The quantum microscopic world view:

I promised an explanation of quantum theory in two steps. In the first step Schrödinger’s wave is something that moves in familiar threedimensional space. You could almost believe, as Schrödinger himself did at first, that there is something real waving that corresponds to the fundamental stuff of matter (electrons, quarks, photons, . . . ), something like Maxwell’s dynamical field of electromagnetism that in the old theory carries what we perceive as light. But alas, the Schrödinger wave function is not even an ordinary number. Its symbol stands for a whole array of numbers, including an amplitude at each point whose square is the probability that a detector will click there, and components in a set of “internal dimensions” that are new features of Nature. Worse, the Schrödinger wave for two detectors moves in six spatial dimensions, three to locate each detector. Adding more detectors requires more dimensions. Whatever it is, the Schrödinger wave is not a “real” field like Maxwell’s.

By 1926, when quantum mechanics finally emerged as a coherent theory, the evidence for an atomic structure in Nature was overwhelming. Today, scanning microscopes allow us to “see” chemical atoms one-by-one. They are real. Since quantum theory makes no reference at all to particles, why then do we have all these particulate atoms, ions, and nuclei? How do we account for the success of those stick-and-ball models of molecules, or crystals, or Crick and Watson’s double-helixed DNA? The answer is that atomic nuclei are made of stuff that interacts with a force law that gives its assemblages an intrinsic tiny size. The parts are sizeless quantum entities, but the things they make can behave – most of the time – like microscopic particles. The force law and its properties are consequences of the fundamental symmetries of the Standard Model which I will now endeavor to describe.

Our large-scale classical view of Nature assumes we can know the current state of things that can be used as an initial condition together with the laws of motion to predict future states (Chapter 2). Newton, Gauss, and others invented procedures for finding the orbits of planets given a few observations of their actual positions in the sky. Since Schrödinger’s equation is a law of motion for the wave function, we might attempt similar applications in quantum theory. But how can we know the current wave function of a system? All we can know empirically of Nature, in the quantum view, is whether a detector clicks. This limitation leads to major differences in strategy for using quantum vs. classical mechanics. Keep in mind that the future of one individual atom out of the trillions of trillions in human-sized matter is rarely significant even when the concept of “individual atom” makes sense. By contrast, predicting the future path of a particular near-Earth asteroid may be urgently important. Quantum mechanics is useful despite its ambiguities because we require very different information in the macroscopic and microscopic regimes.

Bernhard Riemann speculates on the empirical nature of geometry

Our story begins with curiosity about the precise shape of the Earth, which is interesting for the direct evidence it could give that Earth rotates in space. Newton pointed out that the pull of gravity (toward a center) against the centrifugal force of a spinning Earth (away from the axis of rotation) will swell the Earth at its equator, distorting it from a perfect sphere. The effect is small, and at the time no one could look at Earth from space to check its shape.

Digression on the quality of knowledge in a universe of atoms

Notice how cerebral all these advances were of Einstein’s. With little data and few experiments to guide him, he plumbed the deep logic of the formulas that summarized so much of Nature then known to science. At about the same time, however, triggered by technologies whose power had mounted throughout the nineteenth century, a Pandora’s box of new physical phenomena sprang open for which the old formulas failed to account. Subsequent progress in theory required a steady stream of reports from the laboratory to weed out conceptual dead ends and suggest new directions. The new experiments probed matter’s fine structure and its behavior at extremes of temperature and pressure. New theories to explain them conceived matter as a concretion of particles held together by electromagnetic forces. The component particles were supposed to be more or less like those of Democritus, but bound electrically to produce composite atoms corresponding to each of the chemical elements within Mendeleev’s famous periodic table.

The Standard Model of matter and its space-time and quantum foundations emerged during the twentieth century in response to a mounting accumulation of empirical evidence. Crucial experiments led to ideas that were refined by ingenious men and women into a logical pattern. Nature, of course, carries this pattern in her bones. Astronomical observations during the past half-century have convinced most scientists that the Nature we can see today is the result of the grandest experiment of all, the very origin and evolution of the universe. It appears as if Nature blazed forth at the beginning of time in the perfect embodiment of a Platonic form only to clothe herself immediately with obscuring veils. The Standard Model is a remnant of that ideal form, but we have been able to piece together at least part of the grand epic of its evolution.

J. J. Thomson’s 1897 discovery of the electron was the third of three rapid-fire developments at the end of the nineteenth century that launched the campaign, still in progress, to understand the inner structure of atoms. The other two were complete surprises: X-rays, discovered by W. K. Roentgen in 1895, and natural radioactivity, discovered in uranium by H. Becquerel in 1896. Because the X-rays seemed to come from a glowing spot on the glass wall of Roentgen’s apparatus, Becquerel immediately investigated whether phosphorescent materials could be induced to emit similar rays following exposure to the Sun. His discovery that uranium did indeed emit penetrating radiation, without benefit of sunlight, is surely the most profound purely accidental discovery in physics. Until Einstein’s discovery that mass is energy, it was also among the most puzzling: Where did the enormous radiant energy come from? (See Note 27 in Chapter 2.)

I wrote this book for my friends who are not physicists, but who are curious about the physical world and willing to invest some effort to understand it. I especially had in mind those who labor to make the work of physics possible – technical workers in other fields, teachers, science-minded public officials – who read popular accounts but are hungry for a “next step” that might give them a firmer grasp of this puzzling material. Physics gives me great pleasure, more from its beauty than from its usefulness, and I regret that my enjoyment should depend on the effort of so many others who do not share it. Here I have tried to ease my sense of guilt by attempting to disclose in ordinary language what modern physics really is about. Many similar accounts exist. In this one, I attempt to demystify the deep ideas as much as possible in a nonmathematical treatment. Some mathematical ideas are inevitable, and these I try to explain. Physics has entered an exciting phase with talk of new dimensions, exotic matter, and mind-boggling events of cosmic scale. These dramatic ideas rest on a solid conceptual framework, a product of the last century that is now old hat for physicists but remains exotic and impenetrable to most others. This framework, quantum theory and the Standard Model of matter, is an intellectual achievement of the highest order and essential for understanding what comes next. My intention here is to provide a reference and a guide to this known but still regrettably unfamiliar world.

The odd concepts of quantum physics are nearly a century old. They may seem difficult and forbidding at first sight, but the barrier to understanding is not their difficulty but their differentness. Unlike classical models, quantum theory does not attempt to simulate Nature. It gives us information about observations of Nature. But it does contain a “theoretical entity” that claims to contain all information, observation notwithstanding, namely the quantum state vector in Hilbert space. Scientists routinely use the language of Hilbert space, and nonscientists do not use it at all, which is understandable but unfortunate. A sort of baby-talk has become the lingua franca for much of the popular journalism of quantum physics, an awkward patois that combines some of the early groping language of Bohr, de Broglie, Schrödinger, and Heisenberg with more modern words about symmetries and states. Earnest amateurs still ask me about the mystery of Bohr orbits. There is no mystery because there are no orbits, except as historical curiosities. Same for “wave–particle duality.” Same for “quantum jumps.” A friend once asked me to address a philosophy seminar about the “paradoxes of quantum mechanics.” He said he had approached C. N. Yang first, who replied “What paradoxes?” There are some mathematical and logical rough spots in our current best theory of matter, and perhaps some of them can be framed as paradoxes. But the theory is actually rather straightforward, and it has a perfectly clear interpretation linked to experiments that, in their simplest form, anyone can perform (and students do routinely in college laboratories). No empirical observations are known that are inconsistent with the quantum framework.

In Sec. 9.13 we saw how factorization theorems give a lot of predictive power to QCD. They are essential in the analysis of data at high-energy colliders, not just for understanding the QCD aspects but also in searches for new physics, for example.

So far we have seen a genuine proof (Sec. 8.9) only for inclusive DIS, and only in a model theory without gauge fields. In this chapter we will formulate the principles that apply very generally, to other reactions, and when dealing with the full complications of a gauge theory.

The general class of problem concerns the extraction of the asymptotic behavior of amplitudes and cross sections as some external parameter, like a momentum, gets large. In general discussions, we denote the large parameter by Q. As well as factorization theorems in their broadest sense, such asymptotic problems also encompass simpler situations like renormalization, the operator product expansion (OPE), and the IR divergence issue in QED.

There is a common and general mathematical structure in these different problems that could undoubtedly use further codification. Perhaps methods based on Hopf algebras, or some generalization, would provide an appropriate mathematical structure. So far these methods have been applied to renormalization (e.g., Connes and Kreimer, 2000, 2002).

In this chapter, I interleave a general formal treatment with its application to the Sudakov form factor, including explicit calculations at one-loop order. The general treatment will underlie all further work in this book.