The master cart maker makes a cart.
If she removes the axle and wheels,
What is vividly apparent?

– Zen koan

In the cart maker koan we are asked to consider a cart which has been stripped of key features. When the garments of axle and wheels are removed, is some naked, essential “cartness” revealed? Then there’s the cart maker herself. What is vividly apparent in the master maker in this deconstructive act? I’ve been thinking that the work of modern physicists bears some resemblance to the work of the cart maker as described by the old Chan teachers.

A few thousand years ago the idea took hold that there exist fundamental entities, tiny indestructible bits, from which all things are cobbled. These were the primordial atoms—from the Greek atomos, “uncut, unhewn, indivisible.” Perhaps the long-term quest for such uncuttables is fueled by the hope of some categorical security in a world of things breaking down. Our lives lapse into dissolution: pets, parents, friends, cultures, ecosystems, and stars all sliding away. These eternal grains suggest a toehold, a bit of traction in the world, a way to roll through the metaphysical mud.

Giant collider.

 

The electron of modern science is another version of that atom, a bit we have not been able to cleave. It appears to have no internal structure, no parts, whatever chisel we wield. A time-honored method of finding a particle’s parts is to smash it into another particle at enormous energies and see if pieces emerge. The Large Electron-Positron storage ring at CERN has achieved our highest energies of collision, and found no discernible structure within the electron. Neither do we find electron remnants showering down from the heavens as the result of more energetic smash-ups occurring inside supernovae.

 

 

 No Size in Mind

The flip side of the electron’s inviolate wholeness is its lack of size. When we look for the electron’s surface, where it begins, we can’t find its “edge.” If it does have a diameter, experiments show that it’s less than one-ten-thousandth the size of a proton, which is already 10,000 times smaller than a hydrogen atom.1 If one accepts the rather bewildering implication that the electron has no physical size, no extension in space, then it makes (a kind of) sense that we can’t cut it in two. How could we divide an object that is already as small as possible?

Electrodes measure brain activity in Tibetan Buddhist teacher and artist Dru-gu Choegyal Rinpoche. Photo from National Geographic, March 2005:

 

This brings to mind an analogous experiment: the splitting of awareness. Are there constituent parts to awareness? Does it resist being cleaved? Can we find its edge? What things are inside that edge and what things are outside? How big is awareness? A ballpark estimate will do: smaller or bigger than a ballpark? (Don van Vliet, aka Captain Beefheart: “I just looked at myself, and from here to there it ain’t far enough, but from here to here is too short.”)

But the fact that an electron has no size does not imply that it is nothing. If you stick a fork in a toaster that’s toasting, it’s the flow of electrons that cause your convulsive shaking. If your computer is computing, it’s electrons getting the job done.

 

One Size Fits All
Despite its lack of size, when physicists engage in accurately describing this utterly simple particle, the mathematics blooms into a vastness that challenges our imagination as much as its point-like character. The electron seems to embody both the infinite and the infinitesimal. While its dimensions are vanishingly small, its interactions with the world are infinitely variegated. Awareness appears to have similar properties. More on that in a little bit.

Quantum electrodynamics (QED) is our most accurate description of what an electron is. In QED what an electron “is” is what it does, and what it is does is interact. Consider the simplest of cases: a zero-size, zero-structure electron sitting alone in a vacuum. Even in this exquisitely spare situation, the electron is having a rich “conversation” with the surrounding space. In order to describe the electron’s physical interaction with empty space, its conversation with emptiness, I’ll need to describe a few aspects of light.

Light Interlude
We’re all familiar with light. In fact, it’s such a fundamental constituent of our experience that it serves as a primary metaphor: being bright vs. being dim, seeing the light vs. stumbling in the dark, etc. Just as matter contains bits (electrons) that cannot be cleaved, light too has fundamental bits, photons, that cannot be split. The dimmest possible light source is one that emits one photon at a time. Photons also have no structure, no parts, but are quite real. When you see your beloved’s face, that vision of beauty is caused by individual photons interacting with the biochemical molecules (opsins) at the rear of your eye. In fact, the eye is sensitive enough to detect the light from just a handful of photons. Besides absorbing photons, the eye, along with everything else, emits them. If your eyes are blue, that’s because your eyes are radiating photons of the appropriate wavelength.2

Explore All Paths!
Back to the electron. The empty space surrounding the solitary electron is actually filled to the brim. It is not filled with matter or with light, since it’s empty, but this emptiness does have a “structure,” or shape. We know this vacuum structure exists, since it interacts with the electron that’s immersed in it. Physicists describe this interaction as the emission and reabsorption of “virtual photons.” Richard Feynman, one of the inventors of QED, created a graphical language to express the conversation between matter and virtual light, known as “Feynman Diagrams.” Here’s the diagram for the simplest way the electron converses with the vacuum:

 

The electron’s path through time is represented by the arrow pointing from left to right. At a certain point in time it emits a virtual photon (squiggly line) and later absorbs it. One reason the photon is described as virtual is that it is not directly observable. This is a private chat between the electron and photon: no experiment will catch the participants in mid-sentence. But we can observe how the electron is changed by the conversation. Here’s the Feynman diagram for another path the electron can take:

The electron emits a photon, that photon transforms into an electron-positron pair (the circle with two arrows), which then change back into the photon, which is then reabsorbed by the electron.

These are just two paths the electron can take in going from “now” to “later.” In QED there are many more possibilities. In fact, there are infinitely more paths for the electron to pursue. These paths involve an ever greater number of virtual photon emissions and reabsorptions, linked to a growing thicket of electron-positron creations and annihilations. An obvious question to ask is which of these paths is actually taken by the electron? As Feynman said, “The electron explores all paths.” That is, to arrive at an accurate mathematical description of the electron’s existence, one must include in the calculation all possible paths in going from “now” to “later.”

Now consider two electrons interacting with one another, say by colliding. We find this so-called collision to actually be an intense conversation between the electrons, by means of a dense tangle of virtual photons and the accompanying electron-positron loops. Here’s the Feynman diagram for the simplest path of interaction:

Two electrons enter from the left, exchange a virtual photon, and exit to the right. It’s interesting that we cannot say which electron is which after the photon exchange. That is, the electron entering at the bottom of the diagram, could be the one exiting at the top, and the electron entering at the top could be exiting at the bottom. In fact, it’s not really a matter of “could be.” Since these electrons explore all paths, their interaction includes the path where they exchange trajectories and the path where they do not. As with the single electron in the vacuum, there are an infinite number of additional paths in the accurate description of the interaction between two electrons.

A Vehicle of Transformation
The electron is a kind of atomic cart, its virtual photons and electron-positron loops taking the role of axles and wheels. This tiny vehicle initially appears undeconstructible, possessing no size and invulnerable to partition. Upon close examination we find it consists of its interactions with the rest of the universe. It’s a strange kind of reductionism – analyzing a featureless point particle, and finding its nature to be a vast sea of connectivity with the environment. Even nothingness – the vacuum – is a plenum of connection for the electron.

The electron appears before us as nothing other than these connections; it’s all axles and wheels. Removing them leaves us vividly empty-handed. Perhaps we should not be surprised by this empty-handedness, since it was we who built the electron to begin with. Some would (reasonably) argue that we are only describing the electron, not building it. Hence, it is only our description disappearing under the deconstruction, not the electron. But it’s not so straightforward. Being bodily shocked upon fork-fishing for toast is clearly not an event confined to description – we’re jolted! But to speak about that experience in terms of “electron flow” is a profound leap of abstraction. No matter one’s level of mastery with the method of Quantum Electrodynamics, it will never impart the feeling of being shocked. No matter our progress in cognitive science, our understanding will never corral our awareness.

The nature of electrons, independent of our description of them, is mysterious. If one wishes to “demystify” them, which is an interesting and honorable project, one will find that progress has already been made. That progress is known as QED and it’s descriptive. A fancy-pants way to say this is that the progress is epistemological, not ontological. What is an electron really? My intuition says it’s like a weasel.3 And, as Annie Dillard wrote, “A weasel is wild. Who knows what he thinks?”

The Cart Maker Rides Home
I’ve come to think that taking the cart apart reveals more about the cart maker than the cart. The cart maker is also disassembled in her cart deconstruction project. Both cart and cart maker are transformed. This happens because “cart making” itself serves as wheels and spokes for the cart maker. The carts we make are things like pets, parents, friends, cultures, ecosystems, and stars. Just like the electron, I find myself to be constituted by these interactions. Like the electron, I too am all wheels and spokes. Stripping these away leaves empty-handedness. Whose empty hands are these anyway?

The old Chinese teachers knew our passion for the many varieties of cart making; after all, they were cart makers themselves. The koan does not enjoin us to end the cycle of cart construction and destruction, just to notice that we’re doing it. I think the teachers were hopeful that something would become evident for us in the midst of our cart work. Perhaps we’d recognize a kinship with weasels. We’d see beyond our expertise in utility, and find ourselves wild and beyond comprehension.

Return Path: Issue II HavingTaking All Paths – The Electron & The CartmakerFeynman Diagrammatic Analysis – Stars At Noon


  • A hydrogen atom has an enormous diameter of five billionths of a centimeter.
  • If these photons are absorbed by other eyes, which are attached to a human who happens to speak English, the word “blue” will arise in their mind, by way of electrons, just after the photons arrive in their eyes.
  • My intuition also says my intuition is weasel-like.