Analysis and Synthesis I: What Matters for Matter

Query 31 of Isaac Newton’s Opticks was his last word in science. It begins:

After a lengthy sketch of how this concept might lead to explanations of various phenomena in what we would now call chemistry and condensed matter physics, Query 31 concludes with a statement of methodological faith:

“Reductionism” is modern jargon for the program that Newton advocated in Query 31. But this is an ugly and misleading term, and it has become almost a term of abuse in fashionable intellectual circles. So in this series of three columns, I’ll avoid the R-word and stick with the “Analysis and Synthesis.” The phrase has the virtue of emphasizing that both procedures, breaking down and building back up, form essential elements in scientific understanding—and that such understanding is therefore not reduction, but rather enrichment.

Whatever you call it, as a tool for understanding the physical world, the method of analysis and synthesis has been astoundingly successful. In the description of matter, it has worked out more or less along the lines Newton suggested; in cosmology, along lines quite different from anything Newton imagined. (As also appears in Query 31, Newton believed that the world was fashioned by God, whose active intervention is required for its proper functioning.) In both cases, analysis has brought us to foundational models that come close to achieving perfection.

The history of physics emphasizes that synthesis is a challenging and deeply creative activity in its own right. It is more open-ended, however, and less easy to summarize, so I’ll only be able to mention a few illustrative high points.

We shall also see that, in the course of its triumphal advance, ironically the method of analysis and synthesis has itself discovered profound, sharply defined limits to its explanatory power.

Electronic levels of description

It is instructive to work up to our most complete analysis of matter through some intermediate models that are extremely important and useful in their own right.

The first model builds a world in which the only active players are non-relativistic electrons obeying the laws of quantum mechanics. To flesh out this world, we also allow highly localized, static sources of charge, concentrated in positive multiples of |e|, where e is the electron charge. Those sources provide a schematic, “black-box” description of atomic nuclei. This is the world captured in Schrödinger’s original wave equation, using Coulomb’s law for the forces. With hindsight, we realize that this world is an approximation to ours, in which atomic nuclei are regarded as being infinitely heavy compared to electrons (instead of merely thousands of times heavier), and light as moving infinitely faster (instead of merely a hundred times faster).

This model is an extremely economical construction. An objective measure of its parsimony is how few parameters it involves. Superficially, there appear to be three: the charge unit e, Planck’s constant ħ, and the electron mass m _e. But we can swap those parameters for a system of units: length (ħ ²/m _e e ²), time (ħ ³/m _e e ⁴), and mass (m _e). When we express physical results using these units, the original parameters no longer appear at all. So our model contains no genuine parameters whatsoever. Its prediction for any dimensionless quantity is unambiguously fixed by its conceptual structure. Analysis can go no further.

Yet this Spartan framework supports an extremely rich and complex world construction. We can use it to compute what kinds of molecules should exist, and what their shapes are, by identifying local minima of the energy as a function of positions of source charges. (Strictly speaking, we must also specify some symmetry rules to take into account the quantum statistics of electrons.) This framework gives us a parameter-free foundation for the vast subject of structural chemistry.

A second model refines the first by incorporating special relativity, that is, by drawing out the consequences of the finite speed of light, c < ∞. We pass from the Schrödinger to the Dirac equation, and from Coulomb’s law to the Maxwell equations, and thereby introduce real and virtual photons. By passing from Schrödinger-Coulomb to Dirac-Maxwell, we arrive at quantum electrodynamics (QED) with sources. A pure number now enters the game, namely the fine structure constant α = e ² /4πħc. The refined model is in that respect less perfect, but in compensation, it is both more accurate and much more comprehensive. It now includes the dynamical effects of electron spin, the Lamb shift, radiation phenomena, and much more.

These two wondrous models have a big shortcoming, however: The molecules can’t move. Reactions, diffusion, and thermal phenomena are completely missing, as are vibrational and rotational spectra. To allow motion, we must be less schematic about the nuclei. In doing so, however, we open ourselves up to many more input parameters. At a minimum we need the masses of the different nuclei and isotopes, and for accuracy in details we need their spins, magnetic moments, and other properties. Literally hundreds of nuclear parameters are required.

A practical approach is simply to take all the needed quantities from experiment. Of course, with that step we acknowledge that there are many measurements whose results we do not attempt to predict. It is a major compromise in analysis—a strategic retreat. But all is not lost, to say the least, because synthesis can do wonders with this material. Indeed our third model, with its strategic retreat, provides a comprehensive foundation for quantum chemistry and condensed matter physics. Within that scope one can contemplate hundreds of thousands—if not an infinite number—of significant measurements, so a working model containing hundreds of parameters can remain extremely useful and predictive. It is precisely this world construction that Paul Dirac famously described as containing “all of chemistry and most of physics.”

Quark-gluon levels of description

Thanks to advances made over the past 30 years, we now can carry the analysis of matter significantly further, and reconquer most of the lost ground. Now we can base our description of nuclei on a theory whose economy and elegance rivals that of QED. Here, of course, I speak of quantum chromodynamics (QCD).

The fourth model combines the Dirac-Maxwell theory of electrons and photons (as in the second model) with a truncated version of QCD, which I call QCD Lite. QCD Lite builds atomic nuclei using as ingredients only massless particles: the color gluon and two kinds of quarks, up u and down d. There are good reasons to believe that QCD Lite is capable of reproducing the important nuclear parameters at least crudely, perhaps at the 10–20% level. (I’ll explain in a moment why I’ve stated this claim so gingerly.) That is astonishing, because QCD Lite is a parameter-free theory! Indeed, its equations can be formulated using just ħ, c, and a mass Λ _QCD . Those parameters can be exchanged for a system of units, similar to what we did in our first model, the Schrödinger-Coulomb model. (For more on QCD Lite and the origin of mass, see my Reference Frame column “Mass without Mass I: Most of Matter,” Physics Today, November 1999, page 11 .) The fifth and final model is a simple refinement of the fourth where we introduce nonzero masses m _u and m _d for the quarks. This modification produces a much more accurate representation of reality, at some cost in economy.

With this fifth model, the modern analysis of terrestrial matter is essentially complete. A fundamental theory that offers an extremely complete and accurate set of equations governing the structure and behavior of ordinary matter in ordinary conditions with a very liberal definition of “ordinary” requires the parameters ħ, e, m _e, c, Λ _QCD , m _u, and m _d. Three can be traded in for units, and two (m _u and m _d) play a relatively minor role. Thus we have a rough analysis of matter down to two parameters, and an accurate one with four. Two more are required for astrophysics, as I’ll discuss in the next installment.

Here, a confession: Our faith in QCD does not stem from our ability to use it to calculate nuclear parameters—in fact, nobody knows a practical way to do such calculations. The best quantitative tests of the theory occur in an entirely different domain, in ultra-high energy experiments, in which the underlying quark and gluon degrees of freedom and their couplings are exhibited clearly. Our faith relies, for empirical support, on the successful outcome of those tests, together with encouraging but as yet fairly crude results from massive numerical calculations of proton structure and the hadron spectrum. The successes are highly leveraged, because the theory that describes them is extremely rigid. It will not bend without breaking. If we want to stay consistent with requirements of quantum mechanics and special relativity, the possibilities for couplings among quarks and gluons are very restricted. Only a few parameters can possibly appear in QCD—just the quark masses and Λ _QCD . Nothing else is permitted.

So we have rigidly defined equations, tested rigorously and quantitatively at high energy. We know that solving the equations will yield specific values for the nuclear parameters. Thus we are led to believe these values will be accurate even though, at present, we can only carry out the necessary calculations in the simplest cases. I think the expectation is perfectly reasonable, not so different from our faith that QED applies to complicated chemistry. Unfortunately, however, the “reduction” we offer to nuclear physicists, or for that matter to chemists, is not of much use to them in practice. It is a big standing challenge to design better algorithms.

Toolkits and artworks

To include everything we know about matter, the tidy framework of the fifth model must be expanded considerably. A natural next step is to the full-blown standard model. Within that framework, we can accommodate an astonishing variety of phenomena that have been discovered in cosmic rays and at accelerators. On the downside, this step opens us up to many more parameters, about two dozen, mostly describing the masses and weak mixing angles of various elusive or highly unstable particles. It is reminiscent of the earlier step from the second model to the third model. But while that earlier step has now been assimilated into a beautiful and economical theory, as I just discussed, there is at present nothing comparable “Beyond the Standard Model.” At the frontier of analysis, a considerable tension exists between elegance and accuracy.

Tension also is apparent between ease of use and completeness. A profound joke is that one can measure the progress of physics by the problems that can’t be solved. In Newtonian gravity, the three-body problem is difficult, but the two-body problem can be done exactly; in general relativity, two bodies are difficult but one body can be done exactly; in quantum gravity, the vacuum is intractable.

Faced with such choices, wisdom opts for “all of the above.” Different levels of description have different virtues and can be used for different purposes. The Schrödinger-Coulomb theory, for all its limitations, is an amazing work of art. If you doubt it, let me urge you to contemplate the wave-mechanical hydrogen atom, using Dean Dauger’s remarkable shareware “Atom in a Box” ¹ —while remembering that the software animation directly represents a deep aspect of reality. Then try to imagine a carbon atom or a water molecule. And the third model, for all its compromises, is an amazing toolkit. It is used to design new generations of lasers and microelectronic devices, among many other applications.

Real understanding is a much subtler and suppler affair than doing analysis at the finest possible resolution. A “Theory of Everything,” were it constructed, wouldn’t be.

Dirac is sometimes invoked as a mythological patron saint of purity and reductionism. The flesh-and-blood Dirac was trained as an electrical engineer, and in his scientific autobiography ² he pays tribute to the value of that training: