Great piece but if fields are just math, how does Earth’s magnetic field hold its atmosphere? Fields aren’t abstractions. They’re the reason objects behave at all.
You are right at a high overview level. Fields are how physicists account for collective effects of particles. No doubt. But it is all particles all the way down to gravity, only gravity has to be treated as "field-like", in that case spacetime. Of course, some folks want to discretize even gravity to turn it into 'particle-like' constructs (causal sets or whathaveyou) — I would not agree with them that that is necessary.
Note, the gauge field for electromagnetism is the spacetime pseudoscalar, so that is your "field". It is an abstract concept. It is a grade=4 quantity defined at each spacetime location by the curvature and topology of the manifold. Like a 4-form (which the pseudoscalar is functorially related to, in going from Clifford algebra to the exterior algebra of forms) , it is
Consider Jacob Barandes' account of QM. It uses only particle configuration space. No need for fields. Feynman pointed out the same. You can argue with them, but Liam, there is no point in arguing with me. I do not know what base marble reality is composed of, I just use the particle concept for my gauge fields, and spacetime for gravity, which involves the only real "fields" you need, since is presumes a 4D manifold with position + rotation gauging for the physical content (you have to remove dependence on arbitrary choice of coordinate frames to get the physical content). Even then, the GR gauge fields are really fictional, they just account for the gauge independence, and what is "real" is the underlying manifold, the curvature and the topology.
It is possible in principle to give a pure scattering account for the Earth's magnetic field, or any non-gravity field, but no one bothers because it is far too complicated. The dynamics even in classical mechanics need to be at some point treated stochastically or via statmech. You cannot preform any useful calculations any other way.
You can suppose the non-grav fields are "fundamental" but that conflicts with so much empirical evidence it becomes unworkable, you are forced to move to regarding Hilbert space as physical reality. I would reject that as pure Idealism, and nutty. But there is no accounting for metaphysical taste I guess.
I take your point about the algebraic definition of a field. But to me, physics is algebra and not as abstract manipulation, but as the logic of the field itself. The four operators are just the phase operations the field performs, and algebra is our way of writing that down.
That’s why I find it odd to treat manifold/curvature as “real” but fields as “fiction.” We can’t see either because both are descriptions. It’s like saying the Spanish word agua is real but the English word water isn’t. What’s real is the liquid (the recursive structure both words point to). Newton’s forces are the same and are not real things, just words for what the field does.
So for me, the ontology isn’t in “field” or “manifold” as labels, but in the recursion that algebra encodes. Fields aren’t abstractions layered on geometry, they are geometry-in-action. And because reality is always in action, from the smallest particle-field to the largest universal field, it’s all the same field dynamics across scales. One continuous electromagnetic field whose structure you can measure directly in the vacuum.
Fair point. Yes, both are mathematical model, hence fictional.
There is a difference though. The algebra is defined via action on objects. So what are the objects (“elements of reality”)?
I can never accept that the algebra itself is the reality (I have a prejudice against total wholesale relationalism, it is too airy fairy), whereas a spacetime manifold exists right in front of my nose, and my nose is a part of it!
I say more exactly, “a spacetime manifold of some sort or another” because I really confess I do not know what it is, it could be a piece of texture on a hyperbubble gum bubble blown my Mr Zeus or Sir Odin.
So the spacetime manifold view has a “manifest” reality backing it. But you are right, when we write down symbols on paper or in LaTeX for this, that’s not physical reality, it is just physics sciencystuff.
I think I see where we differ? You’re treating the spacetime manifold as something ‘real’ while describing fields as ‘just math.’ But that feels inconsistent to me. The manifold itself can’t be observed directly because it’s an abstract scaffold. Fields, on the other hand, produce measurable effects right in front of our noses: a compass needle moves, a magnet lifts iron, photons hit a detector. If we’re going to grant ‘reality’ status to the manifold, which is never directly measured, it seems at least as reasonable to grant it to the fields that cause the things we actually observe.
Correct, we differ. So let's not waste more time. It will not be a worthy debate, unless you are prepared to change your mind, because I am not just yet prepared to change my mind. Sorry. Fields are not physical in my framework, and I trust my framework. If you trust yours then we part ways, and further discussion is fruitless. But I wish you luck in your theory development. If you can prove that spacetime realism is vamoosed, then great, then I'll have to change my mind. But you cannot prove it, it's just an opinion you have.
As I explained, the fields are algebraic mathematical constructs, and are functions defined on spacetime, not the other way around. What is in front of your nose is at some distance, separated by *space* and *time*. You may use fields to account for the effects of propagation of these into your brains receptors etc., but that is just a language and if you inquire deeply into tit you will find there is no natural way to explain why the fields do what they do, it is external merology. Spacetime topology is not, it has clear structure in 4D and in that structure causality can be primitively defined, intrinsically, hence without appeal to merological causation.
Think about this deeply, you should understand.
So what more can I say? I know how the interactions occur in a spacetime framework. Your field concepts have no such primitive causality. A pure algebraic description has to put in causality ad hoc, merologically. At least with spacetime we have notions of geodesics and whatnot. They are geometrodynamic structure, hence intrinsic, hence non-mreologicalThey give good account of primitive notions of causality, and motivate least action principles (which are again, just models).
But also, I just admitted the spacetime manifold used to describe the physics is still a mathematical model. I do in fact NOT know it is real. I am just telling you I do not need to assume the fields are fundamental, since in my framework they arise as transformation instructions, they are pure abstractions describing symmetries and invariants in the spacetime cobordism. If you do not accept this is more elegant and more natural i cannot help you further. I would say you are not "hearing me" or something like that, you've got your own prejudices and you can fairly say I've got mine.
You can argue spacetime topology is merological too, but I cannot see how you'd make that case, it is just geometry, and in the 4D Block Universe view it is "static" and "eternal" (this does not mean time ins not real, you understand!).
If you want to lift discussion up a level to metaphysics, that is a whole different story, and is not physics, so I can say nothing about lit (like, "so where did spacetime come from?" — I have no idea, I'd just say "God".)
Show me any proof the fields are real? Then there'd be more to say. Why are you ignoring the plain fact that what is in front of my nose is at a few centimeters (space), and takes a few microseconds (time) to propagate sensory signals (mass/energy) into my brain?
I bet you will always be appealing to abstractions. Moreover, what field theory ever has been defined without spacetime as a reference? None that i've seen. Even Finster's 'causal fermionic' theory employs spacetime coordinates, so he is delusional if he thinks the fermions are the base marble.
The space and time in front of my nose is directly measurable with clocks and rulers. By contrast I cannot directly observe any of your fields. No one has even produced a "field meter". Take a look at a compass or voltmeter again, in detail, not superficially like a school kid. These instrument operations are all based on quanta of energy (i.e., particles) deposited in wires and so forth. Your fields are a higher level removed. All observed events ever are detection of particles and energy deposition. In GR all energy is mass. In topological geon theory all mass is nontrivial spacetime topology. If only given a paragraph that's how I rest my case. If you want all the mathematics for that (or most of it) then it is on my github.
I appreciate your response but I think there’s a blind spot here. You say fields are just bookkeeping for particle behavior, while “the real things” are manifold, curvature, and topology. But isn’t that already what a field is? Curvature and topology are the structural features of a field... Electromagnetism, for example, is literally curvature defined on the manifold, with ε₀ and μ₀ baked into the vacuum itself.
If you treat fields as abstractions but curvature/topology as real, you’re just swapping labels. A proton doesn’t exist without the strong field (quantum gravity!) that holds it together, and an electron isn’t a standalone point but a localized field configuration. Particles are fields and their stability is proof of the field’s reality, not its abstraction.
So iif the manifold has measurable structure (ε₀, μ₀, c), why not call that what it is... a real field?
Yes, in a sense, loosley speaking, that is what i wrote. So it is not a blind spot.
However, a manifold is technically different to a field in mathematics. A field is an algebraic structure, a ring with (+,‒,×,÷) which is how I use the term. The spinors and whatnot are really generalized functions on the manifold, and are fields because those binary operations are defined among them. But the manifold is not the field. The fields are entirely abstract algebra, and what is closer to "physical" are the functions so defined, because it is the functions (mappings) that define our models for physical processes.
It is incorrect to say "a proton does not exist without the strong fields" (spinors with color and su(2) and u(1) charges). Because those fields are the abstract algebraic operations between functions on the manifold, not the physical objects themselves. It is correct to say that *our model for a proton* does not exist sensibly without these functions. But it is our model. The model is not reality. It is a model.
This is why the 'abstract nonsense' of Category Theory is "actually useful"🤣 It gives you the frame of mind to separate out objects from relations and reality (non-mathematics) from models (mathematics). Fields are mathematics.
It is not wrong to say "fields are fundamental." But it depends on what you mean. They are pretty fundamental constructs in our mathematical models., since we surely do wish to add, subtract, multiply and divide the spinor-valued functions. Are they physically fundamental? I would say not. The manifolds is "more elementary" and is only loosely speaking colloquially a "field" and not in the technical mathematical definitional sense. We do not (+,‒,×,÷) the spacetime manifold itself (unless you are into some sort of weird Many Worlds ontology).
Think of every object as a self-contained recursive geometry: shells of torsion, aperture, and phase that define its proper motion and inertia. Surrounding that is an external recursive field whose curvature, phase-rate and memory shells impose boundary conditions. Relative motion is simply the dynamical mismatch between those two geometries: the object’s internal phase tries to maintain its own rhythm while the external field drags, locks or phase-shifts it. Which geometry “counts” as primary is scale-dependent. t
The larger, deeper field (with longer memory and stronger boundary coherence) dominates the dynamics but ontologically they are the same type of thing and are nested fields interacting. That’s why inertia, time-dilation, and apparent forces all look like local effects of embedding one recursive geometry inside another.
Great piece but if fields are just math, how does Earth’s magnetic field hold its atmosphere? Fields aren’t abstractions. They’re the reason objects behave at all.
You are right at a high overview level. Fields are how physicists account for collective effects of particles. No doubt. But it is all particles all the way down to gravity, only gravity has to be treated as "field-like", in that case spacetime. Of course, some folks want to discretize even gravity to turn it into 'particle-like' constructs (causal sets or whathaveyou) — I would not agree with them that that is necessary.
Note, the gauge field for electromagnetism is the spacetime pseudoscalar, so that is your "field". It is an abstract concept. It is a grade=4 quantity defined at each spacetime location by the curvature and topology of the manifold. Like a 4-form (which the pseudoscalar is functorially related to, in going from Clifford algebra to the exterior algebra of forms) , it is
Consider Jacob Barandes' account of QM. It uses only particle configuration space. No need for fields. Feynman pointed out the same. You can argue with them, but Liam, there is no point in arguing with me. I do not know what base marble reality is composed of, I just use the particle concept for my gauge fields, and spacetime for gravity, which involves the only real "fields" you need, since is presumes a 4D manifold with position + rotation gauging for the physical content (you have to remove dependence on arbitrary choice of coordinate frames to get the physical content). Even then, the GR gauge fields are really fictional, they just account for the gauge independence, and what is "real" is the underlying manifold, the curvature and the topology.
It is possible in principle to give a pure scattering account for the Earth's magnetic field, or any non-gravity field, but no one bothers because it is far too complicated. The dynamics even in classical mechanics need to be at some point treated stochastically or via statmech. You cannot preform any useful calculations any other way.
You can suppose the non-grav fields are "fundamental" but that conflicts with so much empirical evidence it becomes unworkable, you are forced to move to regarding Hilbert space as physical reality. I would reject that as pure Idealism, and nutty. But there is no accounting for metaphysical taste I guess.
I take your point about the algebraic definition of a field. But to me, physics is algebra and not as abstract manipulation, but as the logic of the field itself. The four operators are just the phase operations the field performs, and algebra is our way of writing that down.
That’s why I find it odd to treat manifold/curvature as “real” but fields as “fiction.” We can’t see either because both are descriptions. It’s like saying the Spanish word agua is real but the English word water isn’t. What’s real is the liquid (the recursive structure both words point to). Newton’s forces are the same and are not real things, just words for what the field does.
So for me, the ontology isn’t in “field” or “manifold” as labels, but in the recursion that algebra encodes. Fields aren’t abstractions layered on geometry, they are geometry-in-action. And because reality is always in action, from the smallest particle-field to the largest universal field, it’s all the same field dynamics across scales. One continuous electromagnetic field whose structure you can measure directly in the vacuum.
Fair point. Yes, both are mathematical model, hence fictional.
There is a difference though. The algebra is defined via action on objects. So what are the objects (“elements of reality”)?
I can never accept that the algebra itself is the reality (I have a prejudice against total wholesale relationalism, it is too airy fairy), whereas a spacetime manifold exists right in front of my nose, and my nose is a part of it!
I say more exactly, “a spacetime manifold of some sort or another” because I really confess I do not know what it is, it could be a piece of texture on a hyperbubble gum bubble blown my Mr Zeus or Sir Odin.
So the spacetime manifold view has a “manifest” reality backing it. But you are right, when we write down symbols on paper or in LaTeX for this, that’s not physical reality, it is just physics sciencystuff.
I think I see where we differ? You’re treating the spacetime manifold as something ‘real’ while describing fields as ‘just math.’ But that feels inconsistent to me. The manifold itself can’t be observed directly because it’s an abstract scaffold. Fields, on the other hand, produce measurable effects right in front of our noses: a compass needle moves, a magnet lifts iron, photons hit a detector. If we’re going to grant ‘reality’ status to the manifold, which is never directly measured, it seems at least as reasonable to grant it to the fields that cause the things we actually observe.
Correct, we differ. So let's not waste more time. It will not be a worthy debate, unless you are prepared to change your mind, because I am not just yet prepared to change my mind. Sorry. Fields are not physical in my framework, and I trust my framework. If you trust yours then we part ways, and further discussion is fruitless. But I wish you luck in your theory development. If you can prove that spacetime realism is vamoosed, then great, then I'll have to change my mind. But you cannot prove it, it's just an opinion you have.
As I explained, the fields are algebraic mathematical constructs, and are functions defined on spacetime, not the other way around. What is in front of your nose is at some distance, separated by *space* and *time*. You may use fields to account for the effects of propagation of these into your brains receptors etc., but that is just a language and if you inquire deeply into tit you will find there is no natural way to explain why the fields do what they do, it is external merology. Spacetime topology is not, it has clear structure in 4D and in that structure causality can be primitively defined, intrinsically, hence without appeal to merological causation.
Think about this deeply, you should understand.
So what more can I say? I know how the interactions occur in a spacetime framework. Your field concepts have no such primitive causality. A pure algebraic description has to put in causality ad hoc, merologically. At least with spacetime we have notions of geodesics and whatnot. They are geometrodynamic structure, hence intrinsic, hence non-mreologicalThey give good account of primitive notions of causality, and motivate least action principles (which are again, just models).
But also, I just admitted the spacetime manifold used to describe the physics is still a mathematical model. I do in fact NOT know it is real. I am just telling you I do not need to assume the fields are fundamental, since in my framework they arise as transformation instructions, they are pure abstractions describing symmetries and invariants in the spacetime cobordism. If you do not accept this is more elegant and more natural i cannot help you further. I would say you are not "hearing me" or something like that, you've got your own prejudices and you can fairly say I've got mine.
You can argue spacetime topology is merological too, but I cannot see how you'd make that case, it is just geometry, and in the 4D Block Universe view it is "static" and "eternal" (this does not mean time ins not real, you understand!).
If you want to lift discussion up a level to metaphysics, that is a whole different story, and is not physics, so I can say nothing about lit (like, "so where did spacetime come from?" — I have no idea, I'd just say "God".)
I'd already given up at the last comment. If you think space is more real than the field in front of your nose, then there's not a lot to be said.
Show me any proof the fields are real? Then there'd be more to say. Why are you ignoring the plain fact that what is in front of my nose is at a few centimeters (space), and takes a few microseconds (time) to propagate sensory signals (mass/energy) into my brain?
I bet you will always be appealing to abstractions. Moreover, what field theory ever has been defined without spacetime as a reference? None that i've seen. Even Finster's 'causal fermionic' theory employs spacetime coordinates, so he is delusional if he thinks the fermions are the base marble.
The space and time in front of my nose is directly measurable with clocks and rulers. By contrast I cannot directly observe any of your fields. No one has even produced a "field meter". Take a look at a compass or voltmeter again, in detail, not superficially like a school kid. These instrument operations are all based on quanta of energy (i.e., particles) deposited in wires and so forth. Your fields are a higher level removed. All observed events ever are detection of particles and energy deposition. In GR all energy is mass. In topological geon theory all mass is nontrivial spacetime topology. If only given a paragraph that's how I rest my case. If you want all the mathematics for that (or most of it) then it is on my github.
Gravity
I appreciate your response but I think there’s a blind spot here. You say fields are just bookkeeping for particle behavior, while “the real things” are manifold, curvature, and topology. But isn’t that already what a field is? Curvature and topology are the structural features of a field... Electromagnetism, for example, is literally curvature defined on the manifold, with ε₀ and μ₀ baked into the vacuum itself.
If you treat fields as abstractions but curvature/topology as real, you’re just swapping labels. A proton doesn’t exist without the strong field (quantum gravity!) that holds it together, and an electron isn’t a standalone point but a localized field configuration. Particles are fields and their stability is proof of the field’s reality, not its abstraction.
So iif the manifold has measurable structure (ε₀, μ₀, c), why not call that what it is... a real field?
Yes, in a sense, loosley speaking, that is what i wrote. So it is not a blind spot.
However, a manifold is technically different to a field in mathematics. A field is an algebraic structure, a ring with (+,‒,×,÷) which is how I use the term. The spinors and whatnot are really generalized functions on the manifold, and are fields because those binary operations are defined among them. But the manifold is not the field. The fields are entirely abstract algebra, and what is closer to "physical" are the functions so defined, because it is the functions (mappings) that define our models for physical processes.
It is incorrect to say "a proton does not exist without the strong fields" (spinors with color and su(2) and u(1) charges). Because those fields are the abstract algebraic operations between functions on the manifold, not the physical objects themselves. It is correct to say that *our model for a proton* does not exist sensibly without these functions. But it is our model. The model is not reality. It is a model.
This is why the 'abstract nonsense' of Category Theory is "actually useful"🤣 It gives you the frame of mind to separate out objects from relations and reality (non-mathematics) from models (mathematics). Fields are mathematics.
It is not wrong to say "fields are fundamental." But it depends on what you mean. They are pretty fundamental constructs in our mathematical models., since we surely do wish to add, subtract, multiply and divide the spinor-valued functions. Are they physically fundamental? I would say not. The manifolds is "more elementary" and is only loosely speaking colloquially a "field" and not in the technical mathematical definitional sense. We do not (+,‒,×,÷) the spacetime manifold itself (unless you are into some sort of weird Many Worlds ontology).
Think of every object as a self-contained recursive geometry: shells of torsion, aperture, and phase that define its proper motion and inertia. Surrounding that is an external recursive field whose curvature, phase-rate and memory shells impose boundary conditions. Relative motion is simply the dynamical mismatch between those two geometries: the object’s internal phase tries to maintain its own rhythm while the external field drags, locks or phase-shifts it. Which geometry “counts” as primary is scale-dependent. t
The larger, deeper field (with longer memory and stronger boundary coherence) dominates the dynamics but ontologically they are the same type of thing and are nested fields interacting. That’s why inertia, time-dilation, and apparent forces all look like local effects of embedding one recursive geometry inside another.