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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.Advancesin both experimentand 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 2P1/2 and 2P3/2 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 surfacesin ways that have no counterpartsin 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 experimentaland theoreticalwork of Aquilantiand co-workers7-12 set the stage for a generation of studies of reactive and inelastic scatteringof open-shellspecies,includinga numberof 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 realworld analogof the classicalfield of a massiveparticleis 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 issueof canonicalversussymmetrizedtensors, each of which can be used with Noether’stheorem, is discussed.

Without doubt, the canonical tensor provides the greatest

† Part of the “Vincenzo Aquilanti Festschrift”. ‡ E-mail: wittig@usc.edu.

10.1021/jp906255u 2009 American Chemical Society Published on Web 12/09/2009 transparency and insight. The Dirac field is treated similarly, again supporting the use of the canonical tensor rather than a symmetrizedone. Insightinto the electromagnetismcase is more straightforward than in the spin-1/2 case. Nonetheless, one sees with minimal math how the spin density is manifest with the classical Dirac field.

Finally, the most important issues are reached. Is the classical

Dirac field useful in the quest for an intuitive understanding of the spin of an elementary particle such as an electron? It is argued that this is not a good way to visualize the spin of a massiveparticle.Unlikeelectromagnetism,where classicalfields can serve as excellent descriptors of nature, massive fields have no real-world classical analogs. An alternate perspective, in which the particle is assigned the privileged role, is recommended. The fermion particle obeys the exclusion rule, and this endows it with what we call spin.

A word about language and symbols is in order. With a few exceptions (which will be identified explicitly), superscripts and subscripts denote contravariant and covariant components, respectively. Standard relativistic 4-vector notation is used throughout: implied summation for repeated indices (one upper, one lower); Greek letters for 4-vectors (e.g., xµ); Roman letters and arrows for 3-vectors (e.g., xi, xb, b∇); covariant and contra-

lower indices, e.g., FεF ) gεµFµνgνF. Except where necessary for clarity, the constants p, c, µ0, and ε0 are each set equal to 1.

I. Background

A theorem due to Emmy Noether provides a straightforward means of examiningthe intrinsicspins of classicalfields. It deals with conserved quantities that are identified through invariance of the action under continuous symmetry transformations. With the end points of the variations held fixed, covariance of the equations of motion is assured through invariance of the Lagrangian density, whose effect on the action is unchanged by the addition of a 4-divergence.

Emmy Noether was a prolific mathematician with important theorems to her credit. The one under consideration here states that if a system has a continuous symmetry there exists a related quantity whose value is conserved, i.e., it is constant in time. The conserved quantity is called the Noether charge and its associated flow is called the Noether current. We are familiar with electric charge, such as that of an electron. Noether charge is a generalization of this.

Though Noether’s theorem is a cornerstone of theoretical physics, most mathematicians have not heard of it, as Emmy Noether’sgreatestcontributionsin mathematicswere in the areas of abstract algebra, group theory, ring theory, group representations, and number theory. She moved from Erlangen to Gottingen in 1916 at the invitation of David Hilbert and Max Klein. The idea was that she would participate with Hilbert, Klein, and Einstein in the resolution of daunting mathematical issues that had arisen in the theory of general relativity. And participate she didsstraightening out the mathematics of what is one of (if not the) greatest physics theories of the twentieth century. It is interesting that, despite its widespread use in quantum physics, Noether’s theorem preceded quantum mechanics by a decade, and it was developed for entirely different purposes.

The present paper is concerned with spins of classical fields rather than aspects of Noether’s theorem per se. Thus, equations relevant to Noether’s theorem will be presented with brief descriptions and justifications, and we will proceed from there. Detailedderivationsare availableelsewhere.The one in Schwabl is excellent and will be referred to a couple times.14 Soper provides truly outstanding insights,15 though the math is less traditional [e.g., the metric (-+++ ) is used rather than (+--- )], and it is often hard to follow. In this paper, the simplest version of Noether’s theorem that is applicable to the fields of interest is used.

To begin, consider the variation of a Lagrangian density L that depends on the fields and their first derivatives, but not explicitlyon spacetime:L (φr,∂µφr). The index r labels the fields. When dealing with a real scalar field, r is irrelevant so it is dropped. For a complex scalar field, r has two labels (for ψ and ψ†). For an electromagnetic field, there are 4 components, so r is assigned the labels 0, 1, 2, and 3. For a Dirac spinor there are 8 components (4 for the spinor and 4 for the adjoint spinor). And so on for other fields and combinations of fields. In principle there is no limit to the number of labels that can be assigned to r.

For an inhomogeneousLorentztransformation,the Lagrangian density(hereafterreferredto simply as the Lagrangian)responds to variations, induced by a change of reference frame, of the spacetime points and the fields. Because the symmetry is continuous,infinitesimaldisplacementsdefine the transformation properties:16

The term δν is an infinitesimal translational displacement, while the term ∆ωνσ is a different kind of infinitesimal displacement.For example,∆ω01 correspondsto an infinitesimal boost, while ∆ω12 corresponds to an infinitesimal rotation. Note that ∆ωνσ is antisymmetric with respect to interchange of its indices: ∆ωνσ )- ∆ωσν. There are 6 independent ∆ωνσ’s: 3 boosts (∆ω0i) and 3 rotations (∆ωij). The term Srsνσ depends on the nature of the fields being transformed; it is also antisym- metric with respect to the interchange of its indices. For the cases of spinor and vector fields, Srsνσ is given by

In eq 3, σνσ ) (i/2)[γν,γσ], where the γ’s are Dirac matrices.

The variation of the Lagrangian with respect to δxν and ∆φr is set equal to zero, and a series of algebraic manipulations yields the continuity equation:16 where with Tµν being the canonical energy-momentum tensor:17

Srsνσ )- i12σrsνσ (spinors) (3) ) δrνδsσ - δsνδrσ (vectors) (4)

Photon and Electron Spins J. Phys. Chem. A, Vol. 113, No. 52, 2009 15321

A consequence of eq 5 is that the spatial integral of g0 is a conserved quantity. To see why this is so, write eq 5 as b∇·gb)

-∂0g0 and integrate over all of three-dimensional space. The left-hand side vanishes because the volume integral is converted to a surface integral and it is assumed that gb falls off sufficiently rapidly at large distances to ensure that the surface integral vanishes. Equation 5 thus yields

and we see that ∫d3xg 0 is conservedsit does not vary in time.

Equations 5-7 summarize the version of Noether’s theorem that will be used below. The existence of a conserved quantity implies a continuity relation for a tensor whose rank is one higher than that of the conserved quantity. Recall that conservation of a scalar charge implies a continuityrelation for a (vector) current: ∂νJν ) 0. Likewise, the conservation of a 4-vector quantity implies a continuity relation for a second rank tensor current. For example, conservation of the 4-momentum T0σ implies ∂νTνσ ) 0. Here the 4-momentum is an example of a Noether charge. And so on for higher rank tensors.

Equation 6 is used to express the conservation of angular momentum. Inserting eqs 1 and 2 into eq 6, with µ ) 0, δν ) 0 (i.e., homogeneous Lorentz transformations), and νσ ) ij

(i.e., rotations, but not boosts) yields

duced, and ∆ωij )- ∆ωji has been used. The 1/2∆ωij in eq 1 is immaterial as far as conserved quantities are concerned,

because the ∆ωij’s are independent of one another. They each representan infinitesimalangulardisplacementabout a Cartesian axis. Thus, the integral of the contents of the large curly bracket is conserved. Namely, it is the ij component (k-direction) of the total angular momentum of the fields. This includes intrinsic spins. The resulting expression is

The parenthetic term is of orbital character. This is obvious when the field yields a massive particle upon its quantization. In the case of electromagnetism,orbital angular momentum can be defined in terms of displacement from an origin of the “center” of a wave of finite transverse extent. The other term is the more important one. It is the spin density:

whose spatial integration yields the spin:

The aboveapproachis a straightforwardmeansof determining intrinsic spins of classical fields. The spins derive directly from the tensor/spinor character of the fields. Once the tensor/spinor nature of the field is set (e.g., scalar, spinor, vector, rank-2 tensor), the spins are automatic, including quantum versions (respectively, 0, (p/2, (p, (2p). The general version of Noether’s theorem is far-reaching, including higher derivatives of the fields, explicit dependence of the Lagrangian on spacetime, and tensor fields of any rank. In the present paper, the Lagrangians depend on only first derivatives of the fields, and they have no explicit spacetime dependence. Only vector and spinor fields are considered.

Photon Spin. The spin of a free electromagnetic field is readily obtained via Noether’s theorem. As used here, the term free electromagneticfield means that no chargesor currentsneed to be taken into account. The Lagrangian is

The contravariant second-rank antisymmetric field strength tensor: Fθπ ) ∂θAπ - ∂πAθ follows the convention in Jackson.18 In terms of Cartesiancomponentsof electricand magneticfields, Fθπ can be expressed as

The covariant tensor Fθπ differs from Fθπ only in the sign of

Eb. In other words, Fθπ is the same as the right-hand side of eq 16, but with Eb replaced with -Eb. The spin density is evaluated using eq 13, with Srsij from eq 4, and πr)- F0r (Appendix A). Thus,

This shows that S0ij is the k-component of Eb × Ab. For example, S012 )- F01A2 + F02A1 ) ExAy - EyAx, which is the z-component of Eb × Ab. The spin density is therefore identified: SbEM ) Eb × Ab. Total spin is obtained by integrating

SbEM over the volume of the field. An important point involves the field’s edge region. Because the spin density is integrated over the volume of the wave, the field’s edge region is not particularly important. Thus, a circularly polarized plane wave propagating in the z-direction can be used to describe Ab:

(Parte **1** de 3)