A conformally invariant derivation of average electromagnetic helicity

The average helicity of a given electromagnetic field measures the difference between the number of left- and right-handed photons contained in the field. In here, the average helicity is derived using the conformally-invariant inner-product for Maxwell fields. Several equivalent integral expressions in momentum space, in $(\mathbf{r},t)$ space, and in the time-harmonic $(\mathbf{r},\omega)$ space are obtained, featuring Riemann-Silberstein-like fields and potentials. The time-harmonic expressions can be directly evaluated using the outputs of common numerical solvers of Maxwell equations. The results are shown to be equivalent to the well-known volume integral for the average helicity, featuring the electric and magnetic fields and potentials.

that is, the projection of the vector |F onto the vector Γ|F . Since Γ is Hermitian, F |Γ|F must be a real number.
The crucial question of which inner product to choose was settled by Gross by requiring the inner product to be invariant under the conformal group [26]. That is: Given any two solutions |F and |G , and their corresponding transformed versions under any transformation in the conformal group, |F and |Ḡ , the inner product must be so that F |G = F |Ḡ .
Gross showed in Ref. [26] that this requirement essentially determines the exact expression of the inner product, which we will use later. The conformal group includes spacetime translations, spatial rotations, and Lorentz boosts, which together form the Poincaré group, plus spacetime scalings, and special conformal transformations [26]. The conformal group is the largest symmetry group of Maxwell equations in free space. Conformally invariant results have hence the maximum possible validity in electromagnetism.
An important distinction is in order at this point. The use of the conformally invariant inner product ensures the maximal validity for average quantities as defined by Eq. (1): The projection of the vector |F onto the vector Γ|F is equal to the projection of |F = T |F onto |ΓF = T Γ|F , i.e., F |T † T Γ|F = F |Γ|F , for any transformation T in the conformal group, where T † is the Hermitian adjoint of T . Satisfying this demand amounts to showing that an inner product exists with respect to which the conformal group acts unitarily (T T † = T † T = I for all T , where I is the identity) on the vector space of solutions of Maxwell equations [26]. Loosely speaking, this means that the value of the averages in Eq. (1) will not change regardless of "the conformal point of view" or "conformal change of coordinate system". This will hold for average helicity, and also for average momentum, average angular momentum, etc. . . . . A different matter is whether the average quantity in a conformally transformed field is the same as the average quantity in the initial field, for all conformal transformations. In this case, we are asking whether |F and T |F have the same average value of a given property Γ, i.e., whether which is often not the case, such as for example when a Lorentz boost simultaneously changes the energy and momentum of a given field. Incidentally, it will be clear later that Eq. (2) is actually met in the case of average helicity.
Writing down an explicit expression for Eq. (1) requires us to choose an explicit representation for the vectors in M and the operators acting on them. We choose the following representation for the vectors in M: where the F ± (k) define the plane-wave components of a version of the Riemann-Silberstein where ǫ 0 , µ 0 , c 0 , and Z 0 = µ 0 /ǫ 0 are the vacuum's permittivity, permeability, speed of light, and impedance, respectively, k is the wavevector, and ω = c 0 k = c 0 √ k · k is the angular frequency. The F ± (k) can be further decomposed as are complex-valued scalar functions andê ± (k) are thek-dependent polarization vectors 1 for each handedness(helicity). We note thatk ·ê ± (k) = 0, which makes the F ± (k)[F ± (r, t))] transverse functions, namely,k · F ± (k) = ∇ · F ± (r, t) = 0. The origin k = 0 is removed from the integral in Eq. (4) because we are considering electrodynamics and excluding electroand magneto-statics, whereby k = ω/c 0 = 0 needs to be excluded.
It important to note that only positive frequencies are included in Eq. (4). This amounts to considering positive energies only, which is possible in electromagnetism since the photon is its own anti-particle. Only one sign of the energy(frequency) is needed because the same information is contained on both sides of the spectrum [5, §3.1] [4]. When only positive frequencies are included, D(r, t), B(r, t), E(r, t) and H(r, t) in Eq. (4) are complex-valued fields. With X standing for D, B, E or H: 1 Theê ± (k) can be obtained by the rotation of (±x − iŷ)/ √ 2, the two vectors corresponding tok =ẑ: We define the complex-valued fields so that the typical real-valued versions are obtained as The restriction to positive frequencies is particularly consequential for the treatment of helicity, the generalized polarization handednesses of the field. One of the advantages of the Riemann-Silberstein vectors is their ability to encode the helicity content of the field. They are the eigenstates of the helicity operator and potentially allow for the splitting of the two polarization handedness in any field, including near and evanescent fields. However, when they are defined by means of real-valued fields, as in D(r,t) √ 2ǫ 0 ± i B(r,t) √ 2µ 0 , their use for splitting the two helicities is not as simple as it becomes when complex-valued fields are used. With real-valued fields we have that the two ± fields determine each other through complex which is at odds with the a priori physical independence of the two helicity components of the electromagnetic field. For example, the complex conjugation connection means that the two ± squared norms D( which could intuitively be thought of as the (r, t)-local helicity intensities, become equal . This contradicts, for example, the fact that there can be electromagnetic fields containing only one of the two helicities, e.g., any linear combination of plane-waves with the same polarization handedness. The Let us go on to computing the average helicity of a given field as a conformally invariant inner product. We will explicitly keep the constants ǫ 0 , µ 0 , c 0 , and Z 0 in the expressions, and use the four fields D, B, E and H. These choices [21,23] facilitate the re-use of the formulas when a description such as the one in Eq. (4) is possible in a non-vacuum background, such as for example in an infinite isotropic and homogeneous linear medium.
We are now ready to focus our attention on the average value of helicity. The helicity operator Λ is defined as the projection of the angular momentum operator vector J onto the direction of the linear momentum operator vector P:  where for electromagnetism, S is the vector of spin-1 matrices 6 .
We start by particularizing Eq. (7) to the helicity operator Λ.
where the last expression contains the explicit form of the helicity operator in our choice of representation 7 . We now use the fact that the F ± (k) are eigenstates of helicity, namely We will now show that Eq. (10) is equivalent to the most common integral expression of the helicity average. To such end, and taking advantage of the fact that k = 0, we define the helicity potentials which in the (r, t) representation, and recalling that −iω → ∂ t , are seen to be related to from where we can use [21, Eq. (2)], namely −∂ t C(r, t) = H(r, t) and − ∂ t A(r, t) = E(r, t), to recognize that these helicity potentials are linear combinations of complex versions of the transverse real-valued "magnetic" A(r, t) and "electric" C(r, t) potentials [2, 8, 11-13, 17, 25]. As previously discussed, this difference is relevant for treating helicity. When real-valued fields are used in the right hand side of Eq. (13), it follows that V + (r, t) * = V − (r, t), which ultimately leads to a zero value of the average helicity as reported in [17].
It is also worth pointing out that the V ± (k) functions are transverse, i.e.,k · V ± (k) = 0, which follows from Eq. (11) and the previously mentioned propertyk · F ± (k). The helicity potentials only contain the transverse degrees of freedom, the same as the free electromagnetic field, which ensures that the results obtained using V ± (k) are gauge independent.
We proceed by using Eq. (11) and the central expression in Eq. (10) to obtain Equation (14) can now be brought to the (r, t) domain as follows. First, we apply the These changes do not affect the result, but allow us to see from Eq. (4) that the F ± (k) exp(−ic 0 k) are the three-dimensional Fourier transforms (r → k) of F ± (r, t). The same relation holds between V ± (k) exp(−ic 0 k) and V ± (r, t). We can now apply applying Parseval's theorem, i.e., the unitarity of the inverse Fourier transform k → r, to each of the two terms in the subtraction in Eq. (15): where the integrand is local in r. We show in Appendix A that when Eq. (10) is brought to the (r, t) domain instead, the 1/|k| term results in the double integral R 3 dr R 3 dr of a manifestly non-r-local integrand including a term such as 1/|r −r| 2 . The inconvenient 1/|k| term is absorbed in the definition of the potentials in Eq. (11).
To further approach the most common expression of the average helicity, we now substi- into Eq. (16) and obtain which is a complex version of the well-known integral for the average helicity featuring real-valued fields, as found e.g., in [21, Eq. (6)]. Appendix B shows that the results of the complex and real versions coincide.
The k-domain expressions in Eqs. (10) and (14) Let us examine the last line of Eq. (19). Because F |Λ|F cannot depend on time, and since the two helicities are independent of each other, it follows that only the ω =ω components 8 Indeed, the simplifying arbitrary choice t = 0 is made by Gross in [26] for evaluating the inner product with integrals featuring (r, t)-dependent integrands.
can contribute to the end result. This allows us to obtain the following three equivalent expressions: where the equalities readily follow from F ω ± (r) = iωV ω ± (r), which follows from Eq. (12). Expressions that are local in r, such as Eqs. (14) and (20) Finally, regarding applications, the electromagnetic helicity is particularly relevant in chiral light-matter interactions. Among these, the interaction of the field with chiral molecules is one of the most researched cases, partly because the optical sensing of chiral molecules is important in chemistry and pharmaceutical applications. In Appendix C we use the above formalism to derive expressions for computing the circular dichroism signal for two different settings of the light-molecule interaction: The 6×6 dipolarizability tensor and the T-matrix.
In conclusion, several equivalent expressions for the average value of the electromagnetic helicity of a given field have been obtained from a starting point featuring maximal electromagnetic invariance, i.e., from an expression whose result is invariant under the conformal group. Some of the obtained expressions can be conveniently evaluated using the outputs of common Maxwell solvers. The use of two potentials, one magnetic and one electric, has a long tradition in the studies of helicity and of the symmetry generated by the helicity operator: Electromagnetic duality [2, 8, 11-13, 17, 25]. Duality can be seen as the underlying reason for adding an electric potential next to the magnetic one.
In the absence of sources, the (real-valued) electric potential C(r, t) is typically defined by first fixing its transverse part where E ⊥ (r, t) is the transverse electric field, and then exploiting the fact that C(r, t) has its own gauge freedom [2,13,22] to fix the longitudinal part by a choice of gauge. When the radiation gauge (∇ · C(r, t) = 0) is chosen, C(r, t) becomes a transverse field, containing the same kind of degrees of freedom as the radiation electromagnetic fields. The electric potential has also been used in the presence of sources [13,22,25]. In the particular context of integral expressions for the average electromagnetic helicity, the introduction of potentials allows r-local integrands. This has been shown in the main text in the derivations leading to the r-local Eq. (16). We will now bring Eq. (10), which does not involve the potentials, to the (r, t) domain and see how precisely the non-r-locality arises. We start hence from Eq. (10), which only contains the F ± (k) fields, and apply the non-result-altering substitutions F ± (k) → F ± (k) exp(−ic 0 k): We will first focus on the first term of the integrand, which we consider as the k-point-wise Using Eq. (A3) and its obvious counterpart for the second term in the integrand of Eq. (A2), we can use Parseval's theorem to write: As explained in the main text, the typically undesired non-locality of the integrand in In this Appendix we show that Eq. (18) of the main text, featuring complex-valued fields is equivalent to the well-known integral for the average helicity featuring real-valued fields, as found e.g. in [21, Eq. 6]: We will use properties of complex-valued vector fields whose Fourier transforms contain only positive frequencies, as defined in Eq. (5) for X standing for A, B, C, D, and E: 9 We note that previously existing non-local expressions for average electromagnetic helicity, like [34, Eq. (65)] and [24,Eq. (36)], can be shown to be equivalent to Eq. (A4).
The real and imaginary parts of X(r, t) = X re (r, t) + iX im (r, t) are related by the Hilbert transform, and then their Fourier transforms, denoted by F {·}, meet [35, p. 49]: We now proceed by writing Eq. (B1) using the real and imaginary parts of each field. Since the end result of the integral must be a real number, we can already discard the imaginary part of the integrand: We will now show that the two expressions in square brackets produce the same contribution.
To such end, let us focus on one of their terms and use the time-harmonic decomposition 10 The same considerations that take the last line of Eq. (19) to the first line of Eq. (20) can be used to write: We now use Eq. (B4) on the last expression in Eq. (B7) where the last equality follows by comparison with Eq. (B7). Equation (B8) shows that the two expressions inside the square brackets in Eq. (B5) produce the same contribution, since 10 The one sided integral in Eq. (6) can be written as a two sided integral over the frequency axis and the familiar result X ω (r) * = X −ω (r) is recovered.
the steps leading to Eq. (B8) can be applied to any of the product terms. We can hence write: Equivalence with Eq. (B2) is shown after considering that the definition of the typical real fields X (r, t) in Eq. (6) implies that X (r, t) = 2X re (r, t). Equation (B2) is finally reached by substituting all the X re (r, t) fields with X (r, t)/2 in Eq. (B9).

Appendix C: Expressions for computing Circular Dichroism
Let us assume that a chiral molecule is located at point r, and embedded in a possibly frequency-dispersive, homogeneous, isotropic, achiral, and non-magnetic medium with per- The wavenumber in such medium is k ω = ω/c ω . In this Appendix, these frequency-dependent quantities are assumed to substitute their constant vacuum counterparts in all the equations in the main text.
One of the most relevant techniques for chiral molecule sensing is Circular Dichroism (CD), which measures their differential absorption upon subsequent illumination with the two helicities. Assuming that the field scattered by the molecule, i.e., the field radiated by the induced dipoles in Eq. (C1), is negligible with respect to the incident field, it is possible to write the rotationally averaged molecular differential absorption as [36].
dω Re{α ω me }2c 2 ω C ω (r), (C2) where Re{α ω me } is the real part of the rotational average of α ω me , C ω (r) is the optical chirality density introduced by Tang and Cohen [37], and the second equality follows from [16,Eq. (5)] and Eq. (4). Besides the dipolarizability model in Eq. (C1), other light-molecule interaction descriptions are possible. For example, the T-matrix of the molecule may be used. The T-matrix is a common object in physics and engineering, which is intrinsically able to include all the multipolar orders of the light-matter interaction, and allows to efficiently compute the coupled electromagnetic response of different objects in a systematic and rigorous way [38].
The conversion between the dipolarizability tensor and the T-matrix of the molecule up to the dipolar order is [39, Eq. (A15)]: where N(M) refers to the electric(magnetic) character. We may now use this conversion to substitute Re{α ω me } in Eq. (C2): In experimental measurements, a solution of chiral molecules is confined in its recipient, which defines a volume D. Assuming uniform concentration of molecules over D, the total CD signal can be computed by the volume integral of any of the expressions in Eq. (C2) or Eq. (C4) over D. For example: where ρ is a constant that depends on the molecular concentration. In the last expression, we recognize one of the integrands from the average helicity in Eq. (20).