Basic Concepts: Fields

In the initial “Basic Concepts” post, I discussed the concept of Force in physics. This time out, I’ll be talking about fields, which is a much dicier proposition. Not only are fields considerably more abstract than forces, but I’ve never lectured on fields in general (specific instances of fields, yes, but not fields as abstract objects). For that matter, I’ve never taken a real field theory class. The chances of my saying something stupid about fields are exponentially greater than the chances of my saying something stupid about forces.

In a certain sense, though, “Fields” is a good topic to follow “Force,” because both are concerned with interactions between objects, or between an object and its environment. In fact, I’ll make that the one-sentence blockquote summary:

Fields are a way of understanding the interaction between an object and its environment.

Fields and forces are intimately linked, and the typical introduction to fields is through force, specifically in the context of electrostatic forces between charged object. In introductory physics classes, we define the electric field of an object in terms of the force on a small test charge placed at a particular point. We define an “electric field vector” at each point in space, which tells you how big the force is, and what direction it’s in. We define “electric field lines” by connecting the vectors together. An electric field line essentially shows you the path that a test charge would follow, if it were released from rest at a point along the line.

We can extend this basic idea to other interactions. Magnetic forces are really only understood through the concept of fields and field lines, and light is understood as a combination of electric and magnetic fields. You’ll sometimes hear gravity discussed in terms of a field, but not that often. The strong and weak nuclear forces can in principle be described in terms of a field, though it’s not that common, because people talking about those forces usually go directly to the quantum picture (about which more later). In all of these cases, the idea is the same– the field is a mathematical object that tells you how to determine the effect of some interaction on an object at some point in space.

So, why do we do this?

The field description is useful for a number of reasons, starting with the fact that it immediately conveys the idea of an interaction filling a region of space. If you bring two charged objects close together, they will interact with one another, no matter how you arrange them. The force between two charges is not a special property of any particular arrangement of those objects, it’s a consequence of the interaction between them, and there will be some interaction no matter where the two are placed. The field gets you that idea.

Fields are also a useful tool for simplifying complicated problems, particularly when it comes to describing interactions between an object and its environment. If you want to talk about a charged particle near a metal surface, you don’t need to actually calculate the electromagnetic force between the particle of interest and each of the atoms making up the surface. Instead, you can calculate a field from the surface, and use that to determine what happens to the particle.

The biggest mathematical advantages of the field picture is that fields are generally governed by simple and elegant mathematical rules. This is most apparent in electromagnetism, where everything you want to know about the interactions between charged particles and their environment can be expressed in four short statements about the properties and behavior of electric and magnetic fields, known as “Maxwell’s Equations.” They fit very nicely on a T-shirt, and are a testament to the mathematical power of the idea.

So are fields just a calculational trick? Yes and no. While fields are fundamentally about describing the way a particular interaction propagates through space, the fields themselves do have a certain degree of physical reality, independent of particular objects. You can see this from, well, from the fact that you can see– the light we use to see by is a combination of oscillating electric and magnetic fields, propagating through space. The fields aren’t really tied to any particular object, and you can talk sensibly about light in terms of fields without worrying at all about what it was that produced the light in the first place. There’s also energy contained in fields, even in a region of space that doesn’t contain any particles.

The other tremendously confusing thing about the concept of fields is that physicists will talk about them as both a continuous mathematical object extending through all of space, and a collection of particles. High-energy theorists are particularly bad about this, talking about a “Higgs field” permeating all of space in one sentence, and then a “Higgs particle” that can be detected in an accelerator in the very next sentence. It can get kind of baffling.

The key idea here is called field quantization, and it’s best explained in terms of light, mostly because I only understand the process in terms of light… I’ll attempt to give a (highly idiosyncratic) explanation of what’s going on when people switch back and forth between particle and field descriptions, but it might be heavy going. I fully expect to have to come back to this idea again, and this first attempt may get ripped to shreds in the comments, but I’ll give it a try anyway.

The theory of electric and magnetic fields was developed in a classical context, and runs into two problems when you try to bring it into line with modern physics. The first of these is a problem with relativity– in principle, the classical description has the fields changing instantaneously. An electron sitting on my desk creates an electric field that extends out to the orbit of Jupiter and beyond. If I move it a centimeter to the right, the whole field changes, including the bit out at the orbit of Jupiter. In a purely classical world, you might expect this to happen instantaneously– that is, the very second I move the electron in Niskayuna, the alien monolith in orbit around Jupiter detects the change in the field. That would involve transmission of information faster than the speed of light, though, which isn’t allowed in relativity.

The solution to the problem is contained within Maxwell’s Equations. Among other things, you can use them to construct oscillating electric and magnetic fields that support one another and propagate through space– light waves. These waves move at the speed of light (duh), and can be used to carry energy and information.

To bring the electric field idea into line with relativity, you need to look in detail and how the field propagates through space, which you can do by describing the field everywhere in terms of these oscillating solutions. From math, we know that any arbitrary pattern can be made up by adding together sufficient numbers of sine and cosine functions with different frequencies and different amplitudes (it’s called the Fourier Theorem), and the same thing works for electromagnetic fields. We can describe the field due to an electron sitting on my desk as the sum of lots of different oscillating electric and magnetic fields, oscillating with different frequencies and traveling at the speed of light. It takes an infinite number of these “modes” of the field to describe any particular pattern (there are an infinite number of possible frequencies, and an infinite number of possible directions, so we’re talking one of the big inifinities, here), but that’s what integral calculus is for. You can think of the electron, in some sense, as constantly emitting all sorts of electromagnetic waves at different frequencies, heading in different directions.

Shifting the position of the electron, then, amounts to changing the distribution of these modes– we have a little more field at one frequency, a little less at another, and so on through the whole infinite number. And when you change the distribution, it takes time for that change to propagate. The alien monolith doesn’t know that there’s more field at one frequency until the electromagnetic wave carrying that frequency reaches Jupiter, and the wave travels at the speed of light. So Maxwell is happy, Einstein is happy, and the rest of us have to wait a few minutes before the aliens turn Jupiter into a star.

(What I describe is mathematically pretty cumbersome, which is why nobody actually describes fields this way, but it’s what’s going on in a conceptual sense. It’s rare to find a situation in which you really need to worry about the propagation delay of electric and magnetic fields, but they do come up, and there are simpler ways to calculate the effect.)

The second problem you hit is with quantum mechanics. Those electric and magnetic fields contain energy (a future Basic Concept), and quantum theory tells us that energy has to come in discrete quantities. If a really weak field contains one unit of energy, we can expect a stronger field to contain two or three times as much energy, but not two and a half times, or π times as much energy.

Mathematically, we deal with this problem by introducing the concept of “photons.” We can describe any arbitrary electromagnetic field as the sum of an infinite number of oscillating modes of the electromagnetic field, each with a particular frequency, a particular direction, and its own amplitude (which determines the amount of energy carried in that mode). If we write this description in quantum-mechanical terms, we find that each mode is quantized– I can have zero, one, or two (or any integer number) units of energy in that field, but not two and a half or π units. This applies for every individual mode of the field, and we can describe the field in terms of the number of units of energy in each mode.

If we take a single mode of the field, and put a single unit of energy in it, and look at its behavior, it turns out to behave exactly like a particle– it carries a small amount of energy, it carries a small amount of momentum, it can be detected at a particular position, and all the other things that we associate with particles. We call these units of energy “photons,” and describe them as particles of light.

(Again, describing a real macroscopic field this way would be incredibly cumbersome. When we actually work with the photon description of electromagnetic fields, we generally assume that the number of available modes is restricted in some way– we have a laser that we describe as a large number of photons in a single mode, for example, and we only worry about what happens to that one mode. In principle, though, the field of an electron sitting on my desk could be described in terms of a constant stream of photons pouring out of the electron all the time. You’d probably go crazy doing it, but that’s the microscopic picture.)

There are two steps from classical fields to quantum particles, then: First, we take the classical description (“The field at this point is so many volts per meter in this direction”) and split it into a sum of oscillating modes (“The field at this position is the sum of these waves headed in these directions with these amplitudes”), and then we treat each mode as a collection of photons (“The field at this position is the sum of these numbers of photons in these modes”). The two descriptions are mathematically perfectly equivalent, so physicists feel free to switch back and forth between them, talking about fields in some contexts, and forces in others.

Why the switching back and forth? Because describing a macroscopic electromagnetic field in terms of photon numbers is an absolute nightmare, and generally not necessary. For big-picture stuff, the classical description works just fine, and we talk about atoms sitting in electric or magnetic fields. When you’re dealing with very weak fields, or particular types of interactions, though, you need to move to the quantum picture, and talk about things in terms of numbers of photons. In either case, though, the fundamental process is the same: you have an object that interacts with its environment via the electromagnetic interaction.

The same thing can be done for the other forces, as well, and for anything else you can describe as a field. The gluons that carry the strong force, and the W and Z bosons that carry the weak force can be thought of as particles of the strong and weak “fields,” though you rarely hear them spoken of that way. Which description you use follows the same basic pattern, too– when talking about the universe in general, physicists will speak of a “Higgs field” as a continuous thing that permeates all of space, just as we talk about electric and magnetic fields filling some region of space. When talking about specific experiments, they’ll talk about a “Higgs boson” as a specific particle that can be created in accelerator experiments, just as we talk about the absorption and emission of single photons in quantum optics experiments. In either case, they’re talking about the same thing– the postulated “Higgs interaction” that gives rise to the observed masses of fundamental particles.

And that brings us back around to the initial point, after some heavy going: “Fields” are a way of dealing with different types of interaction between an object and its environment.

(And yes, I skipped the whole issue of zero-point energy and vacuum fluctuations. That’s a completely fascinating topic, but I’m trying to keep the number of brain-exploding steps per post to a minimum…)

Comments

  1. #1 Markk
    January 22, 2007

    This was pretty good article. As a reference I think the book “The Theory of Almost Everything: The Standard Model, the Unsung Triumph of Modern Physics” by Robert Oerter has one of the best descriptions of fields – both classical and quantum. Its description of classical physics as something like (roughly) -there are particles and fields, particles tell fields where to be and fields tell particles how to move- was very good.

    On the quantum side, I finally got, in some limited sense, virtual particles when I understood them as chopped up pieces of fields. I can still remember 25 years ago in quantum physics class Prof Barschall with a great German accent saying to just calculate! not to think like this (with a wink). Yet I always calculated better when I had some physical intuition. Virtual particles always seemed so weird (how did that virtual photon “know” to go between those two electrons?) and for some reason space filling fields were somehow easier for me to take, so the fact that the photon is just a piece of the field made up of little pieces with accounting, somehow was easier. I never understood why though. I now have this weird, goofy really, and totally non-physical picture in my head of quantum fields being beads on field lines like rope. For example, when one electron interacts with a virtual photon, it pulls the rope and the pulls the photon out of the other electron, pop!

  2. #2 Roy
    January 22, 2007

    Fields can also give us surprising views of things.

    Goldstone Solar System Radar in the Mojave Desert operates at 8560 MHz (‘X’) or 2380 MHz (‘S’) with a 70m Cassegrain antenna. Taking ‘far-field’ as 2*(diameter**2)/wavelength, we get 280 km and 77.8 km for far-field. For the X band, far-field extends out of the atmosphere, something that causes many RF engineers to do a double-take.

  3. #3 Loganayagam R
    January 22, 2007

    Fields are a way of understanding the interaction between an object and its environment.

    Hmm, so what about velocity field/density field/temperature field in a fluid ? Your post/definition is confined almost completely to the fields which mediate the fundamental interactions…

  4. #4 Chad Orzel
    January 22, 2007

    Hmm, so what about velocity field/density field/temperature field in a fluid ? Your post/definition is confined almost completely to the fields which mediate the fundamental interactions…

    The same logic applies to quasi-particle “fields” like phonons, I think, but yes, I was aiming this at the interaction fields.

    I tend to think of “field” in phrases like “velocity field” as being a different term, simply meaning “a mathematical entity that extends through a region of space.” It’s more of a math thing, and not really the same term as “field” in the quantum field theory sense.

    But my knowledge of fluids is practically nonexistent, so I could be way off base on that.

  5. #5 kyle
    January 22, 2007

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  6. #6 gg
    January 22, 2007

    Hmm, so what about velocity field/density field/temperature field in a fluid ? Your post/definition is confined almost completely to the fields which mediate the fundamental interactions…

    Interesting question, and interesting post overall… I’m reminded of a poor graduate TA who once tried to give a thought-provoking lecture on ‘what is energy?’ and got savaged by the students.

    As far as the velocity field goes, one could talk about it in a similar way as an electric field, in saying that it describes the response of a test particle to the flow of the liquid. The big difference seems to be that the velocity field represents a directly measurable quantity, as opposed to an electric field, which is measured indirectly via force. Analogies can, and have been, made between electromagnetic waves and fluid flow, though one can run into problems, as the poynting vector is also not a directly measurable quantity, and can have difficulties in interpretation.

  7. #7 Moshe
    January 22, 2007

    What a nice post! I have lectured about fields to sophomores, the way I introduce this has to do with locality: action at a distance is very counter-intuitive and fields makes that notion unnecessary. Also once you introduce them they acquire a life of their own since they support waves (as you note). Incidentally it is only when introducing quantum mechanics that the non-locality of action at a distance becomes really problematic and necessitates the introduction of local fields (that is my ancient problem with “QM non-locality”). This all applies to fields as replacing the notion of forces.

    The relation between fields and particles is more problematic, and I wish people used more precise and careful language. The reason why point-like particles don’t make sense by themselves can be described by looking at the classical electron theory, and all the paradoxes one gets by discussing self-interaction etc. However, fields can do many more things than describe particles, for example there is no quark “particle” associated with the quark field. Particles are just localized excitations of fields, and whether such excitations exist is a detailed dynamical question.

  8. #8 beldar
    January 22, 2007

    Good definition – the switch between field and boson in articles about the search for the Higgs is less confusing to me now.

  9. #9 Jonathan Vos Post
    January 22, 2007

    “Gravity field” and “gravitational field” are phrases heard in popular culture, albeit wildly misunderstood. One standard misunderstanding is seen behind phrases such as “escaping the gravitational field of the Earth…”

    Not to get ahead of things, but objects affect fields which affect objects. You hinted at that also in Maxwell, with oscillating magnetic field making oscillating electric field, which makes oscillating magnetic field…

    A nice summary of how Einstein’s General Relativity deals with such a feedback between object and environment:

    “Matter tells spacetime how to curve, and curved space tells matter how to move”

    — John Wheeler, Princeton University and the University of Texas at Austin.

  10. #10 Coin
    January 24, 2007

    Okay… so these descriptions are incredibly helpful, but there’s a couple of things that are still confounding me:

    In this article, you seem to be talking about particles almost as if they don’t actually exist– they’re just what we perceive when we try to analyze a single spectral component of a field in isolation, or something. Like, you say: “We can describe the field due to an electron sitting on my desk as the sum of lots of different oscillating electric and magnetic fields, oscillating with different frequencies and traveling at the speed of light… Shifting the position of the electron, then, amounts to changing the distribution of these modes… And when you change the distribution, it takes time for that change to propagate.” You later note that one unit (of oscillation amplitude, I guess?) at one “mode” in this electromagnetic field would basically be a single photon, but really even a single electron creates a field with a distribution of lots of units at lots of different modes, which we can think of as lots of photons pouring out all the time.

    In the previous article, though, the force one, you spoke of the photon as a more concrete kind of thing, saying: “In this picture, any interaction between fundamental particles– two electrons repelling each other, for example– has to be mediated by the exchange of a particle. At the most fundamental level, we understand the repulsion between two electrons as being due to the exchange of a photon: One electron emits a photon, the other absorbs it, and the recoil due to the emission and absorption is responsible for pushing the two apart.”

    I’m trying to figure out if whether there’s difficulty in reconciling these two quotes.

    For starters, it seems a bit confusing because on a plain reading you’ve told us (1) forces don’t exist, just particle interactions (2) particles don’t exist, just fields and (3) fields don’t exist either, just particle interactions.

    The other thing that seems a bit confusing about how you seem to be telling us that photons, as particles, can be abstracted away into just oscillations of these fields: Can all particles be treated like this, or just force carrier particles like photons? Like, electrons. It seems like while photons get abstracted away into values of modes in fields, electrons don’t; when you talk about the electron you talk about it like it’s an actual object– if it too is just a formalism for a value of a mode of a field you haven’t mentioned it, plus the electron gets to interact with things by emitting and receiving other particles instead of just, like, I don’t know, smashing into things or however photons interact. What’s the difference here? I know that electrons are fermions whereas photons and Ws and Zs are bosons, but most of the explanations I’ve seen of that difference, including wikipedia’s and yours, seem to imply that the only real difference between fermions and bosons is that one follows the exclusion principle and the other doesn’t, but it is not obvious to me how that one difference would produce differences in behavior to the point where one doesn’t even seem to “exist”. Do fermions have the same relationship to fields that bosons do after all?

    Overall, which of these things are objects and which are formalisms?

    Also, just for exactness:

    It takes an infinite number of these “modes” of the field to describe any particular pattern (there are an infinite number of possible frequencies

    What does this infinity look like? Are there an infinite number of possible frequencies between 200 and 300 mhz, for example?

    You describe these “modes” like they can be thought of as bins of a discrete fourier transform or something. Are they? If so, what’s the time domain function being transformed, if it even means anything to ask that question?

    Please excuse my stupid questions, thanks :)

  11. #11 Chad Orzel
    January 24, 2007

    For starters, it seems a bit confusing because on a plain reading you’ve told us (1) forces don’t exist, just particle interactions (2) particles don’t exist, just fields and (3) fields don’t exist either, just particle interactions.

    Well, there are particles, and there are particles…

    There’s a useful distinction made between “real” and “virtual” particles in the context of Feynman diagrams. Real particles are the things that we observe out and about in the world– electrons, protons, photons from a laser. Those are actual particles, and they’re subject to the usual conservation laws and the like.

    “Virtual” particles are particles that are never directly observed. Pretty much any time you have a force conveyed by a particle, that particle is a virtual one– it’s emitted by one real particle, and absorbed by another real particle, but nobody ever sees it existing between the two. We can infer its existence from the change in the state of the real paricles, but we don’t directly observe it. Virtual particles aren’t subject to the same rules as real particles– they can pop in and out of existence with very little regard for conservation of energy and the like.

    In a very loose sense, I would say that “virtual” particles are best thought of as excitations of fields, while “real” particles are, well, particles. It’s a sort of fuzzy question, though, because any particle you’d care to name can, in principle, appear as a virtual particle. In general, though, ordinary objects like electrons tend to be real, while force carriers like W bosons are more likely to be virtual. Photons can be either, which is sort of annoying.

    What does this infinity look like? Are there an infinite number of possible frequencies between 200 and 300 mhz, for example?

    Yes.

    You describe these “modes” like they can be thought of as bins of a discrete fourier transform or something. Are they?

    In a sense.
    The enumeration of modes is a sort of calculational trick, just like doing an integral. We set up the problem as if there were a countable number of discrete modes of the electromagnetic field, and then we let the total number of modes go to infinity. If you do this the right way, you still get a sensible answer.

    It’s easier to do it this way, because you can’t directly write down an expression based on having an infinite number of modes. It’s relatively easy to describe a single mode, though, and taking the limit of a sum over many modes is a workable path to the infinite case.

    Most of the time that people talk about photons in an experimental context, they artificially restrict the available number of modes in some way. If you want to talk about an atom interacting with a laser, you only worry about the number of photons in the single mode representing the laser, and assume that all the photons in the other possible modes can be ignored– which is a safe assumption, most of the time.

  12. #12 Aaron Bergman
    January 24, 2007

    In a very loose sense, I would say that “virtual” particles are best thought of as excitations of fields, while “real” particles are, well, particles.

    Eek. Everything is an excitation of a field. Elementary excitations of fields are particles, although not always the ones you think. For example, at low energies, the lightest particle you can form from gluons is a glueball; you never see a single gluon as a particle.

    Virtual particles are computational tricks to describe how particles interact. In truth, its the fields that are interacting, but we can sometimes approximate it by only dealing with the elementary excitations and their interactions and pretending that the rest of the stuff doesn’t matter much. This works great for something electrons and photons because they don’t interact very strongly. For something like gluons, however, this fails miserably because the gluons interact very strongly and the rest of the stuff matters a lot. That’s why to do computations that involve gluons and quarks, one generally has to use large computer simulations.

    Regardless, the moral of the story is that everything is a field; it just so happens that fields can sometimes look like particles.

  13. #13 Coin
    January 25, 2007

    Well, I’m now less confused than I am before, so thanks.

  14. #14 kali
    July 8, 2007

    Hi, You say Virtual particals change the state of real particals, what is that change of state; exactly?
    Thanks from kali

  15. #15 Vladimir Gritsenko
    December 28, 2007

    You said that “the field of an electron sitting on my desk could be described in terms of a constant stream of photons pouring out of the electron all the time.” How does this relate to the conservation of energy? Where do these photons come from?

    Thank you!

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