At the same time as there is this hierarchy of masses with no clear relations, there are other hierarchies that have increadibly clear, but equally unaccountable, relations to each other. The simplest of these, and a paradigm for the others, is the case of electric charge, one of the ultimate mysteries of repeatability. In postponing our understanding of the identical nature of atoms by assuming the indentical nature at least of their building blocks, electrons and nuclei, we assumed a repeatability of several properties. One of these is the mass, which we have already mentioned, repeatable within a class of particles but not obviously related in one class to another. Another was the angular momentum of the constituents, though that may be less worrisome, because the fact that it has a definite value at all seems linked directly and without intermediate steps to the existence of h/, so we can hope that that will have one generic answer that will apply in the same way to all kinds of particles. But electrons also have charge, and in order to explain the fact the ordinary matter does not have ``electric charge effects'' on large scales, we had to expect that nuclei also had charge of an opposite sign.

But unlike angular momentum, there is no obvious relation of electric charge to any of the fundamental constants. Therefore there is no obvious reason to expect that, like the angular momentum, it will come from a single explanation for all kinds of particles. We would more nearly expect it to be like mass, which having no such obvious relation, takes all sorts of values, not remotely the same for electrons as for protons.

But that is not at all what we find. The fact that even
exceedingly large collections of atoms, like the whole earth,
have no overall effects, while the relation of the number of
electrons to nuclei in all the atoms we observe seems reliably
the same, indicates that this balancing of charge between
electrons and nuclei is exceedingy accurate. Certainly, to a
point one can say that the whole reason the nucleus is the
repository for the chemical ``identity'' of an atom is that it
attracts however many electrons are necessary to cancel the
attractive effects of its own charge and thus nuetralize it,
after which point even if there were a small imbalance, an
additional electron would create a worse one. But because both
electrons and all of the building blocks of nucleons never come
in parts, we not only have to require that the gross *number* of the electrons and nucleons cancel, but that within
that number the *exact amounts of charges on each of the
electrons and each of the protons* (the charged building block
of the nucleus) *cancels to exceedingly high accuracy*. And
it does.

But this is stranger than fiction. Electrons are not themselves
obviously ubilt up from anything. Protons and neutrons are made
of quarks and gluons, which themselves are different from
electrons and light, and are also not obviously made of anything
smaller. Furthermore, the combinations of quarks in nuetrons
are different from those in protons, in just such a way that
while the charges from the quarks in the protons add up to
exactly equal the charge on the electron, the charges in the
nuetrons conspire among themselves to cancel to zero net effect.
Each of these claims has been independently tested in
experiments, which is why they are mentioned here. These kinds
of cancellations seem to be a conspiracy of repeatability of the
most accurate and bewildering kind. And to happen, they require
that physical characteristics of two very different things, on
one hand a collection of quarks and on the other another
collection of quarks or an electron, either vanish exactly to
zero or add up to the same thing. In particular in the case of
quarks, this can be shown (with a lot more work and
sophisticated reasoning than is possible to present here) to
require that the electric charges on different kinds of quarks
be exactly 1/3 or 2/3 as large as those on the electron. In
other words, the two amounts of a particular physical
characteristic are observed to have a very particular *dimensionless ratio*.

In some sense, we are bewildered by two expressions of the same
thing. The fact that electric charges are indeed so repeatable,
and the fact that this repeatability comes in exactly the right
amounts to make electrons comparable to certain collections of
quarks, indicates a conspiracy of dimensionless ratios like the
1/3, 2/3 of a particular quantity, that cannot be predicted from
dimensional analysis, because it involves no dimensions. It is
a ratio. At the same time, we see that there are many
elementary particles, like quarks and electrons, which can have
different masses, but that there is only one Planck mass
indicated by the fundamental dimensionful constants that they
all have in common, and the confusion in this case is that there
is no obvious reason for those masses to take one value as
opposed to another *precisely because there is no
dimensionless ratio that we can find* that would predict the
mass of one as the correct fraction of the masses of others or
of the Planck mass.

Thus, assuming that this pattern will continue, and that we will find no more ``kinds of dimension'', and also no new dimensionful constants like h/, G and c, the process of refining our description of physics within its current language, with h/, G and c seeming to define the endpoint or culmination of that language, will consist of discovering either what is responsible for such dimensionless ratios (in the cases like charge equivalence where we know them) or discovering what is responsible for them and then also predicting the values they must take as a result (in cases like the hierarchy problem in which we don't). There are more dimensionless constants in the standard model than those mentiond, for which there is no explanation, but all of them, whether dimensionless in their present description, or represented by dimensionful quantities like an elementary particle mass which is then some fraction of the Planck mass, whether already simply related to integers like the quark electric charges or not obviously related to anything known, represent expressions of essentially this one need.

In the relatively simple and also relatively rare cases like the
quark electric charge, in which the dimensionless ratio is
suspiciously related to integers, as is *the number of
species of quark* it can be hoped that some kind of *counting argument* can be used to relate the number of kinds of
such particles to the amounts like these of the properties that
they possess. But such arguments represent the most extended
forefronts of research at this time, on the border with
as-yet-unrewarded speculations, and are too much to even
approach here.

The other cases, like the hierarchy problem of masses, seem much more confusing and obscure, because nothing about the dimensionless ratios is even recognizeable as special, unlike the ``obvious'' fractions like 1/3 and 2/3. Yet there is one case of a special dimensionless number about which we understand something, which may serve to indicate at least what kinds of patterns dimensionless, but repeatable, numbers can represent.

Thu Aug 31 12:01:42 CDT 1995