Perhaps no concept is more central to modern physics than that of the field. The entire subjects of General Relativity and all of Quantum Mechanics are unintelligible without it. Indeed, no phenomenon in nature that we can explain at this time requires a description that cannot be based on fields. It should not come as a surprise, then, that a lot has been learned about them. Yet we have not mentioned fields before now, having confined our discussions entirely to discrete objects and the paths they travel. Therefore, this will be a tool-building chapter, drawing from our previous discussions as well as new experiments, to understand why fields are necessary, which questions involve them, and what has been learned about them that is useful and interesting. The tools we build will serve as the groundwork in all later development.
We have taken great pains to advertise that the object-laws we have built are very good descriptions of nature. So, why do we now need fields? There are two reasons. Ultimately we will see that nature essentially forces us to use a field paradigm if we are to describe fundamental processes. That conclusion will have to be reached in stages. Sometimes, though, even when we are not forced to use a language of fields, that is simply the most convenient and natural one, and the most physical. We elaborate on the latter possibility first.
Cases in point, which we will use throughout the chapter, are the familiar fields called temperature and pressure, for instance in air or water. We know by now that both the atomic theory and its child, chemistry, are effective descriptions of nature at fairly small scales. Their success in categorizing the repeatability and patterns in physical processes had been firmly established by the work of Mendeleev in the creation of the periodic table of elements. A further, extremely important step was taken by Ludwig Boltzman in (?). He succeeded in showing that not only some properties, but everything about systems like air, water, etc., could be described in terms of the motions, collisions and binding of atoms and molecules alone. This was a triumph for reduction, because it showed that once the behavior of atoms and molecules could be understood, the description was, in a certain sense, complete at that level. (A caution here: Reductionists are most often misunderstood, because the sense of that completeness is a subtle one. Understanding of atoms and molecules does not automatically enable us to predict everything that arises from their interactions; the calculations may be too hard for us to do, or even to understand. The completeness simply means that once we understand the atoms and molecules, we do not need to discover new ``players in the game'' to understand large systems. We simply need to learn how to predict the behavior of the collections of players we have, starting from their individual characteristics. That, of course, can sometimes be almost as bad as ignorance, as we have commented in the first chapter).
Boltzmann's work is curious, though. When we (in everyday life) interact with air or water, we feel effects like temperature or pressure. Yet neither of these is actually a property of any atom or molecule itself. They arise, totally new epistomological objects, from the math of large numbers, and in fact have an identity which is quite independent of any of the details of the atoms or molecules which we happen to choose to consider. All the features of air that we notice on large scales are properties of the large-number interaction, and not of the atoms. (A wind of Helium feels basically like a wind of air; you can still sing when you breathe it, though it sounds a bit different.) Thus, discussing atoms or molecules explicitly is a bit of red herring. It makes more sense to adopt a description which focuses on the large-number properties directly, and simply takes whatever starting conditions it needs from the atomic properties. (For instance, sound travels faster in Helium than in air because of the smaller mass of the atoms. Knowing the respective masses makes it possible to predict the respective speeds.)
The large-number properties are those of smoothness and continuity, the characteristics that we intuitively associate with air or water. Moreover, they explicitly lack reference to any features like individual atoms, or events like their collisions. This smoothness and featurelessness is the characteristic of a field description. It is, in a sense, the exact complement of a description of discrete objects, which treat only ``features''.
The second circumstance in which fields matter is the one that will concern us most in the rest of the chapters. We will show that at the level of the most microscopic, the seemingly fundamental processes, all descriptions that do not involve fields are also not correct, in one way or another.
An appropriate demonstration of this is Young's double-slit experiment with light, which we will introduce in this chapter and treat in much more detail in the chapter on the Superposition Principle and Huygens' construction. It involves an experiment which Fermat's least time principle completely fails to explain, and in fact any description in terms of particles of the familiar, classical type, fails to explain. This may seem surprising, since Fermat's least time principle seemed to work very well for lenses and mirrors, and thus to suggest convincingly that light is ``made of particles''. Yet we will find that Young's double slit experiment makes no sense in terms of particles of a Newtonian sort, while making perfect sense in terms of waves. For the moment, this will leave us with two descriptions of light, each successful in some circumstances and seemingly incompatible with the other. The resolution of this mystery comes several steps later, both logically and historically.
Meanwhile, it is useful to note that Young's double slit experiment uses a language of waves, but does not even attempt to explain waves in what. In that sense it is much like Fermat's description of least time paths, which makes no attempt to describe what particle travels those paths.
The first indication that it was a wave description, and more generally a field description, that was needed for light, came from a seemingly unrelated set of investigations. As we will see four chapters hence, a great deal of data had been accumulated, and partially understood, about Electricity and Magnetism, through the work of Coulomb, Ampere, Michael Faraday and others. For reasons that we will explain in that chapter, these investigations presented their own set of confusions and mutual inconsistencies. Unlike the case of Young versus Fermat, though, a self-contained resolution to the Electricity and Magnetism problems was found by James Clerk Maxwell in 1864. He cast his solution in terms of smooth, featureless, yet (Maxwell claimed) physically ``real'' entities he called the Electric and Magnetic Fields. Most of the equations that described their behavior were simply inherited from Coulomb, Ampere and Faraday, with different interpretation. With these, and one addition of his own, Maxwell could predict all of the phenomena that had been observed, correctly and self-consistently.
It could be said that this date, and this description, mark the point of departure of modern physics from the patterns that had persisted from the oldest times, through the work of Galilleo, Halley, Kepler and culminating in the work of Newton and descriptions like the classical principle of least action drawn from it.
In this one step, three important changes in our understanding of nature took place. First, Maxwell's work comprised the first complete and internally consistent description of all of electric and magnetic phenomena. It even demonstrated that they are two expressions of the same thing. Second, the fact that this description was a field description, in which waves play a natural rôle, enabled Maxwell to make contact with Young's experiment, and to infer for the first time that what we know as light is in fact waves that exist in this `electromagnetic field', whatever that is. Third, the stupendous success of the equations themselves at predicting the behavior of nature brought the field description to prominence as a useful one. Much more than validating Maxwell's contribution alone, this made it a sort of paradigm, on which all of our later successful fundamental field descriptions have been based, to one degree or another.
Maxwell's equations provided the completion of Young's description by telling, in a language of predictive equations, waves in what. It also turned out, as people came to understand better how to compute with waves, that they identified wave behaviors which entirely mimiced the least time paths of Fermat, without requiring particles at all. At least in the sense intended by Fermat and Newton, particles did not seem to be needed, and wave and field descriptions had attained supremacy.
But Maxwell's equations were not the end of the story. By themselves, they show that waves are a very good way to describe light, but they do not even suggest that waves are needed to describe everything. That conclusion comes from yet another direction. Maxwell's equations are not wrong, but they are also not ultimate, in a very subtle sense. The deficiency is not so much in the equations as in the interpretation given to the fields they describe. The classic interpretation leaves out effects that experiments with very small scale interactions have uncovered. The problem, broadly speaking, is this: Maxwell argued that the electric and magnetic fields were ``real'' because he thought that they were physically measurable quantities. (He concluded that the values of these fields were, in fact, what his predacessors had been measuring all along in their experiments.) In that sense they were much like temperature or pressure in air. Such fields, which can be identified because their values can be explicitly measured, are called classical fields. It would have been too much for anyone to accept that Maxwell's fields were ``real'', had it not been implicit that they were measurable.
This assumption ran aground, though, on the work of Max Planck in 1899, concerning the light that shines out of a cavity of any red-hot material. Planck's observations, to make sense, required certain properties that were precisely the properties of classical particles. Specifically, light inside the cavity obeyed exactly the right large-number laws of Boltzmann, but only when there were particular particle-like properties to take as starting conditions. The situation got worse with Albert Einstein's description of a phenomenon called the photoelectric effect, which involved electric currents generated by light shining on pairs of metal plates. This explanation only made sense in terms of particles. Though not in Fermat's sense, the particle description was back, and this time seemed unavoidable. The careful presentation of these issues is the whole subject of Part III.
The only salient result which we will mention here is that a new interpretation, a quantum interpretation of the field, must be recognized. Fields in the quantum interpretation have the same mathematical features as classical fields, and they seem just as physically ``real'' (as opposed to mere calculational conveniences). The difference is that the value of a field, in the quantum sense, cannot be directly measured. It can only be inferred indirectly, from the way it affects the outcomes of other measurements. Justifying (indeed, even explaining) this claim of physical reality without measurability will be the principle work of the sections on Quantum Mechanics.
For now, it is not necessary to ``understand'' the subtleties of the quantum interpretation of fields. Mathematically, we will handle them with the same machinery that we will use for simple classical fields, like temperature or pressure, or the flow fields in air or water. The reason the quantum interpretation of fields needs to be introduced now is that quantum fields have been found to be necessary to explain everything we understand consistently. Everything in nature, it seems, behaves according to the rules of quantum mechanics, and these rules only make sense in a field language, specifically the one employing the quantum interpretation of fields. The conclusion evantually reached from the work of Planck, Einstein and many others, was that Maxwell's Electromagnetic Field was actually a field in the quantum sense, and not a classical one.
In the course of explaining these experiments, Maxwell's discovery, though it had initially missed the details of interpretation, gained a whole new significance. His Electromagnetic Field, while not classical, did appear to be fundamental. That is, unlike the fields of temperature and pressure in air, Maxwell's electromagnetic field does not seem to come from the large-numbers math of some smaller atoms or lumps. The field description, once interpreted correctly, seems to be the whole story. Moreover, in the century and a half since Maxwell's work, it has been found that not only for Electricity and Magnetism, but for everything else as well, a field description, as an ultimate story, seems to be sufficient and consistent.
At this point we can clarify the distinction between the various levels of description at which fields can arise. Ultimately, everything seems to require quantum mechanics, and quantum mechanics requires fields. So Electricity and Magnetism, as well as all the other phenomena that occur in atoms, require a Field Theory at this fundamental level.
In the case of Electricity and Magnetism, as viewed by Young and, under closer inspection, Fermat, the field description is manifest from its most microscopic level all the way to large-scale effects. Along the way, it may be possible to commit certain slips of interpretation, like Maxwell's thinking the field was classical because he had never experienced things like the photoelectric effect, but for experiments like Young's, Coulomb's, Ampere's or Faraday's, the slips create only miniscule errors in prediction. Meanwhile the obvious field-nature of the phenomena is evident at every level.
Somewhat differently, in the case of air or water, the quantum mechanical field description at a fundamental level permits little clump-like formations (atoms and molecules), and we may form consistent intermediate-level descriptions, if we like, which use these as the basic building blocks. Such descriptions are the domain of the atomic theory and chemistry. When we study large numbers of the clumps, however, we find that field behavior arises all over again, though the fields at this level look nothing like the particular fundamental fields that gave rise to the atoms themselves. These new fields are, mathematically speaking, phenomena unto themselves. The relations possible in this epistomological hierarchy are shown in fig. 7.1.
There is another reason to believe that, not only what we understand now, but also everything we will discover, will require a field description. That reason originates in Relativity, from conclusions that will be forced upon us in the next two chapters. It hinges on a principle which, in addition, overcomes a certain awkwardness of our current Newtonian descriptions. This awkwardness has not been a problem to us so far, because the Newtonian descriptions do predict right answers, at least with some accuracy. If, however, we were to look closely at the kinds of laws we have found, we could claim that they vary widely according to `how much sense they make'. Descriptions of objects on the ends of springs make a lot of sense, in that there is a clear reason that the spring is the most important part of the object's external world; the spring is the one part that is actually attached to and pushing on the object. Compared to that, Newton's description of gravity makes `much less sense', since neither the earth, nor any other planet, necessarily touches the object, yet still it falls. How the external world affects the object, and why some parts of that world are more important than others, is not nearly as obvious as it was for the spring.
This aesthetic objection will be raised to the level of a catastrophe when we discover Special Relativity in the next section. No matter by what path we discover it, once the experiments have made clear that we are stuck with Relativity, we have the terrible problem that unless we can describe all interactions as happening at precisely ``some place'', like the place the mass touches the spring, Special Relativity doesn't allow us to know what we are talking about. Nebulous interactions, like a Newtonian Earth affecting a free faller ``at a distance'' are not useable as they stand.
The saving idea, which we have encountered in limited forms while building Newton's equations and the action principle, will become a primary concept in the language of fields. It is the principle of locality, the concept that interactions can only really happen ``at some place''. We will see that all descriptions in terms of fields naturally possess the highest degree of locality possible. This is the reason that, not only have fields been successful in describing Electromagnetism, Gravity, and all the other interactions we have seen, but we believe, based on the requirements of relativity, fields will be necessary elements of any future fundamental theory we will find.
The fact that so many kinds and instances of fields exist, with such a variety of physical interpretations, may make the subject seem more daunting than it is. Just as fields like temperature and pressure, that arise from large-number mathematics, do not depend on most of the details of the atoms they describe, they also don't differ much from one case to another, or really even from fundamental fields. In other words, the properties of a field can be described quite generally. This is very convenient, because it enables us to discover and understand fields where they are familiar, like in air or water, even if the description is not strictly necessary in those cases. Once we have understood their properties, we can simply apply these to new phenomena, even fundamental phenomena, when experiments indicate they are needed. The invariance of the mathematics and the basic language is what gives the subject its unity, making many instances understandable in terms of a few.
With that in mind, the purpose of this chapter is to introduce the basic concepts relevant to classical fields, to get a feel for why they are important, and to build a language for describing them. More refined treatment of particular methods, and issues of interpretation, can then be undertaken later and separately, closer to the time when each is needed to solve some specific problem. We will be concerned with three broad kinds of issues.
The first is what kinds of properties fields in general can have, for which we will use as examples temperature and pressure in air or water. We will see that not all kinds of fields have the same properties, and will discover one of the most important of these, the property of superposition, which the pressure field has and the temperature field does not. A whole chapter in Part III will be devoted to the ramifications of this property, when it applies.
The second area of concern will be, if we are to claim fields are physically real, what behaviors they can have and how those can be predicted. This amounts to asking the same questions, about fields, that we have asked about discrete objects in the preceding chapters.
Last, we will direct our attention to a very particular class of fields, which also exist in water or air, called flow fields. Though they occur in many of the same processes (like sound) as waves in the pressure field, they also occur elsewhere, and are interesting for slightly different reasons. Specifically, flow fields provide an intuitive introduction to the notion of currents. Currents have many interesting properties, and a language of currents turns out to be an exceedingly useful and powerful one to invoke when describing field phenomena. We will make strong use of this when we discuss Electromagnetism, where it will enable us to encode every property of that phenomenon that has been observed, using remarkably few ideas and elementary lines of reasoning.
The general statements made so far are, of course, intended only as guideposts to what follows. A real appreciation of why the concept of the field is so important and so necessary will only come after we have treated its various instances much more thoroughly. This introduction is not meant, in any significant degree, to constitute an explanation. That begins below, as we consider particular examples.