Appendix
Expressions (37) and (38) for the two-body currents associated with the operators \((\textbf{L}\cdot \textbf{S})^2\) and \((\textbf{L}\cdot \textbf{L})\) in non-relativistic interactions are derived in this appendix. The construction of these current operators is similar to the construction used in the \(\textbf{L}\cdot \textbf{S}\) case. The main difference is that these contributions involve products of covariant derivatives. The current is the coefficient of the vector potential.
The interactions before replacing the derivatives by covariant derivatives are
$$\begin{aligned}{} & {} v_{(\textbf{L}\cdot \textbf{S})^2}(\vert \textbf{q} \vert ) (\textbf{L}\cdot \textbf{S})^2 = v_{(\textbf{L}\cdot \textbf{S})^2}(\vert \textbf{q} \vert ) (\textbf{S} \times \textbf{q})\cdot \textbf{p} (\textbf{S} \times \textbf{q})\cdot \textbf{p} \nonumber \\{} & {} \quad = v_{(\textbf{L}\cdot \textbf{S})^2}(\vert \textbf{q} \vert )[ \sum _{ij}(\textbf{S} \times \textbf{q})_i (\textbf{S} \times \textbf{q})_j p_ip_j +i (\textbf{S} \times \textbf{q}) \cdot (\textbf{S} \times \textbf{p})] \end{aligned}$$
(106)
and
$$\begin{aligned} v_{\textbf{L}\cdot \textbf{L}}(\vert \textbf{q} \vert ) \textbf{L}\cdot \textbf{L} = v_{\textbf{L}\cdot \textbf{L}}(\vert \textbf{q} \vert )[ \textbf{q}^2 \textbf{p}^2 - \textbf{q} \cdot (\textbf{q} \cdot \textbf{p}) \textbf{p} + 2 i \textbf{q} \cdot \textbf{p}]. \end{aligned}$$
(107)
where in these expressions the canonical commutation relations are used to move the momentum operators to the right of the coordinate operators.
In order to extract the current it is useful to express (106) and (107) in terms of single-particle variables, where minimal substitution is straightforward:
$$\begin{aligned}{} & {} v_{(\textbf{L}\cdot \textbf{S})^2}(\vert \textbf{q} \vert ) (\textbf{L}\cdot \textbf{S})^2 \nonumber \\{} & {} \quad = \sum _{ij}v_{(\textbf{L}\cdot \textbf{S})^2}(\vert \textbf{q}_1-\textbf{q}_2 \vert )[ (\textbf{S} \times (\textbf{q}_1-\textbf{q}_2))_i (\textbf{S} \times (\textbf{q}_1-\textbf{q}_2))_j {(\textbf{p}_1 - \textbf{p}_2)_i \over 2}{(\textbf{p}_1 - \textbf{p}_2)_j \over 2} \nonumber \\{} & {} \quad +{i\over 2} (\textbf{S} \times (\textbf{q}_1- \textbf{q}_2)) \cdot (\textbf{S} \times (\textbf{p}_1 -\textbf{p}_2))] \end{aligned}$$
(108)
and
$$\begin{aligned}{} & {} v_{\textbf{L}\cdot \textbf{L}}(\vert \textbf{q}_1-\textbf{q}_2 \vert ) \textbf{L}\cdot \textbf{L} \nonumber \\{} & {} \quad = v_{\textbf{L}\cdot \textbf{L}}(\vert \textbf{q}_1-\textbf{q}_2 \vert )[ {1 \over 4}(\textbf{q}_1-\textbf{q}_2)^2 (\textbf{p}_1-\textbf{p}_2)^2 \nonumber \\{} & {} \quad - {1 \over 4}(\textbf{q}_1-\textbf{q}_2) \cdot ((\textbf{q}_1-\textbf{q}_2) \cdot (\textbf{p}_1-\textbf{p}_2))( \textbf{p}_1-\textbf{p}_2) + i (\textbf{q}_1 -\textbf{q}_2) \cdot (\textbf{p}_1- \textbf{p}_2)]. \end{aligned}$$
(109)
These are operator expressions. Replacing the space derivatives by covariant derivatives is equivalent to replacing \(\textbf{p}_i\) by \(\textbf{p}_i -e_i \textbf{A}(\textbf{q}_i)\), maintaining the correct operator ordering. This substitution results in the gauge invariant operators
$$\begin{aligned}{} & {} v_{(\textbf{L}\cdot \textbf{S})^2}(\vert \textbf{q} \vert ) (\textbf{L}\cdot \textbf{S})^2 \nonumber \\{} & {} \quad \rightarrow v_{(\textbf{L}\cdot \textbf{S})^2}(\vert \textbf{q}_1-\textbf{q}_2 \vert ) \left[ {1 \over 4} \sum _{ij} (\textbf{S} \times (\textbf{q}_1-\textbf{q}_2))_i (\textbf{S} \times (\textbf{q}_1-\textbf{q}_2))_j \right. \nonumber \\{} & {} \quad \left. \times (\textbf{p}_1 -e_1 \textbf{A}(\textbf{q}_1) - \textbf{p}_2 +e_2 \textbf{A}(\textbf{q}_2))_i (\textbf{p}_1 -e_1 \textbf{A}(\textbf{q}_1) - \textbf{p}_2 +e_2 \textbf{A}(\textbf{q}_2))_j \right. \nonumber \\{} & {} \quad + \left. {i\over 2} (\textbf{S} \times (\textbf{q}_1- \textbf{q}_2)) \cdot (\textbf{S} \times {(\textbf{p}_1 -e_1 \textbf{A}(\textbf{q}_1) - \textbf{p}_2 +e_2 \textbf{A}(\textbf{q}_2))} \right] \end{aligned}$$
(110)
and
$$\begin{aligned}{} & {} v_{\textbf{L}\cdot \textbf{L}}(\vert \textbf{q}_1-\textbf{q}_2 \vert ) \textbf{L}\cdot \textbf{L} \rightarrow v_{\textbf{L}\cdot \textbf{L}}(\vert \textbf{q}_1-\textbf{q}_2 \vert ) \nonumber \\{} & {} \quad \times \left[ {1 \over 4}(\textbf{q}_1-\textbf{q}_2)^2 {(\textbf{p}_1 -e_1 \textbf{A}(\textbf{q}_1) - \textbf{p}_2 +e_2 \textbf{A}(\textbf{q}_2))}\cdot {(\textbf{p}_1 -e_1 \textbf{A}(\textbf{q}_1) - \textbf{p}_2 +e_2 \textbf{A}(\textbf{q}_2))} \right. \nonumber \\{} & {} \quad - {1 \over 4}(\textbf{q}_1-\textbf{q}_2) \cdot [(\textbf{q}_1-\textbf{q}_2) \cdot {(\textbf{p}_1 -e_1 \textbf{A}(\textbf{q}_1) - \textbf{p}_2 +e_2 \textbf{A}(\textbf{q}_2)) } \cdot {(\textbf{p}_1 -e_1 \textbf{A}(\textbf{q}_1) - \textbf{p}_2 +e_2 \textbf{A}(\textbf{q}_2)) } \nonumber \\{} & {} \quad \left. + i (\textbf{q}_1 -\textbf{q}_2) \cdot {(\textbf{p}_1 -e_1 \textbf{A}(\textbf{q}_1) - \textbf{p}_2 +e_2 \textbf{A}(\textbf{q}_2))} \right] . \end{aligned}$$
(111)
Keeping only the terms that are linear in \(\textbf{A}\) in (110) and (111) and expressing the result in terms of the total and relative momenta and their conjugate coordinates gives:
$$\begin{aligned}{} & {} \quad (110)\rightarrow v_{(\textbf{L}\cdot \textbf{S})^2}(\vert \textbf{q}_1-\textbf{q}_2 \vert ) \nonumber \\{} & {} \quad \times \left[ -{1 \over 2} (\textbf{S} \times \textbf{q})_i (\textbf{S} \times \textbf{q})_j \{ \textbf{p}_i (e_1\textbf{A}(\textbf{Q}+{\textbf{q}\over 2}) -e_2\textbf{A}((\textbf{Q}-{\textbf{q}\over 2}))_j \right. \nonumber \\{} & {} \quad + \left. (e_1\textbf{A}(\textbf{Q}+{\textbf{q}\over 2}) -e_2\textbf{A}(\textbf{Q}-{\textbf{q}\over 2}))_i \textbf{p}_j\} \right. \nonumber \\{} & {} \quad + \left. -{i\over 2} (\textbf{S} \times \textbf{q}) \cdot (\textbf{S} \times (e_1\textbf{A}(\textbf{Q}+{\textbf{q}\over 2}) - e_2\textbf{A} (\textbf{Q}-{\textbf{q}\over 2})) \right] \end{aligned}$$
(112)
and
$$\begin{aligned}{} & {} (111)\rightarrow v_{\textbf{L}\cdot \textbf{L}}(\vert \textbf{q} \vert )\left\{ -{1 \over 2}\textbf{q}^2 \left[ (\textbf{p}\cdot ( e_1\textbf{A}(\textbf{Q}+{\textbf{q}\over 2}) - e_2\textbf{A}(\textbf{Q}-{\textbf{q}\over 2})) + \right. \right. \nonumber \\{} & {} \quad \left. \left. (e_1\textbf{A}(\textbf{Q}+{\textbf{q}\over 2}) - e_2\textbf{A}(\textbf{Q}-{\textbf{q}\over 2})) \cdot \textbf{p}) \right] \right. \nonumber \\{} & {} \quad + {1 \over 2}\{ \textbf{q} \cdot (\textbf{q} \cdot \textbf{p}) \cdot (e_1\textbf{A}(\textbf{Q}+{\textbf{q}\over 2})- e_2\textbf{A}(\textbf{Q}-{\textbf{q}\over 2})) + {1 \over 2}\textbf{q} \cdot (\textbf{q} \cdot (e_1\textbf{A}(\textbf{Q}+{\textbf{q}\over 2})- e_2\textbf{A}(\textbf{Q}-{\textbf{q}\over 2}) \textbf{p} \} \nonumber \\{} & {} \quad \left. \left. -i \textbf{q} \cdot ( e_1\textbf{A}(\textbf{Q}+{\textbf{q}\over 2}) -e_2 \textbf{A}(\textbf{Q}-{\textbf{q}\over 2}) \right] \right\} \end{aligned}$$
(113)
where in (112) \(\textbf{p}_i\) represents the i-th component of the relative momentum rather than the momentum of particle i The next step is to use the commutation relations to move the \(\textbf{p}\) factors to the right of all of the coordinate factors. There are three terms in (112) and (113) where the momenta are on the left of some of the coordinate operators. These terms are:
$$\begin{aligned}{} & {} \textbf{p}_i (e_1\textbf{A}(\textbf{Q}+{\textbf{q}\over 2}) -e_2\textbf{A}((\textbf{Q}-{\textbf{q}\over 2}))_j = \nonumber \\{} & {} \quad (e_1\textbf{A}(\textbf{Q}+{\textbf{q}\over 2}) -e_2\textbf{A}((\textbf{Q}-{\textbf{q}\over 2}))_j \textbf{p}_i - {i \over 2} \partial _{Qi} (e_1\textbf{A}(\textbf{Q}+{\textbf{q}\over 2}) +e_2\textbf{A}(\textbf{Q}-{\textbf{q}\over 2}))_j \end{aligned}$$
(114)
$$\begin{aligned}{} & {} \quad -{1 \over 2}\textbf{q}^2 \textbf{p}\cdot (e_1\textbf{A}(\textbf{Q}+{\textbf{q}\over 2}) - e_2\textbf{A}(\textbf{Q}-{\textbf{q}\over 2})) \nonumber \\{} & {} \quad = -{1 \over 2}\textbf{q}^2 (e_1\textbf{A}(\textbf{Q}+{\textbf{q}\over 2}) - e_2\textbf{A}(\textbf{Q}-{\textbf{q}\over 2})) \cdot \textbf{p} +i{1 \over 4}\textbf{q}^2 \pmb {\nabla }_Q\cdot (e_1\textbf{A}(\textbf{Q}+{\textbf{q}\over 2}) + e_2\textbf{A}(\textbf{Q}-{\textbf{q}\over 2})), \end{aligned}$$
(115)
$$\begin{aligned}{} & {} \quad {1 \over 2}\{ \textbf{q} \cdot (\textbf{q} \cdot \textbf{p}) \cdot (e_1\textbf{A}(\textbf{Q}+{\textbf{q}\over 2})- e_2\textbf{A}(\textbf{Q}-{\textbf{q}\over 2})) \nonumber \\{} & {} \quad = {1 \over 2}\{ \textbf{q} \cdot (e_1\textbf{A}(\textbf{Q}+{\textbf{q}\over 2})- e_2\textbf{A}(\textbf{Q}-{\textbf{q}\over 2})) (\textbf{q} \cdot \textbf{p}) -i{1 \over 4}\{ \textbf{q} \cdot (\textbf{q} \cdot \pmb {\nabla }_Q) (e_1\textbf{A}(\textbf{Q}+{\textbf{q}\over 2})+ e_2\textbf{A}(\textbf{Q}-{\textbf{q}\over 2})). \nonumber \\ \end{aligned}$$
(116)
Using these identities in the operator expressions (112) and (113) above gives
$$\begin{aligned}{} & {} (112) \nonumber \\{} & {} \quad = v_{(\textbf{L}\cdot \textbf{S})^2}(\vert \textbf{q} \vert ) \left[ -{1 \over 2} (\textbf{S} \times \textbf{q})_i (\textbf{S} \times \textbf{q})_j \{ (e_1\textbf{A}(\textbf{Q}+{\textbf{q}\over 2}) -e_2\textbf{A}(\textbf{Q}-{\textbf{q}\over 2}))_j \textbf{p}_i \right. \nonumber \\{} & {} \quad + (e_1\textbf{A}(\textbf{Q}+{\textbf{q}\over 2}) -e_2\textbf{A}(\textbf{Q}-{\textbf{q}\over 2}))_i\textbf{p}_j \nonumber \\{} & {} \quad - {i \over 2} ((e_1\partial _{iQ} \textbf{A}(\textbf{Q}+{\textbf{q}\over 2}) +e_2\partial _{iQ}\textbf{A}(\textbf{Q}-{\textbf{q}\over 2}))_j \} \nonumber \\{} & {} \quad \left. -{i\over 2} (\textbf{S} \times \textbf{q}) \cdot (\textbf{S} \times (e_1\textbf{A}(\textbf{Q}+{\textbf{q}\over 2}) - e_2\textbf{A} (\textbf{Q}-{\textbf{q}\over 2})) \right] \end{aligned}$$
(117)
and
$$\begin{aligned}{} & {} (113) \nonumber \\{} & {} \quad = v_{\textbf{L}\cdot \textbf{L}}(\vert \textbf{q} \vert )\left[ -{1 \over 2}\textbf{q}^2 ( 2 (e_1\textbf{A}(\textbf{Q}+{\textbf{q}\over 2}) - e_2\textbf{A}(\textbf{Q}-{\textbf{q}\over 2})) \cdot \textbf{p}) \right. \nonumber \\{} & {} \quad +i{1 \over 4}\textbf{q}^2 (e_1\pmb {\nabla }\cdot \textbf{A}(\textbf{Q}+{\textbf{q}\over 2})+ e_2\pmb {\nabla }\cdot \textbf{A}(\textbf{Q}-{\textbf{q}\over 2})) \nonumber \\{} & {} \quad + \{ \textbf{q} \cdot (e_1\textbf{A}(\textbf{Q}+{\textbf{q}\over 2})- e_2\textbf{A}(\textbf{Q}-{\textbf{q}\over 2}))(\textbf{q} \cdot \textbf{p}) -i{1 \over 4}\{ \textbf{q} \cdot (\textbf{q} \cdot \pmb {\nabla }_Q) (e_1\textbf{A}(\textbf{Q}+{\textbf{q}\over 2}) e_2\textbf{A}(\textbf{Q}-{\textbf{q}\over 2})) \nonumber \\{} & {} \quad \left. + -i \textbf{q} \cdot ( e_1\textbf{A}(\textbf{Q}+{\textbf{q}\over 2}) - e_2\textbf{A}(\textbf{Q}-{\textbf{q}\over 2}) \right] . \end{aligned}$$
(118)
Since the momentum operators are to the right of the coordinate operators, the operators become numbers in a mixed coordinate-momentum basis. The mixed matrix elements of (117) and (118) are
$$\begin{aligned}{} & {} \langle \textbf{Q},\textbf{q} \vert (117)\vert \textbf{P},\textbf{p} \rangle \nonumber \\{} & {} \quad = v_{(\textbf{L}\cdot \textbf{S})^2}(\vert \textbf{q} \vert ) {1 \over (2\pi )^3} e^{i \textbf{Q}\cdot \textbf{P} + i \textbf{q}\cdot \textbf{p}} \left[ - (\textbf{S}\times \textbf{q})_i (\textbf{S}\times \textbf{q})_j \left\{ p_i (e_1\textbf{A}(\textbf{Q}+{\textbf{q}\over 2}) -e_2\textbf{A}((\textbf{Q}-{\textbf{q}\over 2}))_j \right. \right. \nonumber \\{} & {} \quad \left. {i \over 4} (e_1\partial _{iQ} \textbf{A}(\textbf{Q}+{\textbf{q}\over 2}) +e_2\partial _{iQ}\textbf{A}(\textbf{Q}-{\textbf{q}\over 2}))_j \right\} \nonumber \\{} & {} \quad \left. -{i\over 2} (\textbf{S} \times \textbf{q}) \cdot (\textbf{S} \times (e_1\textbf{A}(\textbf{Q}+{\textbf{q}\over 2}) - e_2\textbf{A} (\textbf{Q}-{\textbf{q}\over 2})) \right] \end{aligned}$$
(119)
and
$$\begin{aligned}{} & {} \langle \textbf{Q},\textbf{q} \vert (118)\vert \textbf{P},\textbf{p} \rangle \nonumber \\{} & {} \quad = v_{\textbf{L}\cdot \textbf{L}}(\vert \textbf{q} \vert ) {1 \over (2\pi )^3} e^{i \textbf{Q}\cdot \textbf{P} + i \textbf{q}\cdot \textbf{p}} \left[ -\textbf{q}^2 ( e_1\textbf{A}(\textbf{Q}+{\textbf{q}\over 2} - e_2 \textbf{A}(\textbf{Q}-{\textbf{q}\over 2})) \cdot \textbf{p} \right. \nonumber \\{} & {} \quad \left. +{i \over 4}\textbf{q}^2 (e_1\pmb {\nabla }_Q\cdot \textbf{A}(\textbf{Q}+{\textbf{q}\over 2})+ e_2\pmb {\nabla }_Q\cdot \textbf{A}(\textbf{Q}-{\textbf{q}\over 2})) \right. \nonumber \\{} & {} \quad + (\textbf{q} \cdot \textbf{p}) \textbf{q} \cdot (e_1\textbf{A}(\textbf{Q}+{\textbf{q}\over 2})- e_2\textbf{A}(\textbf{Q}-{\textbf{q}\over 2})) \nonumber \\{} & {} \quad -{i \over 4}\{ \textbf{q} \cdot (\textbf{q} \cdot \pmb {\nabla }_Q) (e_1\textbf{A}(\textbf{Q}+{\textbf{q}\over 2})+ e_2\textbf{A}(\textbf{Q}-{\textbf{q}\over 2})) \nonumber \\{} & {} \quad \left. -i \textbf{q} \cdot ( e_1\textbf{A}(\textbf{Q}+{\textbf{q}\over 2}) - e_2\textbf{A}(\textbf{Q}-{\textbf{q}\over 2}) \right] . \end{aligned}$$
(120)
Next change the final variables to momentum variables. With this change these expressions become
$$\begin{aligned}{} & {} \langle \textbf{P}',\textbf{p}' \vert (117) \vert \textbf{P}',\textbf{p}' \rangle \nonumber \\{} & {} \quad = \int d\textbf{q} d \textbf{Q} v_{(\textbf{L}\cdot \textbf{S})^2}(\vert \textbf{q} \vert ) {1 \over (2\pi )^6} e^{i \textbf{Q}\cdot (\textbf{P}-\textbf{P}') + i \textbf{q}\cdot (\textbf{p}-\textbf{p}')} \nonumber \\{} & {} \qquad \times \left[ - (\textbf{S} \times \textbf{q})\cdot \textbf{p} (\textbf{S} \times \textbf{q}) \cdot \{ (e_1\textbf{A}(\textbf{Q}+{\textbf{q}\over 2}) -e_2\textbf{A}((\textbf{Q}-{\textbf{q}\over 2})) \right. \nonumber \\{} & {} \qquad -\sum _{ij} {i\over 4}(\textbf{S} \times \textbf{q})_i (\textbf{S} \times \textbf{q})_j (e_1\partial _i \textbf{A}(\textbf{Q}+{\textbf{q}\over 2}) +e_2\partial _i\textbf{A}(\textbf{Q}-{\textbf{q}\over 2}))_j \nonumber \\{} & {} \qquad \left. -{i\over 2}(\textbf{S}^2 \textbf{q}- (\textbf{q}\cdot \textbf{S}) \textbf{S} )\cdot (e_1\textbf{A}(\textbf{Q}+{\textbf{q}\over 2}) - e_2\textbf{A} (\textbf{Q}-{\textbf{q}\over 2})) \right] . \end{aligned}$$
(121)
and
$$\begin{aligned}{} & {} \langle \textbf{P}',\textbf{p}' \vert (118) \vert \textbf{P}',\textbf{p}' \rangle \nonumber \\{} & {} \quad = \int d\textbf{q} d \textbf{Q} v_{\textbf{L}\cdot \textbf{L}}(\vert \textbf{q} \vert ) {1 \over (2\pi )^6} e^{i \textbf{Q}\cdot (\textbf{P}-\textbf{P}') + i \textbf{q}\cdot (\textbf{p}-\textbf{p}')} \nonumber \\{} & {} \qquad \times \left[ -\textbf{q}^2 (\textbf{p}\cdot (e_1\textbf{A}(\textbf{Q}+{\textbf{q}\over 2}) - e_2\textbf{A}(\textbf{Q}-{\textbf{q}\over 2})) +{i \over 4}\textbf{q}^2 (e_1\pmb {\nabla }_Q\cdot \textbf{A}(\textbf{Q}+{\textbf{q}\over 2})+ e_2\pmb {\nabla }_Q\cdot \textbf{A})\textbf{Q}-{\textbf{q}\over 2}) )\right. \nonumber \\{} & {} \qquad + \{ \cdot (\textbf{q} \cdot \textbf{p}) \textbf{q} \cdot (e_1\textbf{A}(\textbf{Q}+{\textbf{q}\over 2})- e_2\textbf{A}(\textbf{Q}-{\textbf{q}\over 2})) {i \over 4}\{ \textbf{q} \cdot (\textbf{q} \cdot \pmb {\nabla }) (e_1\textbf{A}(\textbf{Q}+{\textbf{q}\over 2})+ e_2\textbf{A}(\textbf{Q}-{\textbf{q}\over 2})) \nonumber \\{} & {} \qquad \left. -i \textbf{q} \cdot ( e_1 \textbf{A}(\textbf{Q}+{\textbf{q}\over 2}) -e_2 \textbf{A}(\textbf{Q}-{\textbf{q}\over 2}) \right] . \end{aligned}$$
(122)
In (121) and (122) the derivatives can be removed from
the vector potential by integrating by parts, assuming no contribution from the boundary terms.
The three terms in these expressions with derivatives on the vector potential are
$$\begin{aligned}{} & {} {i \over 4}{1 \over (2\pi )^6}\sum _{ij} v_{(\textbf{L}\cdot \textbf{S})^2}(\vert \textbf{q} \vert ) e^{i \textbf{Q}\cdot (\textbf{P}-\textbf{P}') + i \textbf{q}\cdot (\textbf{p}-\textbf{p}')} (\textbf{S} \times \textbf{q})_i (\textbf{S} \times \textbf{q})_j \nonumber \\{} & {} \quad \times (e_1\partial _i \textbf{A}(\textbf{Q}+{\textbf{q}\over 2}) +e_2\partial _i\textbf{A}(\textbf{Q}-{\textbf{q}\over 2}))_j \} \end{aligned}$$
(123)
$$\begin{aligned}{} & {} \quad {i \over 4}{1 \over (2\pi )^6} \int d\textbf{q} d \textbf{Q} v_{\textbf{L}\cdot \textbf{L}}(\vert \textbf{q} \vert ) e^{i \textbf{Q}\cdot (\textbf{P}-\textbf{P}') + i \textbf{q}\cdot (\textbf{p}-\textbf{p}')} \textbf{q}^2 \nonumber \\{} & {} \quad \times (e_1\pmb {\nabla }\cdot \textbf{A}(\textbf{Q}+{\textbf{q}\over 2})+ e_2\pmb {\nabla }\cdot \textbf{A}(\textbf{Q}-{\textbf{q}\over 2}) ) \end{aligned}$$
(124)
and
$$\begin{aligned}{} & {} -{i \over 4}{1 \over (2\pi )^6} \int d\textbf{q} d \textbf{Q} v_{\textbf{L}\cdot \textbf{L}}(\vert \textbf{q} \vert ) e^{i \textbf{Q}\cdot (\textbf{P}-\textbf{P}') + i \textbf{q}\cdot (\textbf{p}-\textbf{p}')} \{ \textbf{q} \cdot (\textbf{q} \cdot \pmb {\nabla }) \nonumber \\{} & {} \quad (e_1\textbf{A}(\textbf{Q}+{\textbf{q}\over 2})+ e_2\textbf{A}(\textbf{Q}-{\textbf{q}\over 2})). \end{aligned}$$
(125)
The partial derivatives in (123-125) are derivatives of the argument of the vector potential, equivalently with respect to the \(\textbf{Q}\) variables. They can be replaced by \(\pm 2\partial _{q_i}\)
$$\begin{aligned}{} & {} -{ie \over 2}{1 \over (2\pi )^6} \sum _{ij} \int d\textbf{q} d \textbf{Q} (\partial _{q_i} [ v_{(\textbf{L}\cdot \textbf{S})^2}(\vert \textbf{q} \vert ) e^{i \textbf{Q}\cdot (\textbf{P}-\textbf{P}') + i \textbf{q}\cdot (\textbf{p}-\textbf{p}')} (\textbf{S} \times \textbf{q})_i (\textbf{S} \times \textbf{q})_j] \nonumber \\{} & {} \quad \times (e_1\textbf{A}(\textbf{Q}+{\textbf{q}\over 2}) - e_2\textbf{A}(\textbf{Q}-{\textbf{q}\over 2}))_j \end{aligned}$$
(126)
$$\begin{aligned}{} & {} \quad -{i \over 2}{1 \over (2\pi )^6} \int d\textbf{q} d \textbf{Q} \pmb {\nabla }_q [ v_{\textbf{L}\cdot \textbf{L}}(\vert \textbf{q} \vert ) {1 \over (2\pi )^6} e^{i \textbf{Q}\cdot (\textbf{P}-\textbf{P}') + i \textbf{q}\cdot (\textbf{p}-\textbf{p}')} \textbf{q}^2 ]\cdot \left( e_1\textbf{A}(\textbf{Q}+{\textbf{q}\over 2})- e_2\textbf{A}(\textbf{Q}-{\textbf{q}\over 2}) \right) \end{aligned}$$
(127)
$$\begin{aligned}{} & {} \quad {i \over 2}{1 \over (2\pi )^6} \int d\textbf{q} d \textbf{Q} \pmb {\nabla }_q [\cdot \textbf{q} v_{\textbf{L}\cdot \textbf{L}}(\vert \textbf{q} \vert ) e^{i \textbf{Q}\cdot (\textbf{P}-\textbf{P}') + i \textbf{q}\cdot (\textbf{p}-\textbf{p}')} \textbf{q} ]\cdot (e_1\textbf{A}(\textbf{Q}-{\textbf{q}\over 2})- e_2\textbf{A}(\textbf{Q}-{\textbf{q}\over 2})). \end{aligned}$$
(128)
Using (126–128) and the identities
$$\begin{aligned} \left( e_1\textbf{A}\left( \textbf{Q}+{\textbf{q}\over 2}\right) - e_2\textbf{A}(\textbf{Q}-{\textbf{q}\over 2})\right) = \int d\textbf{x} (e_1\delta (\textbf{x} -\textbf{Q}-{\textbf{q}\over 2})- \delta (\textbf{x} -e_2\textbf{Q}+{\textbf{q}\over 2})) A(\textbf{x},0) \end{aligned}$$
(129)
and
$$\begin{aligned}{} & {} \int d\textbf{Q} e^{i \textbf{Q}\cdot (\textbf{P}-\textbf{P}')} (e_1\delta (\textbf{x} -\textbf{Q}-{\textbf{q}\over 2})- e_2\delta (\textbf{x} -\textbf{Q}+{\textbf{q}\over 2})) \nonumber \\{} & {} \quad = e^{i \textbf{x}\cdot (\textbf{P}-\textbf{P}')}( e^{i \textbf{q}\cdot (\textbf{P}-\textbf{P}')/2} - e^{-i \textbf{q}\cdot (\textbf{P}-\textbf{P}')/2}) \nonumber \\{} & {} \quad = e^{i \textbf{x}\cdot (\textbf{P}-\textbf{P}')}[ (e_1-e_2) \cos \left( {\textbf{q}\cdot (\textbf{P}'-\textbf{P}\over 2} \right) +i (e_1+e_2) \sin \left( {\textbf{q}\cdot (\textbf{P}'-\textbf{P}\over 2} \right) ]. \end{aligned}$$
(130)
in the expressions for the currents gives the result below for the three derivative terms. Since the \(\textbf{q}\) in these expressions comes from the vector potential, it does not get touched by derivatives. Since there is no other \(\textbf{Q}\) -dependence these factors multiply everything.
$$\begin{aligned}{} & {} -{i \over 4}{1\over (2\pi )^6} \int d\textbf{q} (\partial _{q_i} [ v_{(\textbf{L}\cdot \textbf{S})^2}(\vert \textbf{q} \vert ) e^{i \textbf{x}\cdot (\textbf{P}-\textbf{P}') + i \textbf{q}\cdot (\textbf{p}-\textbf{p}')} (\textbf{S} \times \textbf{q})_i (\textbf{S} \times \textbf{q})] \nonumber \\{} & {} \quad \times [(e_1-e_2) \cos \left( {\textbf{q}\cdot (\textbf{P}'-\textbf{P}\over 2} \right) +i (e_1+e_2) \sin \left( {\textbf{q}\cdot (\textbf{P}'-\textbf{P}\over 2} \right) ] \end{aligned}$$
(131)
$$\begin{aligned}{} & {} \quad -{i \over 2}{1 \over (2\pi )^6} \int d\textbf{q} \pmb {\nabla }_q\cdot [ v_{\textbf{L}\cdot \textbf{L}}(\vert \textbf{q} \vert ) e^{i \textbf{x}\cdot (\textbf{P}-\textbf{P}') + i \textbf{q}\cdot (\textbf{p}-\textbf{p}')} \textbf{q}^2 ] \nonumber \\{} & {} \quad \times [ (e_1-e_2) \cos \left( {\textbf{q}\cdot (\textbf{P}'-\textbf{P}\over 2} \right) +i (e_1+e_2) \sin \left( {\textbf{q}\cdot (\textbf{P}'-\textbf{P}\over 2} \right) ] \end{aligned}$$
(132)
$$\begin{aligned}{} & {} \quad {i\over 2} {1 \over (2\pi )^6} \int d\textbf{q} \pmb {\nabla }_{q_i}[\cdot \textbf{q} v_{\textbf{L}\cdot \textbf{L}}(\vert \textbf{q} \vert ) e^{i \textbf{x}\cdot (\textbf{P}-\textbf{P}') + i \textbf{q}\cdot (\textbf{p}-\textbf{p}')} \textbf{q}] \nonumber \\{} & {} \quad \times [ (e_1-e_2) \cos \left( {\textbf{q}\cdot (\textbf{P}'-\textbf{P}\over 2} \right) +i (e_1+e_2) \sin \left( {\textbf{q}\cdot (\textbf{P}'-\textbf{P} \over 2} \right) ] \end{aligned}$$
(133)
These derivative expressions go in the expression for the current matrix elements, which is the coefficient of the vector potential.
$$\begin{aligned}{} & {} \langle \textbf{P}',\mathbf {p'} \vert \textbf{J}_{(\textbf{L}\cdot \textbf{S})^2}(\textbf{x},0) \vert \textbf{P},\textbf{p} \rangle = \int d\textbf{q} v_{(\textbf{L}\cdot \textbf{S})^2}(\vert \textbf{q} \vert ) {1 \over (2\pi )^6} e^{i \textbf{x}\cdot (\textbf{P}-\textbf{P}') + i \textbf{q}\cdot (\textbf{p}-\textbf{p}')} \nonumber \\{} & {} \quad \times [ (e_1-e_2) \cos \left( {\textbf{q}\cdot (\textbf{P}'-\textbf{P} \over 2} \right) +i (e_1+e_2) \sin \left( {\textbf{q}\cdot (\textbf{P}'-\textbf{P} \over 2} \right) ] \nonumber \\{} & {} \quad \times \left[ - (\textbf{S} \times \textbf{q})\cdot \textbf{p}(\textbf{S} \times \textbf{q}) -{i \over 2} (\textbf{S}^2 \textbf{q} - (\textbf{S}\cdot \textbf{q})\textbf{S} )\right] \nonumber \\{} & {} \quad -i{1 \over 2}{1 \over (2\pi )^6} \int d\textbf{q} (\sum _i\pmb {\nabla {q}}_i [\cdot (\textbf{S} \times \textbf{q})_i v_{(\textbf{L}\cdot \textbf{S})^2}(\vert \textbf{q} \vert ) e^{i \textbf{x}\cdot (\textbf{P}-\textbf{P}') + i \textbf{q}\cdot (\textbf{p}-\textbf{p}')} (\textbf{S} \times \textbf{q})] \nonumber \\{} & {} \quad \times [ (e_1-e_2) \cos \left( {\textbf{q}\cdot (\textbf{P}'-\textbf{P}\over 2} \right) +i (e_1+e_2) \sin \left( {\textbf{q}\cdot (\textbf{P}'-\textbf{P}\over 2} \right) ]. \end{aligned}$$
(134)
For this interaction the term with the derivative acting on the interaction vanishes. This is because the coefficient is \(\textbf{q} \cdot (\textbf{S}\times \textbf{q})=0\). This means that then the \((\textbf{L}\cdot \textbf{S})^2\) current matrix elements become
$$\begin{aligned}{} & {} \langle \textbf{P}',\mathbf {p'} \vert \textbf{J}(\textbf{x},0)_{(\textbf{L}\cdot \textbf{S})^2} \vert \textbf{P},\textbf{p} \rangle = \int d\textbf{q} v_{(\textbf{L}\cdot \textbf{S})^2}(\vert \textbf{q} \vert ) {1 \over (2\pi )^6} e^{i \textbf{x}\cdot (\textbf{P}-\textbf{P}') + i \textbf{q}\cdot (\textbf{p}-\textbf{p}')} \nonumber \\{} & {} \quad \times [ (e_1-e_2) \cos \left( {\textbf{q}\cdot (\textbf{P}'-\textbf{P}\over 2} \right) +i (e_1+e_2) \sin \left( {\textbf{q}\cdot (\textbf{P}'-\textbf{P}\over 2} \right) ] \nonumber \\{} & {} \quad \times \left[ - {1 \over 2} (\textbf{S} \times \textbf{q})\cdot (\textbf{p}+ \textbf{p}')(\textbf{S} \times \textbf{q}) -{i \over 2} (\textbf{S}^2 \textbf{q} - (\textbf{S}\cdot \textbf{q})\textbf{S}) \right] \end{aligned}$$
(135)
which is equivalent to equation (37).
The part of the \(\textbf{L}^2\) current involving derivatives of the interaction also vanishes. To see this first note the expression for the current is
$$\begin{aligned}{} & {} \langle \textbf{P}',\textbf{p}' \vert \textbf{J}(\textbf{x},0)_{\textbf{L}\cdot \textbf{L}} \vert \textbf{P},\textbf{p} \rangle = \int d\textbf{q} v_{\textbf{L}\cdot \textbf{L}}(\vert \textbf{q} \vert ) {1 \over (2\pi )^6} e^{i \textbf{x}\cdot (\textbf{P}-\textbf{P}') + i \textbf{q}\cdot (\textbf{p}-\textbf{p}')} \nonumber \\{} & {} \quad \times [ (e_1-e_2) \cos \left( {\textbf{q}\cdot (\textbf{P}-\textbf{P}'\over 2} \right) +i (e_1+e_2) \sin \left( {\textbf{q}\cdot (\textbf{P}-\textbf{P}'\over 2} \right) ] \nonumber \\{} & {} \quad \left[ -\textbf{q}^2\textbf{p} +(\textbf{q} \cdot \textbf{p}) \textbf{q} +2 \textbf{q} \right] \nonumber \\{} & {} \quad + {1 \over (2\pi )^6} \int d\textbf{q} [ (e_1-e_2) \cos \left( {\textbf{q}\cdot (\textbf{P}-\textbf{P}'\over 2} \right) +i (e_1+e_2) \sin \left( {\textbf{q}\cdot (\textbf{P}-\textbf{P}'\over 2} \right) ] \nonumber \\{} & {} \quad \times \pmb {\nabla }_q\cdot [ v_{\textbf{L}\cdot \textbf{L}}(\vert \textbf{q} \vert ) e^{i \textbf{x}\cdot (\textbf{P}-\textbf{P}') + i \textbf{q}\cdot (\textbf{p}-\textbf{p}')} \textbf{q}^2 ] \end{aligned}$$
(136)
$$\begin{aligned}{} & {} \quad - {1 \over (2\pi )^6} \int d\textbf{q} [ (e_1-e_2) \cos \left( {\textbf{q}\cdot (\textbf{P}-\textbf{P}'\over 2} \right) +i (e_1+e_2) \sin \left( {\textbf{q}\cdot (\textbf{P}-\textbf{P}'\over 2} \right) ] \nonumber \\{} & {} \quad \pmb {\nabla }[\cdot ( \textbf{q} v_{\textbf{L}\cdot \textbf{L}}(\vert \textbf{q} \vert ) e^{i \textbf{x}\cdot (\textbf{P}-\textbf{P}') + i \textbf{q}\cdot (\textbf{p}-\textbf{p}')})\textbf{q} ]. \end{aligned}$$
(137)
For a rotationally invariant \(v_{\textbf{L}\cdot \textbf{L}}(\vert \textbf{q} \vert )\)
$$\begin{aligned} (\pmb {\nabla } v) \textbf{q}^2 = v' \hat{\textbf{q}} \textbf{q}^2 = v' q \textbf{q} \end{aligned}$$
(138)
while
$$\begin{aligned} \textbf{q} \cdot (\pmb {\nabla }v) \textbf{q} = v'q\textbf{q}. \end{aligned}$$
(139)
These terms come with opposite signs in the expression above so they exactly cancel. What remains after eliminating these terms is
$$\begin{aligned}{} & {} \langle \textbf{P}',\textbf{p}' \vert \textbf{J}(\textbf{x},0)_{\textbf{L}\cdot \textbf{L}} \vert \textbf{P},\textbf{p} \rangle = \int d\textbf{q} v_{\textbf{L}\cdot \textbf{L}}(\vert \textbf{q} \vert ) {1 \over (2\pi )^6} e^{i \textbf{x}\cdot (\textbf{P}-\textbf{P}') + i \textbf{q}\cdot (\textbf{p}-\textbf{p}')} \nonumber \\{} & {} \quad \times [ (e_1-e_2) \cos \left( {\textbf{q}\cdot (\textbf{P}-\textbf{P}'\over 2} \right) +i (e_1+e_2) \sin \left( {\textbf{q}\cdot (\textbf{P}-\textbf{P}'\over 2} \right) ] \nonumber \\{} & {} \quad \times \left[ -\textbf{q}^2\textbf{p} + (\textbf{q} \cdot \textbf{p}) \textbf{q} -i\textbf{q} +{1 \over 2}\textbf{q}^2(\textbf{p}-\textbf{p}') -{1 \over 2} \textbf{q}\cdot (\textbf{p}-\textbf{p}') \textbf{q} + i \textbf{q} \right] \nonumber \\{} & {} \quad = \int d\textbf{q} v_{\textbf{L}\cdot \textbf{L}}(\vert \textbf{q} \vert ) {1 \over (2\pi )^6} e^{i \textbf{x}\cdot (\textbf{P}-\textbf{P}') + i \textbf{q}\cdot (\textbf{p}-\textbf{p}')} \nonumber \\{} & {} \quad \times [ (e_1-e_2) \cos \left( {\textbf{q}\cdot (\textbf{P}-\textbf{P}'\over 2} \right) +i (e_1+e_2) \sin \left( {\textbf{q}\cdot (\textbf{P}-\textbf{P}'\over 2} \right) ] \nonumber \\{} & {} \quad \times {1 \over 2} \textbf{q} \times (\textbf{q} \times (\textbf{p}+\textbf{p}')) \end{aligned}$$
(140)
which is equivalent to equation (38) Again the derivative of the potential cancels in this expression as well. This means that the potentials can be factored.
