∫ The big lie: or, why physics made me think about Plato’s Cave today

In spite of finally getting a full 8 hours of sleep last night, for some reason I was really tired this morning and so I found myself hurrying to class with only about 5 minutes to spare. I did discover a handy little shortcut between the parking lot and the physics building, a little dirt path that cuts through a neat little patch of trees before coming out near the side of the physics building. Still, I was a little worried I might be late for class, because Dr. Van Nieuwenhuizen (thats “van-NEW-wen-hoy-zhun,” since several people asked) has been starting his lecture early the last couple days. That is, until I saw this:

That’s because the guy in the picture is Peter Van Nieuwenhuizen. So, yeah, I figured I would get there on time

What I didn’t figure is that once I got there, my mind would be boggled (is that a valid participle?) by the exposure of one of the biggest lies I’ve ever been told. I mean, this is physics heresy like finding out there is no tooth fairy. Only suppose that didn’t happen until you were 22. Yeah, about like that.

The thing is, Electric and Magnetic fields aren’t real. Okay, I may exaggerate just a little tiny bit. The difference between “real” and “unreal” when you’re talking about things that are already hard to define except in mathematical terms may be slight. Still, the fact is I’ve always taken them pretty seriously, in that, while I knew they were mostly useful as concepts to help you visualize the behavior of charged particles, I also figured they were basically real things. I mean the mathematics tells me that the magnetic field around a bar magnet looks like this:

Theoretical magnetic field around a bar magnet

And after all, if you put a bunch of iron filings around a bar magnet, they look like this:

Iron filings align themselves along magnetic field lines (photo by Dana Mason)

And just to be sure there’s nothing fishy going on, we can check the same thing with a bazillion compasses–although you might have to rob the local Boy Scout equipment store to do it:

Compass needles align along magnetic field lines (photo by Dana Mason)

So, okay, that seems real enough, right? After all, it’s hard to tell what’s “real” and what’s just a mathematical tool in the abstract world of theoretical physics. But it seems reasonable to say that if some mathematical thing (like the field lines in the top picture) has exactly the same properties that “real” space has (like the space inhabited by iron filings and compasses in the lower pictures) then that thing is pretty “real”  itself, right?

Right?

Right. No, that’s really true, even though it sounds like I was setting you up for the answer to be “wrong.” I just wanted to give you a taste of the same head-fake feeling I was experiencing this morning, because you see, even though a good way to define what mathematical things are “real”  is “those things that correspond exactly to what we see in ‘real life’ ,” a magnetic field is not one of those things. Neither is an electric field.

Now I knew there was some controversy about this for some time, because sometimes physicists like to write about things called magnetic and electric potentials, instead of fields, just to make the math easier. They’re just different mathematical objects, but in classical (non-quantum) physics, those two things are actually equivalent. The same information is encoded in each; Tell me the potential and I can tell you the field uniquely, and vice-versa (well, as uniquely as possible. If you’re a nit-picky physicist reading this, pretend I said “up to gauge symmetry). So in that sense, it’s a bit like when you go to the grocery store and you discover you forgot your Safeway Select card. You can give them your phone number, it’s just as good. It allows them to get all the same information.

Unfortunately, as I learned today, the world of quantum mechanics can tell the difference between a potential and a field. And weirder, it likes the potential better. In fact it turns out, in quantum physics there is some information about the “real world”  that only the potential knows about. The magnetic and electric fields can’t be “real”  under the above definition because they actually don’t tell you everything about how electricity and magnetism behave in real life. The potentials, on the other hand, tell you everything the fields tell you, and then a little something extra. So the proper way to think about it is that the potentials are real, and one of the properties of creating a potential is that it makes space look like there’s a field in it. But there is no “field,”  there’s just potential. The “field”  is just a nice mathematical trick to calculate things about where and how particles will move.

I fear I’m getting a bit confusing here– I certainly know I’m a bit lost in thinking about what’s “real”  and what’s not. You may be thinking to yourself “if having a potential around makes it seem like there’s a field around and then a little something extra, then why not just say there are two real things: the field and the something extra?” It’s a good question, and it’s hard to explain without teaching you three years of electrodynamics. But I think there’s a good analogy that can be used here:

Remember Plato’s Allegory of the Cave? In it, our pal Plato tells us a story about people who grow up and spend their whole lives living in a cave, chained up so that all they can see is one dimly lit wall. Behind them is an enormous fire and a passageway across which people occasionally walk. In this way, the chained men only ever see the shadows created by the real people behind them. They come to think of those shadows as reality.

Well I know on some level most of us would be willing to grant that shadows are “real.”  But no one would say, I venture, that they are “real”  in the same way that the people casting them are. The shadows are a mere artifact of the people’s existence: It is obvious to us that the people are what’s “real,”  because once you know what the person looks like, you know exactly what their shadow looks like. On the other hand, the prisoners who see only the shadows are missing out on the full reality of the people who cast them. So while the shadow’s aren’t “unreal”  in the sense of being “figments of our imagination,”  it’s philosophically a lot more comfortable to say “My theory of this cave is that it contains two things: people and fire, and the combination of the two also makes it look like there are shadows,” than to say ” this cave contains three things: fire, shadows, and funny ‘3D’ shadow things that stand in betwee, and have the same shape in profile as the shadows, only they also have some extra attributes like color and texture.” That second theory just doesn’t sound like it’s really describing “reality,” even if it technically includes all the details.

And so it is with fields and potentials. You could think of fields as real if you like, with the potentials filling in the attributes. But you’d be settling for that second description of the cave.

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About Colin West
Colin West is a graduate student in quantum information theory, working at the Yang Institute for Theoretical Physics at Stony Brook University. Originally from Colorado (where he attended college), his interests outside of physics include politics, paper-folding, puzzles, playing-cards, and apparently, plosives.

4 Responses to ∫ The big lie: or, why physics made me think about Plato’s Cave today

  1. Paul West says:

    Are the manifestations, like the iron filings demonstration, detecting potentials? Or are the fields still the manifestation of potentials? Are these potentials the same as what I might think of, such as voltage?

  2. Stephen says:

    Wouldn’t the existence of a field imply an underlying fabric? I remember they used to conceptualize this with “ether” before the idea was completely discredited. But particles, and therefore potentials, must be discrete, no?

  3. Colin West says:

    Sorry it took me so long to comment on this. I was thinking about doing a longer post about how quantum mechanics obscures what’s “real” even further, but I have to hold off on that a bit until I understand it a little more myself.

    Short answers: yes, Stephen correctly points out that in some sense even the potentials aren’t real, being as they are just smooth approximations of something that is actually a discreet, quantized phenomenon. So maybe that takes some of the “oomph” out of the revelation that they’re at least more real than magnetic fields, but I hope it doesn’t. It shouldn’t; the essential point is that there’s nothing fundamental about a field even before quantum mechanics comes in to play, which is certainly not what you’re taught in high school.

    Also, yes, when you see something like an array of iron filings tracing out “field lines,” it’s most technically correct to say they’re responding to the underlying potentials and not due to the fields. In fact, this is one way to see why we dreamt up fields in the first place: because the connection between the potential and the physical phenomenon can be hard to visualize, involving as it does lots of differential vector relations. So yes, a bar magnet creates a magnetic potential, and in the presence of such a potential, an iron filing follows some crazy instructions like “line look at the values of the potential around you, calculate how quickly they’re changing in various directions, and then add those numbers up in a particular way to get the coordinates of the direction you should face. But it just so happens that 99.9 precent of the time, following these directions causes all the iron filings everywhere to all point along nice, smooth lines, which are much easier to describe mathematically. So naturally, physicists’ first attempt at describing things like the motion of iron filings was to say “there are these smooth lines called magnetic field lines, and what happens when you put iron filings near a magnet is they follow those lines.”

    Finally, voltage is indeed the electric potential I’m talking about. Electric voltage is the much easier to visualize of the two, so it’s not entirely foreign to us and a lot of people naturally picture voltage and electric fields as coincident phenomena. But their directly related, and nowadays we know that the relationship is one-way, that the voltage causes things to behave as though there were a field, not the other way around. This is maybe a clearer way to make the point I was trying to make in the previous paragraph: If I have two parallel metal plates (a giant capacitor) and I charge one of them up with positive charge and one of them with negative, there will be high voltage at one plate and low voltage at the other. That’s all that’s “really” there (unless you’re stephen, in which case you’re deciding where to truncate your evaluation of a Feynman path integral) but a charged particle that you place between the plates acts like it’s moving along “electric field lines” that stretch between the two. So it’s convenient to think of al electric field existing between the places, consisting from straight lines from plate to plate, and saying that the law governing the motion of the charged particle in between is that it always moves along these field lines. In reality, though, the more fundamental rule is that it tries to move from low voltage to high (or vice versa, depending on it’s charge).

    What’s weirder is the concept of the magnetic potential, which is like “magnetic voltage” only it is a vector quantity instead of a scalar one. So in addition to asking whether the magnetic “voltage” is high or low, you have to ask which way it’s pointing. This is what makes working with the voltage such a pain when you’re doing simple calculations, and why it’s so much nicer to play along with the lie that there’s just a magnetic field to worry about.

  4. Stephen says:

    It is quite shocking, I agree. Did you know that multiplying negatives isn’t the same as multiplying positives, or that we distributed smallpox to the indians? As a kid I didn’t.

    And the real damage: They teach us in precisely the wrong way.

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