Fields in physics and fields in group theory, are they related?

  1. I just though of this and though "it's abstract math meeting physics, so probably not". After looking up fields in several abstract algebra books I thought that maybe fields in physics were called as such in physics because they share something with the mathematical structure of fields in group theory. I'm highly interested in both physics and abstract mathematics but at the moment don't have time to look this up myself, so I ask this: Do fields in group theory share anything with fields in physics? (Excluding the fact that one is abstract and the other is a physical quantity)
    Please I'd appreciate answers that actually look into this instead of just saying "what does abstract math have to do with physics". I mostly want to prove to my self that the naming is just a coincidence, but I just can't help wondering since group symmetries are used in a lot of aspects of upper level physics.
  2. jcsd
  3. Not really. The word "field" was used in Physics first, I believe, and was introduced by Michael Faraday in the context of electromagnetics. Some fields in physics, like scalar or perhapds tensor fields, may also fit the mathemtical definition of a field, but vector fields (like the electric and magnetic field) do not.

    So they are really independent words, that occasionally be applied to the same thing. I believe the usage in math arose independently, without analogy to physics (but could be wrong on that).
  4. Just look at the definitions of the two and you can see they are not the same. Physical field doesn't need two operations to be defined.
  5. In physics, the word 'field' basically indicates some (scalar-, vector-, operator- or whatever-valued) function that is defined on some space (usually three-dimensional Euclidean space or 4D Minkowski spacetime), rather than a set with two operations (that satisfies some additional properties). Actually we tend to care about how it transforms under various coordinate operations than about whether it is closed or not. (What would closure of the electrical field even mean physically?) Of course, group theory is heavily linked to physics, and used in quite a few notational schemes, but this is not the place to look for one of these connections.

    By the way, I think that dipole is right in saying that the word was used in physics first. According to wikipedia, Maxwell coined the term in physics in 1849 and, apparently, Eliakim Hastings Moore first used the word in mathematics in 1893.
  6. mathman

    mathman 6,754
    Science Advisor
    Gold Member

    From Wikipedia:


    The concept of field was used implicitly by Niels Henrik Abel and Évariste Galois in their work on the solvability of polynomial equations with rational coefficients of degree five or higher.

    In 1857, Karl von Staudt published his Algebra of Throws which provided a geometric model satisfying the axioms of a field. This construction has been frequently recalled as a contribution to the foundations of mathematics.

    In 1871, Richard Dedekind introduced, for a set of real or complex numbers which is closed under the four arithmetic operations, the German word Körper, which means "body" or "corpus" (to suggest an organically closed entity), hence the common use of the letter K to denote a field. He also defined rings (then called order or order-modul), but the term "a ring" (Zahlring) was invented by Hilbert.[2] In 1893, Eliakim Hastings Moore called the concept "field" in English.[3]

    In 1881, Leopold Kronecker defined what he called a "domain of rationality", which is indeed a field of polynomials in modern terms. In 1893, Heinrich M. Weber gave the first clear definition of an abstract field.[4] In 1910, Ernst Steinitz published the very influential paper Algebraische Theorie der Körper (English: Algebraic Theory of Fields).[5] In this paper he axiomatically studies the properties of fields and defines many important field theoretic concepts like prime field, perfect field and the transcendence degree of a field extension.

    Emil Artin developed the relationship between groups and fields in great detail from 1928 through 1942.

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