}}\,,
\label{alg pp}$$ where the field-dependent parameters are given by $$\begin{aligned}
 \Lambda_{\tilde{p}_1\tilde{p}_2}\ =&\
f\xi_0\Big((D_{\tilde{p}_1}f\xi_0 D_{\tilde{p}_2} 
- D_{\tilde{p}_2}f\xi_0 D_{\tilde{p}_1})A_Q
+D_{\tilde{p}_1}f\xi_0[F\Psi, F\Xi D_{\tilde{p}_2}F\Psi]
\nonumber\\
&\
-D_{\tilde{p}_2}f\xi_0[F\Psi, F\Xi D_{\tilde{p}_1}F\Psi]
+\{F\Psi, F\Xi(D_{\tilde{p}_1}F\Xi D_{\tilde{p}_2}
- D_{\tilde{p}_2}F\Xi D_{\tilde{p}_1})F\Psi\}
\nonumber\\
&\
-[F\Xi D_{\tilde{p}_2}F\Psi, F\Xi D_{\tilde{p}_2}F\Psi]\Big)\,,\\
\lambda_{\tilde{p}_1\tilde{p}_2}\ =&\
-X\eta F\Xi(D_{\tilde{p}_1}F\Xi D_{\tilde{p}_2} 
- D_{\tilde{p}_2}F\Xi D_{\tilde{p}_1})F\Psi\,,\end{aligned}$$ and $\Omega_{\tilde{p}_1\tilde{p}_2}$ in (\[Omega IJ\]). The unphysical transformation $\delta_{[\tilde{p},\tilde{p}]}$ is defined by

\[tf pp\] $$\begin{aligned}
A_{\delta_{[\tilde{p},\tilde{p}]}}\ =&\
- f\xi_0\Bigg(Qf\xi_0\big(
QA_{[\tilde{p},\tilde{p}]}+[F\Psi, \,F\Xi D_{[\tilde{p},\tilde{p}]}]F\Psi]\big)
\nonumber\\
&\
+ [F\Psi,\,F\Xi\Big(
QF\Xi D_{[\tilde{p},\tilde{p}]}F\Psi
+[F\Psi,f\xi_0\big(
QA_{[\tilde{p},\tilde{p}]}+[F\Psi, F\Xi D_{[\tilde{p},\tilde{p}]}F\Psi]\big)]
\Big)]\Bigg)\,,
\label{tf pp ns}\\
\delta_{[\tilde{p},\tilde{p}]}\Psi\ =&\
-X\eta F\Xi\Big(
QF\Xi D_{[\tilde{p},\tilde{p}]}F\Psi
+[F\Psi,\,f\xi_0\big(
QA_{[\tilde{p},\tilde{p}]}+[F\Psi, \,F\Xi D_{[\tilde{p},\tilde{p}]}F\Psi]\big)]
\Big)\,.
\label{tf pp ramond}\end{aligned}$$

The first-quantized charge $[\tilde{p},\tilde{p}]$ is defined by $$[\tilde{p},\tilde{p}]\ =\ w_{\mu\nu}[\tilde{p}^\mu,\tilde{p}^\nu]\,,$$ with the parameter $w_{\mu\nu}\,(=-w_{\nu\mu})$, and $[\tilde{p},\tilde{p}]_{12}
=[\tilde{p},\tilde{p}](w_{12}=(v_1v_2-v_2v_1)/2)$ in (\[alg pp\]). At the linearized level, the transformation (\[tf pp\]) becomes $$\begin{aligned}
  \delta_{[\tilde{p},\tilde{p}]}\ \Phi\ =&\
-\xi_0Q\xi_0Q[\tilde{p},\tilde{p}]\Phi\
=\ -\xi_0QX_0[\tilde{p},\tilde{p}]\Phi\,,\\
  \delta_{[\tilde{p},\tilde{p}]}\ \Psi\ =&\
-X\eta\Xi Q\Xi[\tilde{p},\tilde{p}]\Psi\
=\ -X\eta\Xi X[\tilde{p},\ti