In the last post I told the story of my own struggles with quantum field theory and what it is supposed to mean.  In this post (as promised), I want to let someone much more intelligent and eloquent tell the story of quantum fields.

The following are excerpts from Freeman Dyson‘s beautiful essay “Field Theory”, written in 1953, as presented in his book From Eros to Gaia.  I have tried to copy the most essential and visual arguments from the essay and have made no attempt to keep things short or to extract pithy quotations.  Everything below (except for bold section headings) is quoted, even though I have dropped the quotation marks.  I also took some liberties with the paragraph breaks to make it easier to read in online format.

I hope that the picture he is painting is as wonderful and awe-inspiring to you as it was to me.

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On the historical purpose of quantum field theory

Next, a remark about the purpose of the theory. … The point is that the theory is descriptive and not explanatory.  It describes how elementary particles behave; it does not attempt to explain why they behave so.  To draw an analogy from a familiar branch of science, the function of chemistry as it existed before 1900 was to describe precisely the properties of the chemical elements and their interactions.  Chemistry described how the elements behave; it did not try to explain why a particular set of elements, each with its particular properties, exists.

…  Looking backward, it is now clear that nineteenth-century chemists were right to concentrate on the how and to ignore the why.  They did not have the tools to begin to discuss intelligently the reasons for the individualities of the elements.  They had to spend a hundred years building up a good descriptive theory before they could go further.  And the result of their labors … was not destroyed or superseded by the later insight that atomic physics gave.

… Our justification for concentrating attention on the existing theory, with its many arbitrary assumptions, is the belief that a working descriptive theory of elementary particles must be established before we can expect to reach a more complete understanding at a deeper level.  The numerous attempts to by-pass the historical process, and to understand the particles on the basis of general principles without waiting for a descriptive theory, have been as unsuccessful as they were ambitious.  The more ambitious they are, the more unsuccessful.  These attempts seem to be on a level with the famous nineteenth-century attempts to explain atoms as “vortices in the ether”.

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On how to think about classical fields

A classical field is a kind of tension or stress that can exist in empty space in the absence of matter.  It reveals itself by producing forces, which act on material objects that happen to lie in the space the field occupies.  The standard examples of classical fields are the electric and magnetic fields, which push and pull electrically charged objects and magnetized objects respectively.

… What, then, is the picture we have in mind when we try to visualize a classical field?  Characteristically, modern physicists do not try to visualize the objects they discuss.

In the nineteenth century it was different.  Then it seemed that the universe was built of solid mechanical objects, and that to understand an electric field it was necessary to visualize the field as a mechanical stress in a material substance.  It was possible, indeed, to visualize electric and magnetic fields in this way.  To do so, people imagined a material substance called the ether, which was supposed to fill the whole of space and carry the electric and magnetic stresses.  But as the theory was developed, the properties of the ether became more and more implausible.  Einstein in 1905 finally abandoned the ether and proposed a new and simple version of the Maxwell theory in which the ether was never mentioned.  Since 1905 we gradually gave up the idea that everything in the universe should be visualized mechanically.  We now know that mechanical objects are composed of atoms held together by electric fields, and therefore it makes no sense to try to explain electric fields in terms of mechanical objects.

… It is still convenient sometimes to make a mental picture of an electric field.  For example, we may think of it as a flowing liquid which fills a given space and which at each point has a certain velocity and direction of flow.  The velocity of the liquid is a model for the strength of the field.

But nobody nowadays imagines that the liquid really exists or that it explains the behavior of the field.  The flowing liquid is just a model, a convenient way to express our knowledge about the field in concrete terms.  It is a good model only so long as we remember not to take it seriously.  … To a modern physicist the electric field is a fundamental concept which cannot be reduced to anything simpler.  It is a unique something with a set of known properties, and that is all there is to it.

This being understood, the reader may safely think of a flowing liquid as a fairly accurate representation of what we mean by a classical electric field.  The electric and magnetic fields must then be pictured as two different liquids, both filling the whole of space, moving separately and interpenetrating each other freely.  At each point there are two velocities, representing the strengths of the electric and magnetic components of the total electromagnetic field.

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On visualizing the quantum field

Unfortunately, the quantum field is even more difficult to visualize than the classical field.  The basic axiom of quantum mechanics is the uncertainty principle.  This says that the more closely we look at any object, the more the object is disturbed by our looking at it, and the less we can know about the subsequent state of the object.  Another, less precise, way of expressing the same principle is to say that objects of atomic size fluctuate continually; they cannot maintain a precisely defined position for a finite length of time.

… At the risk of making some professional quantum theoreticians turn pale, I shall describe a mechanical model that may give some idea of the nature of a quantum field.  Imagine the flowing liquid which served as a model for a classical electric field.  But suppose that the flow, instead of being smooth, is turbulent, like the wake of an ocean liner.  Superimposed on the steady average motion there is a tremendous confusion of eddies, of all sizes, overlapping and mingling with one another.  In any small region of the liquid the velocity continually fluctuates, in a more or less random way.  The smaller the region, the wilder and more rapid the fluctuations.

… The model does not describe correctly the detailed quantum-mechanical properties of a quantum field; no classical model can do that.  But … the model makes clear that it is meaningless to speak about the velocity of the liquid at any one point.  The fluctuations in the neighborhood of the point become infinitely large as the neighborhood becomes smaller.  The velocity at the point itself has no meaning.  The only quantities that have meaning are velocities averaged over regions of space and over intervals of time.

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On particles, and on the physical world
It is not possible to explain in nontechnical language how particles arise mathematically out of the fluctuations of a field.  It cannot be understood by thinking about a turbulent liquid or any other classical model.  All I can say is that it happens.  And it is the basic reason for believing that the concept of a quantum field is a valid concept and will survive any changes that may later be made in the details of the theory.

The picture of the world that we have finally reached is the following.  Some ten or twenty different quantum fields exist.  Each fills the whole of space and has its own particular properties.  There is nothing else except these fields; the whole of the material universe is built of them.  Between various pairs of fields there are various kinds of interaction.  Each field manifests itself as a type of elementary particle.  The number of particles of a given type is not fixed, for particles are constantly being created or annihilated or transmuted into one another.

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On wonder

Even to a hardened theoretical physicist, it remains perpetually astonishing that our solid world of trees and stones can be built of quantum fields and nothing else.  The quantum field seems far too fluid and insubstantial to be the basic stuff of the universe.

Yet we have learned gradually to accept the fact that the laws of quantum mechanics impose their own peculiar rigidity upon the fields they govern, a rigidity which is alien to our intuitive conceptions but which nonetheless effectively holds the earth in place.  We have learned to apply, both to ourselves and to our subject, the words of Robert Bridges:

Our stability is but balance, and our wisdom lies
In masterful administration of the unforeseen.

August 31, 2010 12:10 am

It is this fact—that the knowing physicist cannot know and knows he cannot know—that bothers the layperson.

As an undergrad student of QM, I struggled with this for a long time. After all, everything up until that point made sense to one degree or another. Thermodynamics took much longer to understand, and in the understanding there had to be a certain period of suspension of disbelief, where I saw an equation, saw the experiment that gave us the equation, and didn’t bother asking why. But eventually, the “why” was produced and TD made sense. Or at least I thought so.

But QM was the crusher. There was nothing to make sense of. Many, many had tried, all of which were smarter than I could ever hope to be, and all had failed. So rather than waste time thinking of why, it’s better to just crunch some more equations and find whatever interesting things we could glean.

Even today, I sometimes stare at Schrodinger’s Equation and think I can see something like a dampened spring equation. But the details pop out and crush any hope I had of making sense of it.

Beyond QM, physicists just learn to stop asking “why” and only focus on the “how”. There is a short period of time where physicists enjoy talking about possible “whys”, but in the end, they grow up and move beyond that.

August 31, 2010 2:16 am

Thank you for the beautiful and brilliant article! It explains in a loving way why it is so hard to talk about these matters.

It reminds me of a personal experience in college taking a quantum physics course. I initially thought I wanted to be a physics major, and took the physics courses. It was partly the quantum mechanics course that taught me physics was not for me (or more accurately: I was not meant for physics). I remember being just flabbergasted by this notion of “spin”: it was not spin in the sense of a rotating object, but rather some indescribable property of a particle. No one could explain to me what precisely was meant by spin or how to visualize the spin of a particle. The attitude was, just calculate and don’t ask such meaningless questions. That philosophy took a little while for me to absorb. 🙂

November 18, 2011 11:09 am

This leads me back to the my comment from your previous article, that the Universe is a giant computer monitor. Imagine if one of the pixels on the monitor somehow became self-aware, how would it interpret the Universe? Can a pixel be self-ware or is the AI running on the computer that’s driving the monitor, controlling the pixel making it appear to be self-aware?

April 1, 2013 4:17 pm

Would they look like a grid of over lapping fields,or at a distance apart from each other..also a field represents a particle,,so is the interaction turn in a particle from a field…?

June 10, 2016 2:01 pm

There was a lack of visualization because Einstein dropped the context (ether) and the rest gladly joined him. Now individuals have difficulties connecting such ideas as pre-quantum pre-field vacuum energy (not potential or kinetic energy) to the contextual space in which all particles travel (such as in Young’s experiment).

June 26, 2016 9:32 am

What does it mean ‹pure energy›?
Is energy something insubstantial or is energy something that take place on an object?
I’ll appreciate your help. Thank you

June 26, 2016 10:55 am

Hi Dario,
The short answer is that energy is stored in fields, which fill all of space. These fields are the most fundamental objects we know of, but it isn’t clear whether they should be thought of made up of physical things.
You might find this post to be useful reading: https://gravityandlevity.wordpress.com/2015/06/23/where-do-electric-forces-come-from/

June 27, 2016 1:25 am

Thank you for your help, Brian. Congratulations for your blog, it is really fantastic.

April 27, 2019 12:04 pm

“it is meaningless to speak about the velocity of the liquid at any one point”: I wonder what Freeman will think about a *complete random field of Quantum Non-Demolition measurement operators* (“complete” in the sense that the random field operators are enough to construct the same Hilbert space as we usually construct using the quantum field operators), for which there are arbitrarily large fluctuations in small regions but there is no measurement incompatibility (a random field: a field of measurements for which [phi(x),phi(y)]=0 everywhere, not just at space-like separation, as for a quantum field). Quantum Non-Demolition measurements have been around for decades, but the idea that there can be a *complete* set of them is new (though it’s very much presaged in the Koopman-von Neumann literature), in my recently published paper “Classical states, quantum field measurement” in Physica Scripta. It’s not quite as meaningless if one can in principle measure something that we can associate with an aritrarily small region ofspace-time and that does evolve.

This understanding of quantum fields, as isomorphic, in a specific sense, to random fields (where of course there has to be a careful accommodation for measurement incompatibility), IMO speaks also to your post “Who am I to say what’s quantum?”, insofar as we don’t have to make such a “sharp distinction between “quantum” objects and “classical” objects”, as you say there, if the state spaces are the same, just differently constructed.

I could have picked out a dozen quotes from this and other posts: thank you very much for them all.

There is one more that I want to focus on here, whether anyone ever sees it or not: “the theory is descriptive and not explanatory”, and then, “a working descriptive theory of elementary particles must be established before we can expect to reach a more complete understanding at a deeper level.” For me, the jump to describing “elementary particles” is too quick: what we should seek to describe in the first instance are the voltages on signal lines that are attached to exotic materials that are driven by exotic electronics, without the assumption that those voltages are caused by particles. Very often, measurements have a continuous sample space of voltages, we have to work hard to create measurements that have discrete sample spaces. In such terms, QFT can be understood as a signal analysis formalism, where representations of the Heisenberg group are as commonplace as they are in quantum theory, then perhaps we can worry quite separately how discrete observables, which are never defined at a point in a field theory, can emerge from an infinite field of measurements, of which we only in fact implement a finite number.

Your almost final comment in your post “A children’s picture-book introduction to quantum field theory”, that the vacuum state is “a boiling sea of random fluctuations, on top of which you can create quantized propagating waves that we call particles”, seems to me to be particularly ready to think about a particular aspect of QFT as signal analysis: that we can think of quantum field operators as applying *modulations* to the vacuum state, so that the statistics of measurements of a modulated state are different from the statistics of measurements of the vacuum state. For a free bosonic field, this allows us to make the very elementary (and obvious, at least in retrospect) observation that a coherent state just shifts a Gaussian probability density for elementary field measurements to a different mean value, whereas a single field operation changes the probability density to be non-Gaussian, and superpositions and mixtures can change the probability density to be anything at all (locally, but not asymptotically at near infinite separation, which would introduce thermodynamics).

I came to your blog because a friend linked on Facebook to your post “What it means, and doesn’t mean, to get a job in physics”. Congratulations and Best Wishes and Good Luck! But I also found a LOT to like in your Physics posts. You are, of course, free to ignore my over-long nonsense, but I love writing something out in response to good stuff, even if no-one else much understands my response.

April 29, 2019 10:43 am

Thank you, Peter, for your remarkable comment, and for the best wishes!

April 29, 2019 11:15 am

Hee hee! Remarkable! Folly is a sweet byway to follow. If you hadn’t replied here I wouldn’t have said, but I wrote to Freeman, though I called him Prof. Dyson to his face, mentioning “complete random fields of QND measurements”, and linking to the paper I mentioned above. Surprisingly, even astonishingly, even though I know him to be gracious, he replied almost immediately, but he said, as I hope he won’t mind me reporting, that “It seems to me a question of taste, whether one prefers the standard QED operators or your commuting QND operators to represent the states of the electromagnetic field. After seventy years I am firmly attached to the standard QED operators. I am sorry that I do not find the QND operators helpful.”
That seems almost right as an assessment, to be honest, that there is nothing *compelling* in Koopman-type approaches, but it’s more that I have some feeling that understanding them *and* their relationship to QFT might cast a little light on QFT, might, as I think of Feynman saying of different mathematical byways, give us an edge, than that random fields might replace QFT. The focus in QM/QFT on representations of the Heisenberg group is too useful and theoretically potent to give up for the weaker structure of a commutative algebra, even if the latter is just as “complete” if we add the vacuum projection operator. I’m all too conscious I can only haltingly communicate that light.