What is a field, really?

I'm going to go with a programmer metaphor for you.

• The mathematics (including "A field is a function that returns a value for a point in space") are the interface: they define for you exactly what you can expect from this object.

• The "what is it, really, when you get right down to it" is the implementation. Formally you don't care how it is implemented.

  In the case of fields they are not matter (and I consider "substance" an unfortunate word to use in a definition, even though I am hard pressed to offer a better one) but they are part of the universe and they are part of physics.

  What they are is the aggregate effect of the exchange of virtual particles governed by a quantum field theory (in the case of E&M) or the effect of the curvature of space-time (in the case of gravity, and stay tuned to learn how this can be made to get along with quantum mechanics at the very small scale...).

  Alas I can't define how these things work unless you simply accept that fields do what the interface says and then study hard for a few years.

Now, it is very easy to get hung up on this "Is it real or not" thing, and most people do for at least a while, but please just put it aside. When you peer really hard into the depth of the theory, it turns out that it is hard to say for sure that stuff is "stuff". It is tempting to suggest that having a non-zero value of mass defines "stuffness", but then how do you deal with the photo-electric effect (which makes a pretty good argument that light comes in packets that have enough "stuffness" to bounce electrons around)? All the properties that you associate with stuff are actually explainable in terms of electro-magnetic fields and mass (which in GR is described by a component of a tensor field!). And round and round we go.

You say:

she said to me that, if I wanted hardcore definitions,

a field is a function that returns a value for a point in space.

Now this finally makes a hell lot of sense to me but I still don't understand how mathematical functions can be a part of the Universe and shape the reality.

You don't have to use super-complicated examples such as electromagnetism. I'll give you two examples which I hope will make it more clear; let me know if this helps.

Example 1: Temperature

You might have come across that the higher you climb (on Earth or somewhere else, but let's think of Earth) the colder the air gets, at an typical rate of about 6ºC per kilometer (it depends on various factors, but this is a ballpark value); in meteorology, this is known as the lapse rate: the rate of temperature drop with altitude.

Now suppose you're observing a large, uniform terrain (e.g. a "flat desert"). If you want to ask:

What is the temperature of the air at a point $(x,y,z)$?

then you'll ascribe a certain value of temperature for each point. But to make a "table" to give the temperature for every point is certainly impractical! You try instead to use a function, an application, that gives the value of the temperature for each point: $$ f : (x,y,z) \mapsto f(x,y,z) $$ I'll use a clearer nomenclature: $$ T : (x,y,z) \mapsto T(x,y,z) $$ So this is a function with arguments in a $\mathcal{R}^3$ space (three-dimensional space, $\mathcal{R}\times\mathcal{R}\times\mathcal{R}$) which gives values in a 1-dimensional $\mathcal{R}$ space. Those values represent the values of the temperature at each coordinate $(x,y,z)$ of $\mathcal{R}^3$. Instead of writing $T(x,y,z)$ you can be more "practical" and write just $T$ as shorthand (especially when you're some calculus in an exercise).

That function represents a field -- the temperature field.

"But what's the use of that?!"

What does it look like? If you have the ideal case of a perfectly flat "desert" and an idealized atmosphere, the temperature field will be something like: $$ T(x,y,z) = T(x,y,z_0) - \frac{dT}{dz} (z-z_0) $$ Some notes:

1. In this situation, the temperature only varies in the vertical; it looks the same at any place over the desert -- there is really no depedence in the coordinates $x$ and $y$. Because of that you could make it easier for you and shorten the expression to just $T(z) = T(z-z_0) - dT/dz$.
2. In case you don't know/forgot: $dT$ is how much the temperature $T$ varies when you increase your height by a small (infinitesimal!) amount $dz$.
3. Don't worry about the minus sign next to the rate. It's put there by hand to have the expected physical meaning. When you go from a height level $z$ to $z+dz$, the temperature should decrease, from $T$ to $T-dT$ where $-dT < 0$, so that $-dT/dz$ is negative (it "takes away" from the temperature as you increase the altitude $z$). Example: from $z=1000$ to $z+dz = 1001$, the temperature should drop from $T$ to $T-0.006$ where $T$ is the temperature at level $z=1000$. Of course, that small value is because $0.006/(1001-1000) = dT/(dz+z-z) = dT/dz = 6$ Celsius per km.
4. I've intentionally abused the expression above to make it easier to understand. A more appropriate expression would be (if you've studied "integrals" in calculus) something like $$T(x,y,z) = T(x,y,z_0) - \int\limits_{z_0}^z \frac{dT}{dz} dz\ .$$

You have to give the temperature at a certain level $z_0$ of your choice to represent a specific case; it can be at the surface, $z_0 = 0\ \mathrm{meters}$. That function you have there represents the temperature field for that situation. If you have a "hot spot" -- e.g. you light up a candle -- then the temperature distribution (the field!) will be different, and the mathematical expression to describe the temperature field will be different (more complicated).

So this temperature field describes what is the temperature over that "desert air". It represents a quantity which has a spatial distribution. You can make it much more shorthanded if you just ignore the frontier condition $T(z_0)$ at a certain vertical level $z_0$ (which is arbitrary!) and write the field as $$-\frac{dT}{dz}\ .$$

Example 2: Wind velocity

The example above illustrates a scalar field: the value of the field at each space point takes a scalar value ("just a number"). Not all fields are scalar. An example is the velocity field, which represents the velocity (direction and magnitude!) of the air at each point.

You can write it as $$\vec v : (x,y,z) \mapsto \vec v(x,y,z)$$ and for each point $(x,y,z)$ it describes what is the direction and magnitude of the air displacement at that point, the vector $\vec v$ at that point.

What does it look like?

(The mathematical expression?) Well, it will depend on the situation of course! The expression can be impossibly complicated to write analytically. You certainly won't write the velocity field (or the temperature field) for the air inside your living room -- it's too complicated to write a mathematical expression! The best you can do is

1. Know a few laws or expressions or (more correctly) models, perhaps deduced from first principles, to describe how the conditions of a tiny piece of air will be influenced by the conditions of the neighbouring regions. Those models can be very simple or more elaborate; in the latter for meteorology, you just use computers to do the complicated ballance for each and every "air cell". In the example 1 with the temperature above, there is no horizontal dependence, but the rate at which the temperature varies vertically depends on the temperature, pressure, etc on top of the "tiny air box/cell/element" and on the bottom -- those are the ones who produce an effect.

2. Make some simplifications about the initial conditions, such as knowing what is the temperature along the walls and assuming (for example) there aren't "hot spots" or if there are, they're too insignificant to spot the difference against the situation where there aren't hot spots.

Example 3: the electromagnetic field

When you put an electrically-charged tiny particle (test particle) near a metalic plate (for example) that has an electric charge itself (like the plate of a large capacitator, for example), in the most general and broad case the force that the particle will feel will depend on where the particle is relative to the charged plate.

The force the test particle feels has a magnitude as well as a direction. If you put the test particle in another position, if will feel the force with a different intensity and direction.

You could put the test particle in many different places around the plate and measure the electric force felt by the test particle. And you collect the direction and intensity of that force. If you are able to condense that description of the magnitudes and directions of the electric force felt by the particle, you're writing it as a field, $$\vec E : (x,y,z) \mapsto \vec E(x,y,z)\ .$$

You can interpret the electromagnetic field as nothing more as a "mash-up" of both the electric force and magnetic force that a test particle will feel at each point of space.

OK, but can you "touch" a field?

As a final note, I'll say the following; this question is more subject to discussion. Personally, I don't quite think about "touching" a field or it being "material"; I don't know how you're supposed to "touch" temperature.

The field represents the set of values for a quantity on a given space, and thus we arrive at your teacher's comment. In the classical physics sense that I've presented above, you can interpret the fields as "our way" of describing something that it's there, in a shorthand (a mathematical expression instead of a "spreadsheet of values"). In that case, I see the concept of field mixing up with the "thing" that it's representing. I won't debate that because I'm not sure I can explain better.

From the way fields are actually used in physics and engineering, and consistent with the mathematical definition, fields are properties of any extended part of the universe with well-defined spatial boundaries. (The latter may be missing in case of infinitely extended objects, e.g., the universe as a whole - if it is infinitely extended.)

Causality is reflected in the fact (that makes physical predictions - and indeed life, which is based on the predictability of Nature - possible) that to a meaningful (and sometimes extremely high) accuracy, changes with time in the complete set of fields relevant for a particular application are determined by the current values of these fields.

Being properties of objects, fields cannot be touched but they can be sensed by appropriate sensors. In particular, several human senses probe properties of fields close the surface of the corresponding sensors:

• Eyes for sensing oscillations of the electromagnetic field passing through the lense,
• ears for (a) sensing oscillations of the pressure field of the air and (b) sensing the direction of the gravitational field,
• the skin for sensing stress fields and temperature fields close to the body surface,
• the tongue for sensing chemical concentration fields close to the surface of the tongue.

More specifically, a field is a numerical property of an extended part of the universe, which depends on points characterized by position and time (though the time dependence may be trivial). It is called a scalar, vector, tensor, operator field etc., depending on whether the numerical values at each point are scalars, vectors, tensors, operators, etc., and a real or complex field depending on whether these objects have real or complex coefficients.

Fields are the natural means to characterize numerically the detailed properties of extended macroscopic objects. This can be seen on a very elementary level. (It also applies to microscopic objects, but there the characterization is much more technical.)

All macroscopic objects possess a number of fields, most of them natural in the sense that all humans in our current technological culture experience in their daily life aspects of these fields either with their own sensors, or with technical gadgets known to be sensitive to these.

• always a scalar mass density field telling how the mass of the object is distributed in space and changes with time,
• in case of uneven composition such as rocks, concentration fields of the various chemical substances it contains.
• in case of nonrigid objects such as fluids, a vector velocity field (or several for each chemical substance), describing the local velocity of the mass flow.
• always a scalar temperature field telling how the temperature of the object is distributed in space and changes with time,

• always a stress tensor field telling how the mechanical forces inside the object are distributed in space and changes with time.

• in case of electrically active objects such as coils or capacitors, a scalar charge density field telling how the charge of the object is distributed in space and changes with time, and a vector current field describing the local velocity of the charge flow.

Not tangible objects such as the space between material objects also have space-time dependent properties, and hence associated fields, namely the (in nonrelativistic case scalar) gravitational field, the (vector) electric field and the (vector) magnetic field.

Hardly visible in everyday life, but very important in physics is an additional field, the (scalar) energy density field telling how the internal energy of the object is distributed in space and changes with time.

Additional fields are employed by physicists whenever the above fields are either not sufficient to give a complete description of the phenomenology they are interested in, or not sufficient to give a tractable theoretical description of the processes.

Causality is implemented by means of parabolic or hyperbolic differential equations relating the derivatives of the fields.