When measuring the wavelength of EM radiation... what's actually being measured?

Submitted by Grand-Tension8668 t3_1125ccr in askscience

So... I have two competing mental models of what EM radiation actually is from a mechanical perspective. (Turns out a "mechanical perspective" might not be possible which is... distressing.)

  1. Photons are not EM radiation, but they have EM "fields" around them in the same way that, say, protons and neutrons have "fields" that interact with each other, and those "fields" are where the "waves" come in. These fields are not governed by particles, but are rather some vague other thing.

OR:

  1. Photons are EM radiation, and they move in a straight line... but only when we're paying close attention, because wave-particle duality. They are the medium propegating the "waves".

I must be wrong either way. Because we can measure these waves in terms of length, specifically in nanometers.

I'm inclined to think that I'm closer to reality with #1 considering that "reducing all forces to collision" is apparently a lost cause which my brain is really having trouble accepting as anything but magical thinking.

If #1 is closer to the correct way of viewing things, you're telling me that this particular spooky action at a distance occurs over great enough distances to be expressed in nanometers? And more importantly, how can we even measure that it occurs in "waves" spanning nanometers? I know it's describing a "vector field" but doesn't that still require some thing that could be detected and measued as having a velocity, and therefore momentum and mass, i.e. a bunch of particles?

And what do photons even do in this context? I've seen them described as essentially a side-effect?

Edit: I love all the discussion occuring but it's also made me realize that the body of my post mostly goes beyond the question. The question being, we're able to describe the physical wavelength in nanometers of these waves that apparently aren't oscillating in space so much as they oscillate between electric and magnetic fields. ...how do you assign a unit of length to that?

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Randywithout8as t1_j8iw2gt wrote

A lot going on here. I think of a photon as a packet of light energy (frequently this is referred to as a "quantum" of light energy). This packet consists of some electromagnetic distortion. This distortion has a wavelength and a frequency. These values are interchangeable due to the speed of light. Additionally, through the speed of light relationship, the energy of the photon is defined by its wavelength and frequency. We can then measure a photon using a photodetector. Think of a photodetector like a solar panel with more precise electronics. When a photon hits the photodetector, it transfers its energy into a semiconductor, the photon energy is absorbed and converted to electrical energy. The electrical energy is measured and then we can work backwards from electrical energy to photon energy to wavelength.

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Randywithout8as t1_j8iycbc wrote

Replying to Randywithout8as (#1,808,989)

As far as model 1 vs model 2. They're both kind of incorrect. A photon is not like a neutron. Photons are massless (not exactly because they have momentum, but that's the way it goes...) a photon is also not the "medium" through which EM radiation travels. A photon is a way to "quantize" (or break down into the smallest possible unit/packet/measurable piece) a beam of light.

Another way to view it: When light is emitted, it is quantized (discrete). It is not continuous. When you turn the dimmer switch on an LED, it doesn't produce a bigger or smaller light beam, it produces a larger or smaller number of photons. Each electron you pass through the LED excites an electron to later decay back to ground state. This decay emits a photon. The more electrons you pass through the LED, the more photons get produced, these photons constructively interfere to produce a higher intensity light.

Try not to think of a photon like a tiny ball of something. It is just a term to describe a unit of a type of energy.

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Laetitian t1_j8izs2w wrote

Replying to Randywithout8as (#1,809,119)

>Another way to view it: When light is emitted, it is quantized (discrete). It is not continuous. When you turn the dimmer switch on an LED, it doesn't produce a bigger or smaller light beam, it produces a larger or smaller number of photons. Each electron you pass through the LED excites an electron to later decay back to ground state.

Though there is also the other way of altering the energy content of a photon, by impacting its frequency (and by extension wavelength.) It's just not useful for affecting the intensity of visible light.

Since we're explaining the photon's essence, I figured making that explicit might help to avoid further confusion.

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hoffmanmclaunsky t1_j8j1h0t wrote

I'm only very casually knowledgeable in QM, but my understanding is that photons are waves that propagate through the electromagnetic field. They are not the medium, the medium is the field.

As for measuring their wavelength, the most classic example is using a double slit to create a wave interference pattern, then measuring that. Lasers make this easiest and most accurate but it's possible to get an idea of the scale of these wavelengths even using white light.

Lastly, wave particle duality and the uncertainty principle don't really work with most "mechanical" interpretations, at least in a general sense. They're helpful if you're thinking about specific aspects of photons and other particles, but the reality is that quantum behavior just has different rules and trying to understand through the lens of more intuitive mechanics just doesn't work all that well a lot of the time.

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PapercuttingTheHell t1_j8j7mri wrote

I could answer but it's primarly borig for anyone else that electricians and electrical and electronical engineer. So just to make it clear, there's no matter involved, only electric fields, which have magnetic properties if used as so. We call it waves because as soon as we want to illustrate them ; models will turn that in something that's most understandable in the form of waves. Flies off

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C-D-W t1_j8jcpgn wrote

I have no clue but it is thought provoking. Lots of discussion about how it isn't a physical phenomenon. But what I find interesting is that when interfacing with the electromagnetic wave the length of your antenna matters a great deal. When talking about wavelength, it certainly has some correlation with a measurable physical property.

Kind of interesting stuff.

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platoprime t1_j8jngne wrote

Replying to Randywithout8as (#1,809,119)

Everything is energy so that's a pretty useless description. Plus photons are self-propagating oscillations in the electromagnetic field they aren't mystery balls of energy any more than any other particle is.

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Randywithout8as t1_j8jo36y wrote

Replying to platoprime (#1,810,582)

I mean, to OP it seems like they're mystery balls. I don't feel "self-propogating oscillations in the electromagnetic field" is very helpful as an explanation... you can critize what I've put down, but maybe try improving on it rather than just listing the definition and feeling superior.

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Laetitian t1_j8jo4vh wrote

Replying to platoprime (#1,810,582)

"In the electromagnetic field" probably just takes it from "mystery balls of energy" to "balls inside a mystery ether" for someone who doesn't understand the subject properly.

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platoprime t1_j8jo8wr wrote

Replying to Randywithout8as (#1,810,620)

Why not? Because you don't think they'd understand it? Because you can't explain it?

I'm struggling to imagine a good justification for, well, misinformation.

> but maybe try improving on it rather than just listing the definition and feeling superior.

I can definitely explain it better if you like.

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platoprime t1_j8jp0pp wrote

>The question being, we're able to describe the physical wavelength in nanometers of these waves that apparently aren't oscillating in space so much as they oscillate between electric and magnetic fields. ...how do you assign a unit of length to that?

Okay I see your confusion. The electric and magnetic fields are overlaid on top of space so they are 100% physically oscillating up and down and side to side as they move through space. If you're talking about motion through one of the fields you're talking about that motion occurring through space.

After all a field is just a value assigned to every point in space.

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shikuto t1_j8ju7er wrote

I’m going to jump into the meat of what you really want to know.

>The question being, we're able to describe the physical wavelength in nanometers of these waves that apparently aren't oscillating in space so much as they oscillate between electric and magnetic fields. ...how do you assign a unit of length to that?

How to assign a length? Light has a speed. 299,800,000 m/s. Now take a frequency. Let’s say 60Hz, because electricity in the US. That means that one cycle has a period of 1/60s. Multiply the speed of light by the period of the wave, and you get the wavelength. In this example, the wavelength of 60hz light is 4,966,666 meters. That’s a super long radio wave.

Visible light? Let’s say 600THz. The period is 1/600,000,000,000,000 of a second. This gives us 2.988e8 / 6e14, evaluating to 4.98e-7 meters. This is 498nm, something blueish.

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Grand-Tension8668 OP t1_j8jz4i7 wrote

Replying to platoprime (#1,810,678)

u/shikuto's comment got me to sort of picture how EM fields are waves (they're traveling through space as they oscillate, after all, which is all a wave is), but it's still surprising to me to say that the change in polarity is actually a locational change, if that's what you're saying.

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Movpasd t1_j8k0w6r wrote

Before I try to answer your question, I think you're suffering from a (very common!) issue with your approach to physics. I've written a lot about this topic on Reddit before, so I'll link some comments: see here, here, here, and here. Though it's not directly connected to your question, I hope that these can be helpful!

Now to get to your question. I'd start by not trying to understand photons right off the bat. It'll be much easier to first try to understand the electromagnetic field classically, and then try to integrate quantum effects into your understanding. The relationship between photons and the classical field is very tricky, and I find it's the cause of much confusion for physics enthusiasts and students.

In classical/Maxwellian electrodynamics, the electric and magnetic fields are vector fields. That is to say, they are mathematical functions which assign a three-dimensional vector to each point in space. These must obey Maxwell's equations (that's part of the theory).

Wave solutions to Maxwell's equations look something like this. (Shockingly, I couldn't find a clean diagram like this on the Internet, so I had to make one in Paint.) Of course, I can't show you every single vector attached to every single point in space on my diagram, because there's an infinite amount of them, so I've just drawn them for a select grid of points. I'm using the notation where vectors going into the plane are drawn with a cross, and those sticking out of the plane with a dot.

The size of the arrows is meant to represent the relative magnitudes of the fields as they evolve through space, but the actual lengths on the diagram don't matter -- the electric and magnetic fields don't have units of distance.

Hopefully this helps you visualise the idea of wavelength -- let me know if you have follow-up questions!

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platoprime t1_j8k2kyv wrote

Replying to Grand-Tension8668 (#1,811,288)

No I don't mean to say polarity is spatial movement. Polarity is a change in the intensity of the electromagnetic field along a line over time.

However it's important to understand that photons are not localized until they interact with something. When they travel through space they don't have definite positions or momentums.

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Laetitian t1_j8k5vf6 wrote

>The question being, we're able to describe the physical wavelength in nanometers of these waves that apparently aren't oscillating in space so much as they oscillate between electric and magnetic fields. ...how do you assign a unit of length to that?

The length is assigned to the distance it takes for the photon energy's wave to travel from one amplitude to the other and back. What that means physically *is* the phaenomena it expresses itself in, in the relevant experiments. If I was to speculate about how it's expressed as a logical principle of physical interactions, I'd say the wave is the photon's potential to bend or change direction, especially in contact with other matter (and the energy that constitutes said matter) as it travels through spacetime. But I'd also acknowledge that there's probably a reason why, even if my logic is correct, physicists would never oversimplify it like that.

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Grand-Tension8668 OP t1_j8k7co9 wrote

Replying to Movpasd (#1,811,383)

Thanks a lot for this reply. I've definitely started recognizing what you're trying to say in those other posts, that things really approach a point where you need to trust the math and coming at it the other way around fails to create an accurate understanding of things. (And that our intuitive understanding of what "stuff" is doesn't really hold water in an absolute sense).

I think I'm coming out of this with a less incorrect "mode C" mental model, at least– EM fields change over time / distance (one in the same in this case but whatever) in a cyclical way, so they're waves. We can measure how long it takes for one "wave cycle" to happen, as in the distance traveled as a point oscillates between the electo- part and the -magnetism part. ...Which is certainly still a pretty wrongheaded explanation and I really need to start learning the math of physics in my spare time.

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karantza t1_j8kcf9s wrote

Replying to Grand-Tension8668 (#1,811,288)

It's definitely not a locational change, but it does have a direction and "intensity", which drawings often represent as if it were a distance.

The electromagnetic field is a vector field, so it points in a real direction, and that gives us polarization. So as light travels in a straight line, oscillating in intensity between the E and B fields over time, those fields do have a direction, but no distance offset from the beam path.

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SilentHunter7 t1_j8kphnf wrote

I'm an EE grad student, who's specializing in microwaves and antenna theory.

So a good way to think about this, is that changes in the Electric and Magnetic fields don't propagate instantly; they travel at the speed of light. So if I turn on an electromagnet, it will take some time for any metal near it to feel that force. Even though that time will probably be measured in nanoseconds, it still takes a nonzero time between the magnet being energized, and for the metal to feel the magnetic force.

Now imagine an antenna as a tiny electromagnet that's being flipped from positive to negative billions of times a second in a sinusoidal pattern (this is an EXTREME oversimplification of what an antenna is, but for the purposes of this discussion, it's enough). The changes in the field will only propagate at the speed of light, but the magnet is changing extremely fast.

This makes it so that if you take a snapshot of the magnetic field at a single instant, you will see the field shift from positive to negative to positive to negative with distance from the antenna. If you measure the distance it takes to go from positive to negative to positive again, that's your wavelength.

Here's a gif of an ANSYS simulation I made of the Electric Field of a simple dipole antenna over time. You can see the wave-like pattern in the field magnitude.

And here's one I made of a Yagi Antenna.

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JTech324 t1_j8l5oiv wrote

Replying to Movpasd (#1,811,383)

I thoroughly enjoyed reading your writing in all those linked comments. You really connect with the reader in their frustration to understand complex topics, and don't sugar coat the fact they very likely will not find comfort in the answers they seek lol.

Do you write anywhere else, like a blog or anything?

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mywhitewolf t1_j8lavct wrote

I think i might be able to answer the question i think you're trying to ask.

Photons aren't independent of the EM Field in the same way waves aren't independent of the ocean. In this analogy the crest of the wave would be considered "the photon". occasionally the crest of the wave has enough energy (is large enough) that it can go over the bow of a ship and cause havoc (analogy of an interaction) we personally see that as a single wave that caused an event on the ship, but the reality is that its just the ocean interacting with its environment.

except the ocean surface is a 2 + 1 dimensional with 1 degree of freedom(energy/height) where as the EM field is 3+1 with many degrees of freedom (spin, charge, etc.) so the interactions are more complex.

its probably not any more helpful than any other reply, but its how i "visualise" the wave/particle duality without it being 2 seperate things. (its an analogy, so it will fail at some point. only the maths will gives you actual insight into behaviour, trust the maths, everything else is an approximation)

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Particular-Rock2036 t1_j8lbl0v wrote

There are a lot of great comments here, but I have a few ideas that might be helpful. First, I would say not to worry too much about building the most “correct” model in your head to understand things. There are lots of ways and models to understand physics, and almost all of them fail sometimes, so all that matters is picking the correct models for appropriate situations. Second, a lot of confusion in physics comes from how certain ideas were developed over time, and I think that’s part of where your confusion comes from. The first solid theory of EM radiation comes from classical electromagnetism, in which electric and magnetic fields permeate all of space (meaning that every point in space has a magnitude and direction for each field) and are affected by each other and by charges according to Maxwell’s equations. Variations in these fields can propagate through space as waves, and the variation over time is described by the wave frequency and the variation over space by the wavelength. So waves and wavelengths make sense in this classical theory, because there is a continuous medium that fills up all of space (the electric and magnetic fields) and a wave is just a periodic variation through that medium, analogous to a wave in water. However, this classical theory isn’t very correct, and in turns out quantum mechanics is necessary to describe some electromagnetic phenomena. One such quantum mechanical idea is that of photons. The simplest theory of photons would be that they are discrete, individual packets of energy which make up EM radiation. Like all quantum mechanical objects, they can have both particle-like and wave-like behavior depending on the circumstance. In situations where a photon is displaying more particle-like behavior, such as with the photoelectric effect where the energy of each photon determines whether current is produced, the photons do not really have any wavelength or frequency to speak of. In situations where they are acting more like waves, such as through the interference patterns found by shining light through two slits, it is no longer useful to think about individual photons going through one slit or another light particles, but it is clear that they have some wavelike nature that allows them to go through both slits and interfere with themselves. Analyzing these interference patterns you can determine the wavelength of the waves that could produce such an interference pattern, and this is the wavelength of the light.

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1ndiana_Pwns t1_j8ltc70 wrote

Replying to wooly_boy (#1,814,888)

It takes energy to create those little EM waves, right? A photon is a little packet of that energy. The exact amount of energy a photon has is determined by the wavelength of the light, with the relationship being a result of all the math behind light propagation and electromagnetism (thank Maxwell and his equations)

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Randywithout8as t1_j8ludfk wrote

Replying to iam666 (#1,815,186)

You might be right. I interpreted it as "when you measure a photon's wavelength, what is actually being measured?" Typically it is the energy of the photon that is being measured. Then the wavelength is calculated. I may have been too literal.

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TwentySevenNihilists t1_j8lznkg wrote

Replying to SilentHunter7 (#1,812,857)

Is what I'm seeing in these animations two/eight smaller antennae arranged such that their wavefronts(?) are in phase with each other, so they amplify to create a much stronger wave with a flatter curve?

I've worked with wireless hardware of various types most of my adult life, but I've still always had that disconnect with trying to visualize a wave in 3d space; especially when looking at some of the antennae I've run across. Would it be accurate enough to say that the "waves" expand in a shell from a point of origin, and each "shell" is more or less the location of the peak field strength at the moment you're measuring it?

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shikuto t1_j8m1s0f wrote

Replying to shikuto (#1,810,999)

u/Grand-Tension8668

Sorry to tag you here in a response to myself like this. I can see the email that Reddit sent me when you made the comment response to my previous comment… but it doesn’t show up on any Reddit client I can find.

Anyhow, something to consider is that the fields that the waves are imposed on (there’s a lot to unpack in that statement) ABSOLUTELY DO exist in the three dimensions of space. That’s the only reason that we’re able to apply spatial dimensions to these waves in the first place. So yes. The things that are waves do exist, spatially.

That’s funky. What do I mean by that? Well… a photon is a particle. But it isn’t matter. It has no mass. It is the carrier of the electromagnetic force. The force itself. Now, electrons and protons that do have mass? They’re matter. They’re both carriers of electrical charge. Charge by itself isn’t a force.

This is all going to be over-simplifying things quite a bit, but bear with me. You need to understand at a simplified, incorrect level before you can understand at a less simplified, slightly more correct level. And that’s how it goes all the way up, since the way science works is that every scientific law or theory includes the implication that it may be wrong, and at best is a tool for providing predictions or analyzing observed data.

On one side, you have a hill of negatively charged electrons. On the other side, a hill of positive protons. In the middle, a flat plain of neutrons, with no electrical charge. The protons “want” to meet up with the electrons, just as much as the electrons “want” to pair with the protons. The power of the “wanting” on both sides is the force. That force is mediated/carried by photons.

That space between the protons and electrons is still extant, in the 3D world. The protons and electrons are both the result of excitations/waves in various different fields. The fields come before the particles, at least so far as we can tell.

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shikuto t1_j8m27fx wrote

Replying to Grand-Tension8668 (#1,811,288)

It looks like you managed to tag someone with a 1-character difference from my username. I wonder what the chances of that are. Anyhow, I’m the one that supplied the assistance, not the innocent u/shikoto - please leave them out of this. I will take the punishment.

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aiusepsi t1_j8miaub wrote

Replying to Grand-Tension8668 (#1,811,768)

EM fields in general aren't cyclical (I would usually say 'periodic' for what you mean by that), for example consider the magnetic field surrounding a magnet; that's completely static, so it doesn't change over time, and it just gets weaker with distance from the magnet.

Electromagnetic waves are, mathematically, a lot like lots of other kinds of waves, like sound waves, or waves on the surface of water. All your need is some kind of field, that is, a property which exists at each point, and for the physics of that field to obey a particular form of equation, and waves will exist in that field. For obvious reasons, that kind of equation is called a 'wave equation'.

For sound waves, the field is air pressure. On the surface of water, it's the height of the water's surface. Those are scalar fields, that is, those properties can be described by a single number. EM is a bit different because it's a vector field with two vectors at each point in space. But ultimately, they end up looking very similar.

If you graph out air pressure on one axis and space on the other axis for a sound wave, you get a sinusoid, i.e. it looks the same as graphing y = sin x. And that's exactly what you get if you graph out just the length (i.e. magnitude) of the electric field vector (rather than worrying about the direction of the vector) along the direction the wave is travelling. Just pretend it's a scalar field like air pressure!

Graphing out the magnitude of the magnetic field is basically the same, the electric and magnetic fields vary in the same way in an EM wave, just that the electric and magnetic field vectors are at right angles to each other, and they're both also at right-angles to the direction the wave is travelling.

Anyway, for all of the above, wavelength is just the distance between adjacent peaks on the graph.

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Grand-Tension8668 OP t1_j8miiow wrote

Replying to mywhitewolf (#1,814,218)

It's certainly helpful. I have started to "get" somewhere along the way that EM waves are oscillating in 3D space and therefore the typical 2D squiggly line isn't sufficient, and that photons are somehow more like an intersecting point.

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SilentHunter7 t1_j8mpg55 wrote

Replying to TwentySevenNihilists (#1,815,615)

So these animations are only showing you a quarter of the picture. It's a plot of the absolute value of the Electric field. It doesn't tell you anything about the direction of the field, nor does it show the Magnetic field.

So for the dipole, all it is is two straight wires, about a quarter wavelength long connected to a transmission line. Imagine the top connected to the center of the coax, and the bottom connected to the shield.

When a wave coming down the coax hits the antenna, it causes a current in the wires. Electrons will be pushed into the top wire and pulled out of the bottom wire. This creates a charge on the wires, negative on top, and positive on the bottom, and you can see that in the animation.

But because waves reverse, soon you'll get a reverse current and the top will become positively charged and the bottom negatively charged. This can happen billions of times a second for something like 2.4GHz wifi.

And also, current creates a magnetic field. So when current is flowing in the wires, there is a magnetic field wrapping around them. This current hits zero when the wires are fully charged, and is at it's maximum right when the wires are neutral.

So now you have Electric and Magnetic fields all swirling around each other at a constant frequency. This is you get EM radiation.

If I haven't lost you yet, you should consider going to school for electrical engineering; antenna theory is some of the most esoteric shit this side of Quantum Mechanics.

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Nescio224 t1_j8mv0p5 wrote

I've always disliked the term "wave-particle duality" and I think Feynman nailed it when he said that quantum objects are neither.

>“Quantum mechanics” is the description of the behavior of matter and light in all its details and, in particular, of the happenings on an atomic scale. Things on a very small scale behave like nothing that you have any direct experience about. They do not behave like waves, they do not behave like particles, they do not behave like clouds, or billiard balls, or weights on springs, or like anything that you have ever seen. Newton thought that light was made up of particles, but then it was discovered that it behaves like a wave. Later, however (in the beginning of the twentieth century), it was found that light did indeed sometimes behave like a particle. Historically, the electron, for example, was thought to behave like a particle, and then it was found that in many respects it behaved like a wave. So it really behaves like neither. Now we have given up. We say: “It is like neither.” There is one lucky break, however—electrons behave just like light. The quantum behavior of atomic objects (electrons, protons, neutrons, photons, and so on) is the same for all, they are all “particle waves,” or whatever you want to call them. So what we learn about the properties of electrons (which we shall use for our examples) will apply also to all “particles,” including photons of light.

I would recommend the Feynman lectures for further reading.

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Grand-Tension8668 OP t1_j8n4uz4 wrote

Replying to Nescio224 (#1,817,270)

That definitely helps me grasp the idea that really we're trying to apply the closest concepts that we have, but that they're sort of just touchstones to use as you work to understand the full picture. ...Of course on some level that's true of how we understand most things in science, you work with what's good enough until it isn't any more.

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Movpasd t1_j8qvfax wrote

Replying to Grand-Tension8668 (#1,811,768)

I'm very glad you found my reply useful!

If you're set on continuing to teach yourself physics (which I think is a very good, though time-consuming idea), I'd start by making sure you're on top of your high school/A-Level maths and physics (KhanAcademy is a great place for this), and then move onto some first-year university introductory textbooks. You don't have to read them back to front -- start with the first chapter, take your time, do the exercises, and when you get bored switch to a different book. (I really like Griffiths' textbooks, but YMMV.)

A good search term is "introduction to [topic]" or "introductory [topic] textbooks". Good topics to start with would be classical mechanics, electrodynamics, and quantum mechanics. You could then move onto special relativity and statistical physics (my favourite!).

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Movpasd t1_j8qwacc wrote

Replying to JTech324 (#1,813,863)

I'm glad you liked my comment! I do have a blog, which I'll link here (assuming there aren't any rules against it -- I can take down my comment otherwise). It's pretty empty at the moment though. I've meant to put down all my thoughts about physics and intuition in one place at some point, but I just haven't gotten around to it. If you liked it, maybe there is an audience for it!

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Movpasd t1_j8qxs6w wrote

Replying to wooly_boy (#1,814,888)

It's quite complicated, but I'll try to give a brief answer --

Maxwellian electrodynamics is a classical field theory. When you canonically quantise such a theory, you find that a (sort of*) conserved, discrete quantity pops out, which can be interpreted as "particle number".

This is in line with observations from a century back around black-body radiation that appeared to show quantisation of energy levels.

It's also a satisfying interpretation because certain calculations in QFT have combinatorial properties which allow them to be represented using Feynman diagrams (which you've probably heard of). Together with the path integral formulation, you get a really useful picture of the physics of scattering. But this is mired in complications (renormalisation, confinement (not a problem for photons but it is for quarks), divergences around every corner, ...).

As a side note, instead of starting from the field theory, you can build up a quantum field theory by starting from a particle-based theory; a common approach for effective theories in condensed matter theory, because it's much more tractable mathematically. The fact that QFT unifies classical many-particle and field theories is an advanced form of the "wave-particle duality" you might have heard of.

*: conserved in the free theory

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