In “The Algebra’s Apprentice” posts I begin writing here, I intend to faithfully tell the story of how I met Macfarlane’s algebra and the journey I have been on ever since. This is an ongoing work in progress, a way to share the narrative arc of a still unfolding exploration.
A Quantum of Curiosity That Killed the Cat – Or Did It?
Many years ago, I first encountered the double slit experiment in a high school class. I found the whole thing so curious and asked so many questions, that my kindly teacher gave me a copy of “In Search of Schrodinger’s Cat”. I read the book cover to cover and thus became determined to study physics at university. Mainly, it was the paradoxes that pulled on me; they seemed to suggest reality might be stranger, more complicated, and more beautiful than I had previously considered.
A Long-Awaited Day
When I began college, I was disappointed to learn that (given the small program size), the quantum mechanics course was only offered every other year. That would place the class during my senior year. I was so eager to learn about the topic that I practically begged the professor to just be allowed to sit in on the class a couple years early. I was so happy that he agreed!
And so, it was with eager anticipation that after years of dreaming about the day I’d learn about this most mysterious subject, I finally sat in a classroom, pencil in hand, captivated by every word.
A Seemingly Innocent Question
Bras and kets, infinite square wells, curious mathematical tricks… At every turn, it was all so tantalizing. But when the professor talked about how the complex phase of the wavefunction allowed constructive and destructive interference, something about this caught me off guard.
“Why complex numbers?” I wondered to myself. Surely, this must be a beautiful and fundamental underlying symmetry of nature! And so, I raised my hand and asked the question:
“Why is the phase complex? What physical thing does that phase represent?”
I felt sure it ought to be something truly and deeply important, and so I was underwhelmed by the reply that real numbers don’t provide the capacity to add constructively AND destructively in the way observation had substantiated.
This wasn’t wrong exactly… But it wasn’t a satisfying response, nor did it truly answer my question. What was the physical meaning in this fundamental symmetry?
A Slow Slide Into Paris Syndrome
It didn’t really start all at once.
At first, I was like an eager tourist arriving at a long sought after destination: utterly enchanted and pulled in a dozen different directions to see all the landmarks I had long dreamed of witnessing firsthand.
It began with a seemingly innocent question:
“Why complex numbers?”
And then another:
“What happens when we change reference frames in flat spacetime?”
Like a snowball picking up speed, aggregating ever more mass, the questions continued piling up:
“But what about curved spacetime? We can’t seriously just send particles whizzing off into the universe and then bring them back and expect that they just add as if they had never left. Won’t things get all turned around and upside down? How can complex numbers track all that?”
“What about the relativity of simultaneity? How do we even know how to integrate a volume over a time if we can’t agree which things even occur at the same time in different reference frames?”
“Noether’s theorem tells us that symmetries yield conservation principles. The fundamental forces of the Standard Model all have their symmetry groups. So, I just know that quantum phase shouldn’t just be a tool to Fourier series anything we wish to model. The symmetry of it should correspond to something more fundamental… So, why doesn’t it?”
And everywhere I walked in that quantum city of lights, I gradually found myself with feelings I had never had about physics before
Yes, I had been confused and hadn’t understood a great many things in the past, but this was different. It wasn’t the temporary difficulty of an idea not quite yet clicking into place or the intrigue of the next concept beckoning me to learn about it. No, it was a whole host of growing contradictions I couldn’t resolve within myself.
And they kept revisiting me over and over again, my mind drawn to them against my better judgement.
There was a deep wrongness in it all.
I couldn’t tell why or what the correct alternative might be, but it was the very first time that I felt something snag in a way I just couldn’t forget.
A Most Productive Dissatisfaction
It might all sound like I was terribly hopeful and then just deeply disheartened by the whole affair, but that isn’t the whole of the story.
Because I wasn’t just disappointed. One might even say I was somewhat incensed.
I recall walking back after class one day with the friend I would eventually marry and going on this diatribe about how cross I was about it all.
He listened patiently (as he is wont to do), unbothered, as I carried on about how we could just not even care about how our reference frames change from underneath us and how I found it deeply suspicious that the phase of the wavefunction didn’t relate to something fundamental and physical like the changing orientation of our reference frame.
It soon became a bit of a habit, really. Me finding something to get all fired up about. Something where the pieces just didn’t quite fit. And then my poor patient husband listening to me carry on about it.
There and Back Again
When that quantum mechanics course came to an end, there were other classes to take, other structures to sketch, and other mysteries to meet.
But I didn’t meet them the same way I once did.
Because something within me had changed, and wherever I walked, questions bubbled up unbidden, and instead of starstruck reverence, I found myself drawn towards the cracks: the inconsistencies, the paradoxes, the places where models frayed at their edges.
And though it might have looked like mere contrarianism from the outside, I wasn’t trying to be pedantic or difficult. Rather, I had begun experiencing a budding conviction that the things which surprise us, the places that don’t mesh… They’re a signal. A signal that things don’t behave how we think they do. They’re a sign that something far more interesting is going on under the surface.
The Empty Places
Here is where I’ll leave my story for tonight. Not somewhere settled. Not somewhere whole.
What I will give you is an invitation to walk with me further. To witness the questions that pulled on me and meet an algebra that time forgot.
Until next time.
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