[geordief (OP)]

As an example an electron and its electric field?

In QFT the former is described as an excitation of the latter,I believe.

Are there other ways of looking at it?

Can we imagine the field as an extension of the object(the two being aspects of the greater whole)?

Or is it just  easier to imagine it the other way round with the object being  some kind of  an "appendage" of the field ?(that would be more  along the lines of the QFT model ,I think)

[alancalverd]

If you start with the field, you need to produce a mechanism for generating mass, and  a reason why the field appears to be radially isotropic from a point. It's a lot easier to assume that the field is a characteristic of the object, since this model also works for large objects like magnets and wires.

[geordief (OP)]

If you start with the field, you need to produce a mechanism for generating mass, and  a reason why the field appears to be radially isotropic from a point. It's a lot easier to assume that the field is a characteristic of the object, since this model also works for large objects like magnets and wires.

Thanks.
Is it normally accepted that the object and the field are two aspects of one "overarching" entity?

Could there be (is there?) a mathematical equation  where the object is defined in terms of the field? (along the lines of energy and mass in relativity)

[Eternal Student]

Hi.

   Original title question:
What is the relationship between an object and its field?

A reasonable reply is much as you ( @geordief ) have already stated.    For QFT,  a particle is just an excitation in the underlying field corresponding to that particle.

In QFT the former (an electron) is described as an excitation of the latter (the electric field),I believe.

   Not quite.   The electron is an oscillation in the electron field, not an oscillation in the electric field.  The electron field is NOT the same as the electric field,   the start of the words looks the same but the ending is very different.    In QFT there isn't an electric field, or at least it's not anything that you need to assume exists as a fundamental object.    The Electric field will just be something that appears on a macroscopic scale.   However, the electron field is fundamental - you must have an underlying field for electrons if you wish to have electrons as fundamental particles in your model.

As an example an electron and its electric field?

    That's possibly where the confusion may start.   In QFT,  an electric field is not something that belongs just to an electron.   On a macroscopic scale, an electrostatic interaction actually involves a coupling between two fundamental particle fields  (the electron field and the photon field).   The overall interaction does look like or emerge as something we can describe as if there was a classical field, the Electric field, in existance.

Are there other ways of looking at it?

    I'm sure there are.    If it's just a change of pictures or vocabulary,  then it's just window dressing.   These sorts of changes are only useful or important if the new vocabulary and pictures are better for human understanding - even then the changes are not always made.
   Examples:  Methanoic acid is a 1 Carbon organic acid you can make in a lab.   Formic acid is a thing you can extract from ants (Formic comes from the Latin name for an ant, Formica).  It turned out that they were the same thing but both names still persist, we never insisted that we all start calling it the same thing. 
     In Physics, we thought that current was the movement of positive charges.  It turned out it's usually electrons that are negatively charged and moving the other way.   We didn't change the vocabulary or diagrams.

    There may also be some other ways of looking at things that would really involve changing the mathematics or mechanics of how things work instead of just changing some words or pictures.

Can we imagine the field as an extension of the object(the two being aspects of the greater whole)?

    Yes ?   I guess that's the opposite way around to how we usually imagine or phrase things with QFT.  The thing is, does it help in some way and will it change the mathematics or mechanics at all?   Is it just a change of vocabulary?

Could there be (is there?) a mathematical equation  where the object is defined in terms of the field? (along the lines of energy and mass in relativity)

     Does it have to be one equation?   QFT is a collection of results, mainly equations, that quite sucessfully treats objects that look like particles as oscillations in fields.   
     If you mean is there some simple and short equation that does the job - well, I don't know but I don't think so.   If we try hard we could fit most of QFT on a T-shirt but it may be a bit longer than E=mc².   Looks like some people have already tried.

[geordief (OP)]

@Eternal Student

Thanks.That was informative.Yes ,I appreciate that different ways of looking at things should really have consequences  to be of interest.
Still it was mostly for my own benefit and curiosity that I was asking.
Hopefully when our notions are disabused we can pick up on another path(as well as sometimes validating  some of our vague suspicions to a degree)

[varsigma]

Can we imagine the field as an extension of the object(the two being aspects of the greater whole)?

You say object. Category Theory says that too.

Instead of "a particle", a category lets you choose a symmetry group. Usually these are from the Cartesian product in the SM: SU(3) x SU(2) x (U1).  These describe interactions between particles, but measuring a state is the secret sauce. You notice that it isn't possible to do this from the quantum end, and a measurement is just a thing a classical device does to the state(s) of your "ensemble".

That is, you describe "particles" in terms of their symmetry groups and conjure up an algebra, see if it obeys the laws of physics. That's about it, in terms of what we know, and of course the heuristics, and the odd ansatz.

A particle becomes a decoration in a graph. Quantum particles are abstractions because we aren't entirely sure what they are, nor are we yet entirely sure about time in this domain, but we do know it's a good idea to assume time is linear.