I get that it’s values. It’s like, you assign a value to every point in space. But what “is” the electron field? It’s… what? I mean like a Kantian “field an sich”. Is the electron field the amount of electron-ness at a given point in space? What does that even mean beyond a calculation?
Are fields “real entities” with an objective physical reality? Or are they just mathematical abstractions that we use for calculation? Can you talk about fields without math? Does that even make sense? Like, I can talk about electrons without math. I can say they’re point particles that carry charge. But can you talk about the electron field outside of math? Or the EM field? Does it genuinely exist outside of an Electrodynamics calculation?
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Your question has already been answered, but, just to fill-in two of the definitions a bit, you have already probably seen the way to calculate the gravitational force on an object of mass m as mg where g is the ‘gravitational acceleration’ of 10 m/s^2 , but the word ‘acceleration’ is wrong for a little technical reason that this holds true even if the object is sitting on a table. It is the Newton equation force=mass x acceleration, but there is no acceleration.
Hence one can make a tiny philosophical transformation and say, g is the acceleration which the object *would have* experienced if we’d removed the table.
If you take account of how the object could be anywhere on the earth, you obtain that g is not a scalar, rather, a vector at each point of space, and if we want it to be a force vector, it is defined to be the hypothetical acceleration vector of any object if it had been released, times its mass, and it is a bit strange that some of the comments say ‘we will never know what it is.’ I mean, I just gave the definition, but perhaps the point is that a mathematical definition does not ‘explain’ what it is.
That is called a ‘vector-field’ and mathematicians have — for many years — over time removed the need of having an ambient vector space like R^3 for this to make sense. There is a rigorous definition of a ‘vector-field’ on any manifold, and it is defined to be ‘a section of the tangent bundle’. Moreover the ‘tangent bunde’ has a nice coordinate-free definition for any manifold as does the `tangent space at a point’ https://en.wikipedia.org/wiki/Tangent_space#Definition_via_derivations
The reason for my replying was the dissonance in my mind between one person saying “consider this space of derivations” and giving a complete definition of it, and another person saying “But we will never know what it is.”
The theory of vector-fields, symmetry groups, Lie algebras etc is also the mathematical language to talk about ordinary differential equations.
It is most natural to visualize a vector-field as being a velocity vector-field, and this is always possible to do, though one has to keep the intuitions a bit separate. If we visualize the gravitational field of the earth as a velocity vector-field, we’d imagine objects drifting down at constant speed.
TL;DR The reason for my replying was the dissonance in my mind between one person saying “consider this space of derivations” and giving a complete definition of it, and another person saying “But we will never know what it is.”
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