Resolution of a conflict between Laser and Elementary Particle Physics

The claim some years ago, contrary to all textbooks, that the angular momentum of a photon (and gluon) can be split in a gauge-invariant way into an orbital and spin term, sparked a major controversy in the Particle Physics community. A further cause of upset was the realization that the gluon polarization in a nucleon, a supposedly physically meaningful quantity, corresponds only to the gauge-variant gluon spin derived from Noether's theorem, evaluated in a particular gauge. On the contrary, Laser Physicists have, for decades, been happily measuring physical quantities which correspond to orbital and spin angular momentum evaluated in a particular gauge. This paper reconciles the two points of view.

A major controversy has raged in Particle Physics recently as to whether the angular momentum (AM) of a photon, andà fortiori a gluon, can be split into physically meaningful, i.e. measurable, spin and orbital parts. The combatants in this controversy 1 seem, largely, to be unaware of the fact that Laser Physicists have been measuring the spin and orbital angular momentum of laser beams for decades! 2 . My aim is to reconcile these apparently conflicting points of view. Throughout this paper, unless explicitly stated, I will be discussing only free fields.
I shall first consider QED, where E, B and A are field operators, and as is customary, employ rationalized Gaussian units. It is usually stated that the momentum density in the electromagnetic field (known, in QED, as the Belinfante version) is proportional to the Poynting vector, i.e.
and it is therefore eminently reasonable that the AM should be given by where the Belinfante AM density is Although this expression has the structure of an orbital AM, i.e. r × p, it is, in fact, the total photon angular momentum density. On the other hand, application * e.leader@imperial.ac.uk 1 For access to the controversy literature see the reviews by Leader and Lorcé [1] and Wakamatsu [2]. Note, though, that I shall criticize some of the statements in [1] 2 For access to the laser literature see the reviews of Bliokh and Nori [3] and Franke-Arnold, Allen and Padgett [4] of Noether's theorem to the rotationally invariant Lagrangian yields the Canonical version which has a spin plus orbital part where the canonical densities are but, clearly, each term is gauge non-invariant. Textbooks have long stressed a "theorem" that such a split cannot be made gauge invariant. Hence the controversial reaction when Chen, Lu, Sun, Wang and Goldman [5] claimed that such a split can be made. They introduce fields A pure and A phys , with where ∇ × A pure = 0, and ∇ · A phys = 0 (7) which are, of course, exactly the same fields as in the Helmholz decomposition into longitudinal and transverse components 3 Chen et al then obtain and since A ⊥ and E are unaffected by gauge transformations, they appear to achieved the impossible. The 3 Indeed the only reason for the new nomenclature was Chen et al's intention to extend these ideas to QCD. explanation is that the "theorem" referred to above applies to local fields, whereas A ⊥ is, in general, non-local.

In fact
In all three versions of AM just mentioned, the integrands differ by terms of the general form ∇ · f , where f is some function of the fields, so that the integrated versions differ by surface terms at infinity, and thus agree with each other if the fields vanish at infinity. For classical fields, to state that a field vanishes at infinity, is physically meaningful, but what does it mean to say an operator vanishes at infinity? The first serious analysis of this question seems to be that of Lowdon [6], utilising axiomatic field theory. I shall comment later on his conclusions. Now the key question is: what is the physical relevance of the various S operators? Can they be considered as genuine spin operators for the electromagnetic field? A genuine spin operator should satisfy the following commutation relations (for an interacting theory these should only hold as ETCs i.e. as Equal Time Commutators) But to check these conditions, manifestly, one must know the fundamental commutation relations between the fields and their conjugate momenta i.e. the quantization conditions imposed when quantizing the original classical theory, yet to the best of my knowledge, with only one exception [7], none of the papers in the controversy actually state what fundamental commutation relations they are assuming. Thus the expressions alone for the operators S are insufficient. Failure to emphasize the importance of the commutation relations in a gauge theory can lead to misleading conclusions. It must be remembered that the quantization of a gauge theory proceeds in three steps: (1) One starts with a gauge-invariant classical Lagrangian.
(3) One imposes quantization conditions which are compatible with the gauge choice. I shall comment on just two cases. In covariant quantization (cq) [8][9][10], for example in the Fermi gauge, one takes and then the Hilbert space of photon states has an indefinite metric. Quantizing in the Coulomb gauge one uses transverse quantization (tq) (see e.g. [11]) and the Hilbert space is positive-semidefinite. There is an important physical consequence of this difference in quantization procedures. Gauge transformations on field operators almost universally utilize classical functions i.e.
where α(x) is a "c-number" function. Clearly this transformation cannot alter the commutators. Or, put another way, gauge transformations are canonical transformations and therefore are generated by unitary operators, which do not alter commutation relations. This means that one cannot go from say Canonically quantized QED to Coulomb gauge quantized QED via a gauge transformation. This point was emphasized by Lautrup [8], who explains that although the theories are physically identical at the classical level, it is necessary to demonstrate that the physical predictions, meaning scattering amplitudes and cross-sections, are the same in the different quantum versions. This is also stressed by Cohen-Tannoudji, Dupont-Roc and Grynberg [12] on the basis that also the Hilbert spaces of the different quantum versions are incompatible. It is not difficult to show that the canonical S can with covariant quantization i.e. S cq can satisfies Eq. (11) and so is a genuine spin operator. However it is not gauge invariant. I shall comment on this presently. For the Chen et al case, since we are dealing with free fields, the parallel component of the electric field is zero i.e. E = 0 so that J chen becomes But this is exactly the expression for J, studied, with transverse quantization, in great detail by van Enk and Nienhuis (vE-N) in their classic paper [13], which is often the basis for statements about spin and orbital angular momentum in Laser Physics i.e one has and where the densities are Now van Enk and Nienhuis show that the commutation relations for S tq gic are very peculiar and not at all like those in Eq. (11). 4 They demonstrate that and stress that the components of S tq gic cannot therefore be considered as the components of a genuine spin vector in general. Moreover, they are careful to refer to this operator as the 'spin' in inverted commas (and similarly L tq gic , is referred to as the 'orbital angular momentum'), but it seems that later papers on Laser Physics have not bothered to respect this convention. Despite all these peculiarities it is claimed, correctly, that the spin and angular momentum of certain types of laser beam can and are regularly measured. 5 So the key question is how is this to be reconciled with the above, where, on the one hand, we have S cq can which looks like a genuine spin operator, but which is not gauge invariant and, on the other, S tq vE-N , which in no way resembles a spin operator, but which is at least gauge invariant. It was shown in [7] that (S cq can · P /|P |), where P is the momentum operator, measures helicity and that its matrix elements between arbitrary physical photon states are gauge invariant. A key step in this proof was to consider the action of (S cq can · P /|P |) on the physical photon state | k, j with transverse polarization. Provided the operators are normal ordered one has (22) and the commutator is then evaluated using the covariant quantization conditions. Acting on a state of helicty λ one eventually finds that (S cq can ·P /|P |) measures helicity: (S cq can · P /|P |) |k, λ = λ |k, λ .
For the case of the helicity based on S tq gic the analogous commutator has to be evaluated using the transverse commutation conditions Eq. (13), but it turns out that the terms k i k j don't contribute, so that also S tq gic · P /|P | measures helicity i.e.
In summary, only the helicity, based either on S can or on S gic , is physically meaningful as a measure of angular momentum. But, interestingly, as van-Enk and Nienhaus [13] show, the other components of s gic , though not angular momenta, are nevertheless measurable quantities. We shall see this concretely in the classical discussion which follows, where, it should be borne in mind that, unlike the QED situation, it 4 Also L tq gic has peculiar commutation relations, but as expected, J tq gic behaves as a perfectly normal total angular momentum. 5 Similar comments apply also to gluons. is straightforward to compare expressions in different gauges.
I turn now to the key question which has remained unresolved in the particle physics discussions, namely, which of the AM densities j bel , j can or j gic is relevant physically. Contrary to the opinion expressed in [1], where it is argued that it is simply a matter of taste, and to [6], which favours the Belinfante version, I shall argue that the laser experiments clearly indicate that it is j gic which plays a direct role in the interaction of classical EM waves with matter and that the Belinfante expression is definitely unacceptable. The criticism that a density should not depend on a non-local field A ⊥ does not apply to the situation of most interest, namely when dealing with monochromatic free fields with time dependence e −iωt , since then E = E ⊥ = −Ȧ ⊥ so that is a local field. Discussing the classical electrodynamics of laser fields, I shall follow custom and switch to SI units. The only effect on all the previous formulae for momentum and AM densities is to multiply them by a factor ǫ 0 . The real, monochromatic physical EM fields (E, B) are, as usual, expressed in terms of complex fields (E, B) The force on, and the torque (about the centre of mass of a small neutral object), in dipole approximation, are given by where the induced electric dipole moment is given by and the complex polarizability is First consider the force acting on the neutral dipole. In [14] it is shown that the total force splits into two terms where, for the cycle average, which I indicate by < >, and for a classical electric dipole with momentum P dipole it is F dissapative that controls its rate of change of momentum (see Chapter V of [15]) Naturally, for the linear momentum, as for the AM, besides the Belinfante version Eq. (1), there exist also the gauge-variant canonical and gauge-invariant gic versions and and as in the AM case the three space-integrated versions are equal if the fields vanish at infinity. Evaluating the cycle average, using Eq. (25), it turns out that so that it is the gauge-invariant canonical version that is physically relevant, and it is, of course, equal to the canonical version evaluated in the Coulomb gauge.
Next consider the torque about the centre of mass of the dipole. One finds that For the cycle average, one finds Now consider the cycle average of s gic given in Eq. (20) so that from Eq. (39) follows the fundamental result The physical torque is thus given by a gauge-invariant expression, as it ought to be, which coincides with the canonical version evaluated in the Coulomb gauge, in accordance with the entire discussion in [3]. At first sight it may seem odd that only the spin vector enters Eq. (41), but it should be remembered that τ is the torque about the centre of mass of the dipole, whereas L is the orbital AM about the origin of the axis system.
Consider now the application of these results to lasers. In the foundation paper on laser angular momentum by Allen, Beijersbergen, Spreeuw and Woerdman [16] the AM is associated with the gauge invariant Belinfante version in Eq. (3). It is therefore important to review some of the properties of the AM density j bel and of the Belinfante linear momentum, whose density is proportional to the Poynting vector. Firstly, for a plane wave propagating in the Z-direction the helicity is the same as the z-component of the angular momentum, and, as shown in Section 2.6.4 of [1], for a left-circularly polarized i.e. positive helicity beam, j bel, z = 0, whereas, per photon j can, z = s can, z = j gic, z = s gic, z = (42) as intuitively expected. Moreover, this result is much more general: j bel obviously has zero component in the direction of the Belinfante field momentum density: Thus the Belinfante AM fails, whereas the gauge invariant canonical version succeeds, in correctly generating the helicity. Secondly, and this seems most surprising in light of the initial comments on the controversy given above, it will be seen presently that for a superposition of polarized plane waves, j bel splits into two terms apparently corresponding to orbital and spin angular momentum [16].
In their analysis Allen et al utilize the paraxial approximation, which corresponds to keeping the first two terms in an expansion in terms of a parameter equal to the beam waist divided by the diffraction length [17], and apply it to a Laguerre-Gaussian laser mode, but their treatment is actually more general and applies to any monochromatic, axially-symmetric vortex beam of finite cross-section. In such a beam propagating in the Z-direction the complex electric field, in paraxial approximation and in the notation often used in laser papers, has the form 6 where and all second derivatives and products of first derivatives are ignored. As in [3] I shall indicate relations that are valid in paraxial approximation by "≃", so for example ω ≃ kc.

For the case of circularly polarization
where σ z = ±1 for left/right circular polarization, and in cylindrical coordinates (ρ, φ, z) the correct physical ones. However, exactly the same functional dependence on ρ follows from the gic expressions. In fact this equivalence is not restricted to Bessel beams. It holds as long as |u| 2 follows a simple power law behaviour |u| 2 ∝ ρ −β . Since the absolute rotation rates depend upon detailed parameters which, according to the authors, were beyond experimental control, it would be incorrect to interpret these results as evidence in favour of the Belinfante expressions. Moreover, in an unpublished paper [19], Chen and Chen have argued that the dependence on l and σ z , of the shift of the diffraction fringes, found by Ghai, Senthilkumaran and Sirohi [20] in the single slit diffraction of optical beams with a phase singularity, implies that the correct expression for the optical angular momentum density is the gic one. And, further, as summarized in the recent review [3] it is the canonical AM in the Coulomb gauge i.e the gic AM that agrees with a wide range of experiments.
For the linear momentum, on the other hand, it seems more difficult to choose experimentally between the Belinfante and gic versions, but I shall give an argument in favour of the gic version for photons. For the cycle averages one finds and in the paraxial case under discussion this becomes Following [15], assuming that the change of momentum of the dipole is due to the momentum of the photons absorbed from the beam, I shall take the number of photons totally absorbed by the dipole per second to be given by 1/ ω times the rate of increase of the dipole's internal energy. For a paraxial beam I then find that Eqs. (36) and (33) are satisfied only if the average photon momentum is taken as where N is the number of photons per unit volume. A similar argument supports the gic version for the AM. Namely, assuming that the change in internal angular momentum of the dipole arises from photon absorption I find that Eq. (41) is satisfied only if In summary, the angular momentum controversy, which has bedevilled particle physicists for some time, is resolved by a host of laser physics experiments which indicate that the Gauge Invariant Canonical linear momentum and angular momentum densities are the physically relevant ones, and that this is not simply a question of taste. 8 Moreover, although there does not exist a genuine spin vector for photons, the van Enk-Nienhuisen=Chen et al= gic 'spin vector' plays a central role in Laser Physics. All of its components can, in principle, be measured, but only one component, strictly speaking the helicity, is a genuine AM. For a paraxial beam propagating in the Z-direction one can show that the Z-component of the gic spin vector coincides with the gic helicity i.e S gic z ≃ gic helicity , so this component is effectively a genuine AM. And finally, recognizing that the fundamental expressions are the gic ones, allows one to avoid the somewhat disturbing claim that what is physically measured corresponds to a gauge-variant quantity evaluated in a particular gauge, i.e. the Coulomb one. I am grateful to K. Bliokh for making me aware of the vast literature on laser angular momentum, to X-S. Chen, C. Lorcé and G Nienhuis for helpful comments, and to J.Qiu and R Venugopalan for hospitality at Brookhaven National Laboratory. I thank the Leverhulme Trust for an Emeritus Fellowship.