The Electron—Getting to the Point

An electron presents a somewhat related challenge. An electromagnetic field exists around any charged particle. If the particle is not in motion, then it is simply an electric field, E, given by Coulomb’s Law. But the same law mandates that the field’s magnitude E increases at locations closer to the particle, approaching infinity in the limit. In addition, the charge density is zero outside the immediate location of the electron, and infinity at that location. Finally, the mass density is also infinite at the location of the electron, and zero elsewhere. These claims are based on the not uncommon assumption that an electron has no spatial extent and is located at a literal mathematical point; experimentally, the electron’s radius is less than 10-22 m [15].[1] A similar situation arises in the case of a black hole, where the mass density becomes infinite at the mathematical point (singularity) of the hole itself.[2]

One way to mathematically represent the situation of an infinite point density is via a Dirac delta function, namely one that is infinite at the point in question, and zero elsewhere. This—usually taken as a convenient fiction as already noted—does the trick really well and surprisingly often, and is now a standard item in the physics toolbox. However, delta functions can quickly turn from convenience to headache, due to the nonlinearity of many applications. That is, the usual way to “precisify” a delta function is as a Schwartz distribution: a linear functional on a space of functions. However—as Wald [25] points out—in many applications (nonlinear ones) delta functions (when viewed as distributions) cannot be sensibly multiplied, and this poses significant difficulties for their use where there are point sources of fields. This is a bit outrageous: why cannot one multiply two functions? The answer is that the Schwartz representation really groups these “fiction-functions” into equivalence classes (ones that provide the same results for certain special integration proper- ties[3]), and integration does not always respect some of the desired characteristics needed for non-linear applications. Yet once ungrouped from each other and treated as genuine functions, delta functions can indeed by multiplied, as we will see in the next section.[4]

Summarizing a bit, one way that infinity arises in physics is as follows: a vector field (such as gravitational or electrostatic force) depends on the spatial separation between one body and another, in a way that increases without bound as that distance decreases to zero. In particular, in these two instances, the force is proportional to the reciprocal of the square of the distance. When that distance is zero, the expression for the force becomes one divided by zero: 1 /0.

Now, division by zero is extremely problematic; it is not simply that it is not defined, but that it is both overdetermined and underdetermined. 0/0 can be set equal to any number (0/0 = x) with impunity, since 0 = 0x. And 1/0 cannot be set equal to any number at all, since 1 = 0x. So there is no non-arbitrary nor even consistent way to define division by zero that respects the basic concept of division: (a/b)b = a, that is, as the inverse of multiplication.

It is tempting to say that this is because the real numbers are too restrictive, and that 1/0 = to. But then what is 2/0? And do we allow 1 = 0 x to? These notions contain hints of a possible solution. In fact, mathematical physics often employs such intuitions, in the form of infinitesimals and infinities; again think of the standard delta function, that is zero at all non-zero reals, yet when infinitesimally close to zero it rises up to infinity.

But mathematicians have invented many sorts of numbers, going well beyond the familiar real and complex fields, including some that explicitly contain infinities as first-class objects. Which fits the physical situation best? We shall not attempt to answer this here, nor even to survey the existing options. Instead, we shall discuss just one such option, with particular application to delta functions and—possibly—to point particles.

  • [1] But see for instance [23].
  • [2] See [27] for an interesting discussion of electrons as black holes. A related set of issues involve theself-force and self-energy of an electron (or any point charge): the field created by a charge affectsnot only space surrounding the charge but also at the charge location(s) as well. Thus an electron’sfield influences it’s own behavior. Similar considerations apply to any particle with non-zero mass:the associated gravitational field should affect the particle itself; see [25].
  • [3] Namely: fi(x)g(x) = “ f2(x)g(x) for all “test” functions g.
  • [4] This is not to say that successful application to non-linear differential equations is an automaticbenefit; as noted, it is not the product per se but rather integration properties of products that is atissue.
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