The Impossibility of Infinitely Small Particles

[This is an essay I wrote in 1989 for folks in a physics discussion group I was part of for a while, led by Steve Bryson as an adjunct to a short course on particle physics he was teaching at the California Academy of Sciences.]

At the first session of our physics discussion group, Steve Bryson asked each of us what we thought of the idea of “point-like” particles, and also more specifically, point-like particles which have mass. I claimed that particles must have a volume, and that in particular it must be absolutely impossible for there to be a particle with mass that is a mere mathematical point (or that is “infinitely small”).

Why must it be impossible? Because anything that has mass (at least) is part of the world, i.e., it is a chunk, or piece of the world. A mathematical point is not part of anything physical. The only things that a mathematical point can possibly be part of are other mathematical “objects”, such as collections of points, lines, spaces (in the mathematical, not physical sense), etc.

Mathematical “objects” are not physical things. The number 2, for example, is not a physical thing in the world. It is a concept, or intellectual abstraction, derived from considering things (either “real” things, or other abstractions) in pairs.

(In Principia Mathematica, Bertrand Russell and Alfred North Whitehead define natural numbers in this way (partially any way). But this is not my argument. You can no doubt “define” numbers in numerous different ways in formal logico-mathematical systems. I am only claiming that our everyday mathematical concepts such as ‘2’, ‘point’, ‘line’, etc., are abstractions from things in the real world (such as pairs, tiny objects, threads or objects-in-a-row, etc.). That is, this is historically how they must have been derived; this is where such concepts were abstracted from.)

Like the number 2, mathematical points, lines, etc., are also not part of the physical world. They are also abstractions from things in the world, very small things in the case of points; threads, or pencil lines, or things-in-a-row, in the case of lines.

Thus my position is that nothing that is part of the world can possibly be a mathematical object like a point. To claim such a thing is to confuse two entirely separate categories of things, things as different as the moon and ideas, or of canvas and paint on the one hand, and abstract patterns on the other.

However, I am not denying that we can (and do) treat physical things in the world as if they were mathematical points. If we did not have reason to do this, the concept of a mathematical point would never have been created in the first place. In particular, it has been a commonplace feature of physics, since at least Newton, to construct physical theories in which physical objects were treated as mathematical points within the theory.

Thus in Newton’s theory of gravity, all the mass of a physical object like the earth is generally considered to be concentrated at a single point at its center (actually at its “center of gravity”). Newton proved that it is acceptable to do this for cases in which the earth’s gravitational attraction is being considered in relation to another object outside of the earth; that mathematically it is equivalent to the real physical situation where the mass of the earth is actually spread out within the whole volume of the earth. It is of course the integral calculus (invented by Newton and Leibniz) which allows you to make such calculations relatively easily.

In the case of the earth, no one is apt to forget that while the mathematical theory (Newton’s equation of gravitation) treats the earth as if it were a single point, actually it is not a single point. But when it comes to particles at the atomic scale or smaller, we do not have this constant reminder that reality is more complex than our mathematical models of it. The tendency is always to assume that there is nothing more to say about these particles—nothing more that can be said—except what is incorporated into the mathematics of our current theories. Thus if the theories treat a particle as if it were point-like, the tendency is to jump to the invalid conclusion that it actually is a point.

The dispute between those (like me) who deny that any part of the world can actually be a mathematical point, and those (like many contemporary physicists) who claim otherwise, is an example of the age-old philosophical dispute between materialism and idealism. Materialists hold that the world actually exists independently of mind and of our ideas and abstractions of it. Idealists think that in some sense ideas or concepts are primary, and that matter (if it really exists at all) is somehow an outgrowth of these ideas.

Mathematicians have always tended to lean towards idealism. “All is number,” said Pythagoras. Mathematicians have a difficult time resisting the view that the mathematical objects that they work with daily must have an existence every bit as “real” as that of tables and chairs. Or as the great English mathematician, G. H. Hardy, put it:

For me, and I suppose for most mathematicians, there is another reality [besides “physical reality” —JSH], which I will call “mathematical reality”.... I believe that mathematical reality lies outside us, that our function is to discover or observe it.1

But while it is true that ideas “exist”, and that patterns and abstractions can be created or made—or “discovered”, if you will—the existence of these “objects” is not the same kind of existence as matter (and energy). Instead the existence of ideas, patterns, abstractions, etc., derives from, and depends upon the existence of matter.

Physics has gotten steadily more mathematical, especially over the past 100 years. To a certain extent this has had the result of turning physicists into mathematicians, and it has also fostered the growth of mathematical idealism among physicists. (I will resist the temptation to also tie the tendencies towards idealism to the continued existence of capitalism, since I know that will offend some people!) Wolfgang Pauli, for example, was a fine mathematician himself, and remarked that “The steady progress of physics requires for its theoretical formulations a mathematics that gets continuously more advanced”.2 And yet Pauli also thought there might be something to astrology, and other screwball metaphysical theories.3 Of the biggest names in physics this past century, Einstein stands almost alone in his staunch materialist tendencies. (But even if Einstein was almost alone, I am quite happy to side with him!)

*       *       *

Since Ernest Rutherford’s experiments around 1910 we have known that atoms have physical volumes consisting mostly of the “electron cloud” (the region where the electrons in the atom may be found), and a very small nucleus where most of the atom’s mass is concentrated. But though the diameter of the nucleus is only about 1/100,000 of that of the atom as a whole, it still takes up a definite volume of space. The nucleus is composed of protons and neutrons, and it has been determined that each of these “nucleons” itself takes up a small but definite volume, and is comprised of 3 quarks within that volume.4

But isn’t there actual evidence that some particles are “point-like”? As Steve Bryson remarked, it is more correct to say that in the case of some particles (electron, muon, neutrinos, quarks, etc.), the only thing known about their size is that it must be below certain values. What evidence there is actually suggests these particles, though indeed very small even by the standard of the diameter of the proton, still must have some size.

Consider for example the scattering experiments which showed that there is a structure to each nucleon, and provided experimental evidence for the quark model. In these experiments beams of highly energetic electrons, muons and neutrinos have been used to bombard protons, and have shown that protons have small internal regions of higher than average mass and charge (which we assume to be the quarks). But the thing to notice here is that these experiments are showing a definite probability that these particles (which many consider to be “points”) will hit and bounce off of other particles (quarks—which these same people also consider to be “points”). If all these particles were actually “points” the probability of their hitting would be zero (since a point is mathematically zero percent of any cross-sectional area).

(Of course, one can quibble a bit here. It is true that the trajectories of the “bullets” and the positions of the target nucleons will be changed somewhat due to electromagnetic (or weak) forces between them, and one could argue that this allows “point” particles to collide. But if this were true the probabilities for collisions would drop off significantly as the bullet energies were increased, and I’ll bet you that this does not happen! (At least once a certain threshhold is reached.) I don’t know if experiments have been done to test this, however.)

That these scattering experiments do show a certain probability of collision between the “bullets” and the target quarks is in fact all that is necessary to estimate the size of the quark. (Remember that it was this same type of experiment that allowed Rutherford to determine the size of the nucleus of the atom.)

The new HERA (Hadron-Electron Ring Accelerator) scheduled for completion at the DESY lab in Hamburg in 1990 is supposed to have enough power to “probe the structure of the proton to a separation of 10-17 centimetres and check whether quarks look pointlike at this close distance.”5 So the dogma of quarks being mere points may soon fall. (Of course the option of jumping to the conclusion that the particles at the next level are “mere points” will still be open for die-hard idealists!)

Most of the discussion I have seen about the these “point-like” particles, however, has been focused on the electron. One recent book described the situation for the electron as follows:

In the 1930s, physicists devoted much attention to the question of the electron dimensions and tried to develop a theoretical formula for its diameter. They arrived at the formula

r = e2/mc2

where e and m are the charge and mass of the electron and c the speed of light. This gives a value of 3 X 10-13 centimetres for the radius. In some phenomena—such as the scattering of light by electrons in the atom—the observations are consistent with this radius, but according to more recent experimental data and the theoretical framework provided by QED, the electron is practically a point particle whose mass and charge are concentrated in a region smaller than 10-16 centimetres.6

It is clear that by “practically a point particle”, all that is really meant here is that the diameter is very small with respect to the volumes under discussion. This quotation also brings out the difficulties—and to some extent the arbitrariness—of the concept of a diameter of a particle.

But I don’t care how small the diameter is, or how arbitrary the electron’s boundaries are; I only wish to claim that the electron is not infinitely small, is not actually a “point”.

The diameter of the earth is also somewhat arbitrary. Should one include the atmosphere? How high up? In point of fact the diameter of any physical object—no matter how big or how small—is ultimately somewhat arbitrary. The reason is the same as before: a “diameter” is only a mathematical abstraction. For some purposes we may choose to consider the diameter of the earth sans atmosphere, and by averaging out the heights of mountains and valleys. For other purposes, we may wish to consider the diameter of the earth to be the average extent of the magnetosphere. The fact that we may now choose to treat the earth as having one diameter, and later choose to treat it as having another diameter, or even to treat it as having no diameter (as a point)—none of these theoretical conveniences shows that the earth is actually a point, or that it does not take up some volume of space.

The same goes for the electron.

*       *       *

“So then, Scott, you think that the electron is a solid little marble.” Not at all. It is invalid logic to assume that the only possibilities are mathematical points or solid marbles, as Abraham Pais does:

...the old quantum theory eased the transition toward an electron without spacial extension.

All the same the classical electron radius remained a concept hovering just barely off-stage, even though special relativity abhors finite-size rigid bodies. To face the alternative, a point electron, was abhorrent for other reasons: a zero radius for the electron means an infinite self-energy [the electrostatic energy of the electron at rest, which is defined as e squared over radius r, and thus tends to infinity as r goes to 0. —JSH]... So, I would think, one rather suffered a particle of finite size than a particle with infinite mass.7

Pais says later that this dilemma has been resolved. Since “ has definitely been established that the electron’s self-energy is not purely electromagnetic in origin”8 the difficulty of the electron’s self-energy going to infinity as the radius goes to zero supposedly no longer presents itself. (I don’t understand this specific claim well enough to take a position on it one way or the other. But isn’t it suspicious that the “resolution” of this problem is left so vague by Pais?) “...the classical answer (for the value of the total self-energy of the electron —JSH) returns if one makes the absurd assumption of an electron radius large compared with the Compton wavelength. Whatever the future may hold, the tiny marble with energy e2/r is gone forever.”9

But I am suggesting that there was never any real dilemma in the first place (perhaps only a trilemma!); there is at least one other possibility besides hard marbles and mathematical points. “Such as??” Such as the possibility that the electron itself has a structure, that it is made up of smaller particles, perhaps. There is no reason to assume that if it has a radius it must be some “solid” or homogenous substance throughout its volume.

“And what of those ‘particles’ that you suppose electrons might be made up of? Are they point particles, or do they have volumes too?” Whether the hypothesized structural components are the sorts of things which should be considered to be particles at all is not something which I can say with any assurance (though I suspect the answer will turn out to be that they are). In any case, no structural component can be a mathematical point (or other mathematical “object”).

“So! Your position leads to an infinite regress of particles, or structures within structures....” This is what Steve Bryson was getting at, I think, when he challenged me in our first discussion session to consider the implications of always dodging solid marbles and mathematical points by hypothesizing further internal structures.

One possibility is that there is an infinite regress of internal structures. I find this possibility unpalatable, but infinitely preferable to the idealist alternative of saying the world is ultimately made up of abstract mathematical ideas, like “points”—a view which I can make no sense of whatsoever. If no other possiblities prove out, then I will fall back on this hypothesis of an infinite regress of internal structures.

But I think there are other alternatives to mathematical points. Perhaps we will find out that at some level matter really does consist of homogenous chunks (“little marbles”, or perhaps “little strings”—but not one dimensional!) and that the old bug-a-boos, such as infinitely fast shock waves, either do not apply at this level, for some reason, or do not cause any insurmountable difficulties. (Perhaps at the quantum, or “sub-quantum”, level there is no way to even initiate shock waves within this homogenous matter.) It would be ironic if old Democritus proved to be correct after all!

Or perhaps there are other possibilities, options other than mathematical points, “little marbles”, or infinite complexity. I admit that the imagination is stressed at trying to discover such possibilities, but this is probably due mostly to our attempts to find everyday analogies for phenomena in the micro-world.

The one thing that is certain, as far as I am concerned, is this: except as a convenience within our theories that describe nature only partially and approximately, there is really no such thing as an infinitely small particle.

Pointless Poems  

Pointless Poem #1  
Some physicists imagine point particles;
Whose diameters are infinitely small.
I’d like to see in their articles
How that differs from not being at all!

Pointless Poem #2  
To physicists sick with idealism,
“Point-like” particles seem like true realism.
   But if they’re infinitely small,
   Can they be there at all?
To me these strange views are surrealism.

—Scott H.
   2/21/89 (with revisions on 3/5/89)


1   G. H. Hardy, A Mathematician’s Apology (Cambridge University Press, 1969), p. 123.

2   Quoted in Abraham Pais, Inward Bound (Oxford University Press, 1988 (1986)), p. 15.

3   For one reference to Pauli’s anti-scientific views, see Hendrik Casimir’s autobiography, Haphazard Reality: Half a Century of Science (NY: Harper & Row, 1983).

4   Yuval Ne’eman and Yoram Kirsh, The Particle Hunters (Cambridge University Press, 1986), pp. 215-6.

5   Ibid., p. 261.

6   Ibid., p. 61.

7   Abraham Pais, op. cit., pp. 371-2.

8   Ibid., p. 389.

9   Ibid., p. 384-5. Pais actually used the letter ‘a’ to represent the radius of the electron; I have changed this to ‘r’ for internal consistency within this essay.

— End —

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