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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A field is very similar to the concept of a function. You give it an input and it gives you an output. The input with a field is a point in space and time, the output could be a number, it could be a vector, or other more complicated things. The field itself is a mathematical abstraction. It is the math we’re using to describes some physical reality.
For example let’s look at an electric field. Charge particles feel an attractive or repulsive force between themselves proportional to the product of their charges and inversely proportional to the distance between them. So when we have a charged particle and we take some point in space and time it’s possible to come up with a number at any given point that represents the force another charged particle would feel at that point.
There’s nothing actually at that point. The electric force doesn’t exist until another particle actually enters the system. The field represents what a particle would feel if it was there. So the field is just a pure mathematical concept.
The field isn’t a real entity, but it does describe real behavior.
And just like if you have a function that describes some behavior of a system you can calculate properties of that system from it. You can do the same with fields. Probably the simplest example would be Gauss’s flux theorem. By finding the flux on an electric field over a closed surface (basically a surface integral of the electric field) you get the charge contained within that surface. The field isn’t real, but the charge is.
So the field is just math, but it’s math that describes real information about your system.
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