test more possibilities with a given number of balls than a conventional computer can. But when you look at the output of the PETE boxes, you only see ordinary white balls or black balls. You never see “negative” balls.

The Square Rule

By now, you might see parallels between Rudolph’s model and more conventional quantum accounts. The balls separated by commas in a little cloud are in superposition with each other. A misty state consisting of, say, a white ball and black ball separated by a comma is a qubit, the equivalent of a bit in a conventional computer. In answer to a yes-or-no question, a qubit says: yes/no. As soon as you look at a misty state, it vanishes, or collapses; that is the infamous measurement problem.

On page 82, Rudolph tells you how to calculate the probability of what you will see when you look at balls in a misty state. The misty state shows all the possible configurations of the balls you have dropped into PETE boxes. What is the likelihood that you will see one particular configuration when you look?

Let’s say you have a misty state consisting of two white balls and three black balls, each separated from the others by commas. Only one ball can emerge from this misty state. You might assume that the odds are 2/5 that you’ll see a white ball and 3/5 that you’ll see a black ball, but you would be wrong. Here is the rule for calculating the odds of seeing any particular configuration in the misty state:

Square the sum of the times the particular configuration appears and divide that square by the sum (over all the configurations that appear) of the squares of the sum of the number of times that they each appear.

This rule isn’t as complicated as it sounds. Let’s take the misty state consisting of two white balls and three black ones. The white ball appears two times, so you square that 2 to get 4. The black ball appears 3 times, so you square that 3 to get 9. The probability that you will see a white ball is 4/(4 + 9) or 4/13; those odds are lower than if there were no square rule. The probability that you will see a black ball is 9/13, higher odds than if there were no square rule. Note that if you add 4/13 and 9/13, you get a probability of 1, because you must see something when you look.

I’ve encountered the square rule before in Feynman, Susskind and Griffiths. The square rule helps you get rid of negative numbers, so you end up with positive probabilities. But nobody knows why the square rule works. “It’s all part of the mystery that you are going to solve for us one day,” Rudolph says. That “you” refers not to an old fogy like me but to the youngsters who are Rudolph’s primary target audience. It’s hard enough for me to see the mystery, let alone solve it.

Quantum Thieves and Pseudo-Psychics

Having set up his misty model in Chapter I, Rudolph employs it in Chapter II to demonstrate what a quantum computer can do. He walks us through two thought experiments. In one, you are a thief trying to steal bars of gold from a bank. The bank has many vaults, each of which contains eight apparently gold bars. In some of the vaults, all the bars are fake. In other vaults, four bars are gold and four are fake.

Each bar is protected by a three-letter code consisting of a combination of black and white, or B and W. B and W can be combined to form eight possible three-letter codes: BBB, BBW, BWB, BWW, WBB, WWB, WBW, WWW. The only way to know whether a bar is genuine and hence that you are in a vault containing real gold is by inputting one of the eight possible codes for that bar into a computer in the vault. You have limited time to guess the code for each bar. If you make random guesses, you will run out of time unless you are very lucky.

If you have PETE boxes with you, however, you can enter black and white balls in misty states into the vault’s computer. In this way, the computer can check all possible codes of all eight bars at once, determining whether any of the codes corresponds to a genuine gold bar. I’m omitting many details of the experiment and all the calculations. The crucial fact is that the PETE boxes, by putting balls in misty states, determine whether you are in a gold-containing vault in a single step.

Rudolph’s other thought experiment features two charlatans, Alice and Bob, trying to convince a skeptic that they have psychic powers so they can win a $1 million prize. [1] Alice and Bob are isolated in two separate rooms with no way of communicating. A tester flips a coin in each room. Alice and Bob win $1 million if both coins come up tails and Alice and Bob both say “black.” Actually, their task is a bit more complicated than that. Let me explain. According to classical physics and probability theory, Alice and Bob have a one in four chance of both saying black when their coins both come up tails. Their challenge is to beat these odds after multiple sessions.

How can Alice and Bob win? There is nothing they can do about the coin flips, which are truly random; Alice can’t know how Bob’s coin has landed, or vice versa. Nor can either know how the other will answer. They can nonetheless beat the odds with the help of PETE boxes. Before the game, Alice and Bob place many pairs of black and white balls into PETE boxes, so that each pair of balls occupies a single misty state. Alice and Bob then separate the balls without looking at them, so the separated balls remain in the misty state.

When Alice and Bob open their PETE boxes, the mist surrounding the balls vanishes, and Alice and Bob see either a black ball or a white ball. If the ball is black, they say, “Black.” After many repetitions, Alice and Bob will both say black when both of their coins say tails at a rate significantly better than chance--or chance in a non-quantum world. Alice and Bob seem to have psychic powers, but they have just exploited the power of entanglement, which occurs when two things occupy a single misty state. When you look at one ball and see its color, you determine the color of the other, entangled ball, even if it is far away.

This “nonlocal correlation,” which seems to violate conventional causality, is what Einstein derided it as “spooky action at a distance.” Common sense told Einstein that physics must be local, meaning that causes and effects propagate in an orderly fashion through space, never exceeding the speed of light. He also assumed that particles have fixed properties all along, not just when you look at them. But in 1964, John Bell proved that entangled particles do not behave as Einstein hoped. Spooky action, which allows Alice and Bob to win $1 million, is real. And it depends in part on the square rule, which changes the odds of what you see when you look, as explained above.

Rudolph’s two experiments help me grasp the difference between superposition and entanglement. The simplest form of superposition involves a single object, such as a ball, in a misty state consisting of two or more possible states of the ball. The gold-bar scenario demonstrates how quantum computing exploits this simple superposition.

Entanglement involves two or more balls in a single misty state. Rudolph shows that this misty state cannot be reduced to separate misty states for each ball, just as a prime number cannot be expressed as the product of two smaller primes. Entanglement, although related to superposition, is a riddle in its own right, one crucial to certain quantum technologies. You can have entanglement without nonlocality but not vice versa. Nonlocality comes into play if the entangled balls are separated.

Rudolph does a good job explaining why entanglement does not allow instantaneous, faster-than-light communication in violation of the theory of special relativity. Entangled, nonlocal correlations are still subject to randomness, which prevents Alice and Bob from sending coherent messages to each other.

Rudolph acknowledges that the rules for both of his thought experiments are “contrived,” that is, oddly complicated and un-straightforward. But the contrivance makes an important point: A quantum computer isn’t a magic wand; it gives you a specific advantage over conventional computers for solving specific problems. Misty computers do not “compute the uncomputable,” Rudolph writes. “They just make previously highly infeasible problems tractable.” And in both of Rudolph’s thought experiments, the magical powers of the PETE boxes depend on those mysterious negative balls.

Einstein Versus Pooh Bear 

Studying something hard affects me physiologically. When I think I’m on the verge of comprehension, especially after multiple failed attempts, I start holding my breath, as though I’m in a room filled with delicate glassware. I do a lot of breath-holding while studying Chapter III of Q Is for Quantum, in which Rudolph confronts the question of whether a misty state is “physically real.” Some physicists say it is “only a tool for calculation” that does not correspond to anything in nature. Others say a misty state is real even though we can never look directly at it.

Rudolph’s grandfather drew attention to the puzzle of mistiness with his famous cat experiment. Quantum mechanics suggests that if you put a cat in a box with a radioactive particle and a bottle of poison gas, the cat is both dead and alive before we look in the box. Are we really expected to believe that? Rudolph lets Einstein and Pooh Bear take different sides of the debate. This section of Q Is for Quantum is light-hearted, whimsical, irreverent. Pooh versus Einstein--cute! And yet this section is deadly serious.

Einstein focuses on the fact that misty states give you only the probability of finding a certain outcome when you look. In classical physics, a probability estimate—for example, that a coin has a fifty percent chance of landing heads up--doesn’t correspond to something real. It’s just a function of your ignorance of unknown factors, commonly called “hidden variables,” that determine precisely how the coin lands. If you knew all those variables, you could predict how the coin lands every time.

Einstein and Pooh argue about food, which matters very much to Pooh. Einstein asserts that a misty state representing a combination of possible food items must be an incomplete approximation of reality, not a precise representation of it. Pooh, in his humble, modest way, shows Einstein that he is wrong. Pooh presents a mathematical proof that the misty state tells us all there is to know about those food items; no non-misty description can account for what you see when you finally look at the food. In my terminology, Einstein argues that a misty state is a synecdoche, because it leaves something out. No, Pooh says, a misty state is not a synecdoche, because it tells us everything there is to know.

Einstein’s argument resembles the one he made in his famous EPR paper, written with Boris Podolsky and Nathan Rosen in 1935. Pooh’s counterargument is a stripped-down version of the PBR theorem, which Rudolph constructed with Pusey and Barrett in 2012. (Pooh Bear, PBR, get it?) Some experts see the PBR theorem, together with Bell’s theorem, as proof that misty states are real.

The Einstein/Pooh debate confronts paradoxes that have baffled me since the beginning of this project. For 10 months now, I’ve been looking at quantum mechanics from this angle and that, narrowing my eyes and opening them wide, trying to make the theory come into focus. Now and then it does, and I allow myself a little, “Aha!” Then I realize I’m looking at it wrong, and it gets blurry again.

The problem is that quantum mechanics itself is blurry--or, rather, the world it describes is blurry. Our eyes, our brains, find the blurriness maddening; we are almost physiologically compelled to reject blurriness and to see, or try to see, in vivid, precise detail. But if we see clearly, we aren’t seeing clearly. When things are in focus, they’re not in focus.

Quantum blurriness is special. The world usually appears blurry to us simply because we don’t know or care what’s going on; the blurriness stems from our ignorance and apathy. Quantum blurriness is mathematically precise, and it stems not from ignorance but from rigorous, hard-won knowledge. Quantum blurriness isn’t subjective, the result of our failings; it’s objective, out there. That’s the hard lesson that Pooh Bear conveys to an incredulous Einstein. 

Magic Tricks

Physicists are like magicians who love revealing how they pull off their tricks. During my quantum experiment, I’ve learned lots of tricks: derivatives and integrals, vectors and matrices, trigonometric functions and complex conjugates. Before I learn a trick, even a relatively simple one, like a derivative, it looks like magic, because I have no idea how it works. Then, ideally, I see the sleight of hand behind the magic. I can’t necessarily perform the trick myself, but I see the sequence of steps leading to Abracadabra! Learning a trick can be both exhilarating and anticlimactic. The magic vanishes, reduced to a mechanical process, an algorithm, something a machine can do.

But here is what keeps happening during my quantum experiment. If I learn the trickery underlying quantum calculations, the magic doesn’t dissipate. It’s as though the magician has showed me exactly how he pulls a rabbit out of the hat. The magician may even help me pull a rabbit out of a hat myself, albeit clumsily. But I still don’t get the trick! I don’t know where the rabbit came from.

That happened with Susskind and Griffiths, and it’s happening again with Rudolph. Rudolph’s mathematical trickery is much simpler than that of Susskind and Griffiths. But even after I perform calculations with lots of white and black balls in misty states, I’m mystified. I find myself staring at those negative balls, -W and -B. I suspect they are the key to the quantum magic, but I don’t know what they are, or how they work.

Rudolph is still mystified too. That’s why he’s such a good teacher. The Lancaster system of education, which my study buddy Dean admires, assumes that teachers become so accustomed to their material that they can’t relate to rookies. Terry Rudolph doesn’t have that problem. Although he’s mastered the mathematical trickery, he hasn’t become habituated to it.

Rudolph’s own theorem “proves” that white and black balls that have tumbled through PETE boxes aren’t white or black before we look at them. But like John Bell, Rudolph has a hard time believing the implications of his theorem. He admits to being in a state of “cognitive dissonance” when he contemplates quantum mistiness. Like Einstein and Bell, Rudolph clings to the idea that the world is as vivid when we’re not looking at it as when we are. “I cannot escape my naïve realistic belief that there is stuff there,” he writes, “and it has physical properties of some form independent of my beliefs.”

Earlier in this chapter, I mention Roger Penrose’s argument that human cognition is not reducible to conventional computation. Penrose relies heavily on Gödel’s theorem, which is a logical argument about the limits of logic, a proof about the limits of proofs. Escher, the artist, delighted in depicting these sorts of self-referential paradoxes. His drawings take you up a staircase step by logical step, up and up and up, back to your starting point—implying that all logic, like all math, is circular, a tautology, reducible to 1 = 1.

The argument of Pooh Bear, the stand-in for Rudolph and his PBR buddies, has this same paradoxical quality. Logic leads Pooh inexorably to a nutty conclusion: Misty states aren’t just useful fictions, they are real. That implies that negative balls are real, wave functions and probability amplitudes are real; they are not just mathematical skullduggery. I distrust Pooh Bear’s logic, although I can’t find a flaw in it.

I’ve been confused from the start of this project about whether quantum mechanics is deterministic or probabilistic. Which is it? It can’t be both. But it is both. Quantum mechanics is deterministic if you accept that, when we’re not looking at things, their properties are undetermined. Accept this indeterminacy, and quantum mechanics is perfectly deterministic. This past leads to this present, and this present to this future. Remember Leonard Susskind’s minus first law, which decrees that information is never lost? Information is never lost, but the information is blurry. The world lurches deterministically from blurry state to blurry state—or, to use Rudolph’s terminology, from mistiness to mistiness.

Fog

Fog shrouds the Hudson this morning. An invisible ship bellows. I can’t see the Freedom Tower, but I know it’s there, across the river. Will we ever find something as concrete as the Freedom Tower behind the quantum mist? Something definite, fixed, un-misty? Its variables there for all to see, like the blinking, colored lights studding the Freedom Tower’s spire?

I hoped Q Is for Quantum would give my quantum project a happy ending: Schrodinger’s grandson helps the old humanities professor get quantum mechanics. My confidence surged as I carried out calculations with Rudolph’s black and white and positive and negative balls. But now I’m not sure what I’ve learned. I accept that the meaning of quantum mechanics is embedded in the math. That’s why I’ve spent so much time staring at Schrodinger’s equation(s), Dirac’s bras and kets, Rudolph’s black and white balls. [2]

But once I get the math, I still face the task of translating that knowledge into terms that I can understand. That’s hard, because my mathematical “understanding” is fleeting. When I look back at my notes on Q Is for Quantum, at all the calculations I carried out with black and white balls, WWs and WBs and -WBs, they seem strange, as if written by someone else. The part of me that scribbled those notes, my mathematical self, remains alien, ghostly, unintegrated with my common-sense, English-speaking self.

Even my faith in English, in ordinary words, is wobbling. Words are my medium. Day after day I use words to get shit done—to get my students to grok the parable of the cave, to mollify my girlfriend, to give myself pep talks. I’m competent with words, but how much comprehension do they give me? Like numbers, words never quite capture the world. Even the most vivid description of nature, like Emily Dickinson’s “A bird came down the walk,” resembles a black and white photo that a dog chewed. A shitty synecdoche.

Someday a young person, perhaps inspired by Q Is for Quantum, might find a more sensible theory beyond quantum mechanics, a theory unburdened by uncertainty, subjectivity, nonlocality. That seems to be Rudolph’s hope, as it was Einstein’s and John Bell’s. My current guess is that if we find such a theory, it will eventually dissolve into mist once again. The mist might differ from the quantum mist, but it will still be mist. It’s mist all the way down.

Notes

1. Magician/skeptic James “The Amazing” Randi established the “One Million Dollar Paranormal Challenge” in the 1960s, which promised a $1 million prize to anyone who could demonstrate psychic powers. The prize ended in 2015 without anyone winning it. Randi died in 2020.

2. After my first time through Q Is for Quantum, I sent Terry Rudolph a bunch of questions about it, which he answered. He posted our exchange on the website for his book.

Table of Contents

ABOUT MY QUANTUM EXPERIMENT

INTRODUCTION
Old Man Gets More Befuddled

CHAPTER ONE
The Strange Theory of
You and Me

CHAPTER TWO
Laziness

CHAPTER THREE
The Minus First Law

CHAPTER FOUR
I Understand That
I Can’t Understand

CHAPTER FIVE
Competence Without Comprehension

CHAPTER SIX
Reality Check

CHAPTER SEVEN
The Investment Principle

CHAPTER EIGHT
Order Matters

CHAPTER NINE
The Two-Body Problem

CHAPTER TEN
Entropy

CHAPTER ELEVEN
The Mist

CHAPTER TWELVE
Thin Ice

CHAPTER THIRTEEN
Irony

EPILOGUE
Thanksgiving

THANKS

DISCUSSION