Photon and Electron Spins † Curt Wittig ‡ Department of Chemistry, UniVersity of Southern California, Los Angeles, California 90089 ReceiVed: July 2, 2009; ReVised Manuscript ReceiVed: October 14, 2009 It is easy to draw an intuitive parallel between the classical free electromagnetic field and its corresponding quantum, the photonsa spin-1 object. The situation with a massive particle such as an electron is less clear, as a real-world analog of the classical field whose quantum is the massive particle is not available. It is concluded that the fermion particle perspective provides the best avenue for an intuitive grasp of the spin of an elementary fermion. I. Introduction The field of chemical dynamics is a mature one, in which it has become the norm for results from detailed experimental studies to be compared to results from high-level theoretical calculations. Advances in both experiment and theory have been motivated by, and are congruent with, ambitious yet sensible goals: deep qualitative understanding of underlying factors that govern chemical transformation, and accurate predictions of even the most nuanced experimental measurements. Scientists in this areasexperimentalists and theoreticians alikeshave been steadfast in their commitment to improve insight and intuition at progressively higher levels of detail. For example, nowadays the chemical dynamics community deals routinely with weak spin-orbit interactions as a system moves from its entrance channels through its exit channels. Moreover, it turns out that such interactions, despite their modest energies, can be important, even decisive, insofar as influencing reaction pathways is concerned. Accord between experiment and theory is sought, but often not easily achieved, at the highest levels of detail. Witness, for example, the reactions of 2 P 1/2 and 2 P 3/2 halogen atoms with simple molecules. Even with light nuclei, and therefore small spin-orbit splittings, the reactivities of these species can differ qualitatively. 1-4 With heavy nuclei, relativistic effects such as spin-orbit interaction can be so important as to alter potential surfaces in ways that have no counterparts in systems comprised of light nuclei. 5,6 The path to success has been arduous, but persistent efforts that have spanned decades are now paying off with remarkable agreement between first-principles theory and exquisite experiments. 1 For example, the pioneering ex- perimental and theoretical work of Aquilanti and co-workers 7-12 set the stage for a generation of studies of reactive and inelastic scattering of open-shell species, including a number of important effects attributed to spin-orbit interaction. This article is also about spin, namely, the intrinsic spin of an elementary entity such as a photon or an electron. It has nothing to do per se with comparisons between theoretical calculations and experimental results, nor does it address spin’s dynamical role in reactive and inelastic scattering. It is about spin itself. The aim is to provide a means whereby intrinsic spin can be understood at an intuitive level. Particular attention is paid to the photon and spin- 1 / 2 particles such as electrons, as these species are of paramount importance, not only in chemical dynamics, but in all of physical science. The intrinsic spin of an elementary particle is a subject rife with subtlety. As used here, the term elementary particle means that the particle is not made of other things. For example, an electron is an elementary particle, but a proton is not, because it is made of quarks bound by gluons. In this sense, the term elementary particle can include things that have no mass, such as photons. Pieter Zeeman discovered spin in the late 1890s, well before there was a theory of quantum mechanics, and to this day spin is integral to the most mathematically rigorous theories of the physical world. We are deft at manipulating spins, as well as dealing with their many applications and inventing new ones. At the same time, chemical dynamicists, for the most part, sidestep obvious but vexing questions: What is spin? What is the best way to visualize it? Is it quantum mechanical? After all, intrinsic spins of elementary things such as the electron and the photon have just two quantum states. Are there classical analogs? Is spin relativistic? After all, spin- 1 / 2 seems to pop out of the Dirac equation. These are hard questions worthy of attention. Though such questions have been pondered for years, they go largely unanswered. For example, Ohanian suggested an intuitive picture for spin. 13 With classical electromagnetic and Dirac fields as examples, it was concluded that spin could be interpreted as a circulation of momentum in the classical wave fields whose quantizations yield a photon and a massive spin- 1 / 2 particle, respectively. Comparison between a classical electromagnetic field and a photon is sensible but requires a more careful look at the field’s spin density. This is facilitated by the application of Noether’s theorem. As discussed below, the case of a massive spinor (Dirac) field is subtler, as no real- world analog of the classical field of a massive particle is known. In what follows, an overview is given of Noether’s theorem as it applies to classical fields. This is needed for the subsequent discussion. Noether’s theorem has been around for almost a century and detailed accounts can be found elsewhere. It is used widely in theoretical physics, much less in chemical dynamics. Its use here is restricted to massive spinor and massless vector (electromagnetic) fields. Its application to the electromagnetic field reveals, without ado, a spin density that yields photon spin straightaway. The issue of canonical versus symmetrized tensors, each of which can be used with Noether’s theorem, is discussed. Without doubt, the canonical tensor provides the greatest † Part of the “Vincenzo Aquilanti Festschrift”. ‡ E-mail: wittig@usc.edu. J. Phys. Chem. A 2009, 113, 15320–15327 15320 10.1021/jp906255u  2009 American Chemical Society Published on Web 12/09/2009