Here’s why we don’t understand what electricity is

This entry is part 1 of 2 in the series Mystifications

In Mary Shelley’s Frankenstein, written in 1818, the young Victor Frankenstein becomes obsessed with the idea that electricity is a kind of fluid that endows living things with their life force. This obsession leads to tragedy.

Shelley’s view of electricity was, in fact, not an uncommon perspective at the time: just a few decades earlier the Italian scientist Luigi Galvani had shown that a shock of static electricity applied to the legs of a dismembered frog would cause the legs to kick. Galvani concluded that there existed a kind of “animal electric fluid” that was responsible for the animation of living creatures.

A diagram from Galvani’s De viribus electricitatis in motu musculari commentarius, 1791.

In the two hundred years since Frankenstein our view of electricity has certainly evolved, as has our ability to generate and control electric currents. But do we really understand what we’re doing? Do we even know what electricity is?

If you ask a teacher “what is electricity?”, the standard answer you’ll get is that “electricity is the movement of electrons.” But there are two glaring problems with this answer.

The first problem is that if you’re going to tell someone “electricity is the movement of electrons,” then you should probably tell the person what an electron is. And it turns out that science struggles to give a satisfying answer to even this obvious question.

For any science student at the college level or below, an electron is always drawn as a black dot and we are told that this black dot has a mysterious property called “charge”. The existence of such charged black dots is supposed to be taken as a given, with no explanation as to where they come from or what they’re made of.

If you happen to keep pursuing such questions to the level of graduate coursework in physics, the buck gets passed one level further, and you are told that an electron is actually a kind of excitation or defect in a pervasive, “electron field” that fills all of space. But don’t try to ask “what is this field?” or “what is it made of?”. You are likely to be told that such questions are nonsensical, and the electron field is simply a mathematical object.

 So, if you’re keeping score, after just two levels of drilling into the question “what is electricity?” we arrive at the answer “it is a mathematical object.” Only a strange kind of person would consider that to be a satisfying form of understanding.

The second big problem with the answer “electricity is the movement of electrons” is that it isn’t really true, at least not any more than saying “the Boston marathon is the movement of eukaryotic cells.”

Inside the copper wires that enable all of your electronic devices, there are many electrons, but they exist in a kind of messy quantum soup of fast-moving and strongly-interacting electrons. This soup is so complicated that any serious chemist or physicist will tell you that there is no hope for ever describing it precisely (even taking the existence of black-dot electrons as a given).

Electricity, then, cannot be carried by just “the motion of electrons”, since electrons are not free to move on their own. Instead, electric currents are carried by complicated wave-like excitations of many electrons simultaneously. These wave-like excitations get called “quasiparticles“, and the primary theory for describing them is called Fermi Liquid Theory.

But for all its successes (including a Nobel prize to its inventor), Fermi Liquid Theory fails the “is this understanding?” test on two levels. First, if you ask even a great physicist to draw you a picture of a quasiparticle so that you can understand what it is, you will either get some obviously farcical cartoon or folksy analogy, or you will get someone once again telling you that it is a “mathematical object”.

An actual diagram from Richard Mattuck’s textbook A guide to Feynman diagrams in the many-body problem, 1992.

But even if you’re willing to accept “mathematical objects” in your life, Fermi Liquid Theory will let you down once again. Fermi Liquid Theory is based on a Taylor series expansion (approximating a complicated function by a straight line), which assumes that a certain number called rs is very small. But even in the most straightforward electrical conductors like copper, which Fermi Liquid Theory apparently describes very accurately, rs is not a small number at all (it is usually between 2 and 5). So in this sense the success of Fermi Liquid Theory in describing electricity looks like a pure accident. Do we really deserve to call it “understanding”?

Mathematically, this embarrassing situation is like claiming that you understand the function sin(x) because you understand the function f(x) = x, and the two are basically the same.

Cringe.

Of course, now that we live in the 21th century, we have accepted the Faustian bargain and electricity is all around us all the time. Electricity and electrical devices have been studied, tested, engineered, and woven so tightly into into the fabric of our everyday lives that nearly 90% of the world’s population makes regular use of them.

But can we really say that we understand electricity, given how much we struggle even to articulate what it even is?

Or are we like the hubristic Victor Frankenstein, toying with forces beyond our comprehension, never stopping to think what dark consequences may await us?

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About Brian Skinner

Brian Skinner is a physicist who specializes in the theory of strongly-interacting many-body systems. He is currently an assistant professor at Ohio State, and he blogs at Gravity and Levity

Comments

  1. I’m sorry to be blunt, but this is just basic philosophical confusion about what it ‘means’ to ‘understand’ something and what it ‘means’ to ‘be’ something. Using the words like this, we also don’t ‘understand’ what water ‘is’, what gravity ‘is’, what anything ‘truly’ ‘is’.

    • “Understanding” is based on the level of abstraction you claim to understand. I understand the motion of a ball as it rolls down the hill, if my level of abstraction is “a rounded object is rolling down an inclined terrain”. I of course don’t understand it at the level of quarks within the ball and the four fundamental forces within the ball that are making this complicated motion possible.

      When we claim to “understand” electricity in school or college, we claim to understand it at the level of electrons. This claim is manifestly false, as demonstrated in the article above. If our claim was just “we understand electricity at the level of flowing into our devices when we plug them into electrical sockets”, our claim would be correct

      • An Engineer says

        This is all premised on who “we” is (and or what the definition of is is). If we is the inclusive humanity with all our various engineers and scientists then yes we do understand electricity. If “we” is the lumpen proles then “nobody knows nuthin”.

        The electron is an asymptotically dense quantum object with negative charge that can be expressed math mathematically as a solution to the electro-weak field equations. The behavior and shape of this particular beasty are well characterized and “we” manipulate them at will in service of our desires.

        So you might not “understand” but I seem to “understand” well enough to succeed at balancing the phase of my local grid and its constituent sites well enough to keep your lights on. Feel free to insinuate that the user base doesn’t capital-K KNOW anything about the underlying substrate that makes their conveniences spin but don’t say the capital-H Humanity doesn’t capital-U Understand what it is doing. We do, I Do, now hold my beer I’ma go and try to face off with a application stack with a complexity level approaching that of a bacteria.

        • To An Engineer:

          Borrego Springs CA has 2 PV farms, 2 diesel generators, battery banks and 3500 people. I asked why 12 engineers were working on phase matching. Yes they knew they had to do it but were having many problems. We have planned outages numerous times a month and unplanned ones as well.

          Why is this so difficult?

          • An Engineer says

            High road:
            There wasn’t enough communication between the principle engineers who designed and oversaw installation of each of these 3 – 5 systems. Either they were installed at different times and the last one in didn’t do enough math or they did the math and the customer (municipality?) didn’t like the real price leading to underestimation of engineering cost or overestimation of gear performance.
            This happens all the time. Customer says “hey this other gear is cheaper lets use that” and the engineers say “oh lets not because it lacks critical feature X” (sometimes that feature is just being the one your local repair men have familiarity with) and the customer doesn’t listen the first three times so the engineers give in; why fight when you will be paid more to fix it later. I personally hate this work flow (it’s wasteful) and have on occasion taken political flack for trying to prevent it, but people only have so much energy to fight. I don’t think the customers are bad guys but they on occasion do lack perspective, it turn out people skills are more important then math skills for engineers but it’s hard for engineers to understand this without living through a few massive failures it’s “the human condition” as they say.

            Low road:
            It’s the PV, and/or the CA rules. I could go on and on about this but it would be petty. I’m not pro-nuclear I’m pro numbers.

            TLDR: If your integration is taking too long don’t blame the gear or the workers or the designer or the government or “electricity”; blame lack of effective communication by all parties. Communication is HARD!!!

            PS: Ok maybe blame the gear a little bit.

    • Ooh, “here’s why we don’t understand what water is” could be a fun entry to this series. Thanks for the suggestion!

      • If what you’re going for is all the ways in which we don’t ‘really’ ‘understand’ things, then water is definitely a candidate on par with electricity. But when *do* you consider something as ‘understood’? What is the ground level we’re aspiring to? Because if we don’t understand anything, the word ‘understand’ has become very vague; it has become some indefinable fictional way of comprehending things to which we can only aspire, but which is unattainable and of which it is unclear whether you can even measurably approach it. A word that can forever be used to wax mystically about things purportedly just out out of reach.

        That life is mysterious is not due to a lack of understanding. Part of the persistent mystery is that no amount of understanding, explanation, unification, dissolves the mysteriousness. It just shifts the goalposts.

        • Sorry to be a troll, Ivo. The series is supposed to be semi-satirical, and part of its goal is to make the point that one must always negotiate with the concept of “understanding”. As you say, one can always define the concept of understanding in such a way that absolutely anything fails to reach the bar.

          The other point is just to geek out about how there is cool and surprising science hidden in even the most familiar things.

          I will say that originally had a disclaimer at the top about this being satire, but Venkat removed it. So I guess the watchword now is: the trolling will continue until morale improves.

          • Ah, I failed to apply Poe’s law and should have given more credit to this being posted on Ribbonfarm. I’m sometimes too firmly stuck on simulacrum level 1.

            Curious to see where this goes :)

      • Yeah, we don’t know/understand things at a fundamental level. It is described best (I’ve read so far) in Erogamer (which starts off as actual porn, but is mostly great because of metaphysics porn; insight crack)

        Quote:

        “Hm,” said Sonia. “I suppose that’s a fair point. Well then, let us suppose instead that even though everything exists—or equivalently and more simply, nothing exists—there is nonetheless some mysterious factor by which certain things exist more.”

        “Bullshit,” Charles said.

        “I suppose we could call it that,” said Sonia.

        “No, I mean the whole idea is bullshit,” Charles said. “It sounds like a metaphysics George Orwell would invent as a parody. ‘All animals are real, but some animals are more real than others.’ How could they tell?”

        “You claimed yourself that Snow White exists as much as we do,” said Sonia. “Then why don’t we run into her at the corner shop? Why do I find chocolate biscuits there, instead of Snow White? There is a story in which the two of us meet, and the people within that story have no way of knowing themselves to be unreal. And yet I find myself discovering chocolate biscuits instead. Clearly, there is some factor that the possibility containing myself and chocolate biscuits possesses in greater quantity, compared to the less real possibility containing myself and Snow White. Even if we reply to the great question by answering that nothing exists, some zeroes are greater than other zeroes and quantitative ratios may be established between them. Like any other self-observing structure, Snow White finds herself to be exactly as real as herself, a ratio of one to one, and in this sense her existence is locally an absolute. But to say this does not say whether Snow White is more or less encounterable than other things. There is some quantitative degree to which our universe is looking more towards the chocolate biscuits. Some essence of how much something is observed or experienced, which the chocolate biscuits have more of than Snow White. We could call it quintessence, or propensity, or mana, or the blood of God.”

        “Or bullshit,” Charles said.

        “I suppose that term is as good as any other,” Sonia said. “By whatever name, it is the single, sole, and only truly universal form of money. And we can extend the same logic further. Having postulated the notion of bullshit, it would follow that things are more real only to the extent that they are, in some sense, more bullshit. Or rather, by definition, anything that makes a possibility more encounterable is exactly what we are calling bullshit. Possibilities are experienced by conscious beings in exact proportion to the total bullshit breathing fire into those possibilities. Then to whatever extent a mathematical model is not describing bullshit, it is mere math. Only to the extent that a mathematical structure is about bullshit does that structure form an encounterable part of its universe. It follows that every sapient species, if they investigate the physics of their universe far enough, will eventually find a level at which physics seems solely to describe the arrangement of pure bullshit—some physical quantity whose apparent meaning is making possibilities more encounterable. By your own argument, the presence or absence of bullshit must be falsifiable for anyone inside the universe to notice a difference. Then the eventually-discovered laws will show that variations in bullshit are experimentally observable and cause other events to proceed differently. Indeed, variations in bullshit will be the only causally potent factor. Across every universe with conscious life, any inquisition into physical law, sufficiently advanced, will render everything into bullshit, which is and must be the sole constituent of experienceable causality.”

        “Can you simplify that?” said Charles.

        “No, but I’ll do it anyways,” said Sonia. “Whatever it is that makes one possibility more encounterable than another, we are calling that bullshit by definition. Then whatever bullshit is, if something isn’t made of it, we won’t encounter it. And if we couldn’t detect something experimentally, we’d have no way of encountering it. So bullshit has to appear to us as a quantity in our equations—in fact, the only thing structured by our equations, because we can’t encounter anything else. ‘By convention there is sweetness, by convention bitterness, by convention color, in reality only atoms and the void.’ And underneath atoms and the void, it’s all just bullshit.”

        “Can you simplify that some more?” said Charles. He felt ashamed, but he wasn’t at his mental best while asleep.

        “The only thing that truly basic physics equations can be about, is the stuff that breathes fire into equations, the stuff that makes things be more real,” said Sonia. “That’s what I’m taking as my character’s second Element, if the DM lets me get away with that.”

        —-

        Also, this:

        “What do you mean, you can’t get to the center of a spiral?” the boy was saying. “I thought I just did.”

        “Not if it’s a hyperbolic spiral,” his mother said. “Those loop around an infinite number of times at the center. If you looked closely at the center of a hyperbolic spiral, you’d just keep seeing more of it. You could keep zooming in and it would keep going around an infinite number of times. So to draw it, you’d have to make an infinite number of circles with your chalk. Which you can’t do.”

        “Because it’s not allowed?” said the boy.

        “Well,” the mother said, “if you tried, I’d make you come inside for lunch before you starved to death. So in that sense, yes, it’s not allowed.”

        “What would happen if I got to the center of the spiral anyways?” said the boy in fascinated tones. “Would it kill me?”

        “It’d wipe out the whole Earth,” the mother said solemnly, in the tones of somebody who didn’t want to discourage this interest in mathematics no matter where it had come from, and would go to any lengths to play along. “It would destroy the rest of the Milky Way too. Enough sidewalk chalk in one place would form a supermassive black hole that pulled everything else into it.”

        “It destroys the whole world if you draw it?” said the boy, looking very impressed by this.

        “That’s right!” said the mother. “A hyperbolic spiral is one of the simplest kinds of entities where it’s easy to draw part of them, but if you drew one in too much detail it would destroy you and your world and everything else you knew existed.”

        “Can you know about the center of a spiral?” said the boy.

        “That’s a very smart question!” said the mother. “You can know about the center in some ways but not others. It’s no problem if you just look at the equation for a hyperbolic spiral without visualizing the figure it draws. But if your brain represented enough of the details inside, it would form a supermassive black hole and then we’re back to destroying galaxies again. It’s like a story that can only be told in metaphor, because showing a movie of it would destroy the movie theater.”

        “Wow!” said the boy. Then he suddenly looked worried. “Can that happen if I think too hard about the centers of spirals?”

        “Honeybuns,” the mother said, “if that could happen, I’d never have hinted to you about the possibility of hyperbolic spirals existing. That you can’t think about the center in enough detail isn’t a rule like taking your medicine is a rule. It’s just something that’s true.”

        “Who made it be true, though?” said the boy. “Was it somebody who didn’t want me destroying galaxies?”

        “We’ll talk about that more when you’re older,” said the mother. “How to divide up responsibility between God and the anthropic principle, I mean. Now finish up drawing as much of the spiral as the truths let you draw, because lunch is almost ready.”

    • It is functional understanding through a model versus understanding of what really exists.

      The fact that we can manipulate a thing is different from having a fundamental understanding of the thing. The two are related, but may never be confused — the author is pointing out that people confuse them all the time. He could have expanded his earlier history of electricity as a “fluid” to include the fact that you can build successful capacitors and do a lot of basic electric things with this fully flawed understanding. Now, as we evolve our models, it is questionable if we are evolving our understanding towards what actually is, or if we are creating something like a mystical story on par with “electric fluids” that we can use to make things work.

      If you believe that there is a “something” that really exists, and your insight into that thing is vital or you worry that missing that could have serious consequences, then you might be bothered by all this, even the idea that as the OP pointed out, these are “mathematical objects” (therefore we ADMIT and ENDORSE that we only understand them as our abstractions of them). If you believe that doesn’t matter as long as you have a functional grasp on what is working for a particular intention, then you might not be bothered by any of it. I could see merit in both positions, even vis-a-vis the ostensible goals of each other.

  2. Respectfully disagree with Ivo. 💧 is pervasive in our natural environment.⚡️ is something we generate, “transmit”, and “utilize”. It’s utility is so great we devote a surprisingly large amount of our infrastructure to it (and Texans get angry when it fails). Any word can be subjected to Socratic inquiry, but as Brian points out we have only the blurriest understanding of what ⚡️ is, much blurrier than 💦, and this is a worthy initial effort at consciousness raising. The diagram of a 🐎 is especially helpful. I look forward to further elaborations.

    • Thanks Larry!
      I’m guessing, though, that if I were to drill down far enough, I could make the case that our understanding of water is similarly blurry to our understanding of electricity.

      • And that would be where I would respectfully agree with Ivo. Socratic inquiry or Aristotelian dissection can be useful on almost anything but are frequently just tedious applications of a general philosophical inquiry into how we use Language as opposed to how we understand a “thing in itself” whether with words or intuition or math.

        ⚡️ is the more worthwhile topic for showing that we really don’t know all that much about something that behaves in such mathematically consistent ways. What do we really understand about “skin effect” even if we can describe it with mathematical precision?

  3. White_Rabbit says

    Around half of this is book/website deals with this problem.
    https://meaningness.com/

  4. An Engineer says

    “Or are we like the hubristic Victor Frankenstein, toying with forces beyond our comprehension, never stopping to think what dark consequences may await us?”

    You are afraid of the wrong thing. Electricity is well within our comprehension and control. First we conquered mater, then energy now we are assaulting the gates of hell itself: information. If we can harness this final member of the holy trinity we will ascend to godhood. But in trying to harness this final beast we court destruction at it’s hands.

    When we stare into the nightmare rectangles they stare into us.

  5. Julien Boyreau says

    I don’t know if the author meant it eventually, but what I found the most interesting in this article is not if electricity can be understood but if “understood” can mean anything objective, proper and self-sufficient.

    Like others, I don’t think so.

    Natural language is the most addictive drug I love.
    As such, we should always enjoy abusing it for poetry.
    But we must always fear to be abused by it as pedantry.

  6. Simaen skolfield says

    Electricity is produced by motion of two opposing forces.

  7. I enjoyed this perspective on the mysteries of electricity. I’m surprised that you didn’t include the weirdness of AC and how wiring diagrams aren’t the concrete mathematical truths you’d expect.

    Also, though you didn’t say it explicitly, I think that you are trying to illustrate the distinction between knowing and understanding. To know something is simply to accept the premise but to understand it is to be able to see all the parts that make up the theory. You can know something is true without understanding why it’s true.

  8. One of the things I like about advanced physics is that it drives me to think about exotic, intuition-breaking phenomenology that breaks anthropocentric conceits in epistemology and ontology. A weakness in philosophical discussions is that they rely too much on familiar human experiences, with regularities in phenomenology that are reflected in our own unconscious behaviors. A question like “particle or wave?” pulls the rug out from under our intuitions and messes with our conceit that we understand shit, even if we can’t define what it means to understand things. It is interesting in a way that, for example, the trolley problem is not.

    On the philosophy end, I’ve found discussions of “intension” (with an s) to be very useful in sorting through meaning/pointing confusions.

    The question of “what does it mean to understand something” is I think ultimately a dull and somewhat bureaucratic one one. The point is to try and understand _specific_ interesting things, and building up a correspondence between what you mean, what you point to, and what you say, that is robust enough to bring the conversation down to object level. So you can talk about what is interesting rather than the middle-management philosophy abstractions. If you get caught up in “what does it mean to mean things” rather than why electrons in particular are interestingly hard to grasp, you’ve allowed yourself to get lost in the philosophy at the expense of what is being pointed to.

    This is in a way a return to one of the oldest topics on ribbonfarm. Two articles I wrote in 2007 get at this same question. In Concepts and Prototypes I talked about how our thinking is shaped by “typical” examples and behaviors, and in How to Define Concepts I discussed the problem of converging on the intension of “convexity” by way of a watertight definition, and how to iterate from intuition towards rigorous characterization. It is notable that in both articles I relied on platonic mathematical concepts rather than empirical physics ones to illustrate my points. What Brian is doing here is the same problem but for the vastly harder domain of physics. In a way, math concepts are easier to philosophize.

    • Lol at “middle-management philosophy abstractions”.

    • Julien Boyreau says

      Well said.
      Meaning alias conception alias understanding is the endless game we play to refine the map of our words to the territory of our experiences, in the 4 possible directions :
      > words to words
      > words to experiences
      > experiences to words
      > experiences to experiences

      The trickiest problem is that we are playing this game each with ourself and between us, thus our maps can conflict.
      The best one is thus to succeed in mapping across people by common ways :
      >> for Words To Words we made logic
      >> for Words To (Experiences To Experiences) we made science

      The hardest one is Words to Words where we should be careful not to produce dead ends by language abuse.

    • Ha!
      Yes, particle vs. wave, indeed!
      I think this hits on the topic of “Understanding”.
      We use words like particle and wave to evoke images of solid stuff and vibrations, but they don’t have any literal correspondence to subatomic phenomena as observed.
      Still we persist in treating the theoretical world as a collection of tiny bowling balls and pond ripples.

      I was having a conversation with one of the co-discoverers of the Higgs boson, and asked her about this wave/particle thing. Was it possible to describe a wave to a third grader in a way that didn’t involve tossing a rock into a pond? Because with that wave traveling from the Sun to Earth, where is the water? The aethers were supposed to solve that problem, right?

      Her answer was really interesting; she said “We treat everything as a particle.”

      And I thought I might be getting somewhere…

      So I am left with the simplest solution being that the evidence of our senses, and the mental models we make of daily phenomena have probably no meaningful relationship to what the world is “Really” made of. Such as infinite vibrations, or the mind of god, or ‘fields’ that we can partially describe with math.

      I refuse to be a solipsist because it is boring.

      Thanks Brian & Venkat!

    • A weakness in philosophical discussions is that they rely too much on familiar human experiences, with regularities in phenomenology that are reflected in our own unconscious behaviors. A question like “particle or wave?”

      As if Einstein wasn’t willing to give up “familiar human experiences”. He was mobbed by many of his colleagues for doing exactly this. He was nevertheless discontent with Bohrs pragmatism and closing of the quantum matters, with viewpoint switching and protocols which work around obstacles.

      The sort of “intuition” physicists and mathematicians reclaim has little to do with “familiar human experiences”. They are intuitions-as-if, “abstract intuitions” as Emmy Noether called them. Still those can break too and quantum physics or set theory witnessed those breakdowns. You can have a smooth mathematical apparatus with entities which are vectors in finite or infinite dimensional Hilbert spaces [1] but then there are those damned particle like events and you have to throw the math at the wall. One can try to avoid this by giving the math an interpretation as “statistics” but that is a statistics-as-if. Alternatively one can interpret the particle events as a branching process of the universe or expect deeper insights from a yet-to-create theory of quantum gravity ( Penrose ).

      There has always been a looming sense that one cannot manufacture normalcy around this, no matter how hard one tries, something which was already captured by Bohr. The weirding is necessary. Physicists are a strange breed who seem to be both frustrated about it – they would rather be (neo)-classicists – but they also deeply enjoy their “weird realism”, which has become the trademark of the moderns.

      [1] I don’t remember exactly who it was who said that he found infinite dimensional Hilber spaces “anschaulich” ( intuitive in a graphical sense ) – maybe Faltings?

  9. In the terms of this argument we don’t understand anything.

    Light
    Gravity
    State changes
    Plasma
    Time

    Anything!

    But when we have theory and formula in the case of many/most of these things which enable us to almost unerringly predict the outcome of their manipulation or action within a sufficiently well defined context, it is surely acceptable to proclame understanding.

    Absolute knowledge is not equivalent to understanding.

  10. Chris Phoenix says

    You pointed out elsewhere* that we can’t understand/describe chemistry in terms of subatomic particles, or biology in terms of chemistry.

    How much of this post is a reflection of a doomed attempt to describe electricity in terms of too-small entities?

    What is the appropriate level to describe electricity at? Are there different sub-fields of electricity with different answers to this question?

    * https://www.ribbonfarm.com/2015/09/24/samuel-becketts-guide-to-particles-and-antiparticles/#comment-243162

  11. The map is not the territory. You can sort of “understand” a map well, but it’s only a representation, one representation, of what “is”. There’s no need to argue the finer details of the map if the map serves the purposes for the time being. However, refining the map, or providing a better, more accurate the map, the more variety of purposes we can use find for other purposes. In other words, a map that worked in a school classroom was sufficient for that context but not necessarily entirely adequate or helpful for landing a rover on Mars.

  12. It’s always interesting to me when people start driving theories “off road”, a lot of neural network results that people use seem to be done in areas very divorced from the cautious domains mapped out by mathematicians who first tried to justify them, and it’s curious how we seem to be able to project out along the lines supposed by a theory, long after the domain of certified approximation has been left, with some kind of confidence of analogy that allows us to treat these localised solutions as helpful in the firstplace.

    So like for copper, what are those terms that would be coming up next in the taylor series? What happened to them, and why are they apparently cancelling?

    My favourite solution would be something elegant and symmetry based, say that there’s a pair of degenerate states equally affected in opposite ways, so that you can superpose the two and end up with a fermi liquid conforming state, though maybe with some localised quirks when it starts scattering.

  13. Great discussion. Would be interesting to incorporate mass of the electron into it because that indicates that electron sometimes behaves as a real and measurable particle.

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