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I think the most precise definition is mathematical: a set of numbers associated with each point that can describe some physical properties on that point, namely, a function with respect to space and time.

In this mathematical point of view, a field is nothing more than a relation, or it’s better to say, it defines some kind of relation. For example, for velocity field, it gives the relation between the velocity of the fluid particles and different positions (and time); for electric field, it gives the relation between the force exerted on a test charge and different positions (and time).

Then the problem is why an electromagnetic field can have energy? It seems to me only entities can have energy, but an electromagnetic field is a kind of relation.

What I mean is, for example, “a rabbit is in my house” ,”rabbit” and ”my house” is a kind of existence, the spatial relation ”…..in……” is also kind of existence, but definitely not the same kind of existence, ”rabbit” and ”my house” are entities, but “…in…” is not. I want to say by the definition of a field, an electromagnetic field is just like “….in…” how can it have energy?!

Even if we can admit an electromagnetic field is also a kind of entity, but indeed we never experience any entity that can define a relation all by itself, which an electromagnetic field does!

What’s more, I heard that in GR, space and time are also kind of entities, for example, gravitational wave, the disturbance of space and time, can have energy. This also confused me in the way that electromagnetic field does. Am I wrong or there really is a duality of a field??

I am struggling with this problem for days, it really confused me. Any help is appreciated.