Projected photon spin and orbital operators

Assume the Axiom of Choice and a constant $\mathrm{\hslash }>0$, interpreted as the angular-momentum scale. Let $M={\mathbb{R}}^3\setminus \{0\}$, $\mathbf{n}(\mathbf{k})=\mathbf{k}\mathrm{/}\mathrm{\mid }\mathbf{k}\mathrm{\mid }$, and use the transverse one-photon Hilbert space

$${\mathcal{H}}_T^{\nu }=\{f\in L^2(M,d\nu ;{\mathbb{C}}^3):\mathbf{k}\cdot f(\mathbf{k})=0\nu \text{-almost everywhere}\},d\nu (\mathbf{k})=\frac{d^3\mathbf{k}}{2\mathrm{\mid }\mathbf{k}\mathrm{\mid }},$$

with the second-variable-linear Hermitian pairing of Transverse one-photon space and helicity projectors. Use its dense smooth transverse domain

$$\mathcal{D}=\{f\in C_c^{\mathrm{\infty }}(M;{\mathbb{C}}^3):\mathbf{k}\cdot f(\mathbf{k})=0\text{ for every }\mathbf{k}\in M\}\mathrm{.}$$

For $\mathbf{n}\in S^2$, let $P(\mathbf{n})=I-\mathbf{n}{\mathbf{n}}^T$ and $h(\mathbf{n})=i(\mathbf{n}\times \cdot )$, where the cross product is extended complex-linearly and ${ϵ}_{123}=1$. These are the transverse projection and helicity multiplier from Transverse one-photon space and helicity projectors; on $P(\mathbf{n}){\mathbb{C}}^3$, $h$ is self-adjoint, $h^2=P$, and its eigenvalues are $\sigma \in \{+1,-1\}$.

Let $e_1,e_2,e_3$ be the standard basis and define the dimensionless spin-one matrices by

$$(S_{i}v{)}_a=-i\sum _{b=1}^3{ϵ}_{iab}{v}_b,i,a\in \{1,2,3\}\mathrm{.}$$

Let $J_i$ be the self-adjoint Stone generator of the rotation subgroup about $e_i$ supplied by Transverse rotations preserve helicity and generate total angular momentum, with convention $U(R_{e_i}(t))=e^{-it{J}_i\mathrm{/}\mathrm{\hslash }}$. Define the bounded projected spin operator and its orbital remainder by

$$S_i^{\mathrm{\prime }}=M_{\mathrm{\hslash }P(\mathbf{n})S_{i}P(\mathbf{n})}=M_{\mathrm{\hslash }{n}_{i}h(\mathbf{n})},L_i^{\mathrm{\prime }}=J_i-S_i^{\mathrm{\prime }}\text{on the exact domain }D(J_i)\mathrm{.}$$

Here $M_{A}f(\mathbf{k})=A(\mathbf{n}(\mathbf{k}))f(\mathbf{k})$ for a matrix field $A$; the proof below establishes the equality of the two formulas for $S_i^{\mathrm{\prime }}$, boundedness, self-adjointness, and invariance of $\mathcal{D}$. The label “projected spin” denotes this operator family and does not assert that its components form a spin-$1$ representation.

Transverse one-photon space and helicity projectors

Transverse rotations preserve helicity and generate total angular momentum

$J_i$ is fixed by the rotation action and is the total one-photon generator. The projected operator $S_i^{\mathrm{\prime }}$ is the helicity vector $\mathrm{\hslash }{n}_{i}h$, while $L_i^{\mathrm{\prime }}$ is defined as the remainder on the exact Stone domain. Their algebra and the self-adjointness of $L_i^{\mathrm{\prime }}$ are established in Projected angular parts are not two independent rotation algebras; no essential-self-adjointness claim is made in this definition.

In this page, “gauge-aware” means that the operators are defined on the transverse subspace using its projector. It does not assert a universal gauge-independent local density, an identity with pointwise products of quantized $A$ and $E$, or equality of canonical Noether and kinetic/Poynting field integrals. Those classical integral comparisons require the separate electromagnetic hypotheses and boundary terms; none is asserted here.