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 $\{x_i\}$ and momenta $\{p_i\}$ is dictated by Lagrangian…” […1 hour of equations…]

Day 2: “In quantum field theory, Lagrangian describes evolution of some field $\phi$ at given point in spacetime…” […1 hour of equations…]

Day 3: “In simplest case, one imagines that field $\phi$ 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??”.

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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 $\infty + \textrm{const.} \neq \infty$.  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.

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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 $\infty + \textrm{const.} \neq \infty$.

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UPDATE:Our stability is but balance” — Freeman Dyson on how to imagine quantum fields.

1. August 30, 2010 6:14 pm

I studied QFT for the first in a class full of PhD students who have done QFT before, so their questions and remarks left me, by the end of each class, wondering what the heck happened!!

& Thanks for sharing your experience

September 4, 2010 12:59 pm

Great post! If it turns out the springs really are the best mechanical metaphor of the composition of reality, then you will be very famous one day

I am also intrigued at your intuition that all of reality is composed of a universal “something”, with different objects simply reflecting different “oscillations” of this universal something. As a student of philosophy, I have learned over the years that every new idea is basically an old idea envisioned in new, more mature way, and you might find the 17th century Dutch philosopher Spinoza very interesting for this reason:

en.wikipedia.org/wiki/Spinoza#Substance.2C_attributes_and_modes

In the middle of the 1600s, Spinoza was speculating that all of reality was comprised of a single, indivisible thing he called “Substance”, “God”, or “Nature”. Furthermore, he argued that there is no ontological difference between thought and matter–they were different “modes” (or “oscillations?”) of this single substance. Everything was fundamentally the same thing: a chair, a house, my body, Nationalism, an apple, God, a racoon’s consciousness, etc.

That’s why I love modern physics, is see it as a blend of philosophy and science.

And if you can ever prove not only that all matter is an oscillation of springs, but that thought, consciousness, and self-awareness is also an oscillation, I would be thoroughly amused.

Which reminds me–I have a book on Hawaiian Shamanism, in which they author tries to argue that what we call “souls” or “spirits”, that is, the energy that animates us, is actually a vibration of a “something” inside us. When you walk into a room and there is a bright and cheerful person, whose spirit is vibrating at a higher frequency, we unconsciously feel happier too. Likewise with some one lethargic and depressed.

I read somewhere else that we are attracted to mates, in part, because our two spirits resonate with each other and share roughly the same frequency. This author did not mean that this “spiritual vibration” reflected a temporary mood, but was rather something more intrinsic and stable to our personality. This oscillation was what made people different from one another, and was what some people refer to as “auras”.

But now I sound like a hippie new-ager, so I’ll stop myself. Keep up the good work!

September 8, 2010 12:17 am

Hi Daniel,

First of all, I should make clear that the ideas in this post aren’t “mine”, even if I am sharing my particular perspective on them. The concept of the quantum field belongs to the heroic pioneers of quantum mechanics and quantum field theory: a rare set of men and women with tremendous imagination and great mathematical ability. I should certainly not be mistaken for someone who has contributed anything to field theory!

That said, I do appreciate your philosophical perspective! The Spinoza reference, I think, is particularly appropriate. Physical science is essentially built around his paradigm. Namely, that the only thing that exists in the universe is physical matter, which follows deterministic laws of motion. To a scientist, what we call “mind” or “consciousness” or “spirit” is just an emergent property of how those physical objects push and pull on each other.

In other words, scientists operate under the assumption that man (like everything else) is just a collection of atoms and nothing more. Those atoms comprise his brain and his organs and everything else about him. Each atom is a “dumb” object in the sense that it simply responds to the motion of all the atoms around it, but somehow all those dumb atoms cooperate to create a man who is (maybe) not dumb.
Call this perspective absolute materialism: scientists assume that everything in existence can be explained by material objects that act through deterministic laws.

At first sight, this assumption sounds ridiculous and over-simplistic. But it has gotten us so tremendously far that we (scientists) have come to have enormous confidence in it. And really, every time it proves to be correct again, we marvel at how something so amazing as life could arise from something so simple as atoms pushing and pulling on each other. Or, in the language of this post, from something so simple as oscillations in a vast network of connected springs.

September 10, 2010 5:47 am

Ah, but a true Spinozist paradigm would actually be very different from the modern scientific one. Think how crazy it would be (and why your talking about oscillations of springs got my gears running):

In Spinoza, mind does not “emerge” from matter. There is no primacy here. For the same reason, it could not be said that matter is an emergent property of thought, as the 18th and 19th century idealists argued. Spinoza’s idea was that this universal Substance is both equally mind *and* matter, and neither emerges from the other.

Of course, now comes the question–“but is it true?” Well, probably not. But likewise, although the materialist foundation of modern science contributes a great deal to what I consider a “true” interpretation of reality, I am hesitant to push it to its furthest conclusions. For instance, if all consciousness really was an illusion caused by our brains (as many influential physicists and cognitive psychologists argue), then there is nothing even remotely close to what we could call free will.

And from what I hear, you are attached to that idea!

Haha, it’s ok, so am I

November 18, 2011 10:55 am

Ha ha, funny story. I think I had the same variant of Russian teacher, for my Theoretical Physics course, Dr. Kovacs. Unfortunately, or fortunately. rather. I graduated with a Computer Science degree instead.

Having the perspective of a little physics background an a lot of Computer Science influence, I’m starting to think that the Universe, is a giant 3D computer monitor with countless pixels. Instead of having 3 components of RGB, the Universal Monitor probably also has some fundamental components that when combined gives us all the sub-atomic particles that build matter, when combined in different levels. And when the components are in balance, we get white, which is in fact the vacuum. Notice that monitors have discrete pixels that, if we sit far enough away, appear to us as continuous. This is analogous to the quantum versus classical mechanics dichotomy.

The only thing to figure out is whether the universe is 32-bit or 64-bits, big endian or little endian.

January 29, 2014 5:07 pm

“Maybe if I had then I could have concentrated on the math and understood why infinity + const. doesn’t equal infinity.” That seems strangely plausible however, I would like to know does infinity – const = infinity.

5. October 20, 2014 10:18 am

I have a question as a lowly nanomaterials experimental physicist who never got around to formally studying QFT when I was in school.

~ Which equation (or equations) suggested to you a “spring fabric” universe model

~ Are the intrinsic oscillations of these springs inherently quantized? Or does quantization come in later somewhere?

November 22, 2014 11:13 am

Hi NN,

The equation that most looks like a “spring fabric” is the equation for the Hamiltonian of a scalar field: http://en.wikipedia.org/wiki/Scalar_field_theory#Quantum_scalar_field_theory . It has a term that looks like kinetic energy of a mass on a spring (the first term in the Hamiltonian at that link), a term that looks like the potential energy of a stretched spring (the third term), and a term that looks like the potential energy of relative stretching between two springs (the second term). Eventually, by the way, I picked up A. Zee’s Quantum Field Theory in a Nutshell, and was very pleased to see that on page 2 of the book he has a picture of “springs in a mattress” as a way of thinking about quantum fields!

The answer to your second question is, essentially, that the quantization of the “springs” is assumed from the beginning. (More specifically, one assumes that the position and momentum of the “springs” follow the same commutation rules as do normal quantum particles, and quantization follows.) So quantum field theory is not a way of explaining where quantum mechanics comes from. It is a way of understanding how essentially everything in our universe can arise from a “field” of quantum objects.

Finally, I find your description of being a “lowly nanomaterials experimental physicist” a little funny. I spend a great portion of my time talking to such “lowly” experimentalists, and I can assure you that they spend more time explaining things to me than I spend explaining things to them!

November 3, 2015 1:01 pm

Hi, I love your posts. Would you recommend any remote QFT course or study material suitable for someone with an undergraduate level understanding of physics?

November 3, 2015 1:04 pm

Thanks!

My favorite QFT book so far is “Quantum Field Theory for the Gifted Amateur” by Lancaster and Blundell.

7. March 10, 2016 5:40 am

Many thanks again, Brian, for explaining Quantum Fields so clearly. Your posts imply, at least to me, that the fundamental fields really do have material reality everywhere in the universe. Which leads me to ask: How can all these fields co-exist at every point? If all the superimposed fields are continually seething with vacuum fluctuations, why does space appear on average to be so ’empty’?

March 10, 2016 10:50 am

I think it’s a little bit dangerous to conclude that “the fundamental fields have a material reality”. Everything that we think of as “material” is made from the fundamental fields. So what does it mean to say that those fields are, themselves, “material”? The way we think of things as “taking up space” comes from the properties of those fields — the concepts of solidness, or space-filling, arise from the properties of the fields. So it’s not necessarily a good question to ask “how do those fields coexist with each other?”

This might seem like a non-answer, but the questions get pretty hard when you start trying to think conceptually about quantum fields, and you start to worry that maybe no conceptual picture can be much better than a colorful analogy.

• March 13, 2016 10:00 am