What is field? A form of matter or just a function of position?

Question

I am new to electromagnetism. A few days ago, I decided to learn about electromagnetism through different books and i came across 2 different definitions about fields:

1. A “field” is any physical quantity which takes on different values at different points in space (Feynman Lecture on Physics, Vol II, part 1-2)

2. A physical entity that acts as mediator of force, conveying the force over the distance from one body to another. This entity is the field. Fields are a form of matter—they are endowed with energy and momentum (Physics for Scientist and Engineer, Vol II, page 722)

The first definition seems to say that the field is a function of location, and the second definition the field is a physical entity ("a form of matter" like some documents say) and they sound rather contradictory.

Can anyone explain to me what the real physics field is? Which of the two definitions above is true, which is false or are they the same? I have searched on the Internet but there seems to be no answer to help me understand physics field well. So I hope someone can answer this question.

Answer 1

In the mathematical sense (1) is true. A field is something which just depends on position. Scalar fields are for example temperature, pressure or energy density. Vector fields are for example a velocity field, a force field, the electric field, the magnetic field or the gradient of a scalar field. Tensor fields are for example the stress-energy tensor or the metric tensor (general relativity).

As you can see the objects that form the field can be almost anything, as long as it is defined for every point in space$^\dagger$. More exotic fields are for example spinor fields, or Grassmann-valued fields.

While (1) is the correct definition, this doesn't make (2) less true. The standard model, which is our current most accurate model of the universe, says that things such as matter, energy and forces are realised as fields. In quantum mechanics matter takes the form of a wavefunction $\psi(\vec r)$, which is a field. In quantum field theory, the upgraded version of quantum mechanics which also includes special relativity, the wavefunction is upgraded to an operator $\hat \psi(\vec r)$. But this is still a field. The field has momentum, energy and spin. The field is matter.

Forces are also carried by fields. The electromagnetic force takes the form of the electric and magnetic fields. Charged particles exchange force by interacting with those fields. In quantum field theory, many fields become "quantized". This means that excitations of a field can only occur in discrete amounts of energy. We call these discrete amounts of energy particles. For the electromagnetic field we call these particles photons.

$\dagger$ Things like temperature are not defined everywhere in space, so strictly speaking it is not a field. By only looking at a small portion of space we can pretend that it is a proper field.

Answer 2

Both definitions are true that you have mentioned. Why? Because there are really just 2 broad definitions of fields:

1. Mathematical fields
2. And physical fields

In physics, we use both for our study of the natural world. Mathematical fields are really just that, math. They can be

• Scalar (number) fields that can describe, for example, temperature
• Vector fields that can describe, for example, magnetic fields
• Tensor fields that can describe, for example, spacetime curvature in General Relativity
• Spinor fields that can describe, for example, Fermions in QFT such as Electrons (Dirac spinors)

To expand on my vector field point above, the magnetic field can be described as (one of Maxwell's equations):

$$\nabla \times \mathbf{B} = \mu_0 \left(\mathbf{J} + \epsilon_0\frac{\partial \mathbf{E}}{\partial t} \right)$$

And the $B$ field is the magnetic field (a physical field that depends on space/location), and we can rewrite it as

$$\mathbf{B} = \nabla \times \mathbf{A}$$

Where $A$ is the vector potential. In other words, we can define the physical field $B$ as a mathematical field $A$, which is just a vector field.

And we basically do the same mathematical prescription for any physical field in physics which we define it as some underlying mathematical field, whatever that may be: spinor, scalar, vector, tensor, etc.

Why do we do this? Well, mathematical fields seem to do a very good job of describing our Universe and we can use them to make predictions and so on. Mathematical fields are models of the physical fields that nature uses and they are just that, models.

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Also, to expand on the 2nd definition that you mentioned, fields are a great way of making sure that interactions do not affect particles and objects instantly, but rather are limited to the speed of light. The effect of the field has to travel outward (at the speed of light or some other velocity) like a wave on water, where it takes some finite time for it to reach its destination.

When you learn about retarded fields in EM theory, my above point (about fields propagating outward to interact with particles) will start to make more sense (if it hasn't already).

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So we can construct a hierarchy of fields:

mathematical fields $\rightarrow$ physical fields $\rightarrow$ what the Universe is.

Answer 3

A field is just a way to describe forces for example to charges or masses, depending from the location the masses or charges are. I can't understand describing them as "physical entity ("a form of matter") They are a physical entity since you can measure them , in measuring the forces, and get always the same results in the same circumstances.

Answer 4

The two definitions are not necessarily contradictory.

When doing physics, we build models, make predictions based on our model and compare those predictions with experimental results. Our definitions and statements refer to the models we built. In this sense the first definition is quite satisfactory: it tells us what we call a field in our model.

The second definition sounds more like a description than a proper definition. It refers to "force" and "matter", so it is only useful if those are already defined. It tries to capture what we mean by field in 'reality'. But we are not sure that fields even exist, as that is a philosophical question and not one about physics. Note that we do not need to know whether fields exist in reality to build a useful theory.