# Photon and Electron Spins

(Parte 3 de 3)

The left-hand side is the usual integration of the spin density. The integrand on the right is recognized as an angular momentum density. For example, with | |2 taken to be like the distribution in Figure 1, the momentum circulates in the gray annulus, and the integrand points in the z direction.

Let us now turn to the spin-1/2 particlerather than the classical field whose quantization yields it. Being a fermion, it obeys the exclusion rule that defines its statistical property. This fundamental property of the particle is encoded into the theory of quantum mechanics by requiring that the exchange of any two identical fermions results in a sign change of the overall wave function, ψ. In practice, this is achieved by expressing ψ in terms of amplitudes that account for particle exchange:

where cij ) (-1)2s, with s the particle spin. Thus, for s ) 1/2 each exchange carries a sign change. In eq 36 the regions between commas (slots) are associated with particles. From left to right: the first slot corresponds to particle 1, the second to

particle 2, and so on. For example, φ(x2,x3,x1,...) is the amplitude for finding particle 1 described by x2, particle 2 described by x3, particle 3 described by x1, and so on. It is understood that the right-hand side of eq 36 includes all possible permutations.

The exchange of two identical massive particles includes a 2π reorientation that is needed to bring the system into registry with the original configuration.2 An angular displacement of 2π might appear to return a system to its starting point, but this is not the case. An angular displacement of 4π does, whereas 2π does not.23,24 To accommodate this 2π into a mathematical framework in which the overall wave function changes sign when any two identical fermions are exchanged requires that the particles are endowed with what we call spin-1/2. In other

words, spin-1/2 is the mathematical way of accounting for the fermion property. For massive particles, spin exists in the particle’s rest frame.

The efficacy of the mathematics that accounts for spin-1/2 is impressive. From the invention of spinors by Cartan in the early

part of the twentieth century, through the introductionof spinors to physics by Pauli and Dirac with their matrices (which are, in fact, representations of algebras introduced a half century earlier by Clifford),25 to the SU(2) and SL(2,C) covering groups,24 the machinery for handling spin-1/2 is a “done deal”. It obeys the same Lie algebra as does integer angular momen-

tum, and so on. The question of why a particle has such a property remains.

The particle perspective can lessen spin’s mystique. For example, one sees right away why spin-orbit interaction is relativistic. It is not because of electron spin, but instead the magnetic field experienced by the electron as it circulates around, e.g., a proton. This follows trivially from a Lorentz transformationof the electron-protonCoulombinteraction.Spin itself is not relativistic. It persists unaltered in the nonrelativistic limit of the Dirac equation.

Invoking momentum flow in a classical field whose subsequent quantization creates fermions is not a good mnemonic. The reason is that spin density is first imported through the tensor/spinor nature of the fields, and after it is in place a manifestation of it is recognized.

To summarize,intuitionregardingspin inevitablycomes back to the property of the fermion particle. It is this property that leads to what we call spin, not the other way around. In other words, the particle’s statistical property is manifest as spin in the mathematics. Spin is not some mysterious thing that determines a particle’s statistics.

Acknowledgment.This research has been supported through a grant from the U.S. National Science Foundation, CHE- 0652830.

Appendix A

) ∂θAπ - ∂πAθ. Indices are lowered: Fθπ ) gθFεgεπ, and deltas are used to keep track of derivatives:

Applying the delta functions to the metrics gives

Appendix B

The integrand on the right hand side is as required.

 ψ ) φ(x1,x2,x3,...) + c12φ(x2,x1,x3,...) + (36)

- 14gθFgεπ{(δµθδνπ - δµπδνθ)Fε + (δµFδνε - δµεδνF)Fθπ}

(gθµgνπ - gθνgµπ)Fθπ}

15326 J. Phys. Chem. A, Vol. 113, No. 52, 2009 Wittig

References and Notes

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Photon and Electron Spins J. Phys. Chem. A, Vol. 113, No. 52, 2009 15327

(Parte 3 de 3)