In my previous posting, we covered the symmetry group U(1) and the electromagnetic force. Today, in Part VI of my symmetry series, we'll take a "watered-down" look at more elaborate symmetry groups and their physical applications. Of particular interest today will be the group SU(3), which is associated with the strong force (also known as Quantum Chromodynamics, QCD, or the quark model). The weak force is associated with the symmetry group SU(2), but I'll leave it for interested readers to consult outside sources to study this connection.
The key player in linking SU(3) symmetry to quark-based particles (and in fact, co-inventing the quark model in the first place) is Murray Gell-Mann, who in several weeks will celebrate his 80th birthday. Gell-Mann's accomplishments in this area occurred in the early 1960s. As noted in this Wikipedia entry, "Throughout the 1950s and 1960s, a bewildering variety of particles were found in scattering experiments. This was referred to as the 'particle zoo'." Gell-Mann's contribution was in developing a classification scheme -- analogous to the periodic table of the elements in chemistry -- that could bring order to the proliferation of newly discovered particles (baryons and mesons).
Two of my favorite physics writers, Dan Hooper and Robert Oerter, discuss Gell-Mann's process of discovery. According to Hooper's book Nature's Blueprint, "In particular, [Gell-Mann] knew that the symmetries behind the theory of quantum electrodynamics... could be described by a mathematical structure called the U(1) symmetry group. Gell-Mann began to wonder whether there might be another symmetry group that described the behavior of the mesons and baryons" (pp. 69-70).
The core concept of symmetry, of course, is that one can perform operations on an object and leave some essential property unchanged. As this set of lecture notes from the University of Utah describes, SU(3) fulfills this requirement in some sort of technical way that is above my mathematical knowledge ("As a symmetry group, SU(3) leaves the inner product of C3 invariant").
Eventually, Gell-Mann learned about the SU(3) groups and began comparing the organizational structures that could be generated from SU(3) with the properties of the newly discovered particles. The key SU(3) structures could either accommodate eight particles, referred to as an "octet" (a hexagon, which has six sides, plus two data points in the center) or ten particles, referred to as a "decuplet" (a tiered, triangular arrangement, with one row have one particle, the next having two, the next having three, and the last having four).
Several websites illustrate octet and decuplet diagrams, such as this one from the Encyclopedia Brittanica, this one from the Wikipedia (which explains Gell-Mann's borrowing of Buddhist terminology to call the octet model the "Eightfold Way") and this one from Florida State University (which makes a humorous distinction between the "quark" and "cork" models). Physicists can be real characters with their terminology, especially Gell-Mann, who also coined the term for "strange" quarks.
The Encyclopedia Brittanica provides a concise summary of how particles can be classified:
With the introduction of strangeness, physicists had several properties with which they could label the various subatomic particles. In particular, values of mass, electric charge, spin, isospin, and strangeness gave physicists a means of classifying the strongly interacting particles -- or hadrons -- and of establishing a hierarchy of relationships between them. In 1962 Gell-Mann and Yuval Neʾeman, an Israeli scientist, independently showed that a particular type of mathematical symmetry provides the kind of grouping of hadrons that is observed in nature.
The location of the data points on these octet and decuplet graphs are determined by plotting properties of the particles along three axes (the "3" in SU(3) stands for the number of dimensions). However, as Tommaso Dorigo explains, depictions on just two dimensions -- electric charge and strangeness -- can suffice (at least for laypersons).
One, labeled by the letter 'Q', describes the electric charge of the states, and goes from -1 to +2, increasing toward the top right corner. The second, labeled by the letter 'S', describes the 'strangeness' of the states. S is the number of strange quarks the baryons contain, and it increases instead as one moves down. Forget the third axis, it is of no use for you.
Instead of electric charge being plotted against strangeness, another particle property known as isospin sometimes is plotted against strangeness. Isospin is a property that, among other aspects, can distinguish a proton from a neutron (Oerter, The Theory of Almost Everything, pp. 144-146).
Dorigo also expresses the symmetry aspect of a decuplet triangle, referring to one on his website as:
...an example of the many possible representations of the symmetry group SU(3), in this case applied to describe the symmetries of quark flavors. You exchange a quark flavor with another by jumping along one of the three directions along the sides of the triangle, and you obtain a new baryon, whose properties are somehow connected to those of the former one.
To recap the key ideas to this point, the mathematical rules of SU(3) symmetry specify where data-points on octet and decuplet diagrams must fall. Gell-Mann found that plotting particles' locations along the dimensions of isospin and strangeness produced visual configurations analogous to those of SU(3). In fact, where there were "empty" locations on SU(3) structures (i.e., data-points without an associated known particle), Gell-Mann predicted that new particles with the necessary characteristics would be discovered. And, lo and behold, they were discovered! (See Oerter, pp. 160-164.)
The Eightfold Way led further to the ultimately successful model of fractionally charged quarks. Oerter explains that Gell-Mann "began with the observation that, although all the known heavy particles could be arranged in SU(3) multiplets, the simplest possible multiplet, a three-particle triplet, was nowhere to be found in the subatomic zoo... Perhaps all the particles in the subatomic zoo were built out of these three fundamental particles. He decided to call the quirky little things quarks..." (164-165). In order for three quarks to form a larger particle, such as a proton (two up quarks and a down) or neutron (two downs and an up), quarks needed to have fractional electromagnetic charges, a controversial idea at the time. The later idea of quark confinement explains why individual, free quarks have not been detected.
Finally, putting together what we know about the electromagnetic, strong, and weak forces' respective Lie groups, we arrive at the conclusion that, "The standard model is based on a Lie group called SU(3) X SU(2) X U(1)," according to this University of Montana document.
In his book Why Beauty is Truth, Ian Stewart gives a crude analogy to the melange of terms comprising the standard model of particle physics:
[I]t just lumps all three gauge groups together as SU(3) X SU(2) X U(1). This construction is simple and straightforward but not terribly elegant, and it makes the standard model resemble something built out of chewing gum and string.
Suppose you own a golf ball, a button, and a toothpick. The golf ball has spherical symmetry SO(3), the button has circular symmetry SO(2), and the toothpick has (say) just a single reflectional symmetry O(1). Can you find some unified object that has all three types of symmetry? Yes, you can: put all three into a paper bag (p. 239).
As Stewart discusses, some Grand Unified Theories of the electromagnetic, strong, and weak forces have attempted to achieve more streamlined configurations using Lie groups such as SU(5) and SO(10).
The Lie groups that have been matched to the physical forces are not the only ones, however. The characterization of one of the "exceptional" Lie groups, E8, was reported in 2007 in the New York Times. Some of the mindboggling properties of the E8 object and how mathematicians studied it are as follows (quoting from the Times article):
It required the work of "[e]ighteen mathematicians spen[ding] four years..." culminating in "77 hours of supercomputer computation..."
E8 "describes the symmetries of a 57-dimensional object that can in essence be rotated in 248 ways without changing its appearance."
"To understand using E8 in all its possibilities requires calculation of 200 billion numbers."
"'You can’t really picture it,' Brian Conrey, executive director of the American Institute of Mathematics, said of E8."
If that last claim isn't the understatement of all time, I don't know what is!