Definition of Field: A Necessary & Sufficient Explanation

[nomadreid]

The definitions I have seen of "field" seem rather unsatisfactory. Wikipedia starts off by saying that a field is any function with spacetime as its domain, but this seems awfully broad, since there are 2^|ℝ| number of functions with spacetime as a domain. Further down, Wikipedia basically says that a field will be the solution to certain differential equations, but that is circular, because in order to know which variable to select to solve for, one must be able to label it as a field. Other sources just cite well-known examples (usually electromagnetic field), which isn't a definition (or even a completely good hint, since one would want to look for a definition to cover scalar, vector or tensor fields). Srednicki's "Quantum Field Theory" doesn't give a definition. Daniel Fleisch ("A Student's Guide to Vectors and Tensors" claims there is no agreed-upon definition, which is odd, given the ubiquity of the concept. Can anyone give me a reasonable definition that is not circular, not too broad, not just listing the major fields, but that is giving a decent definition (giving conditions that are both sufficient and necessary to call a function a field).

 

[Nugatory]

Wikipedia starts off by saying that a field is any function with spacetime as its domain, but this seems awfully broad, since there are 2^|ℝ| number of functions with spacetime as a domain.

Actually many more than that, as the range of the functon doesn't have to be a real number - it can be a vector, a tensor, a spinor, a complex number, ...
Nonetheless, that definition is pretty good. Anything more restrictive will exclude something that you might want to think of as a field, and if you use that definition you'll be able to communicate with other scientists and mathematicians.

Given some of the questions you've been asking in this and other threads, you might want to give Lancaster and Blundell's "Quantum field theory for the gifted amateur" a try. It provides an informal definition of "field" somewhere in the first chapter, one that is pretty much aligned with the definition above.

 

[Dale]

Wikipedia starts off by saying that a field is any function with spacetime as its domain, but this seems awfully broad, since there are 2|ℝ|ℝnumber of functions with spacetime as a domain

What is wrong with that? I cannot fathom what would make you object to that.

 

[nomadreid]

Thanks, Nugatory and Dale. Nugatory, thanks for the literature suggestion. I immediately went and got it. (His informal definition of field is on page two, in the "Ouverture".) I shall indeed be reading it in the coming months. (By the way, I was assuming, for example, that the basis of any relevant vector field was at most of cardinality |ℝ|; |ℂ| =|ℝ|,; I don't think that a tensor product will increase cardinalities; the sum of these possibilities also does not take the range above |ℝ|. However, you are right; if the definition is that general (your "..."), then there is no limit on the cardinality of possible fields.)
Dale: I now accept that definition. It just seemed rather broad. For instance, if F is a defining function from S = the set of spacetime points s to values v, and one now asks questions via a function G about the values v in the range of F, then one has G(v) = G(F(s)). Then the composition of these functions, G°F = H, is also by this definition a field. Or if we have only some spacetime points at the basis of our question, or only space, or only time, all these can also be defined by functions with S as a domain. In other words, it seems to me that almost everything can be a field, which sounds as if it is trivializing the definition. It becomes more difficult to answer "what isn't a field?".

 

[Dale]

It becomes more difficult to answer "what isn't a field?"

Is there some reason that you need there to be non-fields? If everything is a field and you know how to handle fields then you know how to handle everything. Isn't that the goal.

 

[nomadreid]

Dale: But it still rankles, given that the reason for having a word W is to be able to divide the world into W and not-W (and, for intuitionists, everything else). Mathematically this definition would be tricky since it would be based on an unlimited cardinality, so that it would in itself have no meaning until the range was specified, hence my conclusion is that the word "field" by itself has no meaning until one puts a modifier to it: electric field, gravitational field, etc.

 

[jack476]

A field is a function that associates a value (scalar, vector, tensor, operator, etc) with each point in a space (the Cartesian plane, the complex numbers, something more abstract like a Hilbert space, etc). The idea is that a value is associated with a location rather than just any input parameter.

For instance, an equation that shows you the electric potential at every point in the Cartesian xy-plane is a field. On the other hand, an equation that shows you the total potential energy in the field as a function of some parameter like temperature or time is not a field since its input parameter is not a location.

 

[nomadreid]

jack476, thanks.
(1) You state that a field is a function of points in a structure ( I avoid the use of the word "space" here to avoid confusing it with "space" as in "spacetime".), Then you then refer to "location", which has the connotation of being spatial. Your counterexample could be a field in a the first interpretation (domain = any space), although not in the second (domain = spatial locations). I am not quite sure which one you mean. In any case, I have only seen (in physics) the word used with the domain being the set of spacetime points.
(2) Again, I am accepting the general definition, but it does cover a lot of territory, since a function that associated every point in spacetime a random digit of pi would also be considered a field. However, to paraphrase George Orwell, all animals are fields but some animals are more fields than others, so we concentrate on the Napoleons of the field farm, such as electric field, etc. But letting that pass, it raises a new question: wouldn't it be more reasonable to define the domain of a field as all intervals (open and closed, hence including points), since a field spreads out?

 

[Vanadium 50]

Nomadreid, I don't think trying to find a more "reasonable" definition of a field is likelyb to go anywhere. The definition is what it is.

I also don't understand the distinction you are drawing between "space" and :"location".

A field is an object that has a value at every point in space and every instant in time. That's it. Some of these are more useful in physics than others, to be sure.

 

[nomadreid]

...I don't think trying to find a more "reasonable" definition of a field is likely to go anywhere. The definition is what it is...A field is an object that has a value at every point in space and every instant in time. That's it.

I agree, which is why I wrote, a few posts ago, that I accept this definition. (Accepting and liking are two different things, but I have no intention to tilt against windmills. I just wrote why it goes against the grain as an aside.) Anyway, it is clear that the word serves its purpose, since the actual usage of the word does not make use of the full generality of the definition. Pragmatics rule!

I also don't understand the distinction you are drawing between "space" and :"location".
.

This was in reply to the usage of these two words by jack476. I was not too sure of his usage, which is why I asked for clarification. "Space" can mean either physical space (as in the projection of spacetime), or any set of points with a structure (as in Hilbert space), whereas "location" (unless you are talking about memory units in computers) tends to evoke purely spatial ideas. It seemed to me that in one sentence jack467 was saying that the domain of the function would be a space in the more general sense

A field is a function that associates a value ... with each point in a space (. Hilbert space, etc).
.

and then, right afterwards, in the more restricted sense

. ... a value is associated with a location
.

 

[jack476]

This was in reply to the usage of these two words by jack476. I was not too sure of his usage, which is why I asked for clarification. "Space" can mean either physical space (as in the projection of spacetime), or any set of points with a structure (as in Hilbert space), whereas "location" (unless you are talking about memory units in computers) tends to evoke purely spatial ideas. It seemed to me that in one sentence jack467 was saying that the domain of the function would be a space in the more general sense

I suppose I should have been more careful with my choice of words. The only reasoned I mentioned Hilbert spaces was to add a bit of generality, though by "location" I did mean any point in a structure ("location" in a memory array, "location" in phase space, etc), though in practice I don't think that one would invoke the intuitive notion of a physical field when working with very abstract quantities. I am sorry if that caused any confusion.

In physics, I don't think that there exists any truly rigorous definition of field, unless the notion of a physical field somehow comes directly from the formal definition of an algebraic field, so as V50 says I don't know if it's really possible to be any more detailed than that.

 

[nomadreid]

Thanks, jack476. All clear now.
As far as I can make out, there is no direct connection between the mathematical use of the word "field" (nonzero commutative division ring) and the physicist's usage. There are only so many words in the English language ... (as a side note, the Germans, who gave English the word "field" from "Feld", are a bit more discerning, in that they use two different words: "Feld" for physics, and "Körper" for mathematics.) One could torture it a little to find an indirect connection, but you could probably do that for almost any pair of words.

 

[David Lewis]

Originally, a field referred to a network of Michael Faraday's lines of force. Faraday came up with lines of force to replace action-at-a-distance.

"The Sun sends out its attractive message to Earth via a field rather than reaching out to influence the Earth at a distance through empty space. Earth doesn't have to know there is a sun out there. It only knows there is a gravitational field at its own location." -John Wheeler

 

[Omega0]

.

What do you think about this:
A physical field is a field if an interaction leads to an exchange of energy?

 

[Vanadium 50]

A physical field is a field if an interaction leads to an exchange of energy?

I think this is a terrible idea. First, redefining the word "field" can only lead to miscommunication. Second, a magnetic field is not a field by this definition.

 

[David Lewis]

The concept of a field, originated by Lord Kelvin, arose from the hypothesis that empty space was incapable of exerting force on matter. So "field" originally meant a force field (gravitational, electric, magnetic, etc.) and, prior to James Maxwell, conceptualized qualitatively with very little math involved.

 

[Omega0]

I think this is a terrible idea. First, redefining the word "field" can only lead to miscommunication. Second, a magnetic field is not a field by this definition.

Does not this magnetic field allow exchange of engergy? Doesn't we have a charged particle to interact with this field?
Could you name a field where no exchange of energy happens? With no particle?

 

[Nugatory]

Does not this magnetic field allow exchange of engergy? Doesn't we have a charged particle to interact with this field? Could you name a field where no exchange of energy happens? With no particle?

Uncharged particles are unaffected by magnetic fields; charged particles follow curved paths that don't change their energy because the force is always perpendicular to the direction of motion. So there's your example, just as Vanadium 50 said.

 

[Nugatory]

This thread is becoming a bit silly. We have a definition of the word "field", namely a function (not necessarily real-valued) whose domain is points in space or spacetime. That definition is both useful and generally accepted, and if we didn't use the word "field" for such functions we'd have to invent another word for them, because they are ubiquitous in physics.

But it still rankles, given that the reason for having a word W is to be able to divide the world into W and not-W (

But the accepted definition of "field" does exactly that. There are an enormous number of functions whose domain is not points in space or spacetime so count as not-field. They're useful and important, but we don't use them the same way we use fields, which just goes to show that there are fields and not-fields and the distinction matters.

 

[Nugatory]

I'm closing this thread now because the discussion seems to me to have reached the point of diminishing returns. As always, if you see something more to say, PM me or any other mentor and we can reopen the thread.