“So is the universe made of tiny springs, or isn’t it?”
I was very excited on the first day of my first course in quantum field theory. I had no idea what quantum field theory was all about (obviously I wasn’t particularly well-prepared), but I had the strong suspicion that it was going to be awesome. Quantum field theory just looked so cool: all those crazy squiggly diagrams and weird symbols. Apparently they constitute the most important and powerful language for describing the interactions (and even existence!) of matter. So I was stoked. I was ready for a revolutionary, mind-blowing academic experience.
To top it all off, it’s hard to imagine having a more qualified professor than the one I had. He was an excitable, Soviet-school Russian with thick glasses, crazy hair, and a name that rhymes with “Einstein”. A decade or so earlier he had been given the most prestigious award in theoretical particle physics for inventing something called a “penguin diagram”. And even though eloquent English wasn’t his strong suit, I was excited to be his pupil for the next semester.
Here’s how the first few days of lecture played out (I have omitted most of the articles to give you an authentic sense of his Russian-accented English):
Day 1: “In classical mechanics, evolution of some set of coordinates and momenta is dictated by Lagrangian…” [...1 hour of equations...]
Day 2: “In quantum field theory, Lagrangian describes evolution of some field at given point in spacetime…” [...1 hour of equations...]
Day 3: “In simplest case, one imagines that field obeys the following rules…” [...1 hour of equations...]
After about two weeks of this, I was ready to revolt. I wanted to jump up and yell “What the hell are you talking about?? What is this ‘field’ that you keep going on about?” It was the most frustrating thing in the world to me. The man had spent about ten hours writing down equations that described some “field”, but he wouldn’t tell me what those equations actually stood for. What was the field? Was he claiming that some kind of physical material filled all of space? Didn’t we abandon that idea in the 19th century? Was this all an elaborate metaphor?
What I had wanted from this class was for it to start with some kind of positive statement about the makeup of the universe, and only after that to develop a mathematical description of the statement. For example, a class on electromagnetism starts with the statement “All physical objects are made of electric charges which push and pull on each other”. After the students have had a little time to adjust to this sort of mind-blowing idea, then you begin developing a descriptive theory of how those charges push and pull on each other. But my quantum field theory class had started right in with the descriptive theory without telling me what it was describing!
It seemed to me that I wasn’t asking for much. Just tell me what we’re talking about!
Anyway, after a couple weeks of lecture I abandoned any hope that the professor was going to eventually get around to telling me what he was talking about. It was apparently taboo to talk about the subject except through highly-encoded mathematics, something like the rules about pronouncing the name of God. So I developed sort of a weird attitude toward the class and adopted the following approach. Every day I would show up to lecture, without bringing a pencil or paper, and sit in the front row. As the professor proceeded through various long derivations and my classmates furiously took notes, I would just stare at the equations on the board and try to imagine what they could be describing. For me, quantum field theory became a pure exercise in imagination. For three hours a week I would just sit quietly, stare at the blackboard, and imagine the strange world that it must be describing.
Some of these days were pretty boring, but others were absolutely fantastic. Like the day somewhere in the third week when I decided that the professor was telling me the universe was made out of tiny springs. He wrote some equation on the board that described his “field”, and it occurred to me that this equation was describing a “fabric” of tiny springs, all tied together in an enormous network that filled all of space. It seemed to me that he was saying that space-time itself is made of some kind of springy material. And, even more shockingly, that everything we think of as “matter” is just some degree of oscillations in this field.
It was an incredible moment for me. I can only imagine how I must have appeared to my classmates, who could see me sitting at my empty desk, staring wide-eyed at the board, and running my hands through my hair in consternation. It seemed to me that this was the most profound statement that anyone had ever made in a science class. Namely, that the only thing that truly exists in the universe is this infinite fabric, which fills all of space. The fabric is always rippling and contorting; these ripples are what we call “particles”. Some kinds of ripples are pulled together and others are pushed apart. When a whole bunch of them come together and shift jointly across the fabric, we call that a macroscopic object: like a person or a planet. A single particle is a solitary, quantized excitation of the fabric; that’s why tiny particles can pop into and out of existence when the fabric is given a strong, concentrated jolt.
What a crazy idea! Was it true? Did people really believe this? And if so, why weren’t they talking about it all the time?
As the semester progressed, the description of the various fields got a lot more complicated and abstract, and my picture of the “cosmic fabric” got blurrier (apparently it required me to imagine a bunch of different kinds of “fabrics” all occupying the same space simultaneously). I managed to survive the class due to some kind grading (I’m guessing that everyone was given an A), but I certainly never developed the technical competencies that I was supposed to. And I never lost the urge to stand up in the middle of class and shout “So is the universe made of tiny springs, or isn’t it??”.
The class ended with a funny scene. Our take-home final exam required us to perform some long calculation (the amplitude of some process, I guess). Despite my earlier slackery, I gave it my best effort, and finally concluded that the answer was infinity. That seemed strange to me, but I couldn’t see where I had made a mistake, so I went to turn in my exam to the professor. I handed him the paper and he looked it over briefly, then said “Your calculation is good, but there is a finite part, too.”
I gave him a blank look, so he reiterated: “You correctly calculated the main term. But the answer is infinity plus a finite part. You need to calculate the finite correction also.” And he handed the test back to me.
Now, I had never heard of a situation where . Apparently there are some very basic things about quantum field theory that eluded me. But my solution to this dilemma was to wait until the professor left for the day and then walk up to his office and slide the same exam paper under his door. And thus ended my formal training in quantum field theory.
In the end, I only had one purpose in telling this long story. And that’s to say that I finally feel like I have been given a satisfactory answer to my questions about “what is quantum field theory actually describing?”. They came from Freeman Dyson’s phenomenal essay called “Field Theory”, written in 1953 and included in his (strangely-titled) book of essays From Eros to Gaia. I’ve said before that “if you’re in the mood to read a great book about physics, there is no author I can recommend more highly than Richard Feynman.” But after reading through From Eros to Gaia, I might have to amend that statement. Perhaps Freeman Dyson just happens to have a similar way of thinking to me, but I found his essays extremely clear and elucidating. They are great largely because he doesn’t shy away from attempting physical explanations of deep questions, like “what is energy?” or “how do I physically picture the quantum field?”.
The next post (which will be up within a day or two, I promise is up!) will be almost entirely composed of quotes from Dyson’s “Field Theory” essay, which I greatly wish I had read before I walked into that quantum field theory class. Maybe if I had then I could have concentrated on the math and understood why .
UPDATE: “Our stability is but balance” — Freeman Dyson on how to imagine quantum fields.