Fields and quantities

Question

I'm a bit confused about the concept of a field defined as a physical quantity. For example on the Field (physics) Wikipedia page a field is defined as a

physical quantity, represented by a number or another tensor, that has a value for each point in space and time.

It then goes on with this example:

For example, on a weather map, the surface temperature is described by assigning a number to each point on the map;

To my understanding, mathematically at least, according to this definition the field is a function. For example, let $\vec{E}$ be the electric field in some region of space $\Omega$ for the time invariant case. In this case:

1. $\vec{E}:\Omega\mapsto\mathbb{R}^3$ is a vector valued function, which we can refer to as $\vec{E}$
2. $\vec{E}(\vec{r})$ is the value of that function at some position $\vec{r}$
3. $\vec{E}(\Omega)$ is the image of $\Omega$, that is set of all vectors $\vec{E}(\vec{r})$ whose origins are located in $\Omega$.

But which of these is the field and which is the quantity?

It would seem a bit "un-physical" that the function $\vec{E}$ (i.e. the rule with which we associate a vector $\vec{E}(\vec{r})$ te each position $\vec{r}$) would be the quantity, although I sometimes found this interpretation in some sources.

Any help would be appreciated.

Answer 1

The field is actually the whole function. It is not something I would call a "physical quantity" in itself, instead the physical quantity is the value of the field at the position where your measurement apparatus is located. For example the temperature field assigns a temperature to each location in space, but the "physical quantity" is not the whole temperature field, but instead just "the temperature" which you measure at one location.

This is of course not really general and it depends on the kind of field you are looking at. There are fields that don't have a physical meaning on their own but are more auxiliary (for example the electromagnetic potential in classical electrodynamics). There are also fields that are quantized, i.e. quantum fields. A field could also be a property of matter that depends on position, e.g. when a string is vibrating the deviation from the non-vibtrating string is different at each point on the string, so that would also be a field.

You could also see (at least some) fields as a collection of "infinitely many degrees of freedom". Compared to the position of a point particle, which represents only 3 degrees of freedom, a vibrating string has "infinitely many points" (this is just to describe it, in reality it is made up of atoms and so on) and each has their own deviation, all together collectively counting as infinitely many degrees of freedom.

Answer 2

Yes, totally agree. It is a bit confusing and you know the our high schools today are teaching the outdated definitions and our future scientists will be also confused or be repeating the confusion.

Latest WIKI states:

"In physics, a field is a physical quantity, represented by a scalar, vector, or tensor, that has a value for each point in space and time."

However, it is incorrect to say that a field is a field quantity since it was recently abandoned and replaced by "root-power quantity" ref. "Root-power quantity" vs. "field quantity"