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For example, if we really wanted to, could we, at least in principle, model electromagnetism just considering interactions between charged particles without using the EM field? That is, is it possible to predict all the same experimental results without bringing in the EM field (or the gravitational field, in gravitational theory, etc)?

For context, one thing I remember hearing several years ago is that, in general (this was in the context of classical physics), fields are essentially just a way of preserving locality -- of explaining how a particle at one location could have an effect on a particle at a different location. It was pointed out that that we could model these interactions without fields if we really wanted to, but the math would be a lot more complicated. Is that true? If so, would the math be fundamentally different, in the sense of having to invent a whole new framework with different mathematical objects, or would the calculations just be a lot more involved?

The analogy that comes to mind is that of virtual particles vs actual particles -- if I understand correctly, the former are a handy bit of math that make certain QFT calculations much easier, but aren't actually necessary in order to explain anything, whereas the latter are needed in order to explain various observable phenomena. Are fields more like the former or the latter in that sense?

To be clear, I'm not trying to suggest that doing away with fields would be a remotely good idea, regardless of whether they're entirely necessary or not, since they're clearly useful.

Edit: Also, to be clear, I'm not asking about the mathematical definition of fields, as I already understand that. Nor am I asking about whether fields "really" exist, in the sense of corresponding to some sort of physical object, as that opens a whole can of worms about whether anything unobservable in physics "really" exists, which is an unsolved question in philosophy of science. I'm purely asking about whether fields are mathematically necessary to make all the same predictions, without inventing an entirely new mathematical framework.

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    $\begingroup$ Recommended reading: discussion by Kevin Brown, titled Trouton-Noble and the right-angle lever Kevin Brown opens with discussion of two ways of implementing theory of electromagnetism: either in terms of one-on-one interactions of pairs of charged particles, or with the interaction being mediated by a field. (The connection with the Trouton-Noble experiment: that experiment involved charging and discharging a capacitor.) $\endgroup$
    – Cleonis
    Commented May 16, 2023 at 2:46
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    $\begingroup$ A field is simply something that has a measurable physical property in every spacetime point. Do fields exist by that definition? Yes. Can we describe the world without material objects? Yes. Can we describe the world without fields? No. At least I don't know how that should work since I can measure fields everywhere. $\endgroup$ Commented May 16, 2023 at 3:15
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    $\begingroup$ "At least I don't know how that should work since I can measure fields everywhere." In what sense do you measure fields? My understanding of E&M in particular is that we never measure the EM field itself, but rather we measure other related quantities like voltage, force, etc., and calculate field values from those. $\endgroup$ Commented May 16, 2023 at 3:59
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    $\begingroup$ @MikaylaEckelCifrese We do measure fields; mathematically, a field is a scalar or vector value associated with points in space, and we can determine those field values by transducers if our senses don't suffice. A sonic field can be explored by moving your ears around. Calculation is inessential. $\endgroup$
    – Whit3rd
    Commented May 16, 2023 at 8:05

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The analogy that comes to mind is that of virtual particles vs actual particles -- if I understand correctly, the former are handy bit of math that make certain QFT calculations much easier, but aren't actually necessary in order to explain anything, whereas the latter are needed in order to explain various observable phenomena.

I would take issue with that. If you are doing perturbative QFT, then you will be summing up terms in order to compute physical quantities (cross sections, decay rates, etc). Whether you want to associate those terms with cute little diagrams and wrap flowery language like "virtual particle" around them is a matter of personal taste, but they aren't a handy tool - their existence is fundamental to perturbative QFT.

And in fact, my previous paragraph is quite analogous to what I'd say about electromagnetic fields. Would it be possible to compute forces between particles without explicitly referencing an electromagnetic field? Yes. But it will be unpleasant - the force on one particle right now will depend on the state of motion of every other particle at various different times in the past. Momentum and energy would no longer be locally conserved. Actually computing the force on a charged particle would require you to do calculations which are mathematically identical to solving Maxwell's equations (since after all, they are what govern electromagnetism), so there's simply no point in it.


To illustrate the idea, imagine running a computer simulation of a handful of charged particles which interact electromagnetically. With clever choices of plotting scheme, you could watch particles accelerate and release radiation which subsequently affects the motion of the other particles when it arrives at their position. By watching the delicate interplay between the particles and fields, we can make sense of how they move.

Now simply stop printing out the electromagnetic field variables. Now you see particles moving around while making seemingly random movements - they slow down (or start moving) unexpectedly based on no locally identifiable cause. After a while, you start to realize that the forces on the particles don't depend on where the other particles are and what they're doing at that moment - they depend on where they were and what they were doing some time before, with that time delay being proportional to the distance between them.

Clearly we have the capability of formulating these forces without thinking about field theories - our simulation can do it, after all. But when you dig into it, the calculations it does to compute and keep track of all of those forces is precisely what it needs to do to solve a field theory, so by forgetting about the $\mathbf E$ and $\mathbf B$ degrees of freedom you are doing little more than making the universe more philosophically confusing while gaining precisely nothing from a calculational standpoint.

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  • $\begingroup$ "their existence is fundamental to perturbative QFT." But we don't actually need perturbative QFT in particular in order to make accurate predictions. There's also non-perturbative QFT. I'm not familiar with the details and I'm not saying perturbative QFT isn't super useful, but PBS Spacetime did a whole video on how virtual particles shouldn't be thought of as anything more than math because they aren't necessary to explain any observable phenomena, specifically because non-perturbative QFT doesn't use them. $\endgroup$ Commented May 16, 2023 at 5:06
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    $\begingroup$ @MikaylaEckelCifrese Unfortunately as you familiarize yourself with the details, you will learn that with precious few exceptions (e.g. free theories, topological QFTs, low-dimensional CFTs), perturbative QFT is all we have to make predictions at all. The correct proton mass has been calculated - at great computational expense - using QCD on a lattice, but e.g. every decay rate and interaction cross section for the standard model or any of its component theories like QED is calculated perturbatively. $\endgroup$
    – J. Murray
    Commented May 16, 2023 at 5:21
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    $\begingroup$ @MikaylaEckelCifrese Note in particular that even establishing the existence of a non-perturbative Yang-Mills QFT (which is what the standard model / QCD / QFD are) is a millenium problem. $\endgroup$
    – J. Murray
    Commented May 16, 2023 at 5:27
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    $\begingroup$ @MikaylaEckelCifrese There are no algorithms, efficient or otherwise, to perform calculations in non-perturbative QFTs in most cases. We simply do not know how to construct them, outside of toy models like the ones I mentioned above. In principle, if such a theory exists, then its predictions would supplant the predictions of a perturbative approximation to it, but it would be disingenuous to imply that perturbative QFT is a mere tool for convenience - it's usually all that we have. $\endgroup$
    – J. Murray
    Commented May 16, 2023 at 5:37
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    $\begingroup$ @MikaylaEckelCifrese Yes, that's more or less right. For example, in an interacting QFT we don't even know how to construct the Hilbert space of states. $\endgroup$
    – J. Murray
    Commented May 16, 2023 at 5:47
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Electromagnetism can be done without fields, or more precisely without fields as independent degree of freedom. The idea here is that the field is only generated by charges and it is only felt by charges, so if a charge $q_1$ is at the origin at time $0$, and another charge $q_2$ is at distance $r = ct$ from the origin at time $t$, then $q_2$ will feel a force due to $q_1$. This force can be written explicitly in terms of the Liénard–Wiechert potentials which are functions of the distance and velocities of the particles only.

The problem with this approach is that to describe any particle's motion at any time $t_0$, we need to know all particles which were at distance $c (t_0 - t)$ for all times $t < t_0$. In other words, we'd need to solve the equations of motion given boundary conditions on the light cone. The approach can be formally developed and is known as Feynman-Wheeler electromagnetism (NB: the Wikipedia article really doesn't give much detail at all, the original paper is Rev. Mod. Phys. 21 (1949), 425).

In the usual description, we introduce the fields, and they simplify the boundary conditions: giving the values of the fields at time $t_0$ in a small neighborhood of our particle allows us to predict its motion.

So in that sense the fields are not necessary, they just allow us to work with more convenient boundary conditions. Of course, the introduction of the fields as independent degrees of freedom adds so much power to the theory that we really don't want to give up the fields. Imagine having to describe you radio reacting to the electrons in the antenna in the radio station instead of just thinking about the FM wave arriving, and how it's picked up by the antenna.

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  • $\begingroup$ Has anyone tried quantising this? OP is asking about QFT $\endgroup$
    – Jojo
    Commented May 18, 2023 at 19:32
  • $\begingroup$ (I deleted a previous version of this comment, because I thought I was making matters more complicated rather than simpler). @Joe the usual approach to QFT is actually quite close to this: the electromagnetic field spreads by its propagator (a Fourier-transformed analogue of the LW potential). Unlike the classical theory, one has to take all paths into account though. This is how one ends up with path integrals. It is my understanding of history that Feynman's development of QED and specifically perturbation theory was very much inspired by this work on electromagnetism with Wheeler. $\endgroup$
    – tobi_s
    Commented May 22, 2023 at 2:35
  • $\begingroup$ Well, the Lénard-Wiechert potentials $\phi$ and $\vec A$ are solutions to Maxwell's equations, so to me they quite clearly are examples of electromagnetic fields, for certain sources. $\endgroup$
    – Toffomat
    Commented Jul 6, 2023 at 14:22
  • $\begingroup$ Insert the LW $\phi$ and $\mathbf{A}$ into the Lorentz force term into the equations of motions of the particles and these "fields" will disappear, you'll have equations only involving particle positions, particle velocities, and time -- but no fields. In the usual electrodynamics you'd have an independent set of equations for the fields. Since these are two different ways to describe the same phenomena you could of course always argue that the fields exist or don't exist. All the objects we use to describe reality are just tools and not reality itself. $\endgroup$
    – tobi_s
    Commented Jul 7, 2023 at 1:34
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Say for the purpose of OP's question that we are only interested in describing the phenomena of matter and not the fields. Given a local field theory, one can of course integrate out the fields to get the corresponding non-local direct interparticle action.

The issue is the opposite: If we have given up the principle of locality, there are infinitely many non-local direct interparticle models that one can propose, and it's unclear which one is the correct in order to make accurate predictions, cf. OP's title question.

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    $\begingroup$ "If we have given up locality as a principle, there are infinitely many non-local direct interparticle models that one can propose, and it's unclear which one is the correct." Would you mind expanding on this a bit? I'm not sure I understand. $\endgroup$ Commented May 16, 2023 at 5:12
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    $\begingroup$ 1. So you say that there are many non-local models of interaction between particles, but only one which arises from the correct field theory? 2. Do you have a source for that with a more precise statement? 3. Since there would be many different field theories, doesn't this just move the problem to which field theory is the right one to give the correct model of interacting particles? 4. Can you give an example of a non-local particle interaction model which does not arise from considering a suitable chosen field theory? $\endgroup$ Commented May 16, 2023 at 13:15
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    $\begingroup$ 1. Yes, in principle. 3. No, non-local theories are generic; local theories are special in comparison. $\endgroup$
    – Qmechanic
    Commented May 16, 2023 at 13:35
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Light is an electromagnetic field. Our eyes detect light. We don't need to go through a calculation to determine what we see. So, unless you want to go into some philosophical discussion about what it means to "see" something, I would say light is real and therefore fields are real. It would be unthinkably difficult to explain the effects of diffraction and all that if we cannot talk about light itself.

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    $\begingroup$ But light can also be modeled as photons $\endgroup$ Commented May 16, 2023 at 5:00
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    $\begingroup$ Yes, photons can be considered as quantized bits of the field that take part in interactions. How does that affect the issue? $\endgroup$ Commented May 16, 2023 at 11:36
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    $\begingroup$ @flippiefanus you're ignoring the possibility of modelling light (in the case of an eye seeing a source) as a non-local interaction between source and eye $\endgroup$
    – Tristan
    Commented May 16, 2023 at 13:19
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    $\begingroup$ @flippiefanus the case of an eye seeing a source was chosen without loss of generality. Light travelling between A & B could be modelled as a non-local interaction between A & B without a field. Light with only one known end-point could be modelled with one of the partners at infinity $\endgroup$
    – Tristan
    Commented May 17, 2023 at 9:08
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    $\begingroup$ @flippiefanus there's no reason the interaction need be instantaneous $\endgroup$
    – Tristan
    Commented May 18, 2023 at 12:56
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Field theories are often a resort after realizing you cannot work with $10^{23}$ particles...

For the example of electrodynamics, just take any continuous charge distribution, say a charged spherical conductor. How would you be working with the ridiculous number of charged particles in it individually?

Or take the example of deformation of or sound propagation in a solid. You could do it using a lattice theory, but (perhaps after homogenization) instead you can use the continuous description in elasticity, i.e. a classical field theory. High-dimensional vectors and matrices can be much less convenient than functions and PDEs...

would the math be fundamentally different, in the sense of having to invent a whole new framework with different mathematical objects, or would the calculations just be a lot more involved?

And so yes, frameworks differ (although there are mathematical analogies, of course), you might find yourself choosing between analysis and linear algebra.

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Are fields theories necessary to make accurate predictions...

No, they are not necessary to make accurate predictions.

The behavior of a hydrogen atom is predicted pretty accurately via the Quantum Mechanics of a single particle in a potential $e^2/r$.

The behavior of a block sliding down a ramp is predicted pretty accurately via classical Newtonian mechanics.

or do they just make calculations easier?

Sometimes they do, sometimes they don't.

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