“Our stability is but balance” — Freeman Dyson on how to imagine quantum fields

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.

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  And the result of their labors … was not destroyed or superseded by the later insight that atomic physics gave.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”.

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.

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.

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.

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.