Local gauge transformations of the local densities

Let indices $\beta,\gamma,\ldots=1,\ldots,35$ label primary local densities $\rho_{nLvJ}$ (23) listed with stars ($\star$) in Tables 3 and 4, which enter the EDF at N$^3$LO, as shown in Eqs. (30) and (31). Using this notation, the linearized gauge transformation of one of the local densities can be written as

$$\rho_{\beta}'(\vec {r}) = \left\{\left[ K_{nL} \left(1+iG(\vec {r},\vec {r}')\right)\rho_{v}(\vec {r},\vec {r}')\right]_{J}\right\}_{\vec {r}=\vec {r}'}$$

$$= \rho_{\beta}(\vec {r}) + \left\{\left[ K_{nL} iG(\vec {r},\vec {r}') \rho_{v}(\vec {r},\vec {r}') \right]_{J}\right\}_{\vec {r}=\vec {r}'}$$

$$= \rho_{\beta}(\vec {r}) + \rho_\beta^{G}(\vec {r}) ,$$ (37)

where the first term is the untransformed local density and the second term is the part affected by the gauge transformation.

As an illustration, let us begin by considering the simpler case of Galilean transformation, and look at the term with $n=2$, where only two relative momentum operators ${k}$ appear. Operator ${k}$ can be written as ${k}_{\rho}+{k}_{G}$, with the first term acting only on $\rho_{v}(\vec {r},\vec {r}')$ and the second term acting only on $G(\vec {r},\vec {r}')$. Then we have.

$$K_{2L}=\left[{k}{k}\right]_{L} =\left[{k}_{\rho}{k}_{\rho}\right]_{L} +2\left[{k}_{\rho}{k}_{G}\right]_{L} +\left[{k}_{G}{k}_{G}\right]_{L} .$$ (38)

When this is inserted into the expression for $\rho_\beta^{G}$, the last term can be dropped since only the first-order derivatives of $G(\vec {r},\vec {r}')$ survive for the Galilean transformation, and the first term disappears when one takes the limit of $\vec {r}=\vec {r}'$ since $G(\vec {r},\vec {r})=0$. Thus in this case we obtain

$$\rho_\beta^{G}(\vec {r}) = i\left\{2\left[\left[{k}_{\rho}{k}_{G}\right]_{l}G(\vec {r},\vec {r}')\rho_{v}(\vec {r},\vec {r}')\right]_{J},\right\}_{\vec {r}=\vec {r}'}$$

$$= i\sum_{J'}c_{J'}\left[\left\{\left[{k}_{\rho}\rho_{v}\right]_{J'}\right\}_{\vec {r}=\vec {r}'}(\nabla\phi)(\vec {r})\right]_{J} ,$$ (39)

where the second equation results from the vector recoupling (note that $G(\vec {r},\vec {r}')$ is a scalar and $(\nabla\phi)(\vec {r})$ is a vector) and $c_{J'}$ are the ensuing numerical coefficients.

This example illustrates the main features of the derivation, namely, (i) for the Galilean transformations only terms with first-order derivatives of $G(\vec {r},\vec {r}')$ occur in the final expression for $\rho_\beta^{G}$, (ii) local densities appearing in the sum are of one order less in derivatives than the density being transformed, and (iii) the tensor order is preserved so that a local density is transformed into a sum of densities which can couple with a vector to the same tensor order. This leads to the ansatz for the Galilean transformation,

$$\rho_{\beta}^{G}(\vec {r})=i\sum_{\gamma}c(\beta,\gamma)\left[\rho_{\gamma}(\vec {r})(\nabla\phi)(\vec {r})\right]_{J},$$ (40)

where $c(\beta,\gamma)$ are numerical coefficients. Similarly, for the full gauge transformation the corresponding ansatz reads

$$\rho_{\beta}^{G}(\vec {r})=i\sum_{\gamma mI}c_{mI}(\beta,\gamma)\left[\rho_{\gamma}(\vec {r})\left[ D_{mI}\phi \right]_I(\vec {r})\right]_{J}.$$ (41)

In both cases, the numerical coefficients can be found by using the method outlined above, combined with a repeated use of the $6j$-symbols.

However, instead of using this method, it turned out to be more efficient to proceed in another way. First, by using symbolic programming [38], we constructed the transformed densities $\rho_{\beta}^{G}(\vec {r})$ explicitly in terms of derivatives of the density matrices and the gauge angle. Then, from the resulting expression the ansatz (40) or (41) was subtracted, which gave equations for the numerical coefficients by requesting that these differences must be identically equal to zero. Because these equations must hold for all density matrices and gauge angles, we could randomly assume arbitrary values for these quantities and their derivatives. In this way, all linearly independent equations for the coefficients could be obtained and solved analytically, again by using symbolic programming. The solutions were then double-checked by using the full forms of the densities.