Comparison of Different Forms for the “Spin” and “Orbital” Components of the Angular Momentum of Light

Abstract

We compare three attempts that have been made to decompose the angular momentum of the electromagnetic field into components of an “orbital” and “spin” nature. All three expressions are different, and there seems to be no reason to prefer one to another. It appears, on the basis of classical electrodynamics, that there is no unique way of decomposing the angular momentum of the electromagnetic field into orbital and spin components, even in a fixed inertial frame.

1. Introduction

The total angular momentum 𝐉 of the electromagnetic field is given (in SI units) [1] by 𝐉(𝑡)=𝜀₀𝑑³𝑥𝐱×[𝐄(𝐱,𝑡)×𝐁(𝐱,𝑡)]. (1) Henceforth, we will suppress the time coordinate 𝑡 of the fields, all of which depend on time, and also the 𝜀₀ factor.

There has been debate for a long time over whether the total angular moment 𝐉 of the electromagnetic field can be decomposed into an orbital part 𝐋 and a spin part 𝐒 so that 𝐉=𝐋+𝐒. (2) Some authors [2] argue that on the basis of the first principles it is not possible to do this; others [3–5] show that forms can be demonstrated that appear to be, at least algebraically, of a spin and orbital nature.

By means of partial integration Ohanian [4] effected a decomposition with 𝐉=𝐋^′+𝐒^′+𝐉^′_𝑏,𝐒^′=𝑑³𝑥𝐄(𝐱)×𝐀(𝐱), (3)𝐋^′=_𝑖𝑑³𝑥𝐸^𝑖(𝐱)𝐱×∇_𝑥𝐴^𝑖(𝐱), (4) where 𝐀 is the vector potential {𝐁(𝐱)=∇_𝑥×𝐀(𝐱)} and ∇_𝑥 is the gradient operator that operates on functions of 𝐱. Ohanian assumed that the electric charge density 𝜌was zero and deemed (3) and (4) to be the spin and orbital components, respectively, of the electromagnetic field on the basis that the integrand of (3) was not explicitly linear in the 𝐱 coordinate whereas the integrand of (4) was. When the charge density is not zero, a bound term 𝐉^′_𝑏, considered also to be of an orbital nature 𝐉^′_𝑏=𝑑³𝑥𝐱×𝐀(𝐱)∇_𝑥⋅𝐄(𝐱)=𝑑³𝑥𝐱×𝐀(𝐱)𝜌(𝐱), (5) is obtained on whose form all writers agree [6]. Although the sum of (3) and (4) and (5) 𝐉=𝐋^′+𝐒^′+𝐉^′_𝑏 is gauge invariant, the individual terms are not and so have no physical interpretation until the gauge of the vector potential is fixed completely.

Cohen-Tannoudji et al. [3] used the Coulomb (or transverse) gauge, defined by the gauge condition ∇⋅𝐀_𝑡=0, which gives 𝐒^′′=𝑑³𝑥𝐄(𝐱)×𝐀_𝑡(𝐱)𝐋_′′,=_𝑖𝑑³𝑥𝐸^𝑖(𝐱)𝐱×∇_𝑥𝐴^𝑖_𝑡(𝐱)𝐉_𝑏′′,=𝑑³𝑥𝜌(𝐱)𝐱×𝐀_𝑡(𝐱). (6)(7)(8)

The bound component remains the same as (8). It will be shown in Section 2 that the terms that involve the scalar potential in (6) and (7) cancel so that 𝐉=𝐋^′′+𝐒^′′+𝐉_𝑏^′′ in (6), (7), and (8) and 𝐉=𝐋^′′′+𝐒^′′′+𝐉_𝑏^′′ in (9), (10), and (8) but 𝐒^′′ in (6) differs from 𝐒^′′′ in (9) and 𝐋^′′ in (7) differs from 𝐋^′′′ in (10). The forms of (9) and (10) have also been used by van Enk and Nienhuis [5]. The general explicit form for 𝐀_𝑡, given in (13), was not specified by these writers.

On the other hand, Stewart [7] found a decomposition 𝐉=𝐋^′′′′+𝐒^′′′′+𝐉_𝑏^′′ from decomposing the electric field by the Helmholtz theorem [8]: 𝐒^′′′′=1𝑑4𝜋³𝑥𝑑³𝑦𝐁(𝐱)×𝐁̇(𝐲)|𝐱−𝐲|, (11)𝐋^′′′′=1𝑑4𝜋³𝑥𝑑³𝑦𝐁(𝐱)⋅𝐁̇(𝐲)𝐱×𝐲|𝐱−𝐲|³. (12) This decomposition uses the 𝐄 and 𝐁 fields throughout so no issues of gauge arbitrariness arise. Nor is there any complication that arises from using the longitudinal and transverse components of the fields, as 𝐁 is entirely transverse. The expression for 𝐉_𝑏^′′ was again given by (8) [6]. Equations (8), (11), and (12) have been used elsewhere [9] to show that the electromagnetic field makes zero contribution to the angular momentum of the physical electron described by the Lagrangian of quantum electrodynamics and to resolve the paradox concerning the angular momentum of a plane electromagnetic wave [10]. Application has also been made to paraxial rays [11] where it has been shown that the same results are obtained from (11) and (12) as from (9) and (10).

The question addressed in this paper is how (11) and (12) are related to (9) and (10) when the relation below (13) [12, 13] that expresses the vector potential of the Coulomb gauge explicitly in terms of the instantaneous magnetic field is used: 𝐀_𝑡(𝐱)=∇_𝑥×𝑑³𝑦𝐁(𝐲)||𝑑4𝜋𝐱−𝐲=−³𝑦𝐁(𝐲)4𝜋×∇_𝑥1||.𝐱−𝐲 (13) In Section 2 we summarise the derivation [3] leading from (1) to (9) and (10). In Section 3 we obtain an expression for 𝐒^′′′ of (9) and 𝐋^′′′ of (10) using (13) and demonstrate that 𝐋^′′′+𝐒^′′′=𝐋^′′′′+𝐒^′′′′. Section 4 summarises the conclusion of the paper that there is, on the basis of classical physics, no unique decomposition of the angular momentum of the electromagnetic field into spin and orbital components, even in a fixed inertial frame.

2. Derivation of the Standard Version of the Angular Momentum Decomposition

To obtain (3) and (4) from (1) we consider the component of the cross product 𝐄×∇_𝑥×𝐀^𝑖=𝜀^𝑖𝑗𝑘𝜀^𝑘𝑙𝑚𝐸^𝑗𝜕𝐴^𝑚𝜕𝑥^𝑙. (14) By multiplying the Levi-Civita symbols we find [4] 𝐄×∇_𝑥×𝐀=𝐸^𝑛∇_𝑥𝐴^𝑛−𝐄⋅∇_𝑥𝐀. (15) Next, consider the identity 𝜕𝜕𝑥^𝑚𝑥^𝑗𝐸^𝑚𝐴^𝑘=𝛿_𝑗,𝑚𝐸^𝑚𝐴^𝑘+𝑥^𝑗𝐴^𝑘𝜕𝐸^𝑚𝜕𝑥^𝑚+𝐸^𝑚𝜕𝐴^𝑘𝜕𝑥^𝑚 (16) and integrate this over 𝑥^𝑚 from minus infinity to plus infinity. The left-hand side vanishes because the integrand is zero at those points. Integrate over the other two components of 𝐱 to produce a volume integral over all space of d³𝑥 then sum over 𝑚. The result, when multiplied by 𝜀^𝑖𝑗𝑘 and integrated, substituted into the angular momentum obtained from (15), and specialised to the Coulomb gauge, is 𝐉=𝑑³𝑥𝐱×𝐀_𝑡∇_𝑥⋅𝐄+𝐄×𝐀_𝑡+𝐸^𝑛𝐱×∇_𝑥𝐴^𝑛_𝑡. (17) This reproduces (3), (4), and (5) for general gauge and (6), (7), and (8) for the Coulomb gauge.

The next step is to decompose the electric field into its longitudinal −∇𝜙 and transverse 𝐄_𝑡 components. Since the divergence of a transverse vector field is zero, the first term in (17) is unchanged from (8). The term associated with the potential in the third term of (17) is Δ𝐽³|^𝑖=−_{𝑗,𝑘,𝑛}𝑑³𝑥𝜀^𝑖𝑗𝑘𝜕𝜙𝜕𝑥^𝑛𝑥^𝑗𝜕𝐴^𝑛_𝑡𝜕𝑥^𝑘. (18) Consider the identity 𝜕𝜕𝑥^𝑘𝑥^𝑗𝐴^𝑛_𝑡𝜕𝜙𝜕𝑥^𝑛=𝛿_𝑗,𝑘𝐴^𝑛_𝑡𝜕𝜙𝜕𝑥^𝑛+𝑥^𝑗𝜕𝐴^𝑛_𝑡𝜕𝑥^𝑘𝜕𝜙𝜕𝑥^𝑛+𝐴^𝑛_𝑡𝜕²𝜙𝜕𝑥^𝑘𝜕𝑥^𝑛. (19) When (19) is put into (18), the term coming from the first term of the right-hand side of (19) vanishes because of the product of the Kronecker delta and the Levi Civita functions, and we get Δ𝐽³|^𝑖=_{𝑗,𝑘,𝑛}𝑑³𝑥𝜀^𝑖𝑗𝑘𝑥^𝑗𝐴^𝑛_𝑡𝜕²𝜙𝜕𝑥^𝑘𝜕𝑥^𝑛. (20) From the identity 𝜕𝜕𝑥^𝑛𝑥^𝑗𝐴^𝑛_𝑡𝜕𝜙𝜕𝑥^𝑘=𝛿_𝑗,𝑛𝐴^𝑛_𝑡𝜕𝜙𝜕𝑥^𝑘+𝑥^𝑗𝜕𝐴^𝑛_𝑡𝜕𝑥^𝑛𝜕𝜙𝜕𝑥^𝑘+𝐴^𝑛_𝑡𝜕²𝜙𝜕𝑥^𝑘𝜕𝑥^𝑛 (21) and noting that ∇⋅𝐀_𝑡=0, we find Δ𝐽³|^𝑖=−_{𝑗,𝑘,𝑛}𝑑³𝑥𝜀^𝑖𝑗𝑘𝛿_𝑗,𝑛𝐴^𝑛_𝑡𝜕𝜙𝜕𝑥^𝑘=−_𝑗,𝑘𝑑³𝑥𝜀^𝑖𝑗𝑘𝐴^𝑗_𝑡𝜕𝜙𝜕𝑥^𝑘. (22) The angular momentum of the second term of (17) arising from the potential is Δ𝐽²|^𝑖=−_𝑗,𝑘𝑑³𝑥𝜀^𝑖𝑗𝑘𝜕𝜙𝜕𝑥^𝑗𝐴^𝑘_𝑡. (23) Noting the order of the indices, (22) and (23) sum to zero; so we find that the contribution of the longitudinal electric field to the total angular momentum is zero, so 𝐉=𝐋^′′′+𝐒^′′′+𝐉_𝑏^′′ with the components 𝐋^′′′ and 𝐒^′′′ given by (9) and (10). We see already that the angular momentum of the electromagnetic field can be expressed in two different ways, as 𝐉=𝐋^′′′+𝐒^′′′+𝐉_𝑏^′′ given by (6), (7), and (8) and as 𝐉=𝐋^′′+𝐒^′′+𝐉_𝑏^′′ given by (9), (10), and (8).

3. Confirmation of the New Version of the Angular Momentum Decomposition

In this section of the paper we substitute the expression for the vector potential of the Coulomb gauge (13) into (9) and (10) in order to find how (9) and (10) are related to (11) and (12). We will find that the sum of (9) and (10) is equal to the sum of (11) and (12). First we examine 𝐋^′′′ of (10). This may be expressed as 𝐋^′′′=−𝑑³𝑥𝐱×∇_𝑥𝐀̇_𝑡(𝐳)⋅𝐀_𝑡(𝐱)_𝐳=𝐱=−𝑑³𝑥𝐱×𝐀̇_𝑡(𝐱)×𝐁(𝐱)+𝐀̇_𝑡(𝐱)⋅∇_𝐱𝐀_𝑡(𝐱). (24) Using (13), the first term of (24) comes to 𝐋^′′′1=−𝑑³𝑥𝑑³𝑦𝐁(𝐱)×𝐁̇(𝐲)4𝜋𝐱××∇_𝑥1|𝐱−𝐲|, (25) and, by multiplying out the triple vector product, this becomes 𝐋^′′′1=−𝑑³𝑥𝑑³𝑦̇𝐁(𝐲𝐁(𝐱)4𝜋𝐱×)⋅∇_𝑥1|𝐱−𝐲|−∇_𝑥1||𝐁̇(𝐲)(𝐱.𝐱−𝐲⋅𝐁) (26) When the gradient in the second term of (26), 𝐋^′′′12, is expressed explicitly, the term becomes 𝐋^′′′12=1𝑑4𝜋³𝑥𝑑³𝑦𝐁̇(𝐲)⋅𝐁(𝐱)𝐱×𝐲|𝐱−𝐲|³. (27) The first term in (26) is expressed in components as 𝐋^′′′11|^𝑖=−_{𝑗,𝑘,𝑚}𝑑³𝑥𝑑³𝑦𝜀4𝜋^𝑖𝑗𝑘𝑥^𝑗𝐵̇^𝑘(𝐲)𝐵^𝑚(𝐱)𝜕𝜕𝑥^𝑚1|𝐱−𝐲|. (28) By considering the identity 𝜕𝜕𝑥^𝑚𝐵̇^𝑘(𝐲)𝑥^𝑗𝐵^𝑚(𝐱)||=𝐵̇𝐱−𝐲_𝑘(𝐲)𝛿_𝑗,𝑚𝐵^𝑚(𝐱)|𝐱−𝐲|+𝑥^𝑗𝐵^𝑚(𝐱)𝜕𝜕𝑥^𝑚1||+1𝐱−𝐲|𝐱−𝐲|𝜕𝐵^𝑚(𝐱)𝜕𝑥^𝑚, (29) we obtain, with ∇⋅𝐁=0, 𝐋^′′′11|^𝑖=_{𝑗,𝑘,𝑚}𝑑³𝑥𝑑³𝑦𝜀4𝜋^𝑖𝑗𝑘𝐵̇^𝑘(𝐲)𝛿_𝑗𝑚𝐵^𝑚(𝐱)|𝐱−𝐲| (30) or 𝐋^′′′11=1𝑑4𝜋³𝑥𝑑³𝑦𝐁(𝐱)×𝐁̇(𝐲)|𝐱−𝐲|. (31) The second term 𝐋^′′′2 of (24) is simplified by writing it in components and using the relation 𝜕𝜕𝑥^𝑚𝑥^𝑗𝐴̇^𝑚_𝑡𝐴^𝑘_𝑡=𝛿_𝑗𝑚𝐴̇^𝑚_𝑡𝐴^𝑘_𝑡+𝑥^𝑗𝜕𝐴̇^𝑚_𝑡𝜕𝑥^𝑚𝐴^𝑘_𝑡+𝐴̇^𝑚_𝑡𝜕𝐴^𝑘_𝑡𝜕𝑥^𝑚 (32) with the gauge condition ∇⋅𝐀_𝑡=0 to get 𝐋^′′′2=𝑑³𝑥𝐀̇_𝑡(𝐱)×𝐀_𝑡(𝐱). (33) Accordingly, 𝐋^′′′=𝐋^′′′11+𝐋^′′′12+𝐋^′′′2 and 𝐋^′′′=1𝑑4𝜋³𝑥𝑑³𝑦𝐁(𝐱)×𝐁̇(𝐲)||+1𝐱−𝐲𝑑4𝜋³𝑥𝑑³𝑦𝐁̇(𝐲)⋅𝐁(𝐱)𝐱×𝐲|𝐱−𝐲|³+𝑑³𝑥𝐀̇_𝑡(𝐱)×𝐀_𝑡(𝐱). (34) Since 𝐒^′′′ of (9) is given by the negative of the last term of (34), it follows that 𝐋^′′′+𝐒^′′′=𝐋^′′′+𝐒^′′′, confirming the validity of (11) and (12).

4. Conclusion

In this paper we have compared three different expressions for the angular momentum of the electromagnetic field; they are 𝐉=𝑑³𝑥𝐱×𝐀_𝑡𝜌(𝐱)+𝜀₀𝐄×𝐀_𝑡+𝜀₀𝐸^𝑛𝐱×∇_𝑥𝐴^𝑛_𝑡, (35)𝐉=𝑑³𝑥𝐱×𝐀_𝑡𝜌(𝐱)+𝜀₀𝐄_𝑡×𝐀_𝑡+𝜀₀𝐸^𝑛_𝑡𝐱×∇_𝑥𝐴^𝑛_𝑡, (36)𝐉=𝑑³𝑥𝐱×𝐀_𝑡(𝐱)𝜌(𝐱)+𝜀₀𝑑4𝜋³𝑥𝑑³𝑦𝐁(𝐱)×𝐁̇(𝐲)||+𝜀𝐱−𝐲₀𝑑4𝜋³𝑥𝑑³𝑦𝐁(𝐱)⋅𝐁̇(𝐲)𝐱×𝐲|𝐱−𝐲|³, (37) where the vector potential of the Coulomb gauge 𝐀_𝑡 is given by (13) and the 𝜀₀ factor has been restored. The expressions all have the algebraic form of the sum of a spin and an orbital components, and each component of all three expressions is gauge invariant. All three expressions are different, and, on the basis of classical electrodynamics, there seems to be no reason to prefer one to another. The last form (37) [7] has the pedagogical advantage of being derived entirely in terms of fields through the Helmholtz theorem; issues of gauge arbitrariness do not have to be addressed. It appears that there is no unique way of decomposing the angular momentum of the classical electromagnetic field into orbital and spin components, even in an inertial frame.