Cleanest mathematical description of objects which produce fields?

Question

Consider a situation like a magnet which produces a magnetic field, or a point mass which produces a gravitational field. It seems that for practical physical calculation, the physical object's properties are always seen through its fields. Eg See this question.

But is there a clean mathematical description with the objects considering themselves as individual entities and the fields they produce as separate ones? The reason I see this as useful is that sometimes we would want to talk about how the object affects other objects through its fields, and at other times how the objects themselves are affected by other objects' fields.

One could argue, like the above linked question, that electric charges can't be experienced except through the fields they create, but then I'd reply that we very well accept that both the "sun" and the "pull of the sun" on Earth are both real things. So the idea of the particle being identified with the field is not a good cop out.

Answer 1

is there a clean mathematical descriptions of the objects considering themself as an entity and the field they produce as a seperate one?

That is the standard classical mathematics. In classical EM the sources, $\rho$ and $\vec J$, are treated as mathematically distinct from the fields, $\vec E$ and $\vec B$. The objects and fields are separate entities represented by separate variables. The equations establish a relationship between them, but do not identify them.

Answer 2

The way we teach physics is still very much borrowed from the 17th to 19th century when everything was seen from a materialist point of view (probably to a good degree because of the incredible successes of Newtonian mechanics and atomism in chemistry). Some physicists even tried to associate heat with a "Stoff" called phlogiston. Later we insisted that the vacuum has to be made from another "Stoff" called the aether. None of that works. The closer we look at nature, the more it dissolves into fields. Objects don't make fields. Fields make objects.

So when somebody asks "How does a magnet create a magnetic field?", then the short answer is "It doesn't. It's the quantized electromagnetic field that defines the structure of the atoms of the magnet and its external field is merely a remnant of much stronger internal fields.".

Gravity is a very special case on top of that, because it doesn't just bind very large "objects" together, it even defines the geometry of the space between them.

Answer 3

For an electric, a magnetic and a gravitational field it is true that a test specimen is subject to influence in such a field. A force acts on the specimen and it experiences a translatory (electric potential, gravitational potential) or a rotatory (magnetic field) movement.

How are these fields "generated"? For a permanent magnet, by the alignment of the magnetic dipoles of its subatomic particles (primarily electrons). For an electric field, by the separation of electrons. For a gravitational field by the agglomeration of particles.

An electron in itself unites all three fields. But we would not know this if we could not build up a potential or - for the gravitational potential - if nature did not provide us with one. An electron is attracted by the earth, its dipole aligns itself in a magnetic field and it is moved in a voltage difference.

But is there a clean mathematical description with the objects considering themselves as individual entities and the fields they produce as separate ones?

With the understanding of the above, NO, such separation is not possible and does not make sense. Everything is always both object and subject. The test body is only influenced because it is not just a space-occupying object, but has a field and thus interacts with the common field created from other objects.