Our progress so far in understanding the repeatability we see in nature has consisted of beating back our ignorance from one size scale to the next smaller one, when along the way we have been able to figure out the structure that relates them as ``a whole'' to ``its parts''. We started with the characteristics of chemical reactions, and reduced the problem of their repeatability to the problem of the repeatability of atoms, by along the way cataloguing which properties a single atom could have, to what degrees they were posessed by different kinds of atoms, and how the presence of these properties as characteristics of the atoms were sufficient to determine all of what was observed in chemistry. (The details of those properties have not been filled in here, because that is properly the domain of chemistry and is widely presented in many places. The point needed here is that this is how the knowledge is organized.)
Then we attacked the repeatability of atoms, and upon finding that the atoms themselves could be seen as electrons and nucleons held together according to some structure, if we were willing to assume repeatability of the electrons and nucleons, we could explain the repeatability of the atoms if we could account for and describe the structure. The details of how this is done are the subject of the chapters on the quantum theory, and will be addressed there. In the process of trying to account for this structure, we happened upon h/, and because G and c had already been recognized, we constructed the very curious Planck scales of length, mass and time, though we did not see how to use them for anything.
This process of refinement has continued to smaller and smaller scales through several more levels. Without making reference to those details, because again the knowledge is organized in much the same way, we can say that it has been found that the nuclei of atoms, which are in fact the whole repository for the ``identity'' of the atoms (since the electrons around all atoms are the same), are themselves made of collections of only two kinds of smaller building blocks, protons and nuetrons. With this step almost all of nuclear chemistry was reduced, in a way remarkably like that of ``big atomic chemistry'' to a set of rules for combining just two basic nuclear ``elements'', the proton and neutron, of which all particles of each kind are again found to be very much identical to each other. (They also have spin). No similar such sub-decomposition has been found for electrons, so they are still described as if they are not ``built of anything smaller''.
But only ``almost all'' of nuclear chemistry was simple to reduce once the existence of protons and nuetrons was known, and the small amount that was left led to a pandora's box of confusions and new participants in the story. The biggest outstanding problem, even observable on people-sized scales, that was not addressed in the proton-nuetron description was that of natural radioactivity. How it occurred, what it was composed of, or how it related to nuclear structure, were totally unaddressed. In the process of accounting for this set of phenomena, as well as making refinements to the predictions of nuclear properties themselves, it was discovered that there are at least hundreds of other kinds of ``particles'', at least as basic as the proton and nuetron, which can exist and are related to the processes that go on in nuclei, and which can also have important effects on electrons, which is how it has been demonstrated that the spin properties of electrons are not linked to its interaction with light, because they are important in these other interactions as well. But aside from the fairly rare occurrence of natural radioactivity, most of these other particles require exceedingly violent processes to be created, and do not last long once they are, so they are not part of the usual stable, slow world that is familiar to us as animals.
The history of how this bewildering array of new particles was
understood is a fascinating account that spans five decades now,
but unfortunately it is too large and too much tangential to the
main development to even list here (not to mention that I also
don't know most of it). (See Weinberg, fundamental particles,
as reference.) Our important notes from that part of history
are that, in spite of all that was discovered and all the
explanation that was invented to account for it, no new
dimensionful constants like h/, G and c have been
discovered. Furthermore, it has been possible to encapsulate
every phenomenon that has ever been measured about these
particles into a single description, now called the
Standard Model
of elementary particle physics, which has not yet been found to make any wrong prediction about
anything. Beyond the level of structure of protons and
nuetrons and the other particles, previously unsuspected, that
can interact with them, only one more level of structure has
been definitively found. At this level the building blocks,
which constitute protons and nuetrons and many of the other
particles, are a small set of electron-like objects called
quarks, and another small set of a kind of particles called
gluons, which are very much like light in some ways, but whose
collective effects differ greatly. The quarks, like the
electrons, have not been found to be made of anything smaller,
and the same is true for the gluons in a way that also applies
to light. Additionally, there are some other electron-like
particles which are not directly ``present'' as building blocks
of any other particles, but which are necessary to account for
the ways in which they can interact, and which can be seen
independently. These are the neutrinos, which were first
proposed by W Pauli (when?, n.b. how early) long before many of
the confusing particles were even discovered. And, together
with the neutrinos are one last kind of particle, again much
like light and gluons, but different in some very important
respects. The neutrinos and these last kinds of light, called
``weak gauge particles'', while they are not stable building
blocks of anything else we see, are necessary all the way up to
the easily-observable ``big'' world, to account for effects like
radioactivity and the very evenly regulated burning of the sun.
There are hints at additional levels of structure beyond these observed particles, though those have so far been beyond our ability even to infer from anything that we can measure, so many of the details of their proper description remain unknown, though some can be guessed in broad terms.
This synopsis is not intended either as a historical recount or as something that can usefully be ``understood'' as a tool. It is included here to give some sense of how this process of beating back our ignorance by one scale of smallness after another has proceeded beyond the level at which Planck's constant was discovered, and to point out that a remarkably diverse array of phenomena (everything that has ever been measured) has been reduced to rules that can be described by a very simple language that requires the existence and repeatability of very few ``fundamental building blocks'', in this case the elementary particles. It also serves as a context within which to remark that, in spite of how many more levels of structure have been discovered and successfully described, no new dimensionful constants have been found, but also that this refinement of scales has been carried only to a level of smallness far short of the Planck scales. Thus, even within the context of the current language, there are strong indications that there is a large range of scale left to be understood, within which there may be much new structure, even if we assume that all these structures will make use of only the same three fundamental dimensionful constants.
But in this process, we have also unconvered many instances of repeatability that have not been understood basically at all. If people had had the ability to make exceedingly fine measurements of mass, they could have noticed even before the atomic theory that a certain chemical always comes in multiples of a certain mass. The chemical can be subdivided into smaller and smaller portions, but if one tries to divide it into a portion smaller than the basic unit of mass of which it always has some multiple, it becomes a different chemical. Had this been possible, it could have led to a formulation of a kind of atomic theory, as it was recognized that this fundamental unit of mass was the mass of one molecule of whatever is the chemical, and that to obtain a smaller unit of mass than this, that molecule has to be broken into pieces, at which point it ceases to be the same chemical, but instead two pieces of two different ``constituent'' chemicals.
These constituents, though, would also have been found to have
certain intrinsic mass-multiples, which could ultimately be
traced to the masses of the individual atoms that made them up.
A very important way in which this did happen, with the advent
of analytical chemistry, was with the introduction of the notion
of a ``mole'' by Avogadro (actually very early) as an amount of
a chemical that contains a standard number of its basic
molecules (in this case, a very large number, about ). When the masses of moles of the elements are
compared, as was done by Mendeleev (?), it was found that indeed
these are nearly multiples of the mass of a mole of the simplest
one of them, Hydrogen. This has since come to be understood as
the first clue to the existence and repeatability of protons and
nuetrons, because the chemically simplest element, hydrogen, is
the one made of only one proton, and all of the other elements
are made of some definite number of protons and nuetrons, which
themselves are never subdivided in atoms.
Up to this point in the description (historically out of sequence) the reduction is fine. Everything known is expressible as made of some definite number of basic ``nuclear building blocks''. But beyond this point, the same kind of reduction fails completely. Electrons have some definite mass which can be measured to high accuracy, as do protons and nuetrons. These three masses have no obvious precise relation to each other. The proton and nuetron masses are comparable (differing by a few percent), but they are measurably not the same, and they have no obvious relation at all to the mass of the electron. This problem becomes worse when the other newly discovered particles are considered.
The relation of the proton to the nuetron mass, as well as the relation of these to the masses of some of the other newly found particles, turns out to be partly solvable in the following sense. The proton and nuetron and the other particles to which this applies are all those made of the quarks and gluons, which are themselves species of identical particles. While the quarks themselves have some mass, which can be inferred from much more sophisticated measurements (the gluons, like light, have none), the relation of those masses to the actual masses of the ``large'' particles like protons and nuetrons turns out to be somewhat like the relation of the spin of whole atoms to the spin of their parts. Atoms could have many different kinds of angular momentum, including amounts greater than the angular momenta of their parts individually, because the angular momenta turned out also to be characteristics of the way the parts were combined. (This is not too surprising. Referring to one of the few parts of the Rutherford picture that holds true properly, the solar system can have more angular momentum as a whole than the sun and the planets do as a collection of intrinsic individual properties, because of the way the planets are themselves flying around the sun, like disconnected pieces of a spinning top. The same kind of relation applies to the structure of atoms, though qualitatively differently because of the relatively great scale of h/ to atomic processes.) Similarly, it turns out that most of the mass of protons and nuetrons comes not from the quarks themselves as intrinsically massive objects, but from the energy inherent in the way they are combined (for reasons that we will understand when we work through special relativity). While there are difficulties with special aspects of the complexity of this structure that make the numbers hard to compute, those are slowly being overcome, and the predictions strongly suggest that all of the mass of the proton can be accounted from the masses of the quarks and the proportionality constants that describe the interaction itself. The same is true for the other particles in this class.
But this represents only a partial solution. The situation is not as bad as could be imagined, with all of the masses being totally ad hoc and unpredictable, but it is also far from being understood. The masses of the quarks, as well as the masses of the electrons and in some instances of the nuetrinos (which, like anything about nuetrinos, tend to be difficult to measure because they are so hard to catch), have no obvious relation to each other. Also, other numbers which are necessary to describe the interactions, though once measured they correctly relate things like the masses of protons and nuetrons to each other and to the quark masses, have no obvious origin. Moreover, if they have any relation to the Planck mass, it is hopelessly out of reach to measure meaningfully.
For other reasons that we will develop in detail between the chapters on relativity and quantum mechanics, there is an inverse relation between the characteristic mass (or energy) at which something can happen, and the size scale at which it happens. Thus a certain part of our reasoning about building-block descriptions has to be reversed at some point. Down to the levels of chemistry and even nuclear chemistry, physically ``big'' things were made of physically ``smaller'' parts, and the big things also weighed more than their smaller parts, because the masses of the whole are the sum of the masses of the parts. But throughout this realm, that was true because the masses of the wholes containd a negligible contribution from the energy inherent in the way the parts were arranged. We saw the beginning of the reversal of this effect at the level of the proton. For the first time with the proton, knowing the masses of the quarks ``inside'' was not enough to even approximately predict the mass of the whole, because most of that mass came from the energy of the way the parts are arranged and held together. It turns out that beyond the level of the proton, a new problem sets in. If we want to examine parts much smaller than the physical size of the proton, the energies (or equivalently, the masses, for reasons to be discussed in later chapters) of these objects become larger than the mass of the proton itself, and it happens that such objects are not ``required'' to stay in existence. They can be created, but when they are created they quickly vanish (the process is called ``decay''), leaving that energy to be distributed more diffusely in larger bundles like protons, or in light, electrons, nuetrinos, and so forth. That is why, at precisely the level at which the building block estimates for masses became unreliable, we also stopped noticing even the existence of any of the new particles except in those instances when they were explicitly created and watched for their properties before they could disappear again.
All of this is again somewhat outside the main point, but it is needed in order that it make sense to say that, while the Planck size is increadibly smaller than anything we have been able to explore experimentally, the Planck mass is increadibly larger than anything we have been able to use to create a single particle. Thus, in addition to the fact that the elementary particles we know have masses with no obvious relation to each other, if they have any particular relation to the Planck mass, it is for now simply some increadibly small fractional number to which we can assign no particular significance.
To the extent that the relations of these masses to each other,
or to the only thing that they obviously have in common, the
fundamental constants h/, G and c, is obscure, the success of
our building-block description is more or less stalled. This
confusion is known as the hierarchy problem,
because it
refers to the fact that the masses of those particles which seem
for now to be elementary (i.e., not clearly divisible into or
resulting from anything smaller), even though we can measure
them, define a hierarchy of scales whose significance we do not
understand and which we cannot predict. Accounts for the
relations in this hierarchy make up some of the missing steps
between the parts of the description of fundamental processes
that we understand and the distant Planck scales that seem to be
their ultimate source.