Photon and Electron Spins

Photon and Electron Spins

(Parte 2 de 3)

S0ij ) πrSrsij φs (13)

15322 J. Phys. Chem. A, Vol. 113, No. 52, 2009 Wittig

Using Eb )- ∂0Ab, with Ab given by eq 19, and SbEM ) Eb × Ab , yields

The electromagnetic energy density EEM follows just as straightforwardly:

It is understood that the field is of finite transverse extent.

However, because the edge region plays no significant role, eq 19 can be used up to some cutoff distance in the transverse direction. Total spin and energy are obtained by multiplying

SbEM and EEM by the volume V0 occupied by the field. Quantization gives the well-known relations EEMV0 ) pω and

SbEM V0 )( pz. The above approach differs from the one used by Ohanian,13 in which spin is assigned to a circulating current at the edge of a field of finite transverse extent (Figure 1).19 In his paper, the field is taken as circularly polarized with constant magnitude except in its edge region. In expressing the angular momentum densityas rb× (Eb × Bb), a symmetrizedenergymomentumtensor is enlisted, rather than the canonical one given in eq 7. Subsequent (nontrivial) manipulations lead to integration of a density over the wave volume.

As shown above, Noether’stheorem yields the result straightaway, and without the assignment of a privileged status to the field’s periphery region. Spin density is identified, subsequent calculations are trivial, and interpretation is straightforward. Quantizationyieldsthe spin quantaof (p, identifyingthe photon as a spin-1 object. The spin of an electromagnetic field is seen to be of a classical nature: quantization serves only to give (p eigenvalues.

At the same time, there is no doubt that a circulating flow of momentum exists at the edge of a circularly polarized field of finite transverse extent. This follows from Maxwell’s equations. Moreover, the associated angular momentum accounts quantitatively for photon spin. From these unequivocal facts it seems logical to conclude that13 “This angular momentum is the spin of the wave.” However, this overlooks an important point.

The spin of a circularly polarized electromagnetic field is present throughout the field. What happens at the edge is a manifestation of the field’s vector nature, but it is not the spin per se. Nonetheless, the question remains, how can these seemingly disparate viewsseach valid in its own rightsbe reconciled?

The answer lies with the energy-momentumtensor: canonical versus symmetrized. Ohanian used a symmetric tensor, noting that Belinfante derived a recipe for converting the canonical tensor to a symmetric one,20 and arguing that the latter is required by gauge invariance and general relativity. The Lorenz gauge, ∂µAµ ) 0, ensures relativistic covariance, whereas a transversefield is useful for quantization.With source-freespace there is no issue, and eq 19 fixes the gauge. General relativity (gravity) has nothing to do with the issue at hand. Belinfante neededa symmetricenergy-momentumtensorto derivegraviton spin. However, gravity is not relevant here.

The canonical tensor reveals intrinsic spin up front, as seen with eq 12. Of course canonical and symmetrized tensors must each yield the correct answer. Differences can only enter in mathematical ardor and ease with which results are interpreted.

Figure 2 illustrates the distinction between the canonical and symmetrized tensor approaches. It is an adaptation of one given by Soper.21 It is essentially the same as those used to explain the Carnot cycle. Currents are assumed to flow at the edges of

Figure 1. (a) Field strengthversus transversecoordinatefor a circularly polarized electromagnetic wave, e.g., eq 19 truncated at the wave’s edge. (b) The wave’s cross section (xy-plane), is indicated, showing the constantmagnituderegion(yellow)and the edge region(gray)where the magnitude falls to zero. In the edge region, the fields have z-components. Thus, a momentum current circulates around the periphery (blue arrow).

Figure 2. (a) Identical currents flow along the edges of identical squares. Arrows are displaced from the edges for viewing convenience. Each square can be assigned a circulation, i.e., its edge current times its area, that we call its spin. The spin density is uniform throughout. Note: there is no net current in the horizontal or vertical directions for any of the inside edges because of cancellation. For example, red and blue have the same magnitudes but opposite directions. (b) There is no cancellation at the outer edges, so a net current flows on the periphery. As the squares become infinitesimal, the spin density remains constant and the current continues to flow on the periphery.

Photon and Electron Spins J. Phys. Chem. A, Vol. 113, No. 52, 2009 15323 squares. Each square has an amount of circulation given by its edge current times its area (currents are displaced from edges for viewingconvenience).Let us call this circulationthe square’s spinsan odd spin to be sure, but it serves to make a point.

With small squares, it is appropriate to discuss a spin density, which is the edge current for a given square. Because the spin density in Figure 2 is constant, the total spin of an arbitrary area (which is much larger than the area of an individualsquare) is the density times this area. When the density is not constant, its integration over a given area yields the spin. Notice that the edge does not enter in a significant way. Thus, identification of the spin density at the outset enables the spin to be calculated trivially. This is akin to the situation in which the canonical tensor yields the spin density, as in eq 12.

Referring to Figure 2a, it is obvious that, except for the periphery, there is no net current on any vertical or horizontal line because of cancellation. For example, the magnitude of the current that flows upward (red arrows) is equal to that of the current that flows downward (blue arrows). Cancellation does not occur at the outside edges, so a current circulates at the periphery, as shown in Figure 2b. Though this accounts for the spin, it is the tensor/spinor nature of the field (a vector field in the case of electromagnetism) that dictates the underlying spin density. For example, were the field a scalar (e.g., a Schrodinger field), its spin density would be zero everywhereand the angular momentum tensor would be xiT0j - xjT0i. A circulation of momentum around the field’s edge in this case could not be attributed to a spin density.

To see how this fits with Noether’s theorem, consider the symmetrization of the canonical tensor Tµν. A new tensor, Θµν , is created by adding a term:


Because each spin term (SFµν, etc.) is antisymmetric with respect to its last two indices, GFµν is antisymmetricwith respect to its first two indices, i.e., GFµν )- GµFν. To see how this works, exchange F and µ in eq 23. This yields

Now replace the second term inside the parentheses (i.e., -SνµF) with SνFµ. This verifies the antisymmetry property: GFµν )

In addition, Θµν and Tµν each obey the same continuity equation, i.e., ∂µΘµν ) 0 ) ∂µTµν. To verify that this is so, use eq 2 to write ∂µΘµν ) ∂µTµν + ∂µ∂FGFµν. Now note that ∂µ∂FGFµν vanishes because GFµν is antisymmetric with respect to its first

Finally, to show that Θµν is symmetric with respect to interchange of its indices, write

Applying eq 23 yields GFµν - GFνµ ) SFνµ. Thus, Θµν - Θνµ )

Tµν - Tνµ + ∂FSFµν. The right-hand side is seen to vanish, because ∂FJFµν ) 0 ) ∂F(xµTFν - xνTFµ + SFµν) ) Tµν - Tνµ +

∂FSFµν. Thus, the symmetry property: Θµν ) Θνµ is verified. Belinfante devised this procedure.20 With the symmetrized tensor, total angular momentum includes spin, but without its explicit identification.

Spin-1/2. Noether’s theorem again yields an expression for the spin density of a classical field, this time the Dirac field.

Our concern here is solely with a massive spinor field, so we shall forego inclusion of the electromagnetic field. Thus, the Lagrangian is where the γτ are Dirac matrices. Equation 13 is used with Srs ij from eq 3:

The πr term is given by ∂(iψ†γ0γτ∂τψ)/∂(∂0ψR) ) iψR†, where ψR is a spinor component. Note: the ψ†γ0mψ term in eq 26 vanishes when LD is differentiated with respect to ∂0ψR. Also, φr f ψR and φs f ψ ; only τ ) 0 and the ψR spinor component survive differentiation; and γ0γ0 ) 1. Thus,

Spin components are obtained by spatial integration of S0ij. The spin density is

I. Comparison

The results presented in section I are not new, though the facility with which they are obtained using Noether’s theorem is noteworthy. As stated earlier, the goal of this paper is an intuitive grasp of intrinsic spin. Photon and electron spins each deserve attention, and they are sufficiently distinct from one another to warrant comparison.It was pointed out that Noether’s theorem and canonicalenergy-momentumtensors yield intrinsic spins of classical fields in a few steps. To further compare canonical and symmetrized tensor approaches, consider the respective expressions for the ij component of the total angular momentum:

In eq 30 spin is explicit, while in eq 31 it is not, though it can be made so through manipulation. Specifically, put Θ0i ) T0i

15324 J. Phys. Chem. A, Vol. 113, No. 52, 2009 Wittig

The right parenthetic term, by definition, yields Sij upon integration.To see how this works,integratethe right parenthetic term by parts. For xi∂kGk0j only k ) i survives, and for xj∂kGk0i only k ) j survives. Thus,

Applying eq 23 gives Gj0i - Gi0j ) S0ij, showing that eq 3 is Sij ) ∫d3xS 0ij. This illustrates the relationship between the canonical and symmetrized tensor approaches. Though each works, the canonical tensor is preferred, as spin appears at the outset.

Spin’s presence can be subtle when using Θµν. Referring to

Figure 2a, summing individual spins gives a total of 16. This is like using eq 14. The same result is obtainedusing the peripheral current(Figure2b), whichis like usingeq 31. FollowingSoper,21 the origin is the center, and we consider the contribution from the right edge (x1 ) 2). The momentum Θ02 is equal to 1/2, so the contribution from the 4 squares that make up this edge is 4x1Θ02 ) 4. Because the 4 edges that make up the periphery contribute equally, the total spin of 16 is again obtained. This is easily extended to nonuniform density.21 Figure 3a shows the magnitude of the spin increasing left to right, but not changing in the vertical direction. Summing individual spins gives a total of 40. The spin is also obtained using net currents (Figure 3b). Horizontal currents are zero due to cancellation, except for the top and bottom,where they decreasein magnitude right to left. Net currents flow on each vertical edge. Four superposed rectangles are identified, each with a current of 1 unit. Their areas are 4 (blue shading), 8, 12, and 16 squares, again giving a total spin of 40.

IV. Discussion and Summary

Classical fields have intrinsic spins that reflect their tensor/ spinor nature. Even a scalar field can be said to have an intrinsic spin. It just happens to be zero. In the case of electromagnetism the classical field is a vector field, so it has a spin of 1. When quantized, it has spin components of (p along the direction of propagation.

Analogybetweenphotonsand classicalelectromagneticfields is straightforward. The classical field has a spin density of Eb × Ab, which follows immediately from Noether’s theorem and carries over to the quantum case. Spatially nonuniform fields display momentum currents that are manifestations of the spin density. For example, a momentum current can circulate in the edge region of a circularly polarized electromagnetic field that has constant magnitude everywhere except in the edge region. This is not the spin per se. It is a manifestation of an underlying spin density.

Intrinsic spins of massive particles are subtler. Though their classical fields have spins, there are no real-world analogs of the classical fields of massive particles. A delicate issue is whether the particle or the field is more fundamental. In the standard model of physics, fields play the central role. They contain all of the properties that are passed on to their quanta. In this sense the field is more fundamental. However, as far as an intuitive grasp of spin is concerned, it is not clear that anything is gained by focusing on the massive fields as opposed to their particle counterparts. There is no such quandary with the electromagnetic field. Many photons can be placed in a single mode to create the familiarclassicalelectromagneticfield. This is how lasers work: the presence of photons in a resonator mode encourages the creation of additional photons to favor occupancy of this mode.

The classical Dirac field has a spin density that, like its electromagnetism counterpart, is quadratic in field strength. Spatial variationof the field and thereforethe spin density yields a momentum current. This is the same mechanism as in electromagnetism. In each, currents are manifestations of an underlying spin density, like the illustrations in Figures 2 and 3.

The spin of the classical Dirac field enters with the spinor representation. In fact, the relationship between intrinsic spin density and its associated momentum flow is a general feature. Namely, spatial variations of the field result in currents that are consequences of the underlying density. We can see how this works by using a vector identity to express the integration of

SbD as (Appendix B):

A simpleform for the spinorrevealsthe mechanismillustrated in Figures 2 and 3. With σb) σ12z, the nonrelativistic limit of

Figure 3. Two ways of assessing total spin. (a) Currents flow along edges of squares. Line thickness indicates currents whose magnitudes range from 1 to 4. Corresponding spins have magnitudes that range from 1 to 4. Summing the spins for the 16 squares gives a total of 40. (b) Net currents along edges are shown. Four superposed rectangles are identified, each having a current of 1 unit. From right to left their areas are 4 (blue shading), 8, 12, and 16 squares, again giving a total spin of 40.

Photon and Electron Spins J. Phys. Chem. A, Vol. 113, No. 52, 2009 15325

(Parte 2 de 3)