HOBOKEN, HALLOWEEN, 2024.  Things were worse four years ago, I tell my stressed-out friends and students. We were facing the prospect of four more years of you-know-who, plus the Covid pandemic was killing over 1,000 people a day. Below is a revised, updated version of a column I wrote on how I tried to stay sane. –John Horgan

Last spring, after the Covid pandemic scuttled my summer plans, I decided to pursue a long-standing fantasy: learning, or trying to learn, quantum mechanics, math and all. This project has served as a refuge for me lately.

Some physicists claim that quantum mechanics, if you squint at it from the right angle, can be consoling. Noting the crucial role of observation in quantum mechanics, John Wheeler proposed that our existence matters, we were meant to be here; we live in a “participatory” universe.

But the comfort I derive from quantum mechanics stems not from its feel-good implications but from its obscurity, even opacity. I enjoy being immersed in abstractions of abstractions, the meanings of which elude me. As I struggle to grasp wave functions and eigenvectors, I forget about my troubles, and the world’s.

Every now and then, I think I’m getting somewhere, which gives me a fleeting thrill. That happened recently as I muddled through The Structure and Interpretation of Quantum Mechanics by philosopher R.I.G. Hughes, a 1989 book recommended by science writer Jim Holt.

Hughes contends that the mathematics of quantum mechanics isn’t as arbitrary and ad hoc as it seems. Although his treatise baffles me, especially when it devolves into strings of arcane symbols, it gives me moments of (pseudo)illumination.

Take Hughes’s argument that Hilbert spaces are peculiarly suited to representing quantum things. Hilbert spaces aren’t spaces in the ordinary sense with width, length and depth, like your bedroom or the sky. They are habitats for vectors, which you can think of as arrows of a specific length pointing in a specific direction. Vectors can describe anything from a quark to your lover’s face.

Let’s say you represent the state of an electron with a vector, V, in a Hilbert space with three axes, x, y and z. (Remember, these aren’t the dimensions we live in.) Then V equals the sum of three shorter, orthogonal vectors along the x, y and z axes. Call these vectors Vx, Vy and Vz. According to the rules of vector arithmetic, V = Vx + Vy + Vz.

I’m getting this, but not really, when Hughes says the Pythagorean theorem is the key to understanding Hilbert spaces. As always with quantum math, there’s a twist. Instead of the Pythagorean theorem I’m familiar with, Hughes presents one for three dimensions. “Pythagoras’ theorem tells us,” Hughes writes, that |Vx|² + |Vy|² + |Vz|² = V². Really? The Pythagorean theorem works in 3-D? I never knew that.

Those vertical lines mean you’re taking the absolute value of each vector. Why? Shouldn’t squaring the vectors make them positive? Ordinarily, but quantum vectors are often expressed as complex numbers, which contain imaginary numbers, based on the square root of negative one. That’s why you need those vertical lines. Why imaginary numbers? Because they work, is the short answer. Maybe someday I’ll learn the real answer.

Anyway, these squared vector values, |Vx|², |Vy|² and |Vz|², represent the probabilities that the electron will be in a particular state when you measure it. It must be in some state, so all these fractional probabilities must add up to one. So |Vx|² + |Vy|² + |Vz|² = V² = 1. Getting the probabilities to equal one is called normalization.

Now everything is starting to make sense. All these seemingly arbitrary mathematical rules are based on probability theory plus the Pythagorean theorem--which I learned in my childhood!

Sadly, these epiphanies never last. One problem is that Hughes keeps undercutting himself. For example, rather than trying to convince me that imaginary numbers are real, he mischievously suggests that real numbers are imaginary.

“If the inclusion of imaginary numbers is worrying,” he writes, “it is worth considering the sense in which a negative number, –6 say, is real—or, come to that, the sense in which 6 itself is real.”

No wonder that I keep lapsing back into befuddlement, or, at best, a dim quasi-comprehension that eggs me on. Moreover, reality keeps intruding on my quantum reveries. That happens when, eager for fresh air, I stuff Hughes’s book into a backpack, pull on a face mask and leave my 11th-floor apartment.

The elevator stops a few floors below mine, and a young man in a muscle shirt steps on, wearing no mask. Our building requires masks in elevators. I say, You can’t get on without a mask.

Glaring at me, he s-l-o-w-l-y takes a mask from his pocket and pulls it over his face, saying, Happy?

No, I reply, you’re supposed to put it on before you get on the elevator.

My blood is still boiling when I arrive at a park near my building, a grassy pier that juts into the Hudson River. I find a shady spot between several chattering young families, and soon I’m struggling so mightily to comprehend wave functions that I forget about the jerk on the elevator, the election, Covid.

I occasionally look up from Hughes to glance at the Hudson, its surface crisscrossed by waves, and at the exuberant moms, dads and kids around me. It’s the kind of sunny, sparkly day that makes it hard to imagine all the misery out there, all the anger and fear.

I think: What if you could construct a wave function for this fucked- up, real-imaginary country? What would it show? It would show us suspended in a superposition of many possible states, from okay to very, very bad. The election will serve as a test, a measurement, that collapses the wave function of America and reveals who we really are.

If you try to understand quantum mechanics, Richard Feynman warns, you will fail, you will end up going “down the drain.” Hughes and other sages reject Feynman’s stance as defeatist, but I find it heartening. I like the thought of slipping down the drain of a bottomless mystery as the world collapses over and over again around me.

Further Reading:

If you wonder what I learned from my quantum studies, check out my free, online book My Quantum Experiment.