how to define covariant derivative $D_\mu$. To this end, consider for simplicity a free charged scalar field $\phi_\theta(x)$, $$\phi_\theta(x) = \int d\mu(p) (a_p e^{-i p \cdot x} + b^\dagger_p
  e^{i p \cdot x}),\quad {\rm where} \quad d \mu(p) \equiv \frac{d^3
  p}{2p_0}, \quad p_0=\sqrt{\vec{p}^2 + m^2}.$$ We require that the field $\phi_\theta$ obeys twisted statistics in Fock space: $$\begin{aligned}
a_p a_q &=& e^{i p \wedge q} a_q a_p , \quad {\rm where} \quad p
\wedge q \equiv p_\mu \theta^{\mu\nu} q_\nu,   \nonumber \\
a_p a^\dagger _q &=& e^{-i p \wedge q} a^\dagger _q a_p + 2p_0
\delta^{(3)}(p-q)\end{aligned}$$ and similarly for $b(p), b^\dagger(p)$. The twisted operators $a(p),
a^\dagger(p),b(p),b^\dagger(p)$ can be realized in terms of untwisted Fock space operators $c(p),d(p)$ and their adjoints through the well-known “dressing transformations" [@grossefaddeev] $$\begin{aligned}
a_p &=& c_p e^{-\frac{i}{2} p \wedge P}, \qquad \qquad a^\dagger _p= c^\dagger _p
e^{\frac{i}{2} p \wedge P}, \nonumber \\
b_p &=& d_p e^{-\frac{i}{2} p \wedge P}, \qquad \qquad b^\dagger _p= d^\dagger _p
e^{\frac{i}{2} p \wedge P}, {\rm where} \\
P_\mu &=& \int d\mu(p) p_\mu \left(a^\dagger_p a_p + b^\dagger_p
b_p \right) = \int d\mu(p) p_\mu \left(c^\dagger_p c_p + d^\dagger_pd_p \right)\nonumber \\
&=& {\rm total \,
  momentum\, operator}.\end{aligned}$$ Then $\phi_\theta(x)$ may be written in terms of 