# Theory of "fields" in the sense of fields in physics?

Summary:
What's the name for the mathematical theory of fields, in the sense that the word "fields" is used in physics?
The mathematical structure called a "field" is an algebraic structure that doesn't capture the properties of what physicists called a "field". What's the name for a mathematical structure that does represent the general concept of physicist's field?

Is a mathematical version of a physicist's field" just any scalar valued function of several coordinates? That very general definition doesn't express details present in physical fields.

Svein
Manifold?

PeroK
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Summary:: What's the name for the mathematical theory of fields, in the sense that the word "fields" is used in physics?

The mathematical structure called a "field" is an algebraic structure that doesn't capture the properties of what physicists called a "field". What's the name for a mathematical structure that does represent the general concept of physicist's field?

Is a mathematical version of a physicist's field" just any scalar valued function of several coordinates? That very general definition doesn't express details present in physical fields.
A field generally is a quantity defined for every point in spacetime. The quantity may be a real or complex scalar, a vector, a tensor, a spinor etc. In quantum field theory, the field is operator valued. The Quantum Dirac field, for example, is a set of four operator components defined at each point in spacetime.

Manifold?

That does express physical ideas of "space".

A field generally is a quantity defined for every point in spacetime. The quantity may be a real or complex scalar, a vector, a tensor, a spinor etc.
That's true of the physicists "field". It doesn't answer the question in the OP but it does make the question more specific - i.e. what's the situation in mathematics? Is there a different name for the abstraction of each of those cases? - or no particular name for any of them?

In quantum field theory, the field is operator valued..

I think of operators as operating upon something. Does a mathematical abstraction of a quantum field require another mathematical abstraction that tells what the operators operate upon?

PeroK
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I think of operators as operating upon something. Does a mathematical abstraction of a quantum field require another mathematical abstraction that tells what the operators operate upon?
Thay act on Fock space:

https://en.wikipedia.org/wiki/Fock_space

That's true of the physicists "field". It doesn't answer the question in the OP but it does make the question more specific - i.e. what's the situation in mathematics? Is there a different name for the abstraction of each of those cases? - or no particular name for any of them?

They are simply X-valued fields, whatever X is. Scalar, vector and tensor fields are well-known to mathematicians, I imagine.

They are simply X-valued fields, whatever X is. Scalar, vector and tensor fields are well-known to mathematicians, I imagine.

I imagine so too.

I'm just asking a vocabulary question. The typical situation in math is that a general structure (like "ring") has special cases ( e.g. ring-with-unity, commutative ring, Euclidean ring ). Is this the case for the mathematical abstraction of the notion of a "field" in physics? - if such a thing has a name. Or do mathematicians use more lengthy phrases to describe abstractions of different physical fields and their interesting special cases.

For example, "vector space" is mathematical terminology. Is "vector field" mathematical terminology? - perhaps that's a subjective question. A less subjective question is whether a mathematical "vector field" can be defined as a particular kind of "vector space".

We could say a "vector field" is a function that map points on a manifold to vectors in a vector space. Then there can be interesting cases like a "smooth vector field", a "conservative vector field" etc. But I'm unsure if this approach to terminology is common.

I think that the general approach is within the domain of bundle theory. Eg vector field is a section of tangent bundle, phase space of classical mechanics is cotangent bundle, etc.

PeroK
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For example, "vector space" is mathematical terminology. Is "vector field" mathematical terminology? - perhaps that's a subjective question. A less subjective question is whether a mathematical "vector field" can be defined as a particular kind of "vector space".

A vector field isn't a vector space. The focus in physics is different. It's not to categorise things and study their properties but to use those things to model physical phenomena. If we take Newton's second law, for example:
$$\vec F = m \vec a$$
What is going on there precisely mathematically is not the point of physics. It's not immediately obvious how that is defined rigorously. Is the set of all possible accelerations a vector space? Is the mass an element of the real scalar field? Is the set of forces the same vector space as the set of accelerations?

That can all be worked out, I'm sure, but it's not what we dwell on when applying Newton's laws.

The focus in physics is different.

What is going on there precisely mathematically is not the point of physics. It's not immediately obvious how that is defined rigorously.

I interpret your answer to be that there is no concise mathematical terminology that neatly abstracts the zoo of fields that physicts use.

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Svein
From the Wikipedia (https://en.wikipedia.org/wiki/Manifold):

The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described and understood in terms of the simpler local topological properties of Euclidean space. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions.

Manifolds can be equipped with additional structure. One important class of manifolds is the class of differentiable manifolds; this differentiable structure allows calculus to be done on manifolds. A Riemannian metric on a manifold allows distances and angles to be measured. Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model spacetime in general relativity.

Paul Colby
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I think that the general approach is within the domain of bundle theory. Eg vector field is a section of tangent bundle, phase space of classical mechanics is cotangent bundle, etc.

I don't understand why for classical fields this isn't exactly the answer sought in the OP? Are there physically meaningful classical EM fields for example which are not described by a section of a fiber bundle?

weirdoguy
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Answering as a mathematician myself, I think the problem is that math covers several different specialized subjects, and different specialties use the same words in different ways. For example the word "normal" is used in pretty much every specialty with a different meaning: it means a certain kind of subgroup in algebra, it means a certain existence property of real valued functions in set theoretic topology, and it means a space that essentially does not cross itself in algebraic geometry. In linear algebra it sometimes means "perpendicular" for vectors and sometimes "length one", and it means "commutes with its adjoint" for operators. The word "field", as a commutative ring in which all non zero elements have multiplicative inverses, is only used with that meaning in algebra. In differential geometry, or any area in which smooth manifolds are studied, a "field" is pretty much the same as the physicists think it is, i.e. a function defined on the manifold, with values which are either tangent or cotangent vectors, or tensors, or elements of some more general bundle, as weirdoguy said.

You will see this if you look in many textbooks on differential geometry or many geometric treatments of differential equations, e.g. Riemannian Geometry by Do Carmo, or Mathematical Methods of Classical Mechanics by Arnol'd, where the word "field" occurs in the index referring to the word in the physicist's sense, i.e. as a section of a certain bundle. Since the most commonly occurring bundle and the most important one is the tangent bundle, the word field in a differential geometry setting often means by default just (tangent) " vector field". My first mental response to the OP's question was therefore "vector field". In some books, such as Foundations of Differentiable Manifolds by Warner, or Notes on Differential Geometry by Hicks, you will find only "vector field" but not "field" in the index. In Lectures on Differential Geometry by Sternberg however, there is even a section on the "principal bundle" in which he describes how to define a field of "quantities" of any kind at all, as long as the fibers of the bundle in which the values are taken is any differentiable manifold having a certain linear group action.

Although this may make reading math seem confusing, actually all math texts try to be careful to state clearly what the words mean that they will use, just for this reason. I.e. in math we are not careful to always use words in the same way, but we do try to be careful to say in every setting just what the words we use will mean there. Of course we are also careless if we think the reader already knows what we mean. E.g. I don't think Arnol'd bothers to define a "vector field" in his somewhat more advanced book mentioned above, but he does define various types of them, "right invariant", centrally symmetric, etc... But when reading any math text, one should always try to verify what each key word means as used in that book. Hence a "theorem" in any given math book, means exactly a statement that is (hopefully) true as long as the words in that statement have the meanings that are given for them in that same book. This is why definitions are such a crucial part of math writing.

Sometimes you will find a math book in which a theorem you thought was difficult has been given a surprizingly easy proof. The trick is often that the meaning of the words in the theorem have been given in a different way from what you expected, precisely to render the proof easier, at the cost of perhaps proving a less precise statement. A notorious example of this to me is the "Riemann Roch" theorem, the most famous and perhaps most important theorem in algebraic geometry. Sheaf theoretic "proofs" of this theorem are often given so as to make the proof look absolutely trivial. Then at some point one notices that the definition of the "genus" of the complex curve has been given as the dimension of a certain cohomology group, and no attempt has been made to calculate that genus in terms of topology, as Riemann did himself. Thus the theorem obtained is not as strong as the original one. E.g. one may find a statement claiming only that the Riemann Roch theorem states that chi(D) - chi(O) = degree(D), whereas the full statement should include a computation of chi(O) in topological terms. In the 2 dimensional (complex) case, compare Th. 1.6 chapter V of Hartshorne, where the Riemann Roch theorem for complex surfaces is stated in the weaker form. A discussion of the additional information is mentioned in remark 1.6.1, and in the appendix A.4.1.2, a result known as Noether's formula, which implies it was due to Max Noether, hence was actually part of the earliest known form of RR for surfaces.

(Another confusion in this topic, which came up for me while writing the previous paragraph, is the use of the word "surface" to refer both to a complex curve ("Riemann surface"), which has real dimension 2 and complex dimension one, and also to a complex surface, which has real dimension 4 and complex dimension 2.)

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Let's make the counterfactual assumption that I did know something about fiber bundles etc. How would I relate these seemingly pure geometric ideas to a typical approach in physics where a field is defined as a solution to a set of partial differential equations?

If one coordinate of a manifold is interpreted as time then the geometry of the manifold can be interpreted as telling how one set of initial conditions (the other coordinates) will change to another set of conditions as time changes. Is that the general idea?

Well, if you wanna do field theory on fiber bundles you need to invoke jet-bundles in the picture (jets of sections of the bundle representing the field we are considering). Sections of those jet-bundles serve as "infinitesimal" configurations of the field. Partial differential equations can be seen as subsets of the total space of those bundles. There are a lot of papers on that, but not so many books. You can get a gist of the topic by looking up these three papers:

https://arxiv.org/abs/0908.1886
https://arxiv.org/abs/1306.2744
https://arxiv.org/abs/1109.2533

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mathwonk
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RE: #13, when I think of a (vector) "field" on a manifold, I do not think of a solution of some set of differential equations. I.e. at least to me a (vector) field, (on a given manifold) is merely a choice of a tangent vector at each point of the manifold, not satisfying any conditions at all except to vary smoothly perhaps. Thus the vector field itself can be regarded as a differential equation, since one can ask whether there exists say a smooth function whose gradient field agrees with the given field, or whether there is a family of curves filling the manifold, whose velocity vectors agree with the given vectors. So we have some different instincts in this regard it seems. Of course certain fields might satisfy some differential equations, but to me that is not necessary for them to deserve the name "field". Maybe that is the confusion. I.e. whenever there are fields that satisfy a given differential equation, there would also be other fields that do not. But in general I think of a vector field as defining a differential equation rather than as a solution of one. Just my limited knowledge no doubt.

Oh maybe I get it. I presume you are saying you can use a set of partial differential equations to define, in the sense of specify, a particular field, not to define the word "field".

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But in general I think of a vector field as defining a differential equation

Yes, images of a vector fields are particular case of first-order differential equation. But in general (at least in the field (hehe) of geometric foundations of classical physics) a ##k##-th order ordinary differential equation is a subset ##D## of ##k##-th tangent bundle ##D\subset \textsf{T}^kM##. A curve ##\gamma: I\rightarrow M## is said to be a solution of the equation ##D## iff for every ##s\in I## ##k##-th tangent prolongation of this curve belongs to ##D##: ##\textsf{t}^k\gamma(s)\in D##. Not all ##D\subset \textsf{T}M## are images of vector fileds.
Analogously, ##k##-th order partial differential equation is defined as a subset ##D## of ##\textsf{J}^kE## for some bundle ##\varepsilon: E\rightarrow M##. Solutions of this equations are sections of ##\varepsilon## for which their ##k##-th order jet prolongation belongs to ##D##.

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A field generally is a quantity defined for every point in spacetime. The quantity may be a real or complex scalar, a vector, a tensor, a spinor etc.
In mathematical terms, I would think one can describe a field F as a function ##F:\mathbb{R}^{4}\rightarrow X## (I'm not positive, but for relativistic physics I think it would be ##F:\mathbb{R}^{3,1}\rightarrow X##). What X is would then determine the "type" of field (i.e. scalar, vector, etc).

The first examples that come to mind are the electric potential field ##V:\mathbb{R}^4\rightarrow \mathbb{R}##, a scalar field, and the electric field ##E:\mathbb{R}^4\rightarrow \mathbb{R}^3##, a vector field. I don't know if this is far more elementary than the OP is looking for, but this is how I tend to think of fields.

etotheipi and PeroK
PeroK
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In mathematical terms, I would think one can describe a field F as a function ##F:\mathbb{R}^{4}\rightarrow X## (I'm not positive, but for relativistic physics I think it would be ##F:\mathbb{R}^{3,1}\rightarrow X##). What X is would then determine the "type" of field (i.e. scalar, vector, etc).

The first examples that come to mind are the electric potential field ##V:\mathbb{R}^4\rightarrow \mathbb{R}##, a scalar field, and the electric field ##E:\mathbb{R}^4\rightarrow \mathbb{R}^3##, a vector field. I don't know if this is far more elementary than the OP is looking for, but this is how I tend to think of fields.
And in QM you can think of the wavefunction as a complex scalar field. Or, in classical EM the EM field can be a tensor field instead of two vector fields.

etotheipi
A field is a structure which associates a patch of grass to all points in the space. Notable examples include relativistic fields where the cows run around at near the speed of light, and quantum fields where the the cows tunnel though the fences and escape. The pioneer of field theory as it is known today was Hermann Min-cow-ski.

PeroK and Isaac0427