Field
Full Title or Meme
In physics a Field is a mathematical representation of a "real" field.
Context
15-4 B versus A
Is the vector potential just some mathematics useful in making calculations - as the scalar potential is useful in electrostatics- or is the vector potential a "real" field? Isn't the magnetic field the "real" field, because it is responsible for the force on a moving particle? First we should say that the phrase "a real field" is not very meaningful. For one thing, you probably don't feel that the magnetic field is very "real" anyway, because even the whole idea of a field is a rather abstract thing. You cannot put out your hand and feel the magnetic field. Furthermore, the value of the magnetic field is not very definite; by choosing a suitable moving coordinate system, for instance, you can make a magnetic field at a given point disappear.
What we mean here by a "real" field is this: a real field is a mathematical function we use for avoiding the idea of action at a distance. If we have a charged particle at the position P, it is affected by other charges located at some distance from P. One way to describe the interaction is to say that the other charges make some "condition"-whatever it may be in the environment at P. If we know that condition, which we describe by giving the electric and magnetic fields, then we can determine completely the behavior of the particle-with no further reference to how those conditions came about.
In other words, if those other charges were altered in some way, but the conditions at P that are described by the electric and magnetic field at P remain the same, then the motion of the charge will also be the same. A "real" field is then a set of numbers we specify in such a way that what happens at a point depends only on the numbers at that point. We do not need to know any more about what's going on at other places. It is in this sense that we will discuss whether the vector potential is a "real" field.
You may be wondering about the fact that the vector potential is not unique that it can be changed by adding the gradient of any scalar with no change at ll in the forces on particles. That has not, however, anything to do with the question of reality in the sense that we are talking about. For instance, the magnetic field is in a sense altered by a relativity change (as are also E and A). But we are not worried about what happens if the field can be changed in this way. That doesn't really make any difference; that has nothing to do with the question of whether the vector potential is a proper "real" field for describing magnetic effects, or whether it is just a useful mathematical tool. We should also make some remarks on the usefulness of the vector potential A. We have seen that it can be used in a formal procedure for calculating the magnetic fields of known currents, just as o can be used to find electric fields.
In Quantum Mechanics the concept of force is not so helpful as energy (a scaler) and momentum (a vector).
